Production Systems Engineering

Production Systems Engineering Production Systems Engineering Jingshan Li Electrical and Computer Engineering, Unive...

Author: Jingshan Li | Semyon M. Meerkov

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Production Systems Engineering

Production Systems Engineering

Jingshan Li Electrical and Computer Engineering, University of Kentucky Lexington, Kentucky

Semyon M. Meerkov Electrical and Computer Science, University of Michigan Ann Arbor, Michigan

Jingshan Li University of Kentucky Electrical and Computer Engineering Lexington, Kentucky 40506 Semyon M. Meerkov University of Michigan Electrical and Computer Science Ann Arbor, Michigan 48109

ISBN 978-0-387-75578-6

eISBN 978-0-387-75579-3

Library of Congress Control Number: 2008937464 © 2009 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration: In 1932, the famous Mexican artist, Diego Rivera, was commissioned by Edsel Ford, President of Ford Motor Company, to create two murals for the Detroit Institute of Arts. The murals, completed in 1933, are currently located on the Northern and Southern walls of the Detroit Institute of Art’s Rivera Court. The cover image of this book is a fragment of the Northern Wall’s mural. It depicts operations involved in the production of the engine and transmission for the 1932 Ford V8. Although the manufacturing technology has changed dramatically since then, the fundamental principles of production systems did not. This textbook is devoted to these principles. Photograph ©2001 The Detroit Institute of Arts Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

To the memory of SMM’s parents, To JL’s parents, and To our families, whose love and support made this book possible JL and SMM

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“... of making many books, there is no end; and much study is a weariness of the flesh.” Ecclesiastes, 12:12

Preface

Production systems are a major part of modern technology. However, there has not been a single university-level textbook devoted exclusively to this topic. This volume is intended to fill this void. It presents the area of production systems based on the first principles and at the same level of rigor as other engineering disciplines. Therefore, we use the title Production Systems Engineering (PSE). Along with rigor, this book emphasizes practical applications. In fact, every problem considered here originated on the factory floor and, after conceptualizations and analyses, was implemented in practice, leading to productivity improvements and savings. The case studies included in this text are based on these applications. Thus, a rigorous, first-principle-based presentation of PSE problems of practical significance is the main emphasis of this textbook. Preliminary versions of this text have been used by undergraduate and graduate students in courses offered in the US (the University of Michigan and the University of Kentucky), China (Tsinghua University) and Israel (Technion), and found overwhelming approval of students and instructors alike. We hope that the publication of this volume will contribute to engineering education of all students interested in production and, as a result, to an increased efficiency in manufacturing.

Jingshan Li Lexington, Kentucky Semyon M. Meerkov Ann Arbor, MI August 15, 2008

vii

Contents Synopsis I. BACKGROUND MATERIAL AND MATHEMATICAL MODELING 1 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Tools: Elements of Probability Theory . . . . . . Mathematical Modeling of Production Systems . . . . . . . . . .

II. SERIAL PRODUCTION LINES WITH BERNOULLI MODEL OF MACHINE RELIABILITY 4. 5. 6. 7. 8. 9. 10.

Analysis of Bernoulli Lines . . . . . . . . . Continuous Improvement of Bernoulli Lines Design of Lean Bernoulli Lines . . . . . . . Closed Bernoulli Lines . . . . . . . . . . . . Product Quality in Bernoulli Lines . . . . . Customer Demand Satisfaction in Bernoulli Transient Behavior of Bernoulli Lines . . .

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III. SERIAL PRODUCTION LINES WITH CONTINUOUS TIME MODELS OF MACHINE RELIABILITY341 11. 12. 13. 14. 15.

IV. 16. 17.

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Analysis of Exponential Lines . . . . . . . . . . . . Analysis of Non-exponential Lines . . . . . . . . . . Improvement of Continuous Lines . . . . . . . . . . Design of Lean Continuous Lines . . . . . . . . . . . Customer Demand Satisfaction in Continuous Lines

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ASSEMBLY SYSTEMS

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343 391 411 441 471

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Assembly Systems with Bernoulli Model of Machine Reliability . 495 Assembly Systems with Continuous Time Models of Machine Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

SUMMARY, PSE TOOLBOX, AND PROOFS 18. 19. 20.

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545

Summary of Main Facts of Production Systems Engineering . . 547 PSE Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Epilogue Abbreviations and Notations Index ix

653 655 661

Contents Dedication

v

Preface

vii

Contents Synopsis

ix

Foreword

xxiii

I BACKGROUND MATERIAL AND MATHEMATICAL MODELING 1 1 Introduction 1.1 Main Areas of Manufacturing . . . . . . . . . . . . . . . . . . . 1.2 Main Problems of Production Systems Engineering . . . . . . . 1.2.1 Complicating phenomena . . . . . . . . . . . . . . . . . 1.2.2 Analysis, continuous improvement, and design problems 1.2.3 Fundamental laws of Production Systems Engineering . 1.2.4 Techniques used in this textbook . . . . . . . . . . . . . 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . .

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3 3 4 4 6 7 8 9 9 10

2 Mathematical Tools: Elements of Probability Theory 2.1 Random Events . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Terminology . . . . . . . . . . . . . . . . . . . . 2.1.2 Axioms of probability and their corollaries . . . . 2.1.3 Conditional probability . . . . . . . . . . . . . . 2.1.4 Independence . . . . . . . . . . . . . . . . . . . . 2.1.5 Total probability formula . . . . . . . . . . . . . 2.1.6 Bayes’s formula . . . . . . . . . . . . . . . . . . . 2.2 Random Variables . . . . . . . . . . . . . . . . . . . . . 2.2.1 Terminology . . . . . . . . . . . . . . . . . . . . 2.2.2 Discrete random variables . . . . . . . . . . . . . 2.2.3 Continuous random variables . . . . . . . . . . .

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13 14 14 14 16 18 19 20 21 21 22 26

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CONTENTS

2.3

2.4 2.5 2.6

2.2.4 Expected value, variance, and coefficient of variation 2.2.5 Vector random variables . . . . . . . . . . . . . . . . 2.2.6 Asymptotic properties of sums of random variables . Random Processes . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Continuous time, continuous space random processes 2.3.3 Markov processes . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . .

3 Mathematical Modeling of Production Systems 3.1 Types of Production Systems . . . . . . . . . . . . . . . 3.1.1 Serial production lines . . . . . . . . . . . . . . . 3.1.2 Assembly systems . . . . . . . . . . . . . . . . . 3.2 Structural Modeling . . . . . . . . . . . . . . . . . . . . 3.3 Mathematical Models of Machines . . . . . . . . . . . . 3.3.1 Timing issues . . . . . . . . . . . . . . . . . . . . 3.3.2 Machine reliability models . . . . . . . . . . . . . 3.3.3 Notations . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Machine model identification . . . . . . . . . . . 3.3.5 Calculating parameters of aggregated machines . 3.3.6 Machine quality models . . . . . . . . . . . . . . 3.4 Mathematical Models of Buffers . . . . . . . . . . . . . . 3.4.1 Modeling . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Buffer parameters identification . . . . . . . . . . 3.5 Modeling Interactions between Machines and Buffers . . 3.5.1 Slotted time case . . . . . . . . . . . . . . . . . . 3.5.2 Continuous time case . . . . . . . . . . . . . . . 3.6 Performance Measures . . . . . . . . . . . . . . . . . . . 3.6.1 Production rate and throughput . . . . . . . . . 3.6.2 Work-in-process and finished goods inventory . . 3.6.3 Probabilities of blockages and starvations . . . . 3.6.4 Residence time . . . . . . . . . . . . . . . . . . . 3.6.5 Due-time performance . . . . . . . . . . . . . . . 3.6.6 Transient characteristics . . . . . . . . . . . . . . 3.6.7 Evaluating performance measures on the factory floor . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Model Validation . . . . . . . . . . . . . . . . . . . . . . 3.8 Steps of Modeling, Analysis, Design, and Improvement . 3.8.1 Modeling . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Analysis, continuous improvement, and design . 3.9 Simplification: Transforming Exponential Models into Bernoulli Models . . . . . . . . . . . . . . . 3.9.1 Motivation . . . . . . . . . . . . . . . . . . . . . 3.9.2 Exponential and Bernoulli lines considered . . .

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36 38 40 43 43 44 46 56 57 59

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CONTENTS

3.10

3.11 3.12 3.13

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3.9.3 The exp-B transformation . . . . . . . . . . . . . . . . . 3.9.4 The B-exp transformation . . . . . . . . . . . . . . . . . 3.9.5 Exp-B and B-exp transformations for assembly systems Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Automotive ignition coil processing system . . . . . . . 3.10.2 Automotive paint shop production system . . . . . . . . 3.10.3 Automotive ignition module assembly system . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . .

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94 100 101 102 102 106 110 113 115 117

II SERIAL PRODUCTION LINES WITH BERNOULLI MODEL OF MACHINE RELIABILITY 121 4 Analysis of Bernoulli Lines 4.1 Two-machine Lines . . . . . . . . . . . . . . . . . . . . 4.1.1 Mathematical description . . . . . . . . . . . . 4.1.2 Steady state probabilities . . . . . . . . . . . . 4.1.3 Formulas for the performance measures . . . . 4.1.4 Asymptotic properties . . . . . . . . . . . . . . 4.2 M > 2-machine Lines . . . . . . . . . . . . . . . . . . 4.2.1 Mathematical description and approach . . . . 4.2.2 Aggregation procedure and its properties . . . 4.2.3 Formulas for the performance measures . . . . 4.2.4 Asymptotic properties of M > 2-machine lines 4.2.5 Accuracy of the estimates . . . . . . . . . . . . 4.3 System-Theoretic Properties . . . . . . . . . . . . . . . 4.3.1 Static laws of production systems . . . . . . . . 4.3.2 Reversibility . . . . . . . . . . . . . . . . . . . 4.3.3 Monotonicity . . . . . . . . . . . . . . . . . . . 4.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Automotive ignition coil processing system . . 4.4.2 Automotive paint shop production system . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Annotated Bibliography . . . . . . . . . . . . . . . . . 5 Continuous Improvement of Bernoulli Lines 5.1 Constrained Improvability . . . . . . . . . . . . . 5.1.1 Resource constraints and definitions . . . 5.1.2 Improvability with respect to WF . . . . 5.1.3 Improvability with respect to WF and BC 5.1.4 Improvability with respect to BC . . . . . 5.2 Unconstrained Improvability . . . . . . . . . . . . 5.2.1 Definitions . . . . . . . . . . . . . . . . .

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123 124 124 126 131 133 133 133 137 141 143 143 154 154 154 156 156 156 159 161 162 165

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167 168 168 169 172 174 176 176

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CONTENTS . . . . . . . . . . .

179 179 189 190 190 194 194 195 195 197 199

6 Design of Lean Bernoulli Lines 6.1 Parametrization and Problem Formulation . . . . . . . . . . . . . 6.2 Lean Buffering in Bernoulli Lines with Identical Machines . . . . 6.2.1 Two-machine lines . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Three-machine lines . . . . . . . . . . . . . . . . . . . . . 6.2.3 M > 3-machine lines . . . . . . . . . . . . . . . . . . . . . 6.3 Lean Buffering in Serial Lines with Non-identical Bernoulli Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Two-machine lines . . . . . . . . . . . . . . . . . . . . . . 6.3.2 M > 2-machine lines . . . . . . . . . . . . . . . . . . . . . 6.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Automotive ignition coil processing system . . . . . . . . 6.4.2 Automotive paint shop production system . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 202 204 204

5.3 5.4

5.5 5.6 5.7

5.2.2 Identification of bottlenecks in two-machine lines . . 5.2.3 Identification of bottlenecks in M > 2-machine lines 5.2.4 Potency of buffering . . . . . . . . . . . . . . . . . . 5.2.5 Designing continuous improvement projects . . . . . Measurement-based Management of Production Systems . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Automotive ignition coil processing system . . . . . 5.4.2 Automotive paint shop production system . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . .

7 Closed Bernoulli Lines 7.1 System Model and Problem Formulation . . 7.1.1 Model . . . . . . . . . . . . . . . . . 7.1.2 Problems addressed . . . . . . . . . 7.2 Performance Analysis, Monotonicity, Reversibility, and Unimpeding Closed Lines 7.2.1 Two-machine lines . . . . . . . . . . 7.2.2 M > 2-machine lines . . . . . . . . . 7.3 Improvability . . . . . . . . . . . . . . . . . 7.3.1 Two-machine lines . . . . . . . . . . 7.3.2 M > 2-machine lines . . . . . . . . . 7.3.3 Comparisons . . . . . . . . . . . . . 7.4 Bottleneck Identification . . . . . . . . . . . 7.4.1 Two-machine lines . . . . . . . . . . 7.4.2 M > 2-machine lines . . . . . . . . . 7.5 Leanness . . . . . . . . . . . . . . . . . . . . 7.6 Case Study . . . . . . . . . . . . . . . . . .

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207 207 208 216 216 217 218 218 219

221 . . . . . . . . . . . . 223 . . . . . . . . . . . . 223 . . . . . . . . . . . . 223 . . . . . . . . . . . .

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225 225 228 230 230 231 235 236 236 237 239 241

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7.6.1

7.7 7.8 7.9

Modeling and model validation of closed automotive paint shop production system . . . . . . . . . . . . . . . . . . . 7.6.2 S-improvability . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Lean system design . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

8 Product Quality in Bernoulli Lines 8.1 Bernoulli Lines with Non-perfect Quality Machines . . . . 8.1.1 Model and problem formulation . . . . . . . . . . . 8.1.2 Performance analysis . . . . . . . . . . . . . . . . . 8.1.3 Bottlenecks . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bernoulli Lines with Quality-Quantity Coupling Machines 8.2.1 Model and problem formulation . . . . . . . . . . . 8.2.2 Monotonicity properties . . . . . . . . . . . . . . . 8.2.3 Bottlenecks . . . . . . . . . . . . . . . . . . . . . . 8.3 Bernoulli Lines with Rework . . . . . . . . . . . . . . . . . 8.3.1 Model and problem formulation . . . . . . . . . . . 8.3.2 Performance analysis . . . . . . . . . . . . . . . . . 8.3.3 Bottleneck identification . . . . . . . . . . . . . . . 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Annotated Bibliography . . . . . . . . . . . . . . . . . . .

241 242 242 242 243 244

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247 248 248 250 254 259 263 263 264 267 269 271 272 278 287 288 289

9 Customer Demand Satisfaction in Bernoulli Lines 9.1 Modeling and Parametrization . . . . . . . . . . . . . . . . . 9.1.1 Production-inventory-customer (PIC) model . . . . . . 9.1.2 Due-time performance measure . . . . . . . . . . . . . 9.1.3 Parametrization . . . . . . . . . . . . . . . . . . . . . 9.2 Analysis of DT P . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Evaluating DT P in PIC systems with one-machine PS 9.2.2 Evaluating DT P in PIC systems with M -machine PS 9.3 Design of Lean FGB for Desired DT P . . . . . . . . . . . . . 9.3.1 Lean FGB . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Conservation of filtering . . . . . . . . . . . . . . . . . 9.4 Analysis of DT P for Random Demand . . . . . . . . . . . . . 9.4.1 Random demand modeling . . . . . . . . . . . . . . . 9.4.2 Evaluating DT P for random demand . . . . . . . . . 9.4.3 DT P degradation as a function of demand variability 9.4.4 Dependence of DT P on the shape of demand pmf’s . 9.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . .

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293 296 296 297 297 298 298 300 302 302 302 303 303 305 306 308 308 311 312 313

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10 Transient Behavior of Bernoulli Lines 10.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Mathematical description . . . . . . . . . . . . . . . . . . 10.1.2 Second largest eigenvalue problem . . . . . . . . . . . . . 10.1.3 Pre-exponential factor problem . . . . . . . . . . . . . . . 10.1.4 Settling time problem . . . . . . . . . . . . . . . . . . . . 10.1.5 Production losses problem . . . . . . . . . . . . . . . . . . 10.2 Analysis of the Second Largest Eigenvalue . . . . . . . . . . . . . 10.2.1 Two-machine lines . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Three-machine lines . . . . . . . . . . . . . . . . . . . . . 10.3 Analysis of the Pre-exponential Factors . . . . . . . . . . . . . . 10.4 Analysis of the Settling Time . . . . . . . . . . . . . . . . . . . . 10.4.1 Behavior of P R(n) and W IP (n) . . . . . . . . . . . . . . 10.4.2 Analysis of tsP R and tsW IP . . . . . . . . . . . . . . . . . 10.5 Analysis of the Production Losses . . . . . . . . . . . . . . . . . . 10.5.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Percent of loss when the buffers are empty at the beginning of the shift . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Percent of loss when the buffers are not empty at the beginning of the shift . . . . . . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

315 316 316 317 318 319 319 320 320 323 323 326 326 332 332 332 334 336 336 338 339

III SERIAL PRODUCTION LINES WITH CONTINUOUS TIME MODELS OF MACHINE RELIABILITY341 11 Analysis of Exponential Lines 11.1 Synchronous Exponential Lines . . . . . . 11.1.1 Two-machine case . . . . . . . . . 11.1.2 M > 2-machine case . . . . . . . . 11.2 Asynchronous Exponential Lines . . . . . 11.2.1 Two-machine case . . . . . . . . . 11.2.2 M > 2-machine case . . . . . . . . 11.3 Case Studies . . . . . . . . . . . . . . . . 11.3.1 Automotive ignition coil processing 11.3.2 Crankshaft production line . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . 11.5 Problems . . . . . . . . . . . . . . . . . . 11.6 Annotated Bibliography . . . . . . . . . .

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343 344 344 352 361 361 374 381 381 383 386 386 389

12 Analysis of Non-Exponential Lines 391 12.1 Systems Considered . . . . . . . . . . . . . . . . . . . . . . . . . 392 12.1.1 Mathematical description . . . . . . . . . . . . . . . . . . 392 12.1.2 Second-order-similar production lines . . . . . . . . . . . . 393

CONTENTS 12.2 12.3 12.4 12.5 12.6

Sensitivity of P R and T P to Machine Reliability Model Empirical Formulas for P R and T P . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . .

xvii . . . . .

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394 404 409 409 410

13 Improvement of Continuous Lines 13.1 Constrained Improvability . . . . . . . . . . . . . . . . . . . . . 13.1.1 Resource constraints and definitions . . . . . . . . . . . 13.1.2 Improvability with respect to CT . . . . . . . . . . . . . 13.1.3 Improvability with respect to BC . . . . . . . . . . . . . 13.2 Unconstrained Improvability . . . . . . . . . . . . . . . . . . . . 13.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 One-machine lines . . . . . . . . . . . . . . . . . . . . . 13.2.3 Two-machine synchronous exponential lines . . . . . . . 13.2.4 Asynchronous exponential lines . . . . . . . . . . . . . . 13.2.5 M ≥ 2-machine non-exponential lines . . . . . . . . . . 13.2.6 Buffering potency and measurement-based management 13.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . .

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411 411 413 414 418 424 424 429 430 432 433 434 434 437 437 439

14 Design of Lean Continuous Lines 14.1 Parametrization and Problem Formulation . . . . . . . . . . . . . 14.2 Lean Buffering in Synchronous Lines with Identical Exponential Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Two-machine lines . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Three-machine lines . . . . . . . . . . . . . . . . . . . . . 14.2.3 M > 3-machine lines . . . . . . . . . . . . . . . . . . . . . 14.3 Lean Buffering in Synchronous Lines with Non-identical Exponential Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Two-machine lines . . . . . . . . . . . . . . . . . . . . . . 14.3.2 M > 2-machine lines . . . . . . . . . . . . . . . . . . . . . 14.4 Lean Buffering in Synchronous Lines with Non-exponential Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Sensitivity of kE to machine reliability models . . . . . . 14.4.3 Empirical formulas for kE . . . . . . . . . . . . . . . . . . 14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

441 441 443 443 445 445 451 452 452 456 456 458 462 467 468 469

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15 Customer Demand Satisfaction in Continuous Lines 15.1 Modeling and Parametrization . . . . . . . . . . . . . . . . . . . 15.1.1 Production-inventory-customer (PIC) system . . . . . . . 15.1.2 DT P definition . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Parametrization . . . . . . . . . . . . . . . . . . . . . . . 15.2 DT P Evaluation in PIC Systems with Exponential Machines . . 15.2.1 DT P in PIC systems with one-machine PS . . . . . . . . 15.2.2 DT P in PIC systems with M -machine PS . . . . . . . . . 15.3 DT P in PIC Systems with Non-exponential PS . . . . . . . . . . 15.4 Lean FGB and Conservation of Filtering in PIC Systems with Exponential Machines . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Lean FGB . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Conservation of filtering in lean systems . . . . . . . . . . 15.5 DT P in the Case of Random Demand . . . . . . . . . . . . . . . 15.5.1 Random demand modeling . . . . . . . . . . . . . . . . . 15.5.2 DT P for random demand in PIC systems with exponential machines . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 DT P degradation as a function of demand variability . . 15.5.4 Effect of demand pdf on DT P . . . . . . . . . . . . . . . 15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

IV

ASSEMBLY SYSTEMS

471 471 471 473 473 474 474 475 477 479 479 479 482 482 484 485 487 490 490 492

493

16 Assembly Systems with Bernoulli Model of Machine Reliability495 16.1 Analysis of Bernoulli Assembly Systems . . . . . . . . . . . . . . 495 16.1.1 Three-machine assembly systems . . . . . . . . . . . . . . 495 16.1.2 M > 3-machine assembly systems . . . . . . . . . . . . . . 502 16.2 Continuous Improvement of Bernoulli Assembly Systems . . . . . 511 16.2.1 Constrained improvability . . . . . . . . . . . . . . . . . . 511 16.2.2 Unconstrained improvability . . . . . . . . . . . . . . . . . 515 16.3 Lean Buffering in Bernoulli Assembly Systems . . . . . . . . . . 519 16.4 Customer Demand Satisfaction in Bernoulli Assembly Systems . 519 16.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 16.5.1 Automotive ignition module assembly system . . . . . . . 520 16.5.2 Injection molding - assembly system . . . . . . . . . . . . 522 16.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 16.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 16.8 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . . 525 17 Assembly Systems with Continuous Time Models of Machine Reliability 527 17.1 Analysis of Assembly Systems with Exponential Machines . . . . 527 17.1.1 Systems addressed . . . . . . . . . . . . . . . . . . . . . . 527

CONTENTS

17.2 17.3

17.4

17.5 17.6 17.7

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17.1.2 Synchronous case . . . . . . . . . . . . . . . . . . . . . . . 17.1.3 Asynchronous case . . . . . . . . . . . . . . . . . . . . . . Analysis of Non-exponential Assembly Systems . . . . . . . . . . Improvement of Assembly Systems with Continuous Time Models of Machine Reliability . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Constrained improvability . . . . . . . . . . . . . . . . . . 17.3.2 Unconstrained improvability . . . . . . . . . . . . . . . . . Design of Lean Buffering and Customer Demand Satisfaction in Assembly Systems with Continuous Machines . . . . . . . . . . . 17.4.1 Lean buffering . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Customer demand satisfaction . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . .

SUMMARY, PSE TOOLBOX, AND PROOFS

18 Summary of Main Facts of Production 18.1 Individual Machines . . . . . . . . . . 18.2 Serial Lines . . . . . . . . . . . . . . . 18.2.1 Performance analysis . . . . . . 18.2.2 Continuous improvement . . . 18.2.3 Design . . . . . . . . . . . . . . 18.3 Assembly Systems . . . . . . . . . . .

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

528 532 536 539 539 540 540 540 542 542 542 543

545 . . . . . .

19 PSE Toolbox 19.1 Architecture and Functions . . . . . . . . . . . . . . . . . . . . . 19.2 Modeling Function . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Aggregation of parallel machines . . . . . . . . . . . . . . 19.2.2 Aggregation of consecutive dependent machines . . . . . . 19.2.3 Exp-B transformation for serial lines . . . . . . . . . . . . 19.2.4 B-exp transformation for serial lines . . . . . . . . . . . . 19.3 Performance Analysis Function . . . . . . . . . . . . . . . . . . . 19.3.1 Analysis of serial lines with Bernoulli machines . . . . . . 19.3.2 Analysis of synchronous serial lines with exponential machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.3 Analysis of asynchronous serial lines with exponential machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.4 Analysis of closed lines with Bernoulli machines . . . . . . 19.3.5 Analysis of assembly systems with Bernoulli machines . . 19.4 Continuous Improvement Function . . . . . . . . . . . . . . . . . 19.4.1 Unimprovable allocation of WF in Bernoulli serial lines . 19.4.2 Unimprovable allocation of WF and BC simultaneously in Bernoulli serial lines . . . . . . . . . . . . . . . . . . . .

547 547 548 548 551 553 554 555 555 557 557 558 559 559 562 562 563 563 564 566 566 567 568

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19.5

19.6

19.7

19.8 19.9

19.4.3 WF-continuous improvement procedure for Bernoulli serial lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.4 BC-continuous improvement procedure for Bernoulli serial lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.5 S-continuous improvement procedure for closed Bernoulli lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bottleneck Identification Function . . . . . . . . . . . . . . . . . 19.5.1 BN-m and BN-b in serial lines with Bernoulli machines . 19.5.2 c-BN and BN-b in serial lines with exponential machines . 19.5.3 c-BN and BN-b in serial lines with general models of machine reliability . . . . . . . . . . . . . . . . . . . . . . . . 19.5.4 BN-m and BN-b in closed lines with Bernoulli machines . Lean Buffer Design Function . . . . . . . . . . . . . . . . . . . . 19.6.1 Lean buffering for serial lines with identical Bernoulli machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.2 Lean buffering for serial lines with non-identical Bernoulli machines . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Quality Function . . . . . . . . . . . . . . . . . . . . . . 19.7.1 Analysis of Bernoulli lines with non-perfect quality machines and inspection operations . . . . . . . . . . . . . . Customer Demand Satisfaction Function . . . . . . . . . . . . . . 19.8.1 DT P in PIC system with one Bernoulli machine . . . . . Simulation Function . . . . . . . . . . . . . . . . . . . . . . . . . 19.9.1 Simulation of serial lines with Bernoulli machines . . . . . 19.9.2 Simulation of serial lines with exponential machines . . . 19.9.3 Simulation of serial lines with general models of machine reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.9.4 Simulation of assembly systems with Bernoulli machines .

20 Proofs 20.1 Proofs for Part II . . . . . . . 20.1.1 Proofs for Chapter 4 . 20.1.2 Proofs for Chapter 5 . 20.1.3 Proofs for Chapter 6 . 20.1.4 Proofs for Chapter 7 . 20.1.5 Proofs for Chapter 8 . 20.1.6 Proofs for Chapter 9 . 20.1.7 Proofs for Chapter 10 20.2 Proofs for Part III . . . . . . 20.2.1 Proofs for Chapter 11 20.2.2 Proofs for Chapter 12 20.2.3 Proofs for Chapter 13 20.2.4 Proofs for Chapter 14 20.2.5 Proofs for Chapter 15 20.3 Proofs for Part IV . . . . . . 20.4 Annotated Bibliography . . .

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569 570 570 571 572 572 573 575 576 576 577 578 578 579 580 580 581 582 582 584 587 587 587 599 614 616 621 623 623 624 624 638 642 643 645 648 649

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Epilogue

653

Abbreviations and Notations

655

Index

661

Foreword Production Systems Engineering (PSE) is an emerging branch of Engineering intended to uncover fundamental properties of production systems and utilize them for analysis, continuous improvement, and design. This volume describes the main results of PSE. Purpose of this book: This is a textbook for a first year graduate and a senior undergraduate course on PSE. The aim is two-fold. First, it is to present material that is practical and applicable to any production system in large volume manufacturing industries, such as automotive, electronics, appliances, etc. Second, it is to present this material based on the first principles and at the same level of rigor as that in other engineering disciplines, such as Electrical Engineering, Mechanical Engineering, etc. Main Topic: Production systems are machines and material handling devices arranged so as to produce a desired product. Each machine is characterized by three attributes: • Technological operation(s), which it carries out, e.g., cutting or bending metal, washing or heat treating, depositing or etching materials on silicon substrates, etc. • Capacity to produce parts, i.e., the number of parts, which can be produced per unit of time. • Reliability and quality characteristics, e.g., the statistics of machine breakdowns and defective parts produced. Each material handling device is characterized by: • Technology employed for storing and moving parts, e.g., boxes or lift forks, conveyors or automated guided vehicles, etc. • Capacity to store work-in-process between each pair of consecutive operations, i.e., buffer capacity. This textbook does not address the issues of operation technology of either machines or material handling devices. Addressed are the issues of parts flow through a production system. The nature of parts flow is affected by production xxiii

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capacity and reliability of the machines and storing capacity of the material handling devices. In general, numerous issues related to parts flow could be addressed. Included in this volume are only those, which have a clear practical significance. Thus, the rigorous engineering study of practical issues related to parts flow in production systems with unreliable machines and finite buffers is the main topic of PSE and, consequently, of this textbook. Origin of this book: The origin of this volume is also two-fold. First, it is based on 25 years of research and industrial studies, which one of the authors (S.M. Meerkov), together with his students (one of whom is the other author, J. Li), conducted in various automotive plants. Every issue considered in this book originated on the factory floor and, after appropriate conceptualization and analysis, became an application on the factory floor. The case studies included in this volume describe some of these applications. Second, this textbook is based on production systems course that we have been teaching at the University of Michigan for several years. The audience has included both full-time university students and part-time students from manufacturing industries. (In most cases, this course was available through the University of Michigan distance learning system to practicing engineers throughout the US and the world.) Reading materials for this course have consisted of journal papers on the results of our investigations. Both groups of students encouraged us to summarize these results in a textbook. We also felt that such a book would be a contribution to the literature. That is why we undertook the arduous effort of writing this textbook, which lasted for almost five years. Problems addressed: Three groups of problems are considered: • Analysis: Given a production system, i.e., machine and buffer characteristics, calculate its performance measures, such as throughput, work-inprocess, probabilities of blockages and starvations, scrap rate, probability of customer demand satisfaction, duration of the transients, etc. • Continuous improvement: (a) Determine whether limited resources available in the system, e.g., workforce or work-in-process, can be re-distributed so that its performance is improved. (b) Identify the machine and the buffer, which impede the system performance in the strongest manner (i.e., bottleneck machine and bottleneck buffer identification). • Design: Given machine and demand characteristics, determine the smallest buffer capacity (i.e., lean buffering), which ensures the desired throughput and/or level of customer demand satisfaction. This textbook provides analytical methods for solving these problems and illustrates them by case studies. Main difficulties of PSE: Unreliable machines and finite buffers make the problem of parts flow in production systems difficult since the former makes it stochastic and the latter nonlinear. In more specific terms, the difficulties in

FOREWORD

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investigating production systems arise due to mutual interferences of the machines because of breakdowns. Indeed, a breakdown of one machine affects all other machines in the system – by blocking those upstream and starving those downstream. Buffers are supposed to alleviate these perturbations, serving as “shock absorbers” for parts flow. However, having them “infinite” and, hence, efficient for “shock absorbing,” creates economic problems. Thus, the buffers must be finite and the machines, clearly, cannot be absolutely reliable. These features lead to mathematical models of production systems, which are nonlinear and stochastic. Typically, they are difficult to analyze. Thus, the unreliability of the machines and the finiteness of the buffers are the main sources of difficulties in PSE and, consequently, in production systems analysis, continuous improvement, and design. Method of this textbook: Often, production systems are investigated quantitatively based on queuing theory methods. In this book, however, we use a different approach. It is based on recursive equations, which describe production systems at hand. These equations are derived using exact analysis of the simplest systems and subsequent approximate analysis of more complex ones. The properties of the resulting equations characterize the flow of parts and expose the laws that govern production systems behavior. Thus, a system-theoretic analysis, based on recursive equations that describe production systems at hand, is the method of investigation employed in this textbook. Relation to other engineering disciplines: Many engineering disciplines are also concerned with flows. In Electrical Engineering this is the flow of current in a circuit. In Mechanical Engineering, it is fluid or gas flow. Focusing attention on parts flow makes PSE similar to other engineering disciplines. Therefore, rigorous quantitative methods, typical for modern engineering, become possible and, moreover, necessary in this area of technology. As in other disciplines, flows are characterized by their statics (i.e., stationary regimes) and dynamics (i.e., transient regimes). Although this book centers on the statics, results on the dynamics and transients are also included. Solution paradigm: The solutions of the problems considered in this textbook are given in three classes of results. The first class consists of results proved analytically; they are referred to as Lemmas, or Theorems or Corollaries. The second class is comprised of properties, which are shown to exist numerically and approximately; they are referred to as Numerical Facts. Finally, the third class includes results called Improvability Indicators; they are also justified numerically and are intended as guides for production systems improvement in the framework of the so-called Measurement-based Management. Converting the Numerical Facts and Improvability Indicators into Theorems could be a fruitful direction for future research.

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Intended audience, outcomes, and prerequisites: This textbook is intended for senior undergraduate and beginning graduate students in all engineering disciplines. In addition, business and management students may be interested in a course based on this textbook. Finally, practicing production engineers and managers may find it useful to read this book. As an outcome of this course, students are expected to acquire rigorous and practical knowledge on analysis, design, and management of production systems. No specific prerequisites are assumed; however, prior exposure to Probability Theory may be beneficial (although all probability facts, necessary for this course, are described in this textbook). Book organization: The textbook consists of five parts, each comprised of several chapters. The first part includes background and modeling material, while the rest are devoted to various classes of production systems of increasing complexity. Specifically, Part II is devoted to serial lines with so-called Bernoulli machines, Part III treats similar problems for exponential and general models of machine reliability, while Part IV addresses assembly systems. Finally, Part V includes a summary of the main facts of Production Systems Engineering, a description of the PSE Toolbox, which is a suite of C++ programs that implement all methods developed in this book, and the proofs of the theorems presented in this volume. In each part, every chapter begins with a motivating comment, which provides a reason for considering its subject matter. This is followed by an overview, which outlines specific problems addressed and results obtained. Each chapter is concluded with a set of homework problems and a brief annotated bibliography. Almost all chapters include case studies based on industrial applications carried out by the authors in the automotive industry. Special features: While other books on production systems are centered mostly on performance analysis, the present volume has the following special features: • It presents methods for mathematical modeling of production systems. • It provides techniques for designing continuous improvement projects. • It offers quantitative methods for selecting lean buffering and identifying bottlenecks. • It addresses the issue of product quality. • It describes techniques for the analysis of transient behavior of production lines. • It describes a software package, the PSE Toolbox, which implements methods developed. • Finally, it includes numerous case studies, which illustrate the topics considered and, on occasion, serve as problems for homework assignments.

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Chapter contents: Chapter 1 places production systems in the broader context of manufacturing at large. Chapter 2 provides an overview of Probability Theory and derives several facts on machine reliability and performance used throughout this book. Chapter 3 describes the process of mathematical modeling of production systems. Chapters 4 - 10 are devoted to serial lines with the simplest, i.e., Bernoulli, model of machine reliability. Specifically, Chapters 4, 5, and 6 address, respectively, the issues of analysis, continuous improvement, and design. Chapters 7 and 8 are devoted to closed lines and product quality, respectively. Chapter 9 addresses the issue of customer demand satisfaction, and Chapter 10 discusses transient behavior of serial lines. Chapters 11 - 15 are devoted to similar issues in the framework of serial lines with continuous time (exponential and non-exponential) models of machine reliability, while Chapters 16 and 17 consider assembly systems. Chapter 18, summarizes the main facts of Production Systems Engineering. Chapter 19 describes the PSE Toolbox; a demo of the toolbox is available at http://www.ProductionSystemsEngineering.com Finally, Chapter 20 provides proofs of theorems and other formal statements included in this textbook. Advice to Instructors: There are several ways to structure a semester-long course based on this textbook. If the audience has limited background in Probability Theory, the course could cover Parts I and II in detail and a brief overview of Parts III and IV (mainly, the ideas of the aggregation procedures and bottleneck identification techniques). For an audience with a strong background in probability, the emphasis could be on Chapter 3 and Parts II and III; basic ideas of Part IV could also be covered. For a technically mature audience, the course may cover the entire text. Similar approaches can be used for a quarter-based course, but, of course, with a less detailed coverage of the material. The proofs of most mathematical statements included in this volume are typically not covered in class. However, doctoral-level students specializing in manufacturing may find it useful to study these proofs in order to develop their expertise for theoretical research; that is why the proofs are included in Chapter 20. Finally, a solution manual of all problems included in this textbook is available from the Publisher upon instructor’s request. Acknowledgements: The material of this textbook was developed by the authors and the Ph.D. students working under SMM’s supervision. These include (in chronological order) Drs. Jong-Tae Lim, Ferudun Top, David Jacobs, Chih-Tsung Kuo, Shu-Yin Chiang, Emre Enginarlar, and Liang Zhang. The authors are particularly grateful to Liang Zhang who co-created the PSE Tool-

xxviii

FOREWORD

box, contributed to Chapters 7, 8, 10, 11, 13, 17, 20 and spent countless hours proofreading and correcting the manuscript. His contribution is difficult to overestimate. In addition, the following graduate and undergraduate students contributed to this book: S. Ching, A. Hu, J.Z. Huang, A. Khondker, Y. Liu, B. Rumao, H. Shih, and F. Xu. To all of them the authors express their deep gratitude. The stimulating environments of the Department of Electrical Engineering and Computer Science at the University of Michigan, the General Motors Research and Development Center (with which JL was associated) and the University of Kentucky were very conducive to the research that led to this book. Financial support from the National Science Foundation during the time this material was developed and this textbook written was invaluable for the success of this work. Gratitude to the Division of Civil, Mechanical and Manufacturing Innovations of the National Science Foundation is in order. Special thanks are due to many colleagues throughout the world for their advice and encouragement. First of all, the authors are indebted to Professor Pierre Kabamba from the University of Michigan for his technical advice, impeccable scientific taste, and countless hours spent discussing the material included in this textbook. He carefully read every chapter and made valuable suggestions. Professor Stanley Gershwin of the Massachusetts Institute of Technology was a source of inspiration from the very beginning of the authors’ work in the area of manufacturing. He truly can be viewed as the father of the system-theoretic approach to manufacturing, a small part of which – production systems – is addressed in this textbook. Professor Chrissoleon Papadopoulos of the Aristotle University of Thessaloniki, Greece, is acknowledged for both his personal input to our work and for a series of stimulating conferences, which he has organized for many years and which provided an outstanding opportunity to discuss the issues of production systems with numerous colleagues from all over the world. Our industrial partners, who helped to test the methods described in this book in the framework of numerous projects on the factory floor, are gratefully acknowledged. This includes management and production system personnel from • General Motors Hamtramck Assembly Center, Orion Assembly Plant and Livonia Engine Plant, • Ford Livonia Transmission Plant and Ypsilanti and Rawsonville Component Plants, • Chrysler Detroit Axle Plant, Trenton Engine Plant, Dayton Thermal Products Plant, and Jefferson North Assembly Plant. Thanks are due to our Springer Editor, Steven M. Elliot, for his support during the last two years of this project. We also acknowledge the anonymous reviewers who helped to improve the manuscript. While we thankfully acknowledge all of the individuals and organizations mentioned above, needless to say that all errors, which are undoubtedly present

FOREWORD

xxix

in this book, are due to the authors alone. The list of corrections will be maintained at http://www.ProductionSystemsEngineering.com/book/corrections.html Last, but not least, we are deeply indebted to our families for enduring countless hours missing us in family activities. Their love and support truly made this work possible.

Chapter 1

Introduction Motivation: This chapter is intended to place Production Systems Engineering into the general framework of Manufacturing at large and outline the main problems addressed in this textbook. Overview: Five main areas of Manufacturing are described and three main problems of Production Systems Engineering are characterized.

1.1

Main Areas of Manufacturing

The informal definitions and classifications given below are subjective and based solely on the authors’ experience and understanding. Manufacturing – the process of transforming raw materials into a useful product. Everything, which is done at or for the factory floor operations, we view as manufacturing. Manufacturing matters. Indeed, the wealth of a nation can be either taken from the ground (natural resources and agriculture) or manufactured (value added by processing materials). Thus, being one of just two ways of generating national wealth, manufacturing is of fundamental importance. Manufacturing can be classified into two groups: continuous and discrete. Examples of industries with continuous manufacturing are chemical, materials, and power, where core processes evolve continuously in time. Discrete part manufacturing is practically everything else: automotive, electronics, appliances, aerospace and other industries. This textbook is concerned with discrete part manufacturing. Quite informally, manufacturing can be classified into the following five areas: • Machine tools and material handling devices. The main problem here is: Given a desired material transformation and/or relocation, design, implement, and maintain a machine or a material handling device, which 3

4

CHAPTER 1. INTRODUCTION

•

•

•

•

carries out its function in an efficient manner. This is a mature engineering field with numerous achievements to its credit. Production systems. Main problem: Given machines and material handling devices, structure a production system so that it operates as efficiently as the machines in isolation. This can be achieved by maintaining smooth flow of parts throughout the system, so that mutual interference of the machines does not cause losses of production. The term “structure” is used here to include both design of new and improvement of existing production systems. To-date, this field lacks in rigorous quantitative methods and fundamental engineering knowledge. Production planning and scheduling. Main problem: Given a production system and customer demand, calculate a production plan and schedule delivery of materials so that the demand is satisfied in an economically efficient manner. Numerous quantitative methods, often based on optimization, are available in this relatively mature field of manufacturing. Quality assurance. Main problem: Structure and operate the production system so that parts produced are of the desired quality. To-date, statistical quality control is a major quantitative tool for maintaining product quality. Work systems. Main problem: Organize personnel training and operation so that the production process is carried out safely and efficiently. This includes, in particular, designing wage and incentive systems so that the maximum of the utility function of an individual worker coincides with that of the manufacturing enterprise as a whole and, thus, self-interest of the worker leads to high efficiency of the manufacturing enterprise. At present, this field is still in its infancy.

In addition to the above classification, discrete part manufacturing can be subdivided into two groups: job-shop and large volume manufacturing. Job-shop is concerned with manufacturing “one-of-a-kind” products: unique instruments, highly specialized equipment, some aerospace systems, etc. Large volume manufacturing is intented to produce parts and products in multiple copies: cars, computers, refrigerators, and other items of wide use. This textbook is devoted to one area of manufacturing - production systems, although some structural issues of quality assurance are also addressed. While some methods included here may be useful for job-shops as well, the emphasis is on production systems in large volume manufacturing.

1.2 1.2.1

Main Problems of Production Systems Engineering Complicating phenomena

Examples of typical production systems are shown in Figures 1.1 and 1.2, where the circles represent the machines and rectangles are material handling devices

1.2. MAIN PROBLEMS

5

in their function as buffers. The system in Figure 1.1 is referred to as a serial production line while that of Figure 1.2 as an assembly system.

m1

m2

b1

m M−1 bM−1

m3

b2

mM

Figure 1.1: Serial production line

m 11

b 11

m 12

m1M

1

b1M

1

m 01 m

21

b 21

m 22

m

2M 2

b 2M

b 01

m 02

m 0M

0

2

Figure 1.2: Assembly system If machines never broke down, the situation would be simple: to ensure smooth part flow, balance the capacity of the machines and speeds of the material handling devices so that the desired throughput is achieved. (The notions of machine capacity, throughput, etc., are defined precisely in Chapter 3; at this point, an intuitive understanding of these terms is sufficient.) In reality, however, the machines always experience random breakdowns. This leads to a complex phenomenon of perturbation propagation. Indeed, when, say, machine m2 in the serial line fails, machine m3 may eventually become starved for parts and machine m1 blocked by full buffer b1 . If m2 continues staying down, the starvations will propagate downstream reaching eventually the last machine, mM , and the throughput is lost. Similarly, the blockage will propagate upstream, causing all machines to stop, wasting their production capacity. Thus, failure of one machine may affect all other machines in the system – up- and downstream. This is what makes the investigation of production systems difficult (and necessary too – in order to develop methods for alleviating this perturbation propagation). Similar phenomena take place in assembly systems as well: failures of m12 (see Figure 1.2) may lead to the starvation of m13 , . . ., m1M1 , m01 , . . ., m0M0 and to blockage of m11 and m21 , . . ., m2M2 . (Note that it is typically assumed that the first machines m1 and (m11 , m21 ) are never starved and the last machines mM and m0M0 are never blocked.) To alleviate this mutual interference, material handling devices are used not only as means for moving parts but also as buffers – to protect against blockages and starvations. If the buffers are infinite, only starvations are possible, and the parts flow is improved (since the disturbances propagate only downstream). However, infinite buffers are impractical, economically detrimental, and, as it turns out, unnecessary as well.

6

CHAPTER 1. INTRODUCTION

This textbook presents rigorous engineering methods for reducing mutual machine interference by selecting buffer capacities, which are “just right” (rather than just-in-time) for guaranteeing a sufficiently smooth flow of parts and, thus, resulting in an acceptable system behavior. To accomplish this, three main problems of production systems engineering are addressed. These problems are described next.

1.2.2

Analysis, continuous improvement, and design problems

Analysis: The problem of analysis is addressed in this book in two formulations: Problem A1: Given a production system (i.e., the machine and buffer characteristics), calculate its performance measures, such as its throughput, workin-process, and the probabilities of blockages and starvations. The second formulation is concerned with systems having a Finished Goods Buffer (FGB), as shown in Figure 1.3 for the case of a serial line; similarly, assembly systems may have FGBs. The purpose of FGBs is to filter out production and demand randomness and, thus, improve the level of customer demand satisfaction. Demand (D, T) m1

b1

m2

b2

m M−1 b

M−1

mM

FGB

Figure 1.3: Serial production line with a finished goods buffer Problem A2: Given a production system and customer demand specifications (i.e., the shipment size, D, and the shipping period, T ), calculate the probability of demand satisfaction. Continuous improvement: The problem of continuous improvement is also considered in two formulations – constrained and unconstrained improvability. Problem CI1 (Constrained improvability): Given a production system, determine if its performance can be improved by re-allocating its limited resources (such as buffer capacity and/or workforce) and suggest an improved or even optimal allocation. Problem CI2 (Unconstrained improvability): Given a production system, determine the machine and the buffer, which impede system performance in the strongest manner; such a machine and a buffer are referred to as a bottleneck machine (BN-m) and a bottleneck buffer (BN-b), respectively. Design: Design issues are formalized as three problems: Problem D1: Given the machine characteristics and the desired system throughput, determine the smallest (i.e., lean or “just-right”) buffers capacity, which ensures this throughput.

1.2. MAIN PROBLEMS

7

Problem D2: Given the machine and buffer parameters, determine the locations of quality inspection devices so that no defective parts are shipped to the customer and the throughput of non-defective parts is maximized. Problem D3: Given a production system and customer demand specification, calculate the smallest (i.e., lean or “just-right”) capacity of FGB so that the demand is satisfied with the desired probability. Solutions to all problems listed above are provided in Part II of this textbook for the case of serial lines with the Bernoulli model of machine reliability. A generalization for exponential and arbitrary machines is given in Part III. For the case of assembly systems, these problems are addressed in Part IV.

1.2.3

Fundamental laws of Production Systems Engineering

Each engineering discipline is based on a fundamental law that is the foundation for all its analysis and design methods. For instance, in Mechanical Engineering this is Newton’s law: F = ma. Given system parameters, i.e., the force F and mass m, this law allows us to calculate the acceleration a and, subsequently, the behavior, i.e., trajectory, of a moving body. Similarly, in Electrical Engineering, Kirchoff’s law, E=

n X

Vi ,

i=1

allows us to investigate the behavior of an electric circuit, since the voltage drops Vi on each of its elements (i.e., resistors, capacitors, etc.) can be calculated, given the applied voltage E. In Industrial Engineering, Little’s law is often mentioned as a tool for production systems analysis. This law states that, in the steady state of system operation, W IP , (1.1) TP = RT where T P is the average throughput, W IP is the average work-in-process, and RT is the average residence time, i.e., the average time that the part spends in the system – from the moment it enters the first machine to the moment it leaves the last machine. Although this is an important relationship, it does not offer a direct way to investigate production systems. The reason is that all three quantities involved in (1.1) are unknown: neither T P , nor W IP , nor RT are given in advance. Given are only parameters of the machines and buffers and, unless at least two of these quantities are calculated, (1.1) does not offer any quantitative information concerning performance of a production system. In this textbook, methods for calculating T P , W IP and other performance measures, based on machine and buffer parameters, are developed and various relationships among them are established. Some of these relationships, which

8

CHAPTER 1. INTRODUCTION

can be viewed as fundamental facts of production systems, are summarized in Chapter 18.

1.2.4

Techniques used in this textbook

Often, production systems are studied using the methods of Queuing Theory. The classical model of Queuing Theory is illustrated in Figure 1.4. Here, customers arrive into a queue according to some randomly distributed inter-arrival times. There are M servers defined by their processing capacity. After reaching the head of the queue, the customer is processed by the first available server during a randomly distributed service time. Given this model, Queuing Theory develops methods for analysis of its performance characteristics: the average queue length, throughput, residence time, etc.

Customers

1

2

M

Servers

Figure 1.4: Queuing model Unfortunately, production systems typically have a qualitatively different structure. Indeed, instead of servers operating in parallel, they have a serial connection as shown in Figures 1.1 and 1.2. In some cases, however, methods of Queuing Theory may be modified to be applicable to production systems as well. This approach is not pursued here. Instead, methods, which are directly applicable to serial lines and assembly systems, are developed. In most cases, they are based on a three-step approach: First, the simplest systems, consisting of two machines, are investigated using the methods of Markov processes. Second, systems with more than two machines are analyzed using the results of the first step and recursive aggregation procedures; the convergence of these procedures and their accuracy are investigated. Third, these results are extended to non-Markovian cases by discrete event simulations and subsequent analytical approximations. In this framework, the recursive procedures mentioned above are viewed as governing equations of production systems. Their analyses, described throughout this book, lead to formulation of the fundamental facts of Production Systems Engineering.

1.3. SUMMARY

1.3

9

Summary

• This textbook addresses one of the five main areas of Manufacturing described in Section 1.1: Production Systems with unreliable machines and finite buffers. • The fundamental difficulty in studying these systems is due to the phenomena of blockages and starvations caused primarily by machine breakdowns. • The topics covered in this textbook are – analysis, – continuous improvement, and – design of production systems in large volume manufacturing environment. • A system-theoretic approach is used throughout this volume.

1.4

Problems

In all problems listed below, use your common sense to provide and justify your answers. Later in this textbook, you will learn rigorous techniques to solve these and many other problems of the same nature. Problem 1.1 Consider a serial production line with ten identical machines and no buffers. Where do you think a single buffer should be placed so that the throughput of the system is maximized? Problem 1.2 Consider a serial production line with eleven identical machines and ten identical buffers. Assume that one of the machines can be replaced by a more efficient one. Which machine should be replaced so that the throughput is maximized? Problem 1.3 Assume that two types of machines are available: those with short average up- and downtime and those with long average up- and downtime but with the same efficiency e=

1 Tup = , Tup + Tdown 1 + Tdown /Tup

(1.2)

where Tup and Tdown are the average up and downtime, respectively. Which types of machines will you buy for your production line so that the throughput is optimized? Problem 1.4 If the efficiency of a machine in a production line can be improved, is it better to increase its uptime by a factor of α or decrease its downtime by the same factor, so that the system throughput is maximized?

10

CHAPTER 1. INTRODUCTION

Problem 1.5 In a five-machine production line with identical machines and identical buffers, which machine is the bottleneck? Problem 1.6 In a production line where parts are transported on carriers, will the throughput necessarily increase if the number of carriers is increased? Problem 1.7 Is the machine with the smallest throughput in isolation necessarily the bottleneck machine of a production line? Is the smallest buffer necessarily the bottleneck buffer? Problem 1.8 Is an in-process buffer of capacity N = 1000 lean or not? Is a finished goods buffer of capacity N = 1000 lean or not? Problem 1.9 If the only machine that produces defective parts in a serial line is the first one, where should a single quality inspection machine be placed so that the throughput of good parts is maximized? Problem 1.10 Let the term “transients” describe the process of reaching steady state values of either throughput (T P ) or work-in-process (W IP ), or buffer occupancy. Which transients are faster, those of T P , W IP , or buffer occupancy? Problem 1.11 Will the transients of buffer occupancy become faster or slower when machine efficiency is increased? Problem 1.12 What should be the smallest buffer occupancies at the beginning of the shift so that no production losses due to transients take place?

1.5

Annotated Bibliography

There are literally millions of publications in all five areas of Manufacturing. In fact, a quick Google Scholar search returns over five million entries. However, when we started writing this book (2003), the title “Production Systems Engineering” did not appear even once! The references given below are the closest to the subject matter of this textbook. Perhaps, the first publications on quantitative analysis of production lines appeared in Russia: [1.1] A.P. Vladzievskii, “The Theory of Internal Stocks and their Influence on the Output of Automatic Lines,” Stanki i Instrumenty, vol. 21, no. 12, pp. 4-7, 1950 and vol. 22, no. 1, pp. 16-17, 1951 (in Russian). [1.2] A.P. Vladzievskii, “The Probability Law of Operation of Automatic Lines and Internal Storage in Them,” Automatika i Telemekhanika, vol. 13, pp. 227-281, 1952 (in Russian).

1.5. ANNOTATED BIBLIOGRAPHY

11

[1.3] B.A. Sevast’yanov, “Influence of Storage Bin Capacity on the Average Standstill Time of a Production Line,” Theory of Probability Applications, pp. 429-438, 1962. It should be pointed out that in 1957, A.N. Kolmogorov, the founder of Probability Theory, gave a lecture at a meeting of the Moscow Mathematical Society devoted to production systems. Unfortunately, no record of this presentation could be found. (Since Sevastianov was a student of Kolmogorov, it is reasonable to assume that Kolmogorov’s lecture contained ideas close to those of [1.3].) One of the first papers on quantitative analysis of production systems published outside Russia is [1.4] J.A. Buzacott, “Automatic Transfer Lines with Buffer Stocks,” International Journal of Production Research, vol. 5, pp. 183-200, 1967. The decomposition approach to production systems analysis appeared in [1.5] S.B. Gershwin, “An Efficient Decomposition Method for the Approximate Evaluation of Tandem Queues with Finite Storage Space and Blocking,” Operations Research, vol. 35, pp. 291-305, 1987. Among the textbooks and monographs devoted largely, but not exclusively, to Production Systems Engineering, the following are well known: [1.6] J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993. This is an encyclopedic book covering many topics of manufacturing, including production systems, scheduling, and others. Mostly the queueing theory approach is used. [1.7] H.T. Papadopoulos, C. Heavey and J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design, Chapman & Hall, London, 1993. One of the first monographs on the queueing theory approach to production systems. It includes results on serial lines, assembly systems, and flexible manufacturing operations. [1.8] S.B. Gershwin, Manufacturing Systems Engineering, Prentice Hall, Englewood Cliffs, NJ, 1994. A pioneering book as far as the system-theoretic approach to manufacturing is concerned. It addresses the issues of production systems performance analysis and scheduling. The performance analysis is based on a decomposition technique and scheduling uses methods of optimal control. [1.9] D.D. Yao (ed.), Stochastic Modeling and Analysis of Manufacturing Systems, Springer-Verlag, Series in Operations Research, New York, 1994.

12

CHAPTER 1. INTRODUCTION

[1.10 ] H.G. Perros, Queueing Networks with Blocking, Oxford University Press, Oxford, 1994. [1.11 ] T. Altiok, Performance Analysis of Manufacturing Systems, Springer, New York, 1997. Also is based on queuing theory. A characteristic feature is the study of non-exponential reliability models, e.g., co-axial distributions of machine up- and downtimes. Additional results on production systems engineering can be found in [1.12 ] N. Viswanadham and Y. Narahari, Performance Modeling of Automated Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1992. [1.13 ] R.G. Askin and C.R. Standridge, Modeling and Analysis of Manufacturing Systems, Wiley, 1993. [1.14 ] W.J. Hopp and M.L. Spearman, Factory Physics, Irwin/McGraw-Hill, Boston, 1996. [1.15 ] C.L. Curry, Manufacturing Systems Modeling and Analysis, Duxbury Press, to appear. and in the proceedings of the following conferences: [1.16 ] Proceedings of the International Workshop on Performance Evaluation and Optimization of Production Lines, Samos Island, Greece, 1997. [1.17 ] Proceedings of the Second Aegean International Conference on Analysis and Modeling of Manufacturing Systems, Tinos Island, Greece, 1999. [1.18 ] Proceedings of the Third Aegean International Conference on Design and Analysis of Manufacturing Systems, Tinos Island, Greece, 2001. [1.19 ] Proceedings of the Fourth Aegean International Conference on Analysis of Manufacturing Systems, Samos Island, Greece, 2003. [1.20 ] Proceedings of the Fifth International Conference on Analysis of Manufacturing Systems - Production Management, Zakynthos Island, Greece, 2005. [1.21 ] Proceedings of the Sixth International Conference on Analysis of Manufacturing Systems, Lunteren, The Netherlands, 2007. The aphorism, “Manufacturing matters,” cited in Section 1.1, was advanced by Gary Cowger of the General Motors Corporation. He stated it during a seminar at the University of Michigan in 1988.

Chapter 2

Mathematical Tools: Elements of Probability Theory Motivation: As it is described in Chapter 1, the behavior of production systems is strongly affected by random events, such as machine breakdowns or random customer demand. These events lead to other random phenomena, such as unpredictable throughput, defective parts, blockage and starvation of machines, unreliable satisfaction of customer demand, etc. Given this situation, it is no surprise that Probability Theory is the main mathematical tool of Production Systems Engineering. The purpose of this chapter is to review elements of Probability Theory in the framework of production systems and introduce characteristics of unreliable machine performance. Readers familiar with Probability Theory are still encouraged to read this chapter since it contains a number of Production Systems Engineering facts used elsewhere in this book. Overview: Probability Theory consists of three parts: • Random Events: This part deals with events of unpredictable nature, e.g., a machine in a production system is up or down. • Random Variables: Deals with numbers associated with random events, e.g., an interval of time during which a machine is up or down. • Random Processes: Deals with sequences of random variables, e.g., a sequence of time intervals during which the machine is up or down. Each of these three parts consists of formulas for calculating various probabilities. These formulas are: • the formula for the probability of a union of events, • the formula for the probability of an intersection of events, 13

14

CHAPTER 2. MATHEMATICAL TOOLS • the conditional probability formula, • the total probability formula, and • Bayes’s formula.

In this chapter, these formulas are overviewed and discussed in the framework of the simplest production systems.

2.1 2.1.1

Random Events Terminology

Chance experiment (E) – something that can be repeated arbitrarily many times under seemingly identical conditions but with different results, e.g., the status (up or down) of a machine at, say, 9:30 am. Outcome (ζ) – a result of a chance experiment, e.g., the machine is up (or down) at 9:30 am on a particular day. Sample space (S) – the set of all possible outcomes of a chance experiment, e.g., for a two-machine production system S = {ζ1 =(up, up), ζ2 =(up, down), ζ3 =(down, up), ζ4 =(down, down)}. Event (A, B, C, . . .) – a subset of S consisting of one or more outcomes, e.g., for the two-machine system, A = {both machines are in the same state} = {ζ1 , ζ4 }, B = {at least one machine is up} = {ζ1 , ζ2 , ζ3 }. Probability (P ) – a measure of likelihood of various events. Typically, the probabilities stem from practical experience as frequencies of various events, e.g., if during a long period of observation it is determined that the machine is up 90% of the time, then it is assumed that P [{machine is up}] = 0.9.

2.1.2

Axioms of probability and their corollaries

While the probabilities of some events of a chance experiment E might be available from observations, the probabilities of others are not. For example, P [{machine is up}] might be known, while P [{all machines of a production system are up simultaneously}] must be calculated. The axioms of probability are three postulates, which, on one hand, are obvious enough to be accepted as truth and, on the other hand, are powerful enough to be useful for calculating probabilities of practically all events of interest. These axioms are: I. The probability of each event is non-negative: P [A] ≥ 0. II. The probability of the sample space is 1: P [S] = 1. III. If two events, A and B, do not have outcomes in common (i.e., they are non-intersecting or mutually exclusive or disjoint in the sense that A ∩ B = ∅), then P [A ∪ B] = P [A] + P [B]. The first of these axioms states that the likelihood of any event cannot be negative. The second implies that the likelihood of any event cannot be greater than 1 and that something indeed happens as a result of a chance experiment.

2.1. RANDOM EVENTS

15

The third allows us to calculate the probability of the union of two (and, by extension, any finite number) of non-intersecting events. (A generalization of this axiom for an infinite number of events is not of immediate importance for our purpose.) The following are corollaries of the axioms: • If the union of A and B is the sample space S and the intersection is empty, i.e., A ∪ B = S, A ∩ B = ∅, then P [B] = 1 − P [A]. The set B, satisfying above relationships, is called the complement of A and is denoted as Ac . Figure 2.1, where the rectangle represents S, illustrates this corollary.

S

c

B=A

A

Figure 2.1: Illustration of complement • If A is included in or equals to B, i.e., A ⊆ B, then P [A] ≤ P [B]. See Figure 2.2 for an illustration.

S A

B

Figure 2.2: Illustration of inclusion • The probability of the union of two events, A and B in S, is given by P [A ∪ B] = P [A] + P [B] − P [A ∩ B].

(2.1)

The negative term in (2.1) is due to the fact that the outcomes, common for A and B (i.e., the intersection of A and B), are counted twice in the union of A and B (see Figure 2.3).

16

CHAPTER 2. MATHEMATICAL TOOLS

S A

B

Figure 2.3: Illustration of union and intersection Expression (2.1) is the first of the mentioned above important formulas of Probability Theory. It allows us to calculate the probability of the union of two events, mutually exclusive or not, provided that the probability of their intersection is known. A generalization of (2.1) for three events is: P [A ∪ B ∪ C]

= P [A] + P [B] + P [C] − P [A ∩ B] − P [A ∩ C] −P [B ∩ C] + P [A ∩ B ∩ C].

Similarly, for any finite number of events A1 , . . ., An , P [∪nk=1 Ak ]

=

n X

P [Ai ] −

n−1 X

i=1

+

n−2 X n−1 X

n X

P [Ai ∩ Aj ]

i=1 j=i+1 n X

P [Ai ∩ Aj ∩ Al ]

i=1 j=i+1 l=j+1

− . . . + (−1)n+1 P [A1 ∩ A2 ∩ . . . ∩ An ].

2.1.3

Conditional probability

Consider two disjoint events, A and B, shown in Figure 2.4(a). Assume that A takes place (i.e., the chance experiment E results in an outcome ζ, which belongs to A). What is the probability that B takes place? Clearly, the answer is 0.

S

S A

B (a)

A

B

(b)

Figure 2.4: Illustration of conditional probability Consider now the two events shown in Figure 2.4(b). Assume again that A takes place and ask the same question. Clearly, the answer depends on the probability of the outcomes common for A and B, i.e., P [A ∩ B]. But since A

2.1. RANDOM EVENTS

17

has occurred, the sample space for the question at hand is not S but A, and the value of P [A ∩ B] should be viewed relative to P [A] rather than P [S] = 1. Therefore, the probability of B given A, referred to as the conditional probability of B given A and denoted as P [B|A], is defined by P [B|A] =

P [A ∩ B] , P [A]

(2.2)

(assuming, of course, that P [A] > 0). This is the second important formula of Probability Theory. It allows us to calculate the probability of the intersection of two events, if the conditional probability is known, i.e., P [A ∩ B] = P [B|A]P [A] = P [A|B]P [B],

(2.3)

or the conditional probability, if the probability of the intersection is given. Example 2.1 Consider a two-machine line in which each machine is up with probability 0.75 and down with probability 0.25. What is the probability that the line is up (i.e., both machines are up simultaneously), if at least one machine is up? Solution: The sample space for the two machine system is: S = {ζ1 = (up, up), ζ2 = (up, down), ζ3 = (down, up), ζ4 = (down, down)}. The events of interest and their representations as subsets of the sample space are: A = {at least one machine is up} = {ζ1 , ζ2 , ζ3 }, B = {both machines are up} = {ζ1 }. The problem is to calculate P [B|A]. This, according to (2.2), requires the knowledge of A ∩ B, which is given by: A ∩ B = {ζ1 , ζ2 , ζ3 } ∩ {ζ1 } = {ζ1 }. Thus, using (2.2), P [B|A] =

P [{ζ1 }] P [ζ1 ] P [A ∩ B] = = , P [A] P [{ζ1 , ζ2 , ζ3 }] P [ζ1 ] + P [ζ2 ] + P [ζ3 ]

(2.4)

where the last expression is due to Axiom III. Although this expression provides an analytical characterization of the probability that the line is up when at least one machine is up, it does not lead to a specific value for this probability since the problem formulation does not provide the probabilities of the outcomes, ζi , i = 1, 2, 3, 4. However, these probabilities can, in fact, be easily calculated, if it is assumed that the machines are independent. The notion of independence is discussed next.

18

2.1.4

CHAPTER 2. MATHEMATICAL TOOLS

Independence

Two events, A and B in S, are independent if P [B|A] = P [B].

(2.5)

In practical terms, this implies that occurrence of A has no bearing on the likelihood of B. In mathematical terms, (2.5) implies that, as it follows from (2.2), P [A ∩ B] = P [A]P [B]. (2.6) This is the third important formula of Probability Theory - it provides a simple way to calculate the probability of the intersection of two events, provided, of course, that these events are independent. Sometimes, (2.6) is used as the definition of independence of two events, in which case (2.5) is a consequence. This approach is convenient to define the independence of more than two events. Specifically, three events A1 , A2 , and A3 in S are independent if they are pair-wise independent, i.e., P [A1 ∩ A2 ] = P [A1 ∩ A3 ] =

P [A1 ]P [A2 ], P [A1 ]P [A3 ],

P [A2 ∩ A3 ] =

P [A2 ]P [A3 ].

and P [A1 ∩ A2 ∩ A3 ] = P [A1 ]P [A2 ]P [A3 ]. Similarly, n ≥ 3 events are independent if all subsets of n − 1 of them are independent and P [A1 ∩ A2 ∩ . . . ∩ An ] = P [A1 ]P [A2 ] · · · P [An ]. Example 2.2 In addition to the scenario described in Example 2.1, assume that the machines are independent in the sense that they go up and down independently from each other and calculate the same probability. (Note that in practice, the independence assumption may or may not be true, depending, for instance, on the availability of skilled trades - to make the repair of one machine independent from the other.) Solution: Assuming independence, the probability of each outcome in S can be easily calculated using (2.6): P [ζ1 ] = = P [ζ2 ] = =

(0.75)(0.75) = 0.5625, P [{machine 1 is up}]P [{machine 2 is down}] (0.75)(0.25) = 0.1875,

P [{machine 1 is up}]P [{machine 2 is up}]

P [ζ3 ] =

P [{machine 1 is down}]P [{machine 2 is up}]

= (0.25)(0.75) = 0.1875.

2.1. RANDOM EVENTS

19

Therefore, (2.4) results in: P [{two-machine line is up}|{at least one machine is up}] = P [B|A] P [ζ1 ] 0.5625 = = = 0.6. P [ζ1 ] + P [ζ2 ] + P [ζ3 ] 0.9375

2.1.5

Total probability formula

Often, one needs to calculate the probability of an event given its conditional probabilities and the probabilities of the conditions. The instrument to accomplish this is the total probability formula. Consider a set of events Ai in S, i = 1, . . . , n, and assume that they are mutually exclusive and their union is S, i.e., Ai ∩ Aj ∪ni=1 Ai

= =

∅, S.

i 6= j,

Due to the obvious reason, such a set of events is called a partitioning of S (see Figure 2.5).

S

A1

A4 B

A2

A3

Figure 2.5: Illustration of partitioning Let B be another set in S. Clearly {B ∩ A1 } ∪ {B ∩ A2 } ∪ . . . ∪ {B ∩ An } = B. Therefore, using the formula for the probability of a union of non-intersecting events (Axiom III), P [B] = P [B ∩ A1 ] + P [B ∩ A2 ] + . . . + P [B ∩ An ]. Applying the formula for the probability of the intersection of two events (2.3), we obtain: P [B]

= P [B|A1 ]P [A1 ] + P [B|A2 ]P [A2 ] + . . . + P [B|An ]P [An ] n X = P [B|Ai ]P [Ai ]. (2.7) i=1

This is the total probability formula, which is the fourth important formula of Probability Theory.

20

CHAPTER 2. MATHEMATICAL TOOLS

Example 2.3 An automotive assembly plant receives transmissions from three suppliers, A, B, and C. Supplier A has 15% of its transmissions failed within ten years of operation. For suppliers B and C, these numbers are 40% and 25%, respectively. The assembly plant receives 45% of its transmissions from supplier A, 25% from B, and 30% from C. What fraction of cars will have a transmission failure within ten years of operation? Solution: Let F denote the following event: F = {a transmission fails within ten years of operation} Then the example data can be formalized as follows: P [F |A] = 0.15,

P [F |B] = 0.4,

P [F |C] = 0.25,

P [A] = 0.45,

P [B] = 0.25,

P [C] = 0.3.

Thus, by the total probability formula (2.7), the fraction of cars with transmission failure within ten years of operation is: P [F ]

=

P [F |A]P [A] + P [F |B]P [B] + P [F |C]P [C]

= (0.15)(0.45) + (0.4)(0.25) + (0.25)(0.3) = 0.2425.

2.1.6

Bayes’s formula

Continuing the above example, assume that the transmission in a car failed within ten years of operation. What is the probability that this transmission came from supplier A? or from supplier B? or from supplier C? Bayes’s formula is an instrument to answer this question. Let that Ai , i = 1, 2, . . . , n, be a partitioning of S, and F an event in S. Then, by the conditional probability formula, P [Ai |F ] =

P [Ai ∩ F ] . P [F ]

(2.8)

Using the formula for the probability of intersection of two events in the numerator of (2.8) and the total probability formula in the denominator of (2.8), we obtain: P [Ai |F ] =

P [F |Ai ]P [Ai ] P [F |A1 ]P [A1 ] + P [F |A2 ]P [A2 ] + . . . + P [F |An ]P [An ]

(2.9)

This expression is referred to Bayes’s formula and is the fifth important formula of Probability Theory. Example 2.4 Within the scenario of Example 2.3 and given that the transmission in a car that failed within ten years of operation, what is the probability that this transmission came from supplier A? B? or C?

2.2. RANDOM VARIABLES

21

Solution: By Bayes’s formula, P [A|F ] = = P [B|F ] = = P [C|F ] = =

2.2 2.2.1

P [F |A]P [A] P [F |A]P [A] + P [F |B]P [B] + P [F |C]P [C] (0.15)(0.45) = 0.278, (0.15)(0.45) + (0.4)(0.25) + (0.25)(0.3) P [F |B]P [B] P [F |A]P [A] + P [F |B]P [B] + P [F |C]P [C] (0.4)(0.25) = 0.413, (0.15)(0.45) + (0.4)(0.25) + (0.25)(0.3) P [F |C]P [C] P [F |A]P [A] + P [F |B]P [B] + P [F |C]P [C] (0.25)(0.3) = 0.309. (0.15)(0.45) + (0.4)(0.25) + (0.25)(0.3)

Random Variables Terminology

Random variable (X, Y , Z or M , N , . . .) – a function that associates a number with each outcome of a chance experiment, i.e., X(ζ) = x ∈ SX ⊆ R,

ζ ∈ S.

(2.10)

For instance, if ζ1 = {machine A is down}, ζ2 = {machine B is down} and machines A and B stay down 1 and 2 hours, respectively, then X(ζ1 ) = 1,

X(ζ2 ) = 2.

Random variables are typically denoted by capital letters, while their values (referred to as realizations) by their lower case counterparts (say, X and x, as in (2.10) or N and n). Discrete random variable – X(ζ) takes at most countable number of values, e.g., the number of cycles when the machine is up during a shift, where the cycle is the time necessary to process one part. Continuous random variable – X(ζ) takes values on the real line, e.g., the time during which the machine is up within a shift. Mixed random variable – X(ζ) has both discrete and continuous parts. A new capability, offered by random variables in comparison with random events, is that numbers admit arithmetic operations, and this leads to important notions of expected values, variances, correlations, etc. Although theoretically random variables are defined as functions, such as (2.10), in practice they are often introduced by the probabilities (or probability distributions - see below), with which their realizations take place. In this case,

22

CHAPTER 2. MATHEMATICAL TOOLS

their sample spaces are viewed as discrete or continuous sets of numbers, denoted as SX . Below, the three types of random variable are described, along with specific examples, which are used in the subsequent chapters for analysis and design of production systems.

2.2.2

Discrete random variables

A discrete random variable is defined by the probabilities with which it takes various values in its sample space. Assume X takes the following values: x1 , x2 , . . ., xK (i.e., SX = {x1 , x2 , . . . , xK }); then X is defined by P [X = xi ] K X

=

PX (xi ) =

PX (xi ),

i = 1, . . . , K,

1.

i=1

Function PX (xi ), i = 1, . . . , K, is called the probability mass function (pmf). Several useful examples of discrete random variables are given next. Bernoulli random variable. Here SX = {0, 1} and pmf is given by P [X = 0] = 1 − p,

P [X = 1] = p.

This random variable will be used to model the status of a machine. Specifically, we assume that P [{machine is up during a cycle}] = P [{machine is down during a cycle}] =

P [{X = 1}] = p, P [{X = 0}] = 1 − p.

If the status of the machine in each cycle is independent from its status in all other cycles, it is said that the machine is obeying the Bernoulli reliability model and is referred to as a Bernoulli machine. This reliability model is often appropriate for assembly operations where the down time is short enough to be comparable with the length of cycle time (see Chapter 3 for details). Another example of the Bernoulli random variable arises in machines, which can produce either good or defective parts. Indeed, assume that P [{part produced by a machine during a cycle time is good}] = g, P [{part produced by a machine during a cycle time is defective}] = 1 − g. If the quality of parts produced in each cycle is independent from that in other cycles, such a machine is referred to as obeying the Bernoulli quality model. This model is appropriate, for example, for automotive painting operations where the defects are due to independent random events, such as dust, scratches, etc. (see Chapter 8 for more details). Binomial random variable. Consider a Bernoulli machine, which operates during a shift consisting of N cycles. Clearly, the number of cycles, during which

2.2. RANDOM VARIABLES

23

the machine is up, is a random variable taking values on SX = {0, 1, . . . , N }. As it is shown below, the pmf of this random variable contains binomial coefficients and, therefore, it is referred to as binomial. To derive its pmf, consider the event AK = {during N cycles, the machine is up K times}. This event consists of outcomes, which are N -tuples with K 1s and N − K 0s. How many outcomes are in AK ? Since the first 1 can be placed in N different positions, the second in N −1 positions, . . ., and the K-th in N −K +1 positions, ! the total number of various possibilities is N (N − 1) · · · (N − K + 1) = (N N −K)! . But all 1s are identical; therefore, they may be permuted K! times without changing the N -tuple. Thus, the number of distinct N -tuples (or¡the ¢ number N N! . , which is the binomial coefficient K of outcomes in AK ) is (N −K)!K! The probability of each outcome in AK , calculated using formula (2.6) for the probability of the intersection of independent events, is pK (1 − p)N −K . Therefore, based on the formula for the probability of the union of mutually exclusive events (Axiom III), we obtain µ ¶ N K p (1 − p)N −K , K = 0, 1, . . . , N, (2.11) P [AK ] = K which is the pmf of the binomial random variable. This pmf is illustrated in Figure 2.6. 0.4

P[AK]

0.2

P[A ] K

0.18

0.35

0.16

0.3

0.14

0.25

0.12

0.2

0.1

0.15

0.08 0.06

0.1

0.04

0.05 0

0.02

0

2

4

6

8

10

K

(a) p = 0.1, N = 10

0

0

5

10

15

20

K

(b) p = 0.3, N = 20

Figure 2.6: Binomial random variable Example 2.5 What is the probability that a Bernoulli machine produces at least 5 parts during a shift consisting of 7 cycles, if in each cycle the machine is up with probability 0.9? Solution: The event of interest is B

= {at least 5 parts produced in 7 cycles} = {5 parts produced} ∪ {6 parts produced} ∪ {7 parts produced} = A5 ∪ A6 ∪ A7 .

24

CHAPTER 2. MATHEMATICAL TOOLS

Thus, P [B]

µ ¶ µ ¶ µ ¶ 7 5 7 6 7 7 p (1 − p) + p p (1 − p)2 + 5 6 7 7! (0.9)5 (0.1)2 + 7(0.9)6 (0.1) + (0.9)7 = 5!2! = 0.9743.

=

Poisson random variable. Consider a binomial random variable and assume that N → ∞, p→0 in such a manner that N p = ν = const. Then it is possible to show that formula (2.11) reduces to P [AK ] =

ν K −ν e , K!

K = 0, 1, 2, . . . .

(2.12)

This pmf, referred to as Poisson pmf, is convenient to use when a binomial random variable is characterized by a large N . Example 2.6 Assume that a machine obeys the Bernoulli quality model with g = 0.999. Calculate the probability that a shipment of 1000 parts has no more than four defectives. Solution: Since the number of trials (i.e., the parts in a shipment) is large and the probability of success (in this example, defective) is small, we can use Poisson pmf with ν = N (1 − g) = (1000)(0.001) = 1. The event of interest is B = {shipment has no more than 4 defective parts} and, thus, P [B] =

4 X

e−ν

k=0

= =

νk k!

1 1 1 1 1 + + + + ) 0! 1! 2! 3! 4! 0.99634

e−1 (

Geometric random variable. Consider now a machine with a different reliability model. Specifically, assume that the status of the machine in each cycle depends on its status in the previous cycle as shown in the transition diagram of Figure 2.7. In this situation, the machine is a dynamic system (since it has one step memory), and, therefore, its status can be referred to as the state of

2.2. RANDOM VARIABLES

25 P

1− R

down

up

1− P

R

Figure 2.7: Geometric machine the machine. Thus, if α(s) ∈ {0, 1} denotes the state of the machine in cycle (or slot) s, Figure 2.7 implies that, if α(s) = 1, then P [α(s + 1) = 0] = P and P [α(s + 1) = 1] = 1 − P . Similarly, if α(s) = 0, then P [α(s + 1) = 1] = R and P [α(s + 1) = 0] = 1 − R. In terms of conditional probabilities, this can be described as P [α(s + 1) = 0|α(s) = 1] = P,

P [α(s + 1) = 1|α(s) = 1] = 1 − P,

P [α(s + 1) = 1|α(s) = 0] = R,

P [α(s + 1) = 0|α(s) = 0] = 1 − R.

A machine with such a reliability model is called a geometric machine because, as it is shown below, its uptime and downtime are described by a pmf referred to as geometric. Geometric machines are useful to model machining operation where the downtime is typically much longer than the cycle time (see Chapter 3). To derive the pmf of the geometric random variable, assume that before the first cycle the machine is up and calculate the probability that the first breakdown occurs during the t-th cycle, t = 1, 2, . . ., i.e., P [{transition from α = 1 to α = 0 occurs during t-th cycle}]. Clearly, this probability can be expressed as follows: P [{transition from α = 1 to α = 0 occurs during t-th cycle}] = P [{no transition in cycle 1} ∩ {no transition in cycle 2} ∩ . . . ∩{no transition in cycle t − 1} ∩ {transition in cycle t}] (2.13) = (1 − P )t−1 P, where the last expression is due to the formula for the probability of the intersection of independent events. Expression (2.13) defines a valid pmf and is referred to as geometric. A machine with up- and downtime distributed according to this pmf is referred to as obeying the geometric reliability model or just geometric machine. (Similar arguments can be used to introduce the geometric quality model where the machine has two states, producing good and defective parts in each.) With a slight abuse of notations, the uptime and downtime of a geometric machine are denoted by lowercase letters as tup and tdown (since Tup and Tdown are typically used for the expected value of these random variables). Thus, for a geometric machine, P [tup = t] = P (1 − P )t−1 ,

t = 1, 2, . . . ,

(2.14)

26

CHAPTER 2. MATHEMATICAL TOOLS

and, using similar arguments, P [tdown = t] = R(1 − R)t−1 ,

t = 1, 2, . . . .

(2.15)

Graphically, these pmf’s are illustrated in Figure 2.8. P[t =t]

P[tup=t]

up

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

5

10

15

20

25

t

0

0

(a) P = 0.3

5

10

15

20

25

t

(b) P = 0.15

Figure 2.8: Geometric random variable The geometric random variable is viewed as memoryless because, in particular, the probability to go down (or up) during each cycle does not depend on how long the machine has been up (or down) before the transition occurs. Typically, this is not true for real-world machines but often it is an acceptable idealization. The importance of this idealization, however, is in providing a possibility for analytical investigation of many production systems of interest.

2.2.3

Continuous random variables

Since continuous random variables have an uncountable number of outcomes, they cannot be defined by pmf’s and, therefore, are defined by distributions and their densities. To introduce them, consider a continuous random variable, X, taking values on the real line, i.e., SX ⊆ R. Let Ax denote the set Ax = {X ≤ x} (see Figure 2.9). Then the cumulative distribution function (cdf) of X is given by (2.16) FX (x) = P [Ax ] = P [{X ≤ x}] If FX (x) is differentiable, its probability density function (pdf) is given by fX (x) =

dFX (x) . dx

(2.17)

These functions, illustrated in Figure 2.10, have the following properties: • FX (x) is non-decreasing • FX (−∞) = 0 • FX (∞) = 1

2.2. RANDOM VARIABLES

27

Ax R

−

8

x

Figure 2.9: Illustration of set Ax fX(x)

F (x) X

1

0

x

x

(a) cdf

(b) pdf

Figure 2.10: Continuous random variable • fX (x) is non-negative Rx • −∞ fX (x)dx = FX (x) R∞ • −∞ fX (x)dx = 1 • limx→−∞ fX (x) = 0 • limx→∞ fX (x) = 0. Given cdf or pdf, one can calculate the probability of (practically) every set in R. Indeed (see Figure 2.11), Z x2 P {x1 < X ≤ x2 } = FX (x2 ) − FX (x1 ) = fX (x)dx. (2.18) x1

Expression (2.18) provides the physical meaning of the pdf: If δ ¿ 1, then Z x+ δ2 h δi δ = P x− <X ≤x+ fX (x)dx = fX (x)δ + o(δ), 2 2 x− δ2 where o(δ) denotes the higher order terms in δ, i.e., lim

δ→0

o(δ) = 0. δ

Thus, fX (x)δ is the probability that X takes values close to x.

(2.19)

28

CHAPTER 2. MATHEMATICAL TOOLS F (x) X 1

x1

x2

x

Figure 2.11: Probability calculations for a continuous random variable The conditional cdf and pdf are defined by FX (x|A)

=

P [X ≤ x|A] =

fX (x|A)

=

dFX (x|A) , dx

P [{X ≤ x} ∩ {A}] , P [A]

where A is an event, serving as a condition. The total probability formulas for cdf and pdf become: FX (x) fX (x)

= =

n X i=1 n X

FX (x|Ai )P [Ai ],

(2.20)

fX (x|Ai )P [Ai ],

(2.21)

i=1

where Ai , i = 1, . . . , n, is the partitioning. Finally, Bayes’s formulas for cdf and pdf take the form: P [A|x1 < X ≤ x2 ] = P [A|X = x] =

[FX (x2 |A) − FX (x1 |A)]P [A] , P [x1 < X ≤ x2 ] fX (x|A)P [A] . fX (x)

Several useful examples of continuous random variables are given below. Exponential random variable. Consider a machine in Figure 2.12, which is a continuous time analogue of the geometric machine. Namely, if it is up (respectively, down) at time t, it goes down (respectively, up) during an infinitesimal time δt with probability λδt (respectively, µδt). The numbers λ and µ are called the breakdown and repair rates, respectively. They are not probabilities and,

2.2. RANDOM VARIABLES

29 λδ t+o (δt )

1− µδ t − o ( δ t)

down

1− λδ t − o ( δ t)

up

µδ t+o (δ t )

Figure 2.12: Exponential machine therefore, can take any positive value. Note, however, that λδt and µδt are probabilities and, therefore must be between 0 and 1. Using again α(t) as the notation for the state at time t, this machine can be described by the following conditional probabilities: P [α(t + δt) = 0|α(t) = 1]

=

λδt + o(δt),

(2.22)

P [α(t + δt) = 1|α(t) = 0]

=

µδt + o(δt),

(2.23)

where the terms o(δt), lim

δt→0

o(δt) = 0, δt

are added for generality. Denote as tup and tdown the up- and downtime of this machine, respectively, and calculate their pdf’s, ftup (t) and ftdown (t). Since both are evaluated in a similar manner, we calculate below ftup (t). Let P [α, t] = P [{machine is in state α at time t}], t ≥ 0, and assume P [1, 0] = 1,

(2.24)

i.e., the machine initially is up. By total probability formula (2.7), P [1, t + δt] =

P [α(t + δt) = 1|α(t) = 1]P [α(t) = 1] +P [α(t + δt) = 1|α(t) = 0]P [α(t) = 0].

(2.25)

Since by assumption the machine is up at t = 0, the second term in the right hand side of (2.25) is zero. Thus, using (2.22), it follows from (2.25) that P [1, t + δt] = (1 − λδt − o(δt))P [1, t]. Re-arranging the terms, we obtain o(δt) P [1, t + δt] − P [1, t] = −λP [1, t] + . δt δt In the limit as δt → 0, this results in a first order linear differential equation dP [1, t] = −λP [1, t] dt

30

CHAPTER 2. MATHEMATICAL TOOLS

with the initial condition (2.24). Its solution is P [1, t] = e−λt ,

t ≥ 0.

(2.26)

Now, the probability that the transition from α = 1 to α = 0 occurs during (t, t + δt] is: P [{α(t + δt) = 0} ∩ {α(t) = 1}]

=

P [α(t + δt) = 0|α(t) = 1}]P [α(t) = 1]

= (λδt + o(δt))e−λt = λe−λt δt + o(δt). Thus, according to (2.19), the pdf of tup is ftup (t) = λe−λt ,

t ≥ 0.

(2.27)

This pdf is illustrated in Figure 2.13. 0.9

ft (t) up

0.8 0.7 0.6 0.5

λ=0.9

0.4 0.3 0.2 0.1 0 0

λ=0.3

5

10

15

t

Figure 2.13: Exponential random variable A similar argument, applied to tdown , results in ftdown (t) = µe−µt ,

t ≥ 0.

(2.28)

A machine with up- and downtime distributed according to (2.27) and (2.28) is referred to as obeying the exponential reliability model or just exponential. Similar arguments can be used to introduce the exponential quality model. Rayleigh random variable. The exponential random variable, considered above, is memoryless in the same sense as its discrete time counterpart (i.e., the geometric random variable): the breakdown and repair rates, λ and µ are constant, irrespective of how long the machine was up or down. In reality, as it was pointed out before, this is not necessarily the case: empirical evidence shows that in many cases the breakdown and repair rates increase with time. What is the distribution of up- and downtime in this case? We show below that if the breakdown and repair rates are linearly increasing in time, the resulting random variables are characterized by what is known as the Rayleigh distribution.

2.2. RANDOM VARIABLES

31

Accordingly, consider the machine illustrated in Figure 2.14 with breakdown and repair rates λt and µt, respectively, where t is the time that the machine is continuously in one state. Using the arguments analogous to those used in the λ t δ t+o (δt )

1− µt δ t−o ( δ t)

down

up

1− λ t δ t − o ( δ t)

µ t δ t+o (δt )

Figure 2.14: Rayleigh machine derivation of the exponential distribution, we obtain the following differential equation: dP [1, t] = −λtP [1, t] dt with the initial condition P [1, 0] = 1. Its solution is P [1, t] = e−

λt2 2

,

t≥0

and, therefore, ftup (t) = λte−

λt2 2

,

t ≥ 0,

(2.29)

which is the pdf of the Rayleigh random variable (see Figure 2.15 for illustration). Similarly, for the downtime we obtain: ftdown (t) = µte−

µt2 2

,

t ≥ 0.

(2.30)

ft (t) up

0.6

λ=0.9

0.5 0.4 0.3

λ=0.3

0.2 0.1 0 0

1

2

3

4

5

6

7

8

t

Figure 2.15: Rayleigh random variable

32

CHAPTER 2. MATHEMATICAL TOOLS

Rayleigh and other distributions, considered below, might be more realistic than exponential because their most likely up- and downtimes are not necessarily around t = 0. Decreasing transition rates. If one makes a (not very practical) assumption that the breakdown and repair rates are decreasing in time, for instance, as √λt and √µt , then, based on similar arguments, one obtains the following pdf’s: ftup (t) = ftdown (t) =

2λ λ √ √ e− t , t 2µ µ −√ √ e t, t

t > 0,

(2.31)

t > 0.

(2.32)

This pdf is illustrated in Figure 2.16; note the longer “tail” of this distribution in 0.25

f (t) t

up

0.2

λ=0.9 0.15

0.1

λ=0.3

0.05

0 0

5

10

15

20

25

30

t

Figure 2.16: Random variable with decreasing transition rates comparison with that of Rayleigh – a feature that will be important in Chapter 12. Erlang random variable. Consider a serial production line with n identical exponential machines having the breakdown and repair rates λ and µ, respectively. What is the distribution of the sum of their uptimes (respectively, downtimes)? The answer to this question is the Erlang distribution: the sum of n independent exponential random variables each defined by parameter λ has the following pdf: ftup (t)

=

ftdown (t)

=

λ(λt)n−1 e−λt , (n − 1)! µ(µt)n−1 e−µt , (n − 1)!

t ≥ 0,

(2.33)

t ≥ 0.

(2.34)

An illustration of these functions is given in Figure 2.17. Gamma distribution. When n in the above functions is not necessarily an integer, the resulting distribution is called gamma since the gamma function,

2.2. RANDOM VARIABLES 0.9

33

ft (t)

0.9

up

0.8

up

0.7

0.7

0.6 0.5

ft (t)

0.8

0.6 0.5

n=1

0.4

0.4

0.3 0.2

0.3

n=3

n=1

0.2

n=6

0.1 0 0

5

10

n=3

0.1 15

20

25

0 0

t

30

5

n=6 10

(a) λ = 0.9

15

20

25

t

30

(b) λ = 0.3

Figure 2.17: Erlang random variable Γ(Λ), is involved in its representation: ftup (t) = ftdown (t) =

(λt)Λ−1 , Γ(Λ) (µt)M−1 , µe−µt Γ(M) λe−λt

where

Z

∞

Γ(x) =

t ≥ 0,

(2.35)

t ≥ 0,

(2.36)

sx−1 e−s ds.

0

See Figure 2.18 for an illustration; note that the tail of this pdf can be “con0.35

ft (t)

ft (t)

up

up

0.3 0.25

0.3

Λ=2

0.25

0.2

0.2

0.15

0.15

Λ=5

0.1

Λ=2

0.1

Λ=5 0.05

0.05 0 0

5

10

15

20

(a) λ = 0.9

25

30

t

0 0

5

10

15

20

25

30

t

(b) λ = 0.3

Figure 2.18: Gamma random variable trolled” by the value of Λ (respectively, M). Weibull distribution. This type of distribution is widely used in Reliability Theory. Similar to the gamma distribution, it is defined by two parameters, λ and Λ (respectively, µ and M), where Λ and M again control the tail. It is given

34

CHAPTER 2. MATHEMATICAL TOOLS

by ftup (t) ftdown (t)

= =

Λ

λΛ e−(λt) ΛtΛ−1 , M −(µt)M

µ e

Mt

M−1

t ≥ 0, ,

(2.37)

t≥0

(2.38)

and illustrated in Figure 2.19. 1.8

f (t) t

1.8

up

t

1.6

Λ=5

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

Λ=2

0.2 0 0

f (t) up

1.6

1

2

Λ=5 Λ=2

0.2

3

4

5

6

t

0 0

1

(a) λ = 0.9

2

3

4

5

6

t

(b) λ = 0.3

Figure 2.19: Weibull random variable Gaussian or normal random variable. Although this random variable plays a lesser role in the subject matter of this book than in other engineering disciplines, it leads to another pdf, which is of importance in manufacturing and, therefore, is considered here. This random variable characterizes a scaled version of a sum of many identically distributed, independent random variables. The resulting pdf of such a normalized sum, denoted as X, turns out to be defined by two parameters, m and σ, and is given by (x−m)2 1 e− 2σ2 fX (x) = √ 2πσ and is illustrated in Figure 2.20. Log-normal random variable. Assume that tup of a machine is given by tup = eX , where X is a Gaussian random variable. Then the pdf of the uptime turns out to be (lnt−λ)2 1 t > 0, (2.39) ftup (t) = √ e− 2Λ2 , 2πΛt which is referred to as log-normal. Similarly, for the downtime, ftdown (t) = √

(lnt−µ)2 1 e− 2M2 , 2πMt

t > 0.

(2.40)

2.2. RANDOM VARIABLES

35

f (x)

0.1

X

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −20

−15

−10

−5

0

m 5

10

15

20

x

Figure 2.20: Gaussian random variable

5

f (t) t

5

up

4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5

2 1.5

2

0 0

up

Λ=3

1.5

Λ=3

1 0.5

f (t) t

1 0.5

Λ=2 0.2

0.4

0.6

(a) λ = 0.9

0.8

1

t

0 0

Λ=2 0.2

0.4

0.6

(b) λ = 0.3

Figure 2.21: Log-normal random variable

0.8

1

t

36

CHAPTER 2. MATHEMATICAL TOOLS

Having two independent parameters, this pdf also has its tail controlled by their appropriate selection (see Figure 2.21 for illustrations). Remark on mixed random variables: As it is mentioned in Subsection 2.2.1, mixed random variables have both discrete and continuous parts. Similar to continuous random variables, they are defined by their cdf’s, FX (x), however, this function is not continuous anymore. The “jump” of a cdf at a point of the discontinuity is the probability of the argument at which this jump takes place. For instance, in the example shown in Figure 2.22(a),

1

FX(t)

f (t) X

∆

∆

0

x

*

x

0

x

x*

(a) cdf

(b) pdf

Figure 2.22: Mixed random variable P [X = x∗ ] = lim FX (x∗ + ²) − lim FX (x∗ − ²) = ∆. ²→0

²→0

If the derivative of FX (x) at a point of discontinuity is understood as a δfunction, then the pdf of a mixed random variable can be introduced in the usual manner: dFX (x) . fX (x) = dx For instance, if the continuous part of the mixed random variable shown in Figure 2.22(b) is distributed exponentially with parameter λ, then fX (x) = P [X = x∗ ]δ(x − x∗ ) + (1 − P [X = x∗ ])λe−λx ,

2.2.4

t ≥ 0.

Expected value, variance, and coefficient of variation

The expected value, variance, and coefficient of variation of a random variable characterize, roughly speaking, its average value, the width of its pdf, and the level of its “randomness,” respectively. Each of them is discussed below. Discrete random variables: Consider a discrete random variable X defined by its pmf PX (xk ), k = 1, . . . , K. Its expected value, x or E(X), variance, σ 2

2.2. RANDOM VARIABLES

37

or V ar(X), and the coefficient of variation, CV (X), are given, respectively, by x = E(X) =

K X

xk PX (xk ),

k=1

σ2

= V ar(X) =

K X

(xk − x)2 PX (xk ),

k=1

σ . CV (X) = x While the meaning of x and σ 2 is intuitively clear, CV requires the following comment: Small σ 2 obviously implies that the values that X takes are mostly close to x and, therefore, X is not “very random.” Assume, on the other hand, that σ 2 is large. Does this mean that X is very random? Not necessarily: if x is also large, X might still be viewed as relatively close to its mean. Therefore, the coefficient of variation, CV , which is a ratio of σ and x, is a measure of “randomness” of a random variable. Typically, long-tailed pmf’s (and pdf’s – see below) have CV > 1; for short-tailed pmf’s, CV < 1. The formulas for x, σ 2 , and CV of the Bernoulli, geometric, binomial, and Poisson random variables, introduced in Subsection 2.2.2, are given in Table 2.1. Note that for a geometric random variable, CV < 1, while for others CV can be either larger or smaller than 1, depending on the values of the parameters p, n, or ν. Table 2.1: Expectation, variance, and coefficients of variation of discrete random variables Random variable

Expectation

Variance

Bernoulli

p

p(1 − p)

Binomial

np

np(1 − p)

Geometric Poisson

1 P

1−P P2

ν

ν

qCV q √

1−p p 1−p np

1−P √1 ν

Continuous random variables: While the meaning of x, σ, and CV remains the same as above, the analytical expressions change by having integrals instead of the sums and pdf’s instead of pmf’s. Namely, if X is a continuous random variable and fX (x), x ∈ (−∞, ∞), is its pdf, then Z ∞ x = E(X) = xfX (x)dx, −∞ Z ∞ σ 2 = V ar(X) = (x − x)2 fX (x)dx, CV (X)

=

σ . x

−∞

38

CHAPTER 2. MATHEMATICAL TOOLS

These characteristics for all random variables, discussed in Subsection 2.2.3, are given in Table 2.2. Note that CV for the exponential random variable is 1, for Rayleigh and Erlang it is less than 1, and for the random variable with the decreasing transition rate it is greater than 1. The CV s of the gamma, Weibull, and log-normal random variables may be either greater or less than 1, depending on how parameter Λ has been selected. There is empirical evidence that in many production systems the coefficients of variation of up- and downtime of the machines on the factory floor are less than 1. Is this a coincidence? It turns out that it is not: It is possible to show that if the breakdown (respectively, the repair) rate of a machine is an increasing function of time, then CVup < 1 (respectively, CVdown < 1); if the breakdown (respectively, the repair) rate of a machine is a decreasing function of time, then CVup > 1 (respectively, CVdown > 1). Therefore, since in most practical situations, the breakdown and repair rates are increasing in time, CV s of manufacturing equipment are less than 1. Table 2.2: Expectation, variance, and coefficients of variation of continuous random variables Random variable Exponential Rayleigh Decreasing transition rate Erlang Gamma Weibull Gaussian Log-normal

Expectation

Variance

1 λ

1 λ2 4−π 2λ

p

π 2λ

1 2λ2

5 4λ4

n λ Λ λ

n λ2 Λ λ2

1 Γ(1 λ

+

1 ) Λ

1 [Γ(1 λ2

+

2 ) Λ

Λ2 2

2

4−π π

√

− Γ2 (1 +

σ2

m eλ+

q

CV 1

2

e2λ+Λ (eΛ − 1)

1 )] Λ

= 0.523

5 = 2.236

√1 n √1 Λ √ 2 )−Γ2 (1+ 1 ) Γ(1+ Λ Λ 1) Γ(1+ Λ σ m

p

eΛ2 − 1

This result has important implications for production systems analysis and design: It is shown in Chapters 12 and 14 that if CVup and CVdown are less than 1, there exist simple approximate (but sufficiently precise) methods for analysis and design of production systems with arbitrary distributions of upand downtime.

2.2.5

Vector random variables

In some situations, more than one random variable is necessary to describe a system. For instance, a production line with M machines is described by 2M random variables, each defining the up- and down time of a machine. Below, the issue of vector random variables is addressed and the notion of correlation is introduced. If it does not lead to loss of generality, we confine the discussion to

2.2. RANDOM VARIABLES

39

two random variables. Also, only continuous random variables are considered; the case of discrete random variables is quite similar. Consider two continuous random variables, X and Y , both having the real line, R, as their sample space. Clearly, this pair cannot be defined by their individual cdf’s or pdf’s since this would leave the issue of their possible interrelation unspecified. Therefore, a pair of random variables is defined by what is called the joint cdf given by ∀(x, y) ∈ R2 .

FXY (x, y) = P [X ≤ x, Y ≤ y],

If FXY (x, y) is differentiable with respect to both variables, the joint pdf of X and Y is: ∂ 2 FXY (x, y) . fXY (x, y) = ∂x∂y Given the joint pdf, the individual pdf’s of X and Y (referred to now as marginal pdf’s) can be obtained as Z ∞ fXY (x, y)dx = fY (y), −∞ Z ∞ fXY (x, y)dy = fX (x). −∞

Having two random variables, the conditional cdf’s and pdf’s can be introduced (in a manner similar to conditional probabilities of random events – see Subsection 2.1.3). For instance, the conditional pdf of X given Y is fX|Y (x|y) = fX (x|Y = y) =

fXY (x, y) . fY (y)

Analogously, the conditional pdf of Y given X is fY |X (y|x) = fY (y|X = x) =

fXY (x, y) . fX (x)

Therefore, the Bayes’s formula for pdf’s becomes fY |X (y|x) =

fX|Y (x|y)fY (y) . fX (x)

Similar to random events, two random variables are independent if and only if their joint pdf is the product of their marginals, i.e., fXY (x, y) = fX (x)fY (y). Again, similar to random events, n ≥ 3 random variables X1 , . . ., Xn are independent if n − 1 of them are independent and fX1 ,...,Xn (x1 , . . . , xn ) = fX1 (x1 )fX2 (x2 ) · · · fXn (xn ).

40

CHAPTER 2. MATHEMATICAL TOOLS

The expected value, variance, and the coefficient of variation of each component of a vector random variable can be calculated using its marginal pdf and the expressions given in Subsection 2.2.4. In addition, having more than one random variable allows for the introduction of the so-called mixed moments. For instance, the second mixed moment of X and Y is Z ∞Z ∞ xyfXY (x, y)dxdy. E(X, Y ) = xy = −∞

−∞

This particular moment is called the correlation of X and Y . The central second mixed moment, given by E[(X − x)(Y − y)] = xy − x · y, where x and y are the expected values of X and Y , is called the covariance of X and Y and is denoted as Cov(X, Y ). Clearly, Cov(X, X) = V ar(X),

Cov(Y, Y ) = V ar(Y ).

Finally, the correlation coefficient of X and Y is defined as ρ=

Cov(X, Y ) . σX σY

This coefficient is a measure of linear dependency between X and Y . It is possible to show that −1 < ρ < 1. For independent random variables, X and Y , Cov(X, Y ) = 0

and

ρ = 0,

which is due to the fact that for independent random variables xy = x · y. Any two random variables, X and Y , such that Cov(X, Y ) = 0, are called uncorrelated (note that they may not be independent). For uncorrelated random variables, V ar(X + Y ) = V ar(X) + V ar(Y ).

2.2.6

Asymptotic properties of sums of random variables

While all the results described so far amount merely to “probability bookkeeping,” this subsection describes the main theoretical results of Probability Theory. They show, in particular, that, under mild assumptions, the frequencies

2.2. RANDOM VARIABLES

41

of various events converge to their probabilities when the number of observations tends to infinity. This makes the interpretation of probability as frequency indeed valid. The asymptotic results can be classified into two groups: laws of large numbers and central limit theorems. Several versions of these results, depending on assumptions made, are available in the literature. Two of them are described below. Strong law of large numbers: Let X1 , X2 , . . ., be a sequence of independent identically distributed random variables with expected value x. Then h P

n i 1X Xi = x = 1. n→∞ n i=1

lim

(2.41)

As an application of this law, consider a sequence of independent repetitions of a random experiment, and let the random variable Ij be the indicator that event A occurred in the j-th trial., i.e., ½ 1 if A takes place in the j-th trial, Ij = 0 otherwise. Clearly, the expected value of Ij is P [A]. The relative frequency of A in n repetitions of the random experiment can be represented as n

1X Ij . πA (n) = n j=1

(2.42)

Therefore, using the strong law of large numbers, we conclude that i h P lim πA (n) = P [A] = 1, n→∞

i.e., the frequency of the event indeed converges to its probability. Example 2.7 Consider a machine with up- and downtime characterized by random variables tup and tdown , respectively, and assume that up- and downtime occur independently. The pdf’s or pmf’s of tup and tdown may be arbitrary but it is assumed that they have finite expected values Tup and Tdown , respectively. Calculate the expected value of the number of parts produced during a cycle time (which is the time necessary to process a part). This expected value is referred to as the machine efficiency, e. Solution: Assume that tup and tdown are measured in units of cycle time and let tup,i and tdown,i , i = 1, . . . , n, denote the i-th occurrence of up- and downtime, respectively. Clearly, the average number of parts, produced during Pn the interval i=1 (tup,i + tdown,i ), is a random variable, en , given by Pn Pn 1 tup,i i=1 tup,i i=1 n = 1 Pn . en = Pn i=1 (tup,i + tdown,i ) i=1 (tup,i + tdown,i ) n

42

CHAPTER 2. MATHEMATICAL TOOLS

In the limit as n tends to infinity, Pn i=1 tup,i lim en = lim Pn n→∞ n→∞ i=1 (tup,i + tdown,i ) Pn limn→∞ n1 i=1 tup,i P Pn = . n limn→∞ n1 i=1 tup,i + limn→∞ n1 i=1 tdown,i According to the strong law of large numbers, with probability 1 the following takes place: n

1X tup,i n→∞ n i=1 lim

=

Tup ,

=

Tdown .

n

1X tdown,i n→∞ n i=1 lim

Thus, the efficiency, e, of the machine is: e = lim en = n→∞

Tup . Tup + Tdown

(2.43)

Note that this formula is applicable to machines with arbitrary distributions of up- and downtimes, as long as they have finite expected values, which is generically true in practice. Formula (2.43) is used extensively throughout this book. Central limit theorem: Let X1 , X2 , . . ., be a sequence of independent identically distributed random variables with expected value x and variance σ 2 . Then, as n → ∞, the pdf of the random variable Pn i=1 Xi − nx √ , σ n is Gaussian with the expected value 0 and variance 1. This theorem explains why many natural phenomena are characterized by Gaussian distributions. In addition, it offers a possibility to approximately evaluate probabilities of various events, which are difficult to calculate directly. Example 2.8 Consider a Bernoulli machine defined by the parameter p. Assume that we observe the status of this machine (up or down) at every cycle time and calculate the frequency, πup (n), of the machine being up during n cycles. How large should n be so that the parameter p is identified with high accuracy and large probability, e.g., P [|πup (n) − p| < 0.01] ≥ 0.95?

2.3. RANDOM PROCESSES

43

Solution: Since πup (n) has the expected value p and variance p(1 − p), according to the central limit theorem, the random variable π (n) − p p up p(1 − p)/n for n sufficiently large has an “almost” Gaussian distribution with expected value 0 and variance 1. Therefore, ³ 0.01n ´ , P [|πup (n) − p| < 0.01] ≈ 1 − 2Q p p(1 − p) where

1 Q(x) = 2π

Z

∞

exp(− x

(2.44)

t2 )dt. 2

The value of the right hand side of (2.44) cannot bepcalculated directly since p is unknown. However, given that 0 < p < 1 (i.e., p(1 − p) ≤ 0.5) and that Q(x) is a decreasing function of x, (2.44) results in the inequality: √ P [|πup (n) − p| < 0.01] > 1 − 2Q(0.02 n). From the problem formulation, we have: √ 1 − 2Q(0.02 n) = 0.95, resulting in

√ Q(0.02 n) = 0.025.

Function Q takes the value 0.025 when its argument is approximately 1.95. Thus, √ 0.02 n = 1.95, which implies that we need n = 9506 observations to identify p with the required accuracy (0.01) and probability (0.95).

2.3 2.3.1

Random Processes Terminology

Random process (X(t), Y (t), Z(t) or M (t), N (t), . . .) – a mapping that associates a function of t with each outcome of a chance experiment, i.e., X(ζ, t) = x(t),

x ∈ SX ⊆ R, ζ ∈ S, t ∈ R.

(2.45)

Random processes are typically denoted by capital letters, while their realizations are denoted by their lowercase counterparts (e.g., X(t) and x(t), as in (2.45)). These realizations are called sample paths or trajectories of a random process.

44

CHAPTER 2. MATHEMATICAL TOOLS

As it follows from (2.45), for each fixed t, the random process is a random variable, while for each fixed ζ, the random process is a (usual, i.e., non-random) function of t. Thus, a random process can be understood either as a “collection” of random variables (parameterized by t) or a “collection” of functions of t (parameterized by ζ). Formally, this means that X(ζ, t = t∗ ) = X(ζ = ζ ∗ , t) =

X(ζ), x(t).

Since both X(ζ) and t may be either continuous or discrete, there are the following types of random processes: Discrete time, discrete space random process – t is discrete and, for each t, X(ζ, t) takes at most a countable number of values; e.g., the number of parts produced by a production system per hour during a shift. Continuous time, discrete space random process – t is continuous and, for each t, X(ζ, t) takes at most a countable number of values; e.g., continuously tracked status of a machine (up = 1 or down = 0) in a production system during a shift. Discrete time, continuous space random process – t is discrete and, for each t, X(ζ, t) takes values on the real axis; e.g., temperature in the oven of an automotive paint shop measured at the beginning of each hour. Continuous time, continuous space random process – t is continuous and, for each t, X(ζ, t) takes values on the real axis; e.g., continuously measured temperature in an oven of an automotive paint shop. Continuous (or discrete) time, mixed space random process – t is continuous (or discrete) and, for each t, X(ζ, t) is a mixed random variable, e.g., buffer occupancies in the so-called “flow model” of a production system (see Chapter 11). For the sake of brevity, the general discussion of random processes is carried out below in terms of the continuous time, continuous space case only. However, a specific random process, which is central for analysis and design of production systems, i.e., the Markov process, is discussed for all important cases.

2.3.2

Continuous time, continuous space random processes

Consider a continuous time, continuous space random process X(ζ, t) at time t = t1 . This results in a random variable X(ζ, t1 ), which we denote as X(t1 ). As a random variable, it can be defined by its cdf or pdf, denoted as FX (x1 , t1 ) and fX (x1 , t1 ), i.e., FX (x1 , t1 )

=

fX (x1 , t1 )

=

P [X(t1 ) ≤ x1 ], ∂FX (x1 , t1 ) . ∂x1

These are referred to as the first order cdf and pdf of a random process, respectively. Similarly, fixing two time moments, t1 and t2 , we obtain two random

2.3. RANDOM PROCESSES

45

variables, X(t1 ) and X(t2 ), which are defined by their joint cdf or pdf referred to as the second order cdf and pdf: FX (x1 , x2 ; t1 , t2 )

=

fX (x1 , x2 ; t1 , t2 )

=

P [X(t1 ) ≤ x1 , X(t2 ) ≤ x2 ], ∂ 2 FX (x1 , x2 ; t1 , t2 ) . ∂x1 ∂x2

In general, the n-th order cdf and pdf of X(t) are defined by FX (x1 , . . . , xn ; t1 , . . . , tn ) = fX (x1 , . . . , xn ; t1 , . . . , tn ) =

P [X(t1 ) ≤ x1 , . . . , X(tn ) ≤ xn ], ∂ n FX (x1 , . . . , xn ; t1 , . . . , tn ) . ∂x1 ∂x2 · · · ∂xn

In the same manner as a random variable is defined by its cdf or pdf, a random process is defined by all its finite dimensional cdf’s or pdf’s. Obviously, this is not a constructive definition and, therefore, in most practical cases random processes are defined by other means, e.g., by “rules” that generate their sample paths; examples of such rules are given in Subsection 2.3.3. The expected value, variance, correlation, and covariance of random processes are introduced in the same manner as for random variables. Namely, Z ∞ xfX(t) (x; t)dx, E(X(t)) = x(t) = −∞ Z ∞ V ar(X(t)) = σ 2 (t) = (x − x(t))2 fX(t) (x; t)dx, −∞ Z ∞Z ∞ RX(t) (t1 , t2 ) = E(X(t1 )X(t2 )) = x1 x2 fX(t) (x1 , x2 ; t1 , t2 )dx1 dx2 , −∞

CX(t) (t1 , t2 )

=

−∞

RX(t) (t1 , t2 ) − x(t1 )x(t2 ).

Naturally, the expected value and variance are functions of time (rather than constants, as in the case of random variables) and the correlation and covariance are functions of two independent variables – the time moments that generate the two random variables for which these characteristics are calculated. A random process X(t) is called wide sense stationary (wss) if its expected value is a constant and its correlation is a function of one argument only, which is, in fact, the difference of t2 and t1 : E(X(t))

=

RX(t) (t1 , t2 ) =

x = const, ∀t, RX(t) (t2 − t1 ) = RX(t) (τ ),

∀t1 , t2 , τ = t2 − t1 .

Clearly, the sample paths of such a process are “homogeneous” in time, and therefore the term “stationary” is used. Let X(t) be a wss random process and x(t) a sample path. Consider the time average value of x(t), denoted as E(X(t)) and called the sample path mean, and the time average value of the product x(t)x(t + τ ), denoted as RX(t) (τ ) and

46

CHAPTER 2. MATHEMATICAL TOOLS

called sample path correlation: E(X(t)) = RX(t) (τ )

=

1 lim T →∞ 2T 1 T →∞ 2T

Z

T

x(t)dt, −T Z T

lim

x(t)x(t + τ )dt. −T

If E(X(t)) = =

RX(t) (τ )

E(X(t)), RX(t) (τ ),

the process X(t) is called ergodic. Roughly speaking, ergodicity implies that the trajectories of a random process do not contain random additive components and, therefore, averaging in time and averaging in probability gives the same result. An example of a continuous time, continuous space random process is the so-called white noise, which is a process with the following properties: it is wss, ergodic, the first order pdf is Gaussian, and the covariance function is the δfunction. The last property implies that for any t1 and t2 , X(t1 ) and X(t2 ) are uncorrelated. If all properties take place but X(t1 ) and X(t2 ) are correlated, the process is called colored noise. The term “white noise” comes from the fact that, as it turns out, its power is distributed uniformly over all frequencies – similar to white light. In colored noise, the power distribution is not uniform.

2.3.3

Markov processes

Significance: As it is well known, most dynamic systems in engineering can be described by differential or difference equations, say, dx = Φ(x), t ∈ [0, ∞); x ∈ Rn , Φ : Rn → Rn , dt or

x(n + 1) = Φ1 (x(n)), n = 0, 1, . . . ; x ∈ Rn , Φ1 : Rn → Rn .

This description works well if all processes are strictly predictable (i.e., deterministic). If unpredictable effects take place, these descriptions must be modified to include random effects. This can be accomplished, for instance, by dX = Φ(X, ξ(t)), t ∈ (0, ∞); X ∈ Rn , ξ ∈ Rm , Φ : Rn × Rm ∈ Rn , dt or X(n+1) = Φ1 (X(n), ξ(n)), n = 0, 1, . . . ; X ∈ Rn , ξ ∈ Rm , Φ1 : Rn ×Rm ∈ Rn , where ξ(t) and ξ(n) are, respectively, continuous and discrete time random processes. In this situation, X(t) and X(n) are also random processes and, thus, the dynamics of engineering systems are random.

2.3. RANDOM PROCESSES

47

If ξ(t) and ξ(n) are of a general nature, the solutions of the above equations are all but impossible to derive, even in the case of linear systems (i.e., when Φ and Φ1 are linear functions of their arguments). However, if ξ(t) is white noise or ξ(n) is a sequence of independent random variables, the resulting X(t) and X(n), which are referred to as Markov processes, can be analyzed relatively easily. Thus, Markov processes are instruments for analysis of stochastic engineering systems, at least in the simplest situation. Clearly, they play a central role in analysis of production systems where unpredictable phenomena are in abundance. Although they allow for analysis of only the simplest situations, they often lead to important practical conclusions. In addition, in some problems these conclusions may be (empirically) extended to non-Markovian situations as well (see, for instance, Chapters 12 and 14). Below, several types of specific Markov processes and methods for their analysis are discussed. Markov chains: Markov chain X(n), n = 1, 2, . . ., is a discrete time, discrete space random process with the sample space SX consisting of a finite or countable set of points i = 1, 2, . . . , S (or i = 0, 1, . . . , S − 1) and satisfying the following property: P [X(n + 1) = i|X(n) = j, X(n − 1) = r, . . . , X(n − l) = s] = P [X(n + 1) = i|X(n) = j] =: Pij , ∀i, j, r, . . . , s ∈ SX . This implies that the conditional probability of any state of the process at time n + 1 is independent of its states at time n − 1, n − 2, . . ., and depends only on the state of the process at time n. In particular, this means that the probability of transition from j to i is independent of how long the process was in j. Thus, X(n) is a Markov chain if it has one-step memory. Clearly, therefore, X(n + 1) = Φ1 (X(n), ξ(n)), where ξ(n), n = 0, 1, . . ., is a sequence of independent random variables, defines a Markov process if the pmf’s of ξ(n) depend, at most, on the state of the process at time n. The probabilities Pij , defined above, are referred to as the transitions probabilities of the Markov chain. Obviously, S X

Pij

≥ 0,

∀i, j,

Pij

= 1,

∀j.

i=1

It is convenient to arrange the transition  P11 P12  P21 P22  P= ... ... PS1 PS2

probabilities as a square matrix  . . . P1S . . . P2S  , (2.46) ... ...  . . . PSS

48

CHAPTER 2. MATHEMATICAL TOOLS

which is referred to as the transition matrix. The sum of the elements of each column of this matrix is 1 and, therefore, it is referred to as a stochastic matrix. Equivalently, the transition probabilities can be represented as labels of the links in a directed graph, as shown in Figure 2.23; such a graph is referred to as the transition diagram of a Markov chain. P 21 P22

2

1

P 12

P32

P31

P23

P11

P13

3 P33

Figure 2.23: Transition diagram of a Markov chain If the transition diagram is such that there is a path from each state to any other state (i.e., there is a nonzero probability to go from any state i to any state j in a finite number of steps), the Markov process is called irreducible. If, in addition, there is a state with a “self-loop” (i.e., there is a nonzero probability to remain in the same state at n and n + 1, n ∈ {0, 1, . . .},), the Markov process is ergodic. Analysis of Markov chains amounts to calculating the probabilities of each of its states for every n and, in particular, for n = ∞. This can be carried out as follows: Using the total probability formula, the probability of state i can be expressed as P [X(n + 1) = i] =

S X

P [X(n + 1) = i|X(n) = j]P [X(n) = j],

i = 1, . . . , S.

j=1

(2.47) Introducing the notation Pj (n) := P [X(n) = j] and taking into account that the above conditional probabilities are the transition probabilities of a Markov chain, (2.47) can be re-written as a set of linear discrete time equations: Pi (n + 1) =

S X j=1

Pij Pj (n),

i = 1, . . . , S,

(2.48)

2.3. RANDOM PROCESSES

49

constrained by S X

Pi (n) = 1.

i=1

Solving these equations, we obtain a complete characterization of the Markov chain. In many instances, it is sufficient to know just the steady state values of these probabilities, i.e., Pi := Pi (∞). If the process is ergodic, these probabilities exist, are unique (i.e., independent of the initial probability of each state), and are defined by the following linear equations: Pi =

S X

Pij Pj ,

i = 1, . . . , S,

(2.49)

j=1 S X

Pi = 1,

i=1

referred to as the balance equations. Their solution describes the steady state behavior of a Markov chain. It turns out that the Pi ’s, i = 1, . . . , S, are the components of the eigenvector of the transition matrix P of (2.46) associated with the eigenvalue equal to 1. Example 2.9 Consider a machine defined by the geometric reliability model with breakdown and repair probabilities P and R, respectively. What are the steady state probabilities for the machine to be up and to be down? Solution: The transition diagram of a geometric machine is shown in Figure 2.7. Since P and R are constant, all states are “connected” and “self-loops” exist, it represents an ergodic Markov chain with the transition probabilities P01 = P and P10 = R. Therefore, the balance equations can be written as P1

=

P10 P0 + P11 P1 = RP0 + (1 − P )P1 ,

P0

=

P00 P0 + P01 P1 = (1 − R)P0 + P P1 ,

constrained by P0 + P1 = 1. Thus, the steady state probabilities that the machine is up and down, respectively, are P1

=

P0

=

R , R+P P . R+P

(2.50)

Note that, since the machine produces parts only when it is up, it follows from (2.50) that the average number of parts produced per cycle time in the steady state of system operation, i.e., the machine efficiency e, is e=

R . R+P

(2.51)

50

CHAPTER 2. MATHEMATICAL TOOLS

Taking into account that, as it is shown in Table 2.1, the average up- and downtime of a geometric machine are Tup =

1 , P

Tdown =

1 , R

(2.51) can be re-written as e=

Tup , Tup + Tdown

which, of course, coincides with expression (2.43) obtained for the arbitrary pmf’s of up- and downtime. Continuous time, discrete space Markov processes: A continuous time, discrete space Markov process is a random process X(t), with t ∈ [0, ∞), sample space consisting of a finite set of points i = 1, 2, . . . , S (or i = 0, 1, . . . , S − 1), and satisfying the following property: P [X(t + δt) = i|X(t) = j, X(s) = x(s), ∀s < t] = P [X(t + δt) = i|X(t) = j] ∀i 6= j. = νij δt + o(δt),

(2.52)

As in the discrete time case, this implies that the conditional probability of any state of the process at time t + δt depends only on the state of the process at time t and is independent of how it came to and how long it stayed at the latter state. Moreover, this conditional probability is proportional to δt, and the coefficient of proportionality, νij , is a constant referred to as the transition rate. The analysis of this process again amounts to calculating the probability of each state. To derive an equation, which defines these probabilities, by the total probability formula, we write: P [X(t + δt) = i] =

S X

P [X(t + δt) = i|X(t) = j]P [X(t) = j].

j=1

Using the Markov property, this can be re-written as P [X(t + δt) = i] =

S X

νij δtP [X(t) = j]

j=1,j6=i

+P [X(t + δt) = i|X(t) = i]P [X(t) = i] + o(δt). (2.53) Since

S X j=1

P [X(t + δt) = j|X(t) = i] = 1,

2.3. RANDOM PROCESSES

51

the last term in the above expression can be expressed as P [X(t + δt) = i|X(t) = i]

=

S X

1−

P [X(t + δt) = j|X(t) = i]

j=1,j6=i S X

= 1−

νji δt + o(δt).

j=1,j6=i

Thus, from (2.53) we obtain: S h i X νij δtP [X(t) = j]+ 1− νji δt P [X(t) = i]+o(δt).

S X

P [X(t+δt) = i] =

j=1,j6=i

j=1,j6=i

In terms of the notations Pj (t)

:=

νii

P [X(t) = j], S X

:= −

νji ,

(2.54)

j=1,j6=i

the above expression becomes Pi (t + δt)

=

S X

νij δtPj (t) + [1 + νii δt]Pi (t) + o(δt)

j=1,j6=i

= Pi (t) +

S X

νij Pj (t)δt + o(δt),

i = 1, . . . , S,

j=1

which can be re-arranged as S

o(δt) Pi (t + δt) − Pi (t) X = , νij Pj (t) + δt δt j=1

i = 1, . . . , S.

In the limit as δt → 0, this becomes a linear differential equation S

dPi (t) X = νij Pj (t), dt j=1 with the constraint

S X

i = 1, . . . , S,

(2.55)

Pi (t) = 1.

i=1

The solution of this equation defines the probability of each state at every time moment. Note the similarity of this equation with (2.48) for the Markov chain.

52

CHAPTER 2. MATHEMATICAL TOOLS

Clearly, the role of the transition  ν11  ν21   ... νS1

matrix is played now by the matrix  ν12 . . . ν1S ν22 . . . ν2S  , ... ... ...  νS2 . . . νSS

(2.56)

which is referred to as the infinitesimal generator. The process is ergodic if the infinitesimal generator admits a flow from each state to any other state (perhaps, through some intermediate states). If the process is ergodic, there exists a unique steady state of (2.55), independent of the initial conditions. Therefore, the steady state probabilities, Pi := Pi (∞), of all states satisfy S X

νij Pj

= 0,

i = 1, . . . , S,

(2.57)

j=1 S X

Pi

= 1,

i=1

which, using (2.54), imply that S X

S X

νij Pj + νii Pi =

j=1,j6=i

S X

νij Pj −

j=1,j6=i

νji Pi = 0.

j=1,j6=i

In other words, in the steady state, Pi

S X

νji

=

S X

νij Pj

j=1,j6=i

j=1,j6=i

flow from state i

flow into state i

(2.58)

and S X

Pi = 1.

i=1

The solution of these equations provides a complete characterization of the steady states or stationary behavior of the random process at hand. Similar to the discrete time case, Pi ’s, i = 1, . . . , S, are the components of the eigenvector of the infinitesimal generator (2.56) corresponding to the eigenvalue 0. Example 2.10 Consider an exponential machine with breakdown and repair rates λ and µ, respectively. Determine the steady state probabilities that the machine is up and is down.

2.3. RANDOM PROCESSES

53

Solution: The transition diagram of an exponential machine is shown in Figure 2.12. Since the rates λ and µ are constant, it represents a continuous time, discrete space Markov process with transition rates ν01 = λ and ν10 = µ. Therefore, the balance equations (2.58) can be written as µP0 − λP1 P0 + P1

= 0, = 1.

Thus, the steady state probabilities that the machine is up and down, respectively, are P1

=

P0

=

µ , µ+λ λ . µ+λ

(2.59)

Since the machine produces parts only when it is up, it follows from (2.59) that the machine efficiency is µ . e= µ+λ Taking into account that the average up- and downtime of an exponential machine are 1 1 Tdown = , Tup = , λ µ this can be re-written as e=

Tup , Tup + Tdown

which again coincides with (2.43) obtained for arbitrary pdf’s of up- and downtime. Continuous time, continuous space Markov processes: with t ∈ [0, ∞], X ∈ R is Markov if P [X(t) ∈ L|X(s), s ≤ τ ] = P [X(t) ∈ L|X(τ )],

Process X(t)

L ⊂ R, τ < t.

(2.60)

As before, this implies that for a Markov process the future depends only on its present state. Since X(t) is defined on a continuum, its constructive definition through (2.60) is hardly possible. Therefore, such processes are often introduced by differential equations, which generate their sample paths. For instance, such an equation may be of the form ˙ X(t) = Φ(X(t)) + Ψ(X(t))w, ˙

Φ : R → R, Ψ : R → R,

where w˙ is a white noise process with variance equal to 1. There are mathematical subtleties in the precise meaning of this equation (referred to as Ito equation), but they are not of importance for our discussion.

54

CHAPTER 2. MATHEMATICAL TOOLS

As in the case of Markov chains, to analyze a continuous time, continuous space Markov process means to probabilistically characterize its states at every t or at least in the limit as t → ∞. Since the states occupy a continuum, such a characterization must be carried out in terms of densities rather than probabilities. Accordingly, let fX (x, t) be the density of X at time t. Then, it has been shown (using again the total probability formula – this time for the densities) that the fX (x, t) obeys the so-called Fokker-Plank equation: ∂ 1 ∂2 2 ∂fX (x, t) = − Φ(x)fX (x, t) + Ψ (x)fX (x, t). ∂t ∂x 2 ∂x2 If the deterministic equation x˙ = Φ(x) is globally asymptotically stable, then under mild condition on Ψ(x) there exists a steady state solution of the Fokker-Plank equation defined by ∂ Φ(x)fX (x) ∂x Z

=

∞

fX (x)dx =

1 ∂2 2 Ψ (x)fX (x), 2 ∂x2

(2.61)

1.

−∞

Solving this with appropriate boundary conditions, the density f (x), which characterizes the stationary behavior of X(t), is obtained. Continuous time, mixed state Markov processes: Although continuous space Markov processes described above are important in electrical, mechanical and other engineering disciplines, Production Systems Engineering is more concerned with mixed state Markov processes. The reason is that while buffer occupancies might be described by continuous random variables (when a flow model is used – see Chapter 11), the states of the machines are always discrete – up or down. This, along with other reasons (e.g., finite capacity of buffers), leads to a mixed state description of production systems. Fortunately, the equations that arise here are not of complete generality, which makes them simpler. Indeed, in production systems considered in this book, randomness comes from machine breakdowns and, given the status of the machine, the buffer occupancy evolves deterministically. This leads to the following model of a continuous time, mixed space Markov process: P [α(t + δt) = i|α(t) = j] = νij δt + o(δt), dX(t) = Φ(X(t), α(t)), dt

i 6= j,

(2.62) (2.63)

where α can be viewed as the state of the machine, α ∈ {0, 1}, and X as the buffer occupancy. In more general situations, α can be viewed as taking S values, α ∈ {1, . . . , S} and X being an n-dimensional vector. The important feature is that α is evolving on a discrete state space while X on a continuum.

2.3. RANDOM PROCESSES

55

Dynamical system (2.62), (2.63) can be described by S first order cdf’s of X, one for each of α ∈ {1, . . . , S}: P (x, i, t) := P [X(t) ≤ x, α(t) = i],

i = 1, . . . , S, X ∈ Rn , x ∈ Rn .

In terms of densities, this becomes P [x1 < X1 (t) ≤ x1 + δx1 , · · · , xn < Xn (t) ≤ xn + δxn ; α(t) = i] = f (x, i, t)δx1 · · · δxn + o(δx1 , . . . , δxn ). To evaluate these densities, the total probabilities formula for densities (2.21) is used: f (x, i, t+δt) =

S X

S X

f (xj , j, t)νij δt+f (xi , i, t)[1−

j=1,j6=i

νji δt]+o(δt), (2.64)

j=1,j6=i

where xj denotes x at time t and evolving under α(t) = j. As it follows from (2.63), xj (t) = x(t + δt) − Φ(x(t), j)δt + o(δt),

j ∈ {1, . . . , S}.

Using this expression and the notation (2.54), from (2.64) we obtain f (x, i, t + δt)

S X

=

f (x − Φ(x, j)δt, j, t)νij δt

j=1,j6=i

+f (x − Φ(x, i)δt, i, t)(1 + νii δt) + o(δt). Expending all the deviating terms in Taylor series and keeping only first order terms results in: thef (x, i, t) +

∂f (x, i, t) δt = ∂t

=

S X

f (x, j, t)νij δt

j=1,j6=i

i h ∂f (x, i, t) Φ(x, i)δt (1 + νii δt) + o(δt) + f (x, i, t) − ∂x S X f (x, j, t)νij δt + f (x, i, t)

j=1,j6=i

−

∂f (x, i, t) Φ(x, i)δt + f (x, i, , t)νii δt + o(δt). ∂x

Dividing by δt, in the limit as δt → 0, we have: S

∂f (x, i, t) X ∂f (x, i, t) = Φ(x, i), f (x, j, t)νij − ∂t ∂x j=1

i ∈ {1, . . . , S}.

56

CHAPTER 2. MATHEMATICAL TOOLS

In the steady state as t → ∞ (if it exists), this becomes the following equation in partial derivatives: S

X ∂f (x, i) Φ(x, i) = f (x, j)νij , ∂x j=1 S Z X i=1

Z

∞

∞

··· −∞

(2.65)

f (x1 , . . . , xn , i)dx1 . . . dxn = 1. −∞

Note that the main difference between this equation and the similar equation (2.57) for the continuous time discrete space Markov process is the partial derivative term in the left-hand side of (2.65).

2.4

Summary

• The three main parts of Probability Theory – Random Events, Random Variables, and Random Processes – are all based on five fundamental formulas (although they may have a different “appearance” in each part): – the probability of the union, – the conditional probability, – the probability of intersection (including that in the case of independence), – the total probability, and – the Bayes’s formula. • The exponential pdf arises naturally as up- and downtime distributions of the machines with constant breakdown and repair rates. • If the breakdown and repair rates are linearly increasing in time, the resulting pdf’s are Rayleigh distributions. • If the breakdown and repair rates are monotonically increasing (respectively, decreasing) in time, the coefficients of variation on the resulting pdf’s is less (respectively, larger) than 1. • For any pdf of up- and downtime, machine efficiency is given by e = Tup /(Tup + Tdown ), where Tup and Tdown are the average values of up- and downtime, respectively. • If the breakdown and repair rates are constant, the production system, consisting of several machines, can be described by a Markov process. Otherwise, the description is non-Markovian, which makes analytical investigations, in most cases, prohibitively difficult. • Under the assumption that up- and downtime of the machines are exponential, production systems can be described by a continuous time, mixed state Markov processes, where the continuous dynamics of the buffer occupancies are modulated by discrete (and random) states of the machines.

2.5. PROBLEMS

2.5

57

Problems

Problem 2.1 Consider a production system consisting of three Bernoulli machines. Each machine is up during a cycle time with probability 0.9 and down with probability 0.1. Machines are up or down independently from each other. (a) What is the probability that all three machines are up simultaneously given that at least two machines are up? (b) What is this probability, given that at least one machine is up? (c) Explain the reason for the difference between these two numbers. Problem 2.2 Consider a production system consisting of three Bernoulli machines and a controller, which also obeys the Bernoulli reliability model. This production system is considered up if the controller and at least two machines are up. During each cycle, the controller is up with probability 0.8 and each machine is up with probability 0.9. The controller and the machines fail independently. (a) Calculate the probability that the production system is up. (b) Assume that a second controller, identical to the first one, is added to the system, and the re-designed system is up if at least one controller and two machines are up. Calculate the probability that the re-designed production system is up. (c) Explain the reason for the difference between the two numbers you calculated. Problem 2.3 Assume that in each cycle, the machine can produce either a good or a defective part according to the Bernoulli quality model. The good part is produced with probability 0.99. Calculate the probability that the number of defectives in 10 parts produced by this machine is less than or equal to 2. Problem 2.4 A car assembly plant uses axles from two suppliers, A and B. The axles manufactured by A and B fail within 10 years with probabilities 0.2 and 0.1, respectively. The assembly plant uses 50% of axles from each of the suppliers. (a) Find the probability that the axle in a car fails within 10 years. (b) What is the probability that the failed axle came from supplier A? (c) What is the probability that the failed axle came from supplier B? Problem 2.5 Consider a machine with the geometric reliability model. Assume that the breakdown probability, P , is 0.1 and repair probability, R, is 0.6. (a) Calculate the probability that the machine is up for more than 10 cycles. (b) Calculate the probability that the machine is down for less than 3 cycles. Problem 2.6 Consider a machine with the exponential reliability model. Assume that the breakdown rate, λ, is 1 and repair rate, µ, is 2.

58

CHAPTER 2. MATHEMATICAL TOOLS

(a) Calculate the probability that the machine is up for more than 10 units of time. (b) Calculate the probability that the machine is down for less than 3 units of time. Problem 2.7 Prove that the exponential distribution satisfies the memoryless property, i.e., prove that if tup is distributed exponentially, then the following equality holds: P [tup > t + h|tup > t] = P [tup > h]. Also, interpret the physical meaning of this equality. Problem 2.8 Assume that the breakdown rate of a machine is λ(t) = αβtβ−1 , where α and β > 0. What is the pdf of the uptime? Problem 2.9 A production system has two spares of a critical component that has average uptime 1/λ = 1 month. Find the probability that the three components (the operating one and the two spares) will last more than 6 months. Assume that the component lifetimes are exponentially distributed. Problem 2.10 Derive the expressions for the expected value and the variance of the uniform and exponential continuous random variables. Problem 2.11 Assume that X1 , X2 , . . ., is a sequence of independent, identically distributed discrete random variables. Consider the sum process defined as follows: n = 1, 2, . . . . Sn = X1 + X2 + . . . + Xn , Determine (by calculations) if Sn is a Markov process. Problem 2.12 Assume that X1 , X2 , . . ., is a sequence of independent Bernoulli random variables with p = 0.5. Consider the sequence Yn defined by n = 2, 3, . . . . Yn = 0.5(Xn + Xn−1 ), Determine (by calculations) if Yn is a Markov process. Problem 2.13 Consider a machine, which has two failure modes, 1 and 2. When it is up, it can go down during a cycle time to failure mode 1 with probability p1 , to failure mode 2 with probability p2 , or it can stay up with probability 1 − p1 − p2 . If the machine is in failure mode i, it can be repaired during a cycle time with probability ri . When up, the machine produces one part per cycle time; when down, no production takes place. (a) Can this machine be described by a Markov process?

2.6. ANNOTATED BIBLIOGRAPHY

59

(b) If so, draw its transition diagram. (c) Calculate the steady state average throughput of this machine, assuming that no blockages and starvations take place. Problem 2.14 Prove that the stationary probabilities Pi , i = 1, . . . , S, defined by (2.49) form the eigenvector of the transition matrix (2.46) corresponding to the eigenvalue 1. Problem 2.15 Prove that the stationary probabilities Pi , i = 1, . . . , S, defined by (2.57) form the eigenvector of the infinitesimal generator (2.56) corresponding to the eigenvalue 0.

2.6

Annotated Bibliography

Probability Theory is covered in a plethora of textbooks. Among them, the following are recommended for students of Manufacturing in general and Production Systems in particular: [2.1] S. Ross, Introduction to Probability Models, Eighth Edition, Elsevier, Amsterdam, 2003. A more detailed coverage can be found in [2.2] S. Ross, Stochastic Processes, Second Edition, John Wiley & Sons, New York, 1996. [2.3] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1994. [2.4] C. W. Helstrom, Probability and Stochastic Processes for Engineers, Second Edition, Macmillan Publishing Company, New York, 1991. The derivation of pdf’s of up- and downtime for machines with constant breakdown and repair rates, discussed in Section 2.2, is similar to that given in [2.5] S.B. Gershwin, Manufacturing Systems Engineering, Prentice Hall, Englewood Cliffs, NJ, 1994. The derivation of pdf’s of up- and downtime for machines with monotonically increasing and decreasing breakdown and repair rates is reported in [2.6] J. Li and S.M. Meerkov, “On the Coefficients of Variation of Up- and Downtime of Manufacturing Equipment,” Mathematical Problems in Engineering, pp. 1-6, 2005. The properties of coefficients of variation for distributions defined by monotonically increasing and decreasing transition rates, cited in Subsection 2.2.4, have been derived in

60

CHAPTER 2. MATHEMATICAL TOOLS

[2.7] R.E. Barlow, F. Proschan and L.C. Hunter, Mathematical Theory of Reliability, John Wiley & Sons, New York, 1965. The idea of the proof for formula (2.43) (see Example 2.7 of Subsection 2.2.6) was suggested by S. Gershwin (oral communication). Using a different approach, this formula has been derived in [2.2].

Chapter 3

Mathematical Modeling of Production Systems Motivation: All methods of analysis, continuous improvement, and design described in this textbook are model-based, i.e., their application requires a mathematical model of the production system under consideration. Therefore, the issue of mathematical modeling is of central importance. The main difficulty here is that no two production systems are identical. Even if they were designed identically, numerous changes and adjustments, introduced in the course of time by engineering and equipment maintenance personnel, force them to evolve so that they become fundamentally different. Thus, there are, practically speaking, infinitely many different production systems. Nevertheless, it is possible to introduce a small set of standard models to which every production system may be reduced, perhaps at the expense of sacrificing some fidelity of the description. The purpose of this chapter is to discuss these standard models and indicate how a given production system can be reduced to one of them. The issue of parameter identification is also addressed. Overview: The mathematical model of a production system is defined by the following five components: • Type of a production system: It shows how the machines and material handling devices (or buffers) are connected and defines the flow of parts within the system. • Models of the machines: They quantify the operation of the machines from the point of view of their productivity, reliability, and quality. • Models of the material handling devices: They quantify their parameters, which affect the overall system performance. • Rules of interactions between the machines and material handling devices: They define how the states of the machines and material handling devices affect each other and, thus, facilitate uniqueness of the resulting mathematical description. 61

62

CHAPTER 3. MATHEMATICAL MODELING • Performance measures: These are metrics, which quantify the efficiency of system operation and, thus, are central to analysis, continuous improvement, and design methods developed in this book.

This chapter describes each of these components and comments on parameter identification and model validation.

3.1 3.1.1

Types of Production Systems Serial production lines

Serial production line – a group of producing units, arranged in consecutive order, and material handling devices that transport parts (or jobs) from one producing unit to the next. Figure 3.1 shows the block diagram of a serial production line where, as in Chapter 1, circles represent producing units and rectangles are material handling devices.

Figure 3.1: Serial production line The producing units may be either individual machines or work cells, carrying out machining, washing, heat treatment, and other operations. If assembly operations are performed, the parts to be attached to the one being processed are viewed as produced by another production system and, therefore, the line is still serial (rather than an assembly system – to be considered in Subsection 3.1.2). The producing units may also be departments or shops of a manufacturing plant. For instance, they may represent the body shop, paint shop, and the general assembly of an automotive assembly plant. Finally, the producing units may even be complete plants, representing various tiers of a supply chain. However, since the emphasis of this book is on parts flow rather than on the technology of manufacturing, we refer to all producing units as machines. The material handing devices may be boxes, or conveyors, or automated guided vehicles, when the producing units are machines or work cells or shops in a plant. They may be trucks, trains, etc., when the producing units are plants. Whatever their physical implementation may be, we refer to them as buffers, since the most important feature of material handling devices, from the point of view of the issues addressed in this textbook, is their storing capacity. The buffers, discussed above, are called in-process buffers. In addition, serial production lines may have finished goods buffers (FGB). The purpose of the latter is to filter out production randomness and, thereby, ensure reliable satisfaction of customers demand by unreliable production systems. An example of a serial line with a FGB is shown in Figure 3.2.

3.1. TYPES OF PRODUCTION SYSTEMS

63 Demand

FGB

Figure 3.2: Serial production line with a finished goods buffer In some cases, parts within a serial line are transported on carriers, sometimes referred to as pallets, skids, etc. Such lines are called closed with respect to carriers (see Figure 3.3). Here, raw materials must be placed on a carrier, and the finished parts must be removed from the carrier, returning the latter to the empty carrier buffer. Thus, the performance of such lines may be impeded, in comparison to the corresponding open lines, since the first machine may be starved for carriers and the last machine may be blocked by the empty carrier buffer. Too many carriers lead to frequent blockages of the last machine; too few carriers lead to frequent starvations of the first machine. Thus, an additional problem for closed lines is selecting a “just right” number of carriers.

Empty carrier buffer

Figure 3.3: Closed serial line Along with producing units, serial lines may include inspection operations intended to identify and remove defective parts produced in the system. Such a line is shown in Figure 3.4 where the shaded circles are the machines, which may produce defective parts, and the black circles are the inspection machines; the arrows under the inspection machines indicate scrap removal.

Figure 3.4: Serial line with product quality inspection Another variation of serial lines is production lines with rework. Here, if a defective product is produced, it is repaired and returned to an appropriate operation for subsequent re-processing. An example of a serial line with rework is shown in Figure 3.5. Such lines are typical, for instance, in paint shops of automotive assembly plants. A generalization of lines with rework are the so-called re-entrant lines, illustrated in Figure 3.6, where some of the machines are represented by ovals to

64

CHAPTER 3. MATHEMATICAL MODELING

Figure 3.5: Serial line with rework better indicate the flow of parts. Here, each part may visit the same machine multiple times. Typically, this structure arises in semiconductor manufacturing where, on the one hand, equipment costs are extremely high, and, on the other hand, the products have a layered structure, which necessitates/permits the utilization of the same equipment at various stages of the production process. Clearly, these lines may have even more severe problems with blockages and starvations and, therefore, their performance is typically inferior to corresponding “untangled” serial lines. In addition, since each machine serves several buffers, priorities of service become an important issue.

Figure 3.6: Re-entrant line The serial production line is a “work horse” of manufacturing. It is hardly possible to find a production system, which would not include one or more serial lines. Moreover, all other production systems may be broken down into serial lines connected according to a certain topology. Thus, the study of serial lines is of fundamental importance in Production Systems Engineering, and it is a major component of this textbook (Parts II and III).

3.1.2

Assembly systems

Assembly system – two or more serial lines, referred to as component lines, one or more merge operations, where the components are assembled, and, perhaps, several subsequent processing operations performed on an assembled part. Figures 3.7 and 3.8 show the block diagrams of typical assembly systems where, as before, the circles represent the machines and rectangles are the buffers. Systems similar to that of Figure 3.8 are typical in automotive en-

3.1. TYPES OF PRODUCTION SYSTEMS

65

gine plants where the horizontal line represents the general engine assembly (with engine blocks as “raw materials”), while the vertical lines are various departments producing engine parts, such as crank shaft, camshaft, etc.

Figure 3.7: Assembly system with a single merge operation

Figure 3.8: Assembly system with multiple merge operations Clearly, assembly systems may be viewed as several serial production lines connected through their finished goods buffers. Each of these component lines may have all other variations described above, e.g., being closed with respect to carriers or re-entrant. In this book, assembly systems are studied in Part IV.

Figure 3.9: Complex production system While it is highly desirable that a production system under consideration be reduced to either a serial line or an assembly system, it is possible to carry out some analyses (for instance, performance evaluation) for more complex modelsTypes of production systems!complex lines. Figure 3.9 shows an example of such a model.

66

CHAPTER 3. MATHEMATICAL MODELING

3.2

Structural Modeling

It is quite seldom that production systems on the factory floor have exactly the same structure as one of those shown in Figures 3.1 - 3.9. For instance, a serial line may have multiple machines in some operations, as shown in Figures 3.10 and 3.11. The situation in Figure 3.10 typically happens because no machines of the desired capacity are available for some technological operations. Figure 3.11 exemplifies the situations where a machine performs several synchronous dependent operations in the sense that all operations are down if at least one of them is down. In all these cases, the production systems must be reduced to one of the standard types discussed above (see Figure 3.12) in order to carry out their analysis and design using the tools described in this book. We refer to this process as structural modeling.

Figure 3.10: Serial production line with parallel machines

Figure 3.11: Serial production line with synchronous dependent machines aggregated machine

aggregated machine

Figure 3.12: Structural model of serial production lines of Figures 3.10 and 3.11 The general approach to structural modeling is based on the maxim attributed to Einstein: “The model should be as simple as possible, but not simpler.” The last clause makes the process of modeling more an art than engineering and, like the arts, must be taught through examples and experience. A few examples described below illustrate how this process is carried out, while Subsection 3.3.5 shows how the characteristics of the aggregated machines of Figure 3.12 can be calculated. Case studies in Section 3.10 offer additional examples. Consider an automotive ignition module production system shown in Figure 3.13, which operates as follows: The raw materials for parts A1 and A2 are loaded on conveyors at operations 1 and 9, respectively, and then transported to other operations. At operation 8, parts A1 are unloaded into the buffer, which is another conveyor and which transports them to the mating (or merge) operation 13, where the assembly of A1 and A2 takes place. Operations 14 - 18 perform additional processing.

3.2. STRUCTURAL MODELING

67

As it follows from this description, this system can be modeled as shown in Figure 3.14, which is a standard assembly system.

Load A2 9

11

10

12

18

Load A1 16

17

15

14

13 Buffer conveyor

1

8

7

3

2

6

5

4

Figure 3.13: Layout of automotive ignition module assembly system 1

2

3

4

5

6

7

8

A1

13

A2

9

10

11

14

15

16

17

18

12

Figure 3.14: Structural model of the automotive ignition module assembly system of Figure 3.13 The situation with the system of Figure 3.15 is more complex. Here, 13 injection molding machines produce seven different part types necessary for the assembly. Which part is produced by a specific injection molding machine depends on scheduling. A physical model of this system is shown in Figure 3.16. To simplify it, we note that from the point of view of the in-process buffers, it is not important which particular machine is producing a specific part type at each time moment. What is important is the rate of parts flow into each buffer. Therefore, it is possible to substitute the 13 real machines by 7 virtual machines (see Figure 3.17), each producing a specific part type. Also, the additional processing operations can be aggregated into one assembly machine. If it is possible to calculate the parameters of the virtual machines,

68

CHAPTER 3. MATHEMATICAL MODELING

based on the parameters of the real machines and scheduling procedures (which, in fact, can be done with a certain level of fidelity), then the production system of Figure 3.16 is reduced to a standard assembly system, shown in Figure 3.17.

Injection molding

Storage

17 18

13

19 9

12 10

17

13

14 15

16

11

12 11

10

19 9

17

13 18

16

14 15

12 11

10

19 9 8

8

8

2

Assembly and additional processing

7

7

7

18

16

14 15

6 5

4

4

3

3

6 5

6 5

4 3 2

2

20

20

20

Storage

Figure 3.15: Layout of injection molding - assembly system

The development of a simple, but still relatively precise structural model of a production system, is one of the most important stages of production systems analysis, continuous improvement, and design. Since no formal methods for such modeling exist (or, perhaps, are even possible), production system engineers and managers must develop these skills through practical experience.

3.3 3.3.1

Mathematical Models of Machines Timing issues

Cycle time (τ ) – the time necessary to process a part by a machine. The cycle time may be constant, variable, or random. In large volume production systems, τ is practically always constant or close to being constant. This is the case in most production systems of the automotive, electronics, appliance, and other

3.3. MATHEMATICAL MODELS OF MACHINES

Injection molding

69

In-process buffer Assembly and additional processing

Finished goods buffer

Figure 3.16: Physical model of injection molding - assembly system

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

b6

m7

b7

m0

b

0

Figure 3.17: Structural model of injection molding - assembly system with virtual machines

70

CHAPTER 3. MATHEMATICAL MODELING

industries. Variable or random τ takes place in job-shop environments where each part may have different processing specifications. In this book, we consider only machines with a constant cycle time; similar developments, however, can be carried out for the case of random (e.g., exponentially distributed) processing time. Machine capacity (c) – the number of parts produced by a machine per unit of time when the machine is up. Clearly, in the case of constant τ , c=

1 . τ

Machines in a production system may have identical or different cycle times. In the case of identical cycle time, the time axis may be considered as slotted or unslotted. Slotted time – the time axis is slotted with the slot duration equal to the cycle time. In this case, all transitions – changes of machines’ status (up or down) and changes of buffers’ occupancy – are considered as taking place only at the beginning or the end of the time slot. Production systems satisfying this convention are called synchronous. Unslotted or continuous time – the above mentioned changes may occur at any time moment. If the cycle times of all machines are identical, such a system, with a slight abuse of the definition, is still referred to as synchronous. If the cycle times are not identical, the system is called asynchronous. Production systems with machines having different cycle times are typically considered as operating in unslotted time. In the unslotted case, production systems can be conceptualized as discrete event systems or as flow systems. Discrete event system – a job (i.e., part) is transferred from the producing machine to the subsequent buffer (if it is not full) only after the processing of the whole job is complete. In this case, the buffer occupancy is a non-negative integer. Flow system – infinitesimal parts of the job are (conceptually) transferred from the producing machine to the subsequent buffer if it is not full. Similarly, an infinitesimal part of a job is taken by a downstream machine from the buffer, if the machine is not down and the buffer is not empty. In this case, there is a continuous flow of parts into and from the buffers. Clearly, the buffer occupancy in this situation is a non-negative real number. Obviously, the discrete event conventions are closer to reality. However, flow systems are sometimes easier to analyze and often lead to reasonable conclusions. In this textbook both conventions are addressed.

3.3.2

Machine reliability models

Machine reliability model – the probability mass functions (pmf’s) or the probability density functions (pdf’s) of the up- and downtime of the machine in the slotted or unslotted time, respectively. The reliability models considered in this book are listed below.

3.3. MATHEMATICAL MODELS OF MACHINES Reliability models for the slotted time case:

71 Two models are addressed.

• Bernoulli reliability model (B) – at the beginning of each time slot, the status of the machine – up or down – is determined by a chance experiment, according to which it is up with probability p and down with probability 1−p, independently of the status of this machine in all previous time slots. This is the simplest reliability model. Indeed, first, it is static, i.e., no past status of the machine affects its status in the upcoming slot and, second, its pmf is very simple. Nevertheless, it is still practical, especially for describing assembly operations where the downtime is typically very short and comparable with the cycle time of the machine. Most of the analysis, continuous improvement, and design problems, considered in this book, are first solved using this simplest case and then extended to more complex scenarios. • Geometric reliability model (Geo) – uptime and downtime pmf’s are given by the geometric pmf’s (2.14), (2.15), i.e., Ptup (t) = Ptdown (t) =

P [tup = t] = P (1 − P )t−1 , P [tdown = t] = R(1 − R)

t−1

t = 1, 2, . . . , ,

t = 1, 2, . . . .

(3.1)

As it is clear from the discussion in Chapter 2, these pmf’s are generated by the transition diagram of Figure 2.7, which implies that the state of the machine (up or down) in any time slot is defined by that of the previous time slot with the probabilities of breakdown and repair P and R, respectively. Clearly, in this case the machine is a dynamic system with one step memory and, as it follows from Chapter 2, it can be described by a Markov chain. Methods of analysis of production systems with this reliability model are more complex than in the memoryless case. In comparison with the Bernoulli model, this is a more realistic description of a machine. Production lines with Bernoulli and geometric reliability models are usually considered as discrete event systems, while all reliability models for the continuous time case (listed below) are viewed in the framework of flow systems. Reliability models for the continuous time case: The continuous time case is, perhaps, more realistic than the slotted time and, therefore, a larger set of reliability models is addressed. They are as follows: • Exponential reliability model (exp) – the uptime and downtime pdf’s of the machine are given by the exponential distributions (2.27), (2.28), i.e., ftup (t) = ftdown (t) =

λe−λt , µe−µt ,

t ≥ 0, t ≥ 0.

(3.2)

The transition diagram, generating this reliability model, is shown in Figure 2.12, which implies that, if up, the machine may go down in each infinitesimal interval δt with rate λ and, if down, it may go up during δt with rate µ.

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CHAPTER 3. MATHEMATICAL MODELING

Clearly, this is also a dynamical system, and it can be described by a continuous time, discrete space Markov process. Methods of analysis of production systems with this reliability model are roughly of the same complexity as those for the geometric case and can be applied to the same technological operations. The main drawback of this model is that the breakdown and repair rates are constant, which is hardly true in reality. Therefore, other reliability models are introduced. • Rayleigh reliability model (Ra) – the uptime and downtime pdf’s of the machine are given by Rayleigh distributions (2.29), (2.30), i.e., ftup (t) ftdown (t)

= =

λte− µte

λt2 2

2 − µt2

,

t ≥ 0,

,

t ≥ 0.

(3.3)

The breakdown and repair rates of a Rayleigh machine, as it has been shown in Chapter 2, are λt and µt respectively, implying that both are linearly increasing in time. The resulting system is, of course, still dynamic but it is not described by a Markov process anymore, and no rigorous analytical methods for analysis of such systems are available. In this book, we present empirical methods for analysis of this and other non-Markovian situations (see Chapter 12). A deficiency of both exponential and Rayleigh reliability models is that their coefficients of variation cannot be placed at will – they are fixed, respectively, at 1 and 0.52, for all values of λ and µ. In reality, however, machines on the factory floor may have widely different coefficients of variation. The experimental evidence indicates, however, that in most cases they take values between 0 and 1. This necessitates considering other reliability models. • Weibull reliability model (W) – the uptime and downtime pdf’s of the machine are given by Weibull distributions (2.37), (2.38), i.e., ftup (t) = ftdown (t) =

Λ

λΛ e−(λt) ΛtΛ−1 , M −(µt)M

µ e

Mt

M−1

t ≥ 0, ,

t ≥ 0.

(3.4)

The Weibull distribution is commonly used in reliability theory. Being defined by two parameters, λ and Λ or µ and M, it allows us to place both its expected value and variance and, thus, the coefficient of variation, at will. The next two reliability models, also with the ability of placing their CVs at will, are used for the sake of generality: • Gamma reliability model (ga) – the uptime and downtime pdf’s of the machine are given by the gamma distributions (2.35), (2.36), i.e., ftup (t) = ftdown (t) =

(λt)Λ−1 , Γ(Λ) (µt)M−1 , µe−µt Γ(M) λe−λt

t ≥ 0, t ≥ 0,

(3.5)

3.3. MATHEMATICAL MODELS OF MACHINES where

Z

∞

Γ(x) =

73

sx−1 e−s ds.

0

When Λ and M are positive integers, these distributions coincide with the Erlang distribution. Although there are some analytical methods for analysis of production systems with Erlang reliability models, they are so computationally intensive that the analysis of even two machine lines with buffer of capacity more than 5 is practically impossible. Therefore, these methods are not included in this textbook. • Log-normal reliability model (LN) – the uptime and downtime pdf’s of the machine are given by the log-normal distributions (2.39), (2.40), i.e., ftup (t) = ftdown (t) =

(ln t−λ)2 1 e− 2Λ2 , 2πΛt (ln t−µ)2 1 √ e− 2M2 , 2πMt

√

t ≥ 0, t ≥ 0.

(3.6)

This model is considered since, unlike all others, it does not coincide with the exponential case when its CV = 1. All reliability models described above imply that both up- and downtime are distributed according to the same type of pdf. In reality, of course, this is not necessarily the case. Therefore, we introduce • Mixed reliability model (M) – the uptime and downtime are defined by different distributions selected from the set {exp, Ra, W, ga, LN }. One more reliability model is considered in this book – for situations when no data on the type of the distributions involved is available: • General reliability model (G) – the up- and downtime may be distributed according to arbitrary pdf’s. Clearly, rigorous analysis in this situation is impossible. However, in the framework of a number of problems addressed in this book, we show that the solutions practically do not depend on the distributions involved and are defined mostly by their first two moments. In this way, the results obtained for specific pdf’s are extended to the general model of machine reliability. Time-dependent vs. operation-dependent failures: As it was pointed out on a number of occasions above, machines in a production system can be blocked or starved. If a machine is, indeed, blocked or starved, it is forced down and does not perform its technological operation. In this situation, is its uptime “ticking” or not? The answer depends on the nature of the machine and its technological operation. For instance, tool wear does not occur when the machine is idle, while power failures are practically independent of the machine

74

CHAPTER 3. MATHEMATICAL MODELING

status. To distinguish between these two situations, the following notions are introduced: Operation-dependent failures – machine breakdowns cannot occur while it is blocked or starved. Time-dependent failures – machine breakdowns may occur even while it is blocked or starved. It turns out that all performance measures, considered in this book, take practically identical values under either time- or operation-dependent failures: in most cases the difference is within 1% - 3% (in particular, when buffers are not too small). Therefore, selection of a failure mode – time- or operationdependent – can be made on the basis of convenience for subsequent analyses. Since, as it turns out, time-dependent failures are simpler for analysis, this is the convention considered throughout this book. Extensions to operation-dependent failures are possible.

3.3.3

Notations

In the slotted time case, each machine is denoted in this book by a pair [Ptup (t), Ptdown (t)], where Ptup (t) and Ptdown (t) are the pmf’s of up- and downtime, respectively. For instance, a machine can be denoted as [Bup , Bdown ] or [Geoup , Geodown ] or [Bup , Geodown ]. Similarly, in the continuous time case, each machine is denoted as [ftup (t), ftdown (t)], where the first symbol defines the uptime pdf and the second the downtime pdf, each belonging to the set {exp, Ra, W, ga, LN, G}. Examples of this notation can be given as [expup , expdown ], [Wup , gadown ], [Gup , LNdown ], etc. Analogous notations are used for production systems consisting of several machines. For example, a serial line with M machines is denoted as {[ftup (t), ftdown (t)]1 , [ftup (t), ftdown (t)]2 , . . . , [ftup (t), ftdown (t)]M }, where [ftup (t), ftdown (t)]i denotes the reliability model of the i-th machine in the system. If all machines in the system have identical reliability models, the serial line is denoted as {[ftup (t), ftdown (t)]i , i = 1, . . . , M }. Similar notations are used for production systems operating in slotted time. The above notations are extended to machines with non-identical cycle times as follows: Each machine is denoted by a triple [τ, ftup (t), ftdown (t)], and a serial production line by the expression {[τ, ftup (t), ftdown (t)]1 , [τ, ftup (t), ftdown (t)]2 , . . . , [τ, ftup (t), ftdown (t)]M }.

3.3. MATHEMATICAL MODELS OF MACHINES

3.3.4

75

Machine model identification

Machine model identification – the process of determining the machine characteristics described above. These characteristics include the machine cycle time, τ , and its reliability model, i.e., Ptup (t), Ptdown (t) or ftup (t), ftdown (t). The cycle time in most cases can be easily identified using a stop watch and measuring the time used by the machine to process a part. If manual loading and unloading operations take place, the loading and unloading time must be included in the machine cycle time. In some production systems, manual loading and unloading take more time than part processing. In these cases, the machine has a large cycle time even if its “own” (i.e., technological) cycle time is small. Also, it should be pointed out that the “official” cycle times of the machines, recorded in an appropriate log, may be far from the real one. This happens because equipment maintenance personnel often makes adjustments, which modify the cycle time, without properly recording it or even realizing that a change has occurred. In the course of time, these changes are accumulated and lead to the situation mentioned above. Nevertheless, machine cycle time can be viewed as a rather “stable” or slowly changing machine characteristic. However, in every application it must be carefully measured. The machine reliability model is much more difficult to identify. Strictly speaking, it requires the identification of the histograms of up- and downtime, which, in turn, require a very large number of measurements during a long period of time. The result is that the pmf’s or pdf’s of up- and downtime of the machines on the factory floor are, practically, never known. What is typically known is the average up- and downtime of the machines, Tup and Tdown , often referred to as the mean time to failure (MTTF) and mean time to repair (MTTR), respectively. These characteristics are determined by measuring the durations of randomly occurring up- and downtime, tup,i and tdown,i and then calculating their averages according to Pn i=1 tup,i , (3.7) Tup = Pn n i=1 tdown,i Tdown = , (3.8) n where number n is sufficiently large to guarantee statistically reliable estimates (see Problem 2.8 of Chapter 2). It is advisable to continually monitor Tup and Tdown since they may (and often do) change in time. In this case, Tup (s) and Tdown (s) are calculated, where s = 1, 2, . . . , is the index of the period of observation, consisting of n occurrences of up- and downtime, i.e., Pn i=1 tup,i (s) , (3.9) Tup (s) = Pn n i=1 tdown,i (s) Tdown (s) = , (3.10) n s = 1, 2, . . . .

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CHAPTER 3. MATHEMATICAL MODELING

If Tup (s) and/or Tdown (s) exhibit undesirable trends (i.e., Tup (s) decreasing and/or Tdown (s) increasing), appropriate actions must be taken, for instance, re-evaluation of preventative maintenance procedures. As it will be shown in Parts II - IV, the knowledge of Tup and Tdown is important and, moreover, necessary for the production line management technique referred to as measurement-based management (MBM). It should be pointed out that values of Tup and Tdown are often available from equipment manufacturers. However, they may be quite different from their real values. The same can be said for some operator log data. Therefore, for any analysis, design, and continuous improvement project, it would be prudent to accurately identify Tup and Tdown . As it is shown in Parts II - IV, MBM requires the knowledge of not only Tup and Tdown but also the coefficients of variation, CVup and CVdown . Fortunately, they can be calculated based on the measurements, which are used to calculate Tup and Tdown . Indeed, since Pn 2 i=1 (tup,i − Tup ) , V ar(tup ) = n−1 Pn 2 i=1 (tdown,i − Tdown ) , V ar(tdown ) = n−1 the CV s can be calculated as CVup

=

CVdown

=

p V ar(tup ) , Tup p V ar(tdown ) . Tdown

(3.11) (3.12)

If continuous monitoring of up- and downtime takes place, CVup (s) and CVdown (s) may be calculated as follows: p V ar(tup (s)) , s = 1, 2, . . . , (3.13) CVup (s) = Tup (s) p V ar(tdown (s)) CVdown (s) = , s = 1, 2, . . . . (3.14) Tdown (s) Realistically speaking, Tup , Tdown and CVup , CVdown (or Tup (s), Tdown (s) and CVup (s), CVdown (s)) may be the only characteristics of reliability models available from the factory floor. Fortunately, they are also sufficient, as it is shown in Parts II - IV, to solve most of the analysis, continuous improvement, and design problems of practical importance, while the knowledge of Tup and Tdown alone is not sufficient for these purposes. Finally, it should be pointed out that even when Tup , Tdown and CVup , CVdown (or Tup (s), Tdown (s) and CVup (s), CVdown (s)) are available, in most cases it cannot be assumed that these values are, in fact, precise. Experience shows that in most realistic cases, these data may have up to 5% - 10% errors

3.3. MATHEMATICAL MODELS OF MACHINES

77

as compared with their real values. This discrepancy is largely due to the fact that machine characteristics are changing in time and the measurement process of the realizations tup,i and tdown,i is not fault free.

3.3.5

Calculating parameters of aggregated machines

As it was mentioned in Section 3.2, structural modeling of a production system may require “combining” parallel machines (as in Figure 3.10) or consecutive dependent machines (as in Figure 3.11) into one aggregated machine (as in Figure 3.12). Below, methods for calculating parameters of aggregated machines are given. Aggregating parallel machines: This aggregation process is illustrated in Figure 3.18. Below, we provide expressions for the parameters of the aggregated par par par machine mpar agg , i.e., {τagg , Tup,agg , Tdown,agg }. m1 par

m2

m agg

mS

Figure 3.18: Aggregating parallel machines (a) Identical machines. Assume that each machine mi , i = 1, . . . , S, is characterized by τi = τ = 1/c, Tup,i = Tup , and Tdown,i = Tdown . Then cpar agg is selected as cpar agg = Sc, which implies that par τagg =

1 cpar agg

=

τ . S

(3.15)

The average uptime and downtime of the aggregated machine are selected as par Tup,agg par Tdown,agg

=

Tup ,

(3.16)

=

Tdown .

(3.17)

When machines mi , i = 1, . . . , S, are exponential with parameters λ and µ, these expressions become λpar agg µpar agg

=

λ,

(3.18)

=

µ.

(3.19)

78

CHAPTER 3. MATHEMATICAL MODELING

(b) Nonidentical machines. Assume that each machine mi , i = 1, . . . , S, is characterized by τi = 1/ci , Tup,i , and Tdown,i . Then cpar agg is selected as cpar agg =

S X

ci ,

i=1

i.e., 1 par τagg = PS

1 i=1 τi

.

(3.20)

The average up- and downtime of the aggregated machine are selected based on the following considerations. The parallel system has the capacity cpar agg only when all S machines are up; when less than S machines are up, the capacity is lower. To model this situation, we assume that the capacity of the aggregated machine remains the same while its uptime is reduced appropriately. For instance, in the case S = 2, the following four events can occur: both machines are up, both are down, the first is up and the second down, and the first is down par can be defined as: and the second is up. Then, Tup,agg par Tup,agg

=

Tup,1 Tup,2 +

τ2 τ1 +τ2 Tup,1 Tdown,2

1 2 (Tup,1

+

τ1 τ1 +τ2 Tup,2 Tdown,1

+ Tdown,1 + Tup,2 + Tdown,2 )

.

Here, the first term in the numerator indicates the case when both machines are up, and the latter two terms refer to the situation where one machine is up and the other down, while the denominator is the average (up+down)-time of par is selected so that the throughput of the the machines. Given this, Tdown,agg aggregated machine is the same as that of the parallel system, i.e., par := cpar T Pagg agg

2 par X Tup,agg Tup,i = ci . par par Tup,agg + Tdown,agg Tup,i + Tdown,i i=1

In other words, par = Tdown,agg

Tdown,1 Tdown,2 + 1 2 (Tup,1

τ2 τ1 +τ2 Tdown,1 Tup,2

+

τ1 τ1 +τ2 Tdown,2 Tup,1

+ Tdown,1 + Tup,2 + Tdown,2 )

.

In the case S > 2, these arguments lead to the following expressions: par Tup,agg

=

par Tdown,agg

=

i PS h QS 1 1 1 par ( + ) τagg i=1 τi Tdown,i j=1,j6=i Tup,j Tdown,j i , (3.21) PS h QS 1 1 1 1 ( + ) i=1 Tup,i Tdown,i j=1,j6=i Tup,j S Tdown,j h i P Q S S 1 1 1 par τagg i=1 τi Tup,i j=1,j6=i ( Tup,j + Tdown,j ) i . (3.22) PS h QS 1 1 1 1 ( + ) i=1 Tup,i Tdown,i j=1,j6=i Tup,j S Tdown,j

3.3. MATHEMATICAL MODELS OF MACHINES

79

If all S machines are exponential with parameters λi and µi , these expressions become i QS PS h λi µi j=1,j6=i (λj + µj ) i=1 h i, = λpar (3.23) agg QS par PS 1 Sτagg µ (λ + µ ) i j j i=1 τi j=1,j6=i i QS PS h λi µi j=1,j6=i (λj + µj ) i=1 h i. = µpar (3.24) agg QS par PS 1 Sτagg i=1 τi λi j=1,j6=i (λj + µj ) Clearly, the above expressions are only approximations of the real system. However, in most cases, they lead to accurate estimates of the original system performance. For instance, consider a system shown in Figure 3.19 and assume that the machines are exponential with τ Tup Tdown

= {1.2, 2, 2.6, 1, 3.4, 2.5, 3, 1.4}, = {100, 33.3, 52.6316, 20, 50, 100, 33.3, 100}, = {10, 5, 6.6667, 8.3333, 10, 5.8824, 4, 12.5},

i.e., e = {0.91, 0.87, 0.89, 0.71, 0.83, 0.94, 0.89, 0.89}. Then the exponential machines of the aggregated system, shown in Figure 3.20,

m5

m2 m1

b1

b2

m3

m4

b3

m6 m7

b4

m8

Figure 3.19: Serial production line with parallel machines are defined by the following parameters: par τagg par Tup,agg par Tdown,agg

i.e.,

= = =

{1.2, 1.13, 1, 0.97, 1.4}, {100, 40.8558, 20, 50.8103, 100}, {10, 5.7091, 8.3333, 5.9039, 12.5},

epar agg = {0.9091, 0.8774, 0.7059, 0.8959, 0.8889}.

The throughput of these two lines, T P and T Pagg , has been evaluated by simulations for buffer capacities varied from N = 5 to N = 100. The accuracy of the aggregation has been quantified by ²T P =

T P − T Pagg · 100%. TP

80

CHAPTER 3. MATHEMATICAL MODELING

m1

par

b1

magg1

b2

m4

b3

par

magg2

b4

m8

Figure 3.20: Aggregated version of the serial line of Figure 3.19 The results, along with 95% confidence intervals, are given in Table 3.1 and illustrated graphically in Figure 3.21. Clearly, the accuracy is sufficiently high, especially when N can accommodate at least one downtime of the machines. Table 3.1: Accuracy of aggregation of parallel machines N 5 10 20 30 40 50 100

0.5388 0.5874 0.6202 0.6292 0.6326 0.6339 0.6347

TP ± 0.0002354 ± 0.0002444 ± 0.0001873 ± 0.0002383 ± 0.0001769 ± 0.0002363 ± 0.0001998

0.5105 0.5699 0.6138 0.6272 0.6318 0.6332 0.6346

T Pagg ± 0.0002409 ± 0.0002215 ± 0.0002238 ± 0.0002062 ± 0.0002004 ± 0.0002182 ± 0.0002008

²T P -5.25% -2.98% -1.03% -0.32% -0.13% -0.11% -0.02%

Aggregating consecutive dependent machines: This aggregation is illustrated in Figure 3.22. (a) Identical machines. Again, it is first assumed that machines are identical and each machine, mi , i = 1, . . . , S, is characterized by {τ, Tup , Tdown }. The con con con parameters of the aggregated machine mcon agg , i.e., {τagg , Tup,agg , Tdown,agg }, are selected as follows: con τagg

=

con Tup,agg

=

con Tdown,agg

=

τ, ³

(3.25)

´S−1 Tup Tup , Tup + Tdown ´S i h ³ Tup (Tup + Tdown ). 1− Tup + Tdown

(3.26) (3.27)

In the case of exponential machines with parameters λ and µ, we obtain λcon agg

=

µcon agg

=

λ(λ + µ)S−1 , µS−1 λµ ´S i . h ³ µ (λ + µ) 1 − λ+µ

3.3. MATHEMATICAL MODELS OF MACHINES

81

−2

ε

TP

(%)

0

−4

−6 0

50 N

100

Figure 3.21: Illustration of accuracy of parallel machines aggregation

m1

m2

con

m agg

mS

Figure 3.22: Aggregating consecutive machines (b) Nonidentical machines. Assume each machine mi , i = 1, . . . , S, is characterized by {τi , Tup,i , Tdown,i }. The cycle time of the aggregated machine mcon agg , con , is selected as follows: i.e., τagg con = max τi . τagg i

(3.28)

The average up- and downtime of the aggregated machine are selected based on the following considerations. The consecutive dependent system is working con can be only when all S machines are up. For example, when S = 2, Tup,agg understood as the time that both machines are up within an average (up+down)time of the machines, i.e., con = Tup,agg

1 Tup,1 Tup,2 (Tup,1 + Tdown,1 + Tup,2 + Tdown,2 ) · . 2 Tup,1 + Tdown,1 Tup,2 + Tdown,2

con Given this, Tdown,agg is again selected so that the throughput of the aggregated machine is the same as that of the consecutive dependent system, i.e.,

econ agg :=

con Tup,agg Tup,1 Tup,2 = · . con con Tup,agg + Tdown,agg Tup,1 + Tdown,1 Tup,2 + Tdown,2

In other words, con = Tdown,agg

´ ³ 1 Tup,1 Tup,2 (Tup,1 +Tdown,1 +Tup,2 +Tdown,2 ) 1− . 2 (Tup,1 + Tdown,1 )(Tup,2 + Tdown,2 )

82

CHAPTER 3. MATHEMATICAL MODELING In the case S > 2, these arguments lead to the following expressions: con Tup,agg

=

S S ³ ´ Y 1X Tup,i , (Tup,i + Tdown,i ) S i=1 Tup,i + Tdown,i i=1

con Tdown,agg

=

S S ³ ´i h Y 1X Tup,i .. (3.30) (Tup,i + Tdown,i ) 1 − S i=1 Tup,i + Tdown,i i=1

(3.29)

If all S machines are exponential with parameters λi and µi , these expressions become λcon agg

=

PS

³

i=1

µcon agg

=

PS i=1

³

S 1 λi

1 λi

+

1 µi

+

1 µi

´Q

S µi i=1 λi +µi

,

S ´h QS ³ i ´i . 1 − i=1 λiµ+µ i

(3.31) (3.32)

The accuracy of this aggregation is of the same order of magnitude as that for parallel machines. PSE Toolbox: The process of aggregating parallel and consecutive dependent machines is implemented as the first two tools of the toolbox function Modeling. For a description and illustration of these tools, see Section 19.2.

3.3.6

Machine quality models

In some manufacturing operations, machines can produce defective parts, along with non-defective ones. To formalize this situation, we introduce machine quality models – the pmf or pdf of time intervals during which the machine produces good or defective parts. Examples of quality models are listed below: • Bernoulli quality model - each part produced during a cycle time is good with probability g and defective with probability 1 − g, independent of the quality of parts produced during previous cycles. • Exponential quality model - when up, the intervals of time during which a machine produces good or defective parts are distributed exponentially with parameters γ and β, respectively. • General quality model - when up, the intervals of time during which a machine produces good or defective parts are distributed according to arbitrary pdf’s. The Bernoulli quality model is appropriate when the defects are due to independent random events (for instance, dust or scratches in automotive painting operations). The exponential and general quality models are appropriate when the defects are due to deterioration in the machine operation, such as wearing of the cutting tools or vibration of the workpiece.

3.4. MATHEMATICAL MODELS OF BUFFERS

3.4 3.4.1

83

Mathematical Models of Buffers Modeling

For the purposes of this textbook, the mathematical model of a buffer is very simple – it is its capacity, N , i.e., the maximum number of parts that it can store. It is assumed throughout that N < ∞, implying that buffers are finite. The number of parts contained in a buffer at a given time is referred to as its occupancy. Since in a production system, the occupancy of a buffer at a given time (slot or moment) depends on its occupancy at the previous time (slot or moment), buffers are dynamical systems with the occupancy being their states. If the machines are modeled as discrete event systems, the state of the buffer is an integer between 0 and N . In flow models, states are real numbers between 0 and N . It is assumed that a part, produced by a machine, is immediately placed in the downstream buffer, if it is not full. Similarly, it is assumed that a part is immediately available for processing by a machine, if the upstream buffer is not empty. Although these assumptions are introduced to simplify the analysis, most production systems are designed and operated so that they hold. In some cases, where they do not hold but the times for placing a part in the buffer and transferring a part to the subsequent machine are relatively constant, this assumption might be “compensated” by an appropriate increase of the machine cycle time. In any case, experience shows that these assumptions do not lead to erroneous results as far as steady state performance measures are concerned. In some cases, buffers are relatively complex material handling devices, e.g., robots, automated guided vehicles, etc., and, therefore, may experience breakdowns in the same manner as machines do. In these cases, a material handling device can be modeled as a machine followed by a buffer, as illustrated in Figure 3.23. MHD

= mechanical part of MHD

buffering capacity of MHD

Figure 3.23: Modeling material handling devices with breakdowns

3.4.2

Buffer parameters identification

Buffer model identification – the process of determining the capacity of the buffer, N .

84

CHAPTER 3. MATHEMATICAL MODELING

As it has been mentioned above, buffers on the factory floor may take the form of boxes, which store the work-in-process, or conveyors, or silos, or robotic storage devices, or automated guided vehicles. In most cases, their capacity can be relatively easily identified. For instance, in a kanban system, N is determined by the number of kanban cards between each pair of consecutive machines. In robotic storage devices, the capacity also can be evaluated through the analysis of the storage spaces available and part size. In the case of conveyors or similar material handling devices (e.g., silos), where the parts are transported from one operation to another on carriers, the identification of buffer capacity is carried out as follows:

Conveyor mi+1

mi carriers

v l

Figure 3.24: Conveyor material handling Assume that the length of the conveyor between machines mi and mi+1 is l, its speed is v, and the cycle time of the machines is τ (see Figure 3.24). Then the time to travel from mi to mi+1 is l , v and the number of carriers necessary to sustain continuous operation is m lT travel , N0 = τ Ttravel =

where dxe is the smallest integer larger than or equal to x. If only N0 carriers are available between mi and mi+1 , this conveyor provides no buffering capabilities. However, if there is space for Ki > N0 carriers on the conveyor between mi and mi+1 , the buffer capacity is Ni = Ki − N0 . Similar arguments can be used for identification of buffering capacity of other material handling devices when the cycle times of the machines are not identical.

3.5

Modeling Interactions between Machines and Buffers

This section defines conventions, which specify how the states of the machines and buffers affect each other. The purpose of these conventions is to ensure uniqueness of the mathematical description of the production systems at hand.

3.5. MODELING OF INTERACTIONS

3.5.1

85

Slotted time case

State changing convention: Machine state (or status in the Bernoulli reliability case) is determined at the beginning of each time slot. This implies that a chance experiment, carried out at the beginning of each time slot, determines whether the machine is up or down during this time slot. Buffer state is determined at the end of each time slot. This implies that buffer occupancy may change only at the end of a time slot. For instance, if the state of the buffer was 0 at the end of the previous time slot, then the downstream machine does not produce a part during the subsequent time slot, even if it is up. If the buffer state was h 6= 0 and the downstream machine was up and not blocked, then the state of the buffer at the end of the next time slot is h if the upstream machine produces a part, or h − 1 if the upstream machine fails to produce. Blocking and starvation conventions: Blocked before service (BBS) – a machine cannot operate if it is up (as determined at the beginning of the time slot), the downstream buffer is full (as determined at the end of the previous time slot), and the downstream machine does not take a part from this buffer at the beginning of this time slot. In other words, the part, which is being processed by a machine, is viewed as being already in the subsequent buffer. Blocked after service (BAS) – if the machine is up (as determined at the beginning of the time slot) and the upstream buffer is not empty (as determined at the end of the previous time slot), the machine operates on the part, even if the downstream buffer is full. The machine becomes blocked if the same conditions persist during the next time slot. Thus, the capacity of buffers under BBS and BAS conventions are related as i = 1, . . . , M − 1. NiBBS = NiBAS + 1, In other words, while NiBAS ≥ 0, NiBBS ≥ 1. All performance measures under these conventions are quite similar. In this textbook, we use the blocked before service convention, since it leads to a simpler description. As far as starvations are concerned, it is assumed that a machine is starved during a time slot if it is up (as determined at the beginning of this time slot) and the upstream buffer is empty (as determined at the end of the previous time slot).

3.5.2

Continuous time case

In analytical investigations of flow models, i.e., when state transitions of the machines and buffers may take place at any time moment, the state changing conventions (either before or after service) do not affect system analysis or performance. In numerical investigations of flow models, the slotted time conventions are used with slot duration δt ¿ τ .

86

3.6

CHAPTER 3. MATHEMATICAL MODELING

Performance Measures

The mathematical models of machines and buffers, described above, are necessary, in particular, for calculating performance measures of production systems at hand. Although some of them have been mentioned in Chapters 1 and 2, below we provide formal definitions.

3.6.1

Production rate and throughput

Production rate (P R) – average number of parts produced by the last machine of a production system per cycle time in the steady state of system operation. This metric is appropriate for production systems with all machines having identical cycle times. In the asynchronous case, this metric is referred to as Throughput (TP) – average number of parts produced by the last machine of a productions system per unit of time in the steady state of system operation. Clearly, T P can be used in the synchronous case as well; in this case, T P = c · P R, where c is the machine capacity. Since P R and T P are steady state performance measures, due to conservation of flow, the same average number of parts is produced by any machine in the system. Both P R and T P are functions of machine and buffer parameters. For instance, in the case of a serial line with M Bernoulli machines, i.e., {[Bup , Bdown ]1 , . . . , [Bup , Bdown ]M }, this function can be denoted as P R = P R(p1 , . . . , pM , N1 , . . . , NM −1 ). In the case of {[expup , expdown ]1 , . . . , [expup , expdown ]M }, this function becomes P R = P R(λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 ). In the asynchronous exponential case, this function is T P = T P (τ1 , λ1 , µ1 , . . . , τM , λM , µM , N1 , . . . , NM −1 ). Due to the complex nature of interference among the machines and buffers (through blockages and starvations), these functions are extremely complicated, and their closed-form expressions are all but impossible to derive, except for the case of two-machine systems. For M > 2, several approximation techniques, based on decomposition or aggregation of longer lines into two-machine systems,

3.6. PERFORMANCE MEASURES

87

have been developed. Specific aggregation techniques, used in this book, are described in Parts II - IV. The situation becomes even more complex for systems with non-Markovian machines. For instance, if the machines are characterized by the Weibull reliability model (3.4), the production rate has the form P R = P R(λ1 , Λ1 , µ1 , M1 , . . . , λM , ΛM , µM , MM , N1 , . . . , NM −1 ), and this function cannot be expressed in closed form even for M = 2. Therefore, the approach to non-Markovian systems, used in this book, is based on empirical expressions, derived through extensive experimentation with systems at hand (see Part III). All of the above remarks remain valid in the case of more complex serial lines and assembly systems. For example, in the closed serial line of Figure 3.3 with Bernoulli machines, this function is P R = P R(p1 , . . . , pM , N1 , . . . , NM −1 , S, N0 ), where S is the number of carriers and N0 is the capacity of the empty carrier buffer. For the assembly system of Figure 1.2 with Bernoulli machines, PR

=

P R(p11 , . . . , p1M1 , N11 , . . . , N1M1 ; p21 , . . . , p2M2 , N21 , . . . , NM2 ; p01 , . . . , p0M0 , N01 , . . . , N0(M0 −1) ).

(3.33)

In production lines that include non-perfect quality machines, P R denotes the production rate of non-defective parts. In this case, for the Bernoulli reliability and quality models, P R = P R(p1 , . . . , pM , g1 , . . . , gM , N1 , . . . , NM −1 , S, N0 ). The average number of parts used by the first machine per cycle time is another performance measure, referred to as the consumption rate (CR). The difference between CR and P R is the scrap rate (SR).

3.6.2

Work-in-process and finished goods inventory

Work-in-process of the i-th buffer (W IPi ) – average number of parts contained in the i-th in-process buffer of a production system in the steady state of its operation. Total work-in-process (W IP ) – average number of parts contained in all inprocess buffers of a production system in the steady state of its operation. For instance, for serial lines with M machines, W IP =

M −1 X

W IPi .

i=1

Finished goods inventory (FGI) – average number of parts contained in the finished goods buffer of a production system in the steady state of its operation.

88

CHAPTER 3. MATHEMATICAL MODELING

Clearly, each of these performance measures is a function of system parameters. For instance W IPi for the line {[Bup , Bdown ]1 , . . . , [Bup , Bdown ]M }, can be denoted as W IPi = W IPi (p1 , . . . , pM , N1 , . . . , NM −1 ). For the line {[expup , expdown ]1 , . . . , [expup , expdown ]M }, this function is W IPi = W IPi (λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 ). As in the case of the production rate, there are no closed-form expressions for these functions, except for M = 2 and machines with Bernoulli, geometric, or exponential reliability models. Again, for M > 2, aggregation techniques are available (see Parts II - IV). A similar situation takes place in the case of the finished goods inventory. The function of interest here is F GI = F GI(λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 NF GB ), which corresponds to serial lines with exponential machines. Unfortunately, there are no closed-form expressions for this performance measure, even for M = 2, and only approximations are available.

3.6.3

Probabilities of blockages and starvations

While P R and W IP are performance measures widely used in manufacturing, both in practice and theory, the metrics discussed in this subsection are used quite rarely, at least in practice. Nevertheless, as it is shown in this textbook, they play a central role in measurement-based management of production systems. While Tup ’s and Tdown ’s characterize the reliability of the machines in isolation, and N ’s characterize the storing efficacy of the buffers, the blockages and starvations characterize the production systems as a whole, i.e., machines and buffers placed in specific positions within a production system. As a result, these metrics, as shown in Parts II - IV, define the system bottlenecks and other system-wide characteristics. Consequently, they may and should be used for managing production systems. While the discussion of measurement-based management is included in Parts II - IV, below we define these performance metrics (for the blocked before service convention). Blockage of machine i (BLi ) – steady state probability that machine i is up, buffer i is full, and machine i + 1 does not take a part from the buffer. Starvation of machine i (STi ) – steady state probability that machine i is up and buffer i − 1 is empty.

3.6. PERFORMANCE MEASURES

89

For the case of serial lines in slotted time, these performance measures can be expressed as BLi

=

P [{mi is up at the beginning of the time slot} ∩ {bi is full at the end of the previous time slot} ∩ {mi+1 does not take a part from bi at the beginning of the time slot}], i = 1, . . . , M − 1,

STi

=

P [{mi is up at the beginning of the time slot} ∩ {bi−1 is empty at the end of the previous time slot}], i = 2, . . . , M.

For the case of serial lines in continuous time, these performance measures can be expressed as BLi

= P [{mi is up at time t} ∩ {bi is full at time t} ∩ {mi+1 does not take material from bi at time t}], i = 1, . . . , M − 1,

STi

= P [{mi is up at time t} ∩ {bi−1 is empty at time t}], i = 2, . . . , M.

It is typically assumed that m1 is never starved and mM is never blocked. Each of the above probabilities are functions of the machine and buffer parameters and their locations within the system. For instance, for {[expup , expdown ]1 , . . ., [expup , expdown ]M } these functions become BLi STi

= =

BLi (λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 ), STi (λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 ).

Again, closed-form expressions for these performance metrics can be derived only for M = 2 and Bernoulli, geometric, or exponential machines. For M > 2, the aggregation procedures mentioned above lead to estimates of these quantities (see Parts II - IV).

3.6.4

Residence time

Residence time (RT) – average time a part spends in the system in the steady state of its operation. As it follows from the Little’s Law, RT can be easily evaluated, if P R (or T P ) and W IP are known. Indeed, RT =

W IP [cycle time] PR

or

RT =

W IP [units of time]. TP

The residence time is important for quoting product delivery time to the customer. PM Note that RT can be quite large even if the total processing time, i.e., i−=1 τi , is small. Note also that in industry RT is sometimes referred to as the flow time or the system cycle time.

90

CHAPTER 3. MATHEMATICAL MODELING

3.6.5

Due-time performance

Due-time performance (DTP) – steady state probability to ship to the customer the desired number of parts during a given time period. Assume that the customer requires D parts to be shipped during each shipping period T . Then DT P = P [{ship to the customer D parts every shipping period T }]. This performance metric is typically used for production systems with finished goods buffers (FGB). For instance, for the line {[expup , expdown ]1 , . . ., [expup , expdown ]M } with a FGB of capacity NF GB this function can be denoted as DT P = DT P (λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 , NF GB , D, T ). Analytical expressions for this function are derived only for a single-machine production system, and lower bounds are available for M > 1 (see Parts II IV).

3.6.6

Transient characteristics

Transient characteristics – a group of performance measures that describe how P R, W IP , and the probabilities of buffer occupancy reach their steady state values. These properties are described in terms of several metrics. Perhaps, the most important of them is the settling time, ts , which is the time necessary to reach and remain in a ±5% neighborhood of the steady state, starting from zero initial conditions (i.e., all empty buffers). Thus, in this textbook, we characterize the transient behavior by tsP R and tsW IP , (i.e., the settling times of P R and W IP , respectively). In addition, we investigate the properties of the second largest eigenvalue of the transition matrix of Markov chains that characterize production systems at hand, which, as it turns out, defines the behavior of tsP R and tsW IP .

3.6.7

Evaluating performance measures on the factory floor

Clearly, all the performance measures mentioned above can be evaluated by monitoring the appropriate metrics on the factory floor and calculating their average values. In current practice, P R or T P are monitored in most production systems (but, astonishingly, not in all). W IP is monitored, on a continuous basis, less often. The blockages and starvations are monitored very rarely. We show in this book that all of them must be monitored continuously in order to exercise measurement-based management of production systems.

3.7. MODEL VALIDATION

3.7

91

Model Validation

Model validation – process of assessing accuracy of the mathematical model of the production system. Typically, this process is carried out by comparing predictions of the model d with factory floor measurements. For instance, let P R and P R be the production rates identified on the factory floor and calculated using the model, respectively. Then the value of the error, defined as

²P R =

d |P R − P R| · 100%, PR

(3.34)

gives a measure of model fidelity. The values of ²P R , which can be viewed as acceptable, are of the same order of magnitude as the accuracy with which parameters of the model are identified. Typically, as it was mentioned above, machine parameters are identified with a 5% - 10% accuracy (mostly due to the low accuracy of the data available on the factory floor). Thus, an accuracy of model prediction at the level of 5% - 10% is often viewed as acceptable. In cases where the errors are outside of this “fuzzy” region, the process of modeling must be repeated anew. In other words, the process of production systems modeling is iterative in nature, as it is the case in all engineering disciplines. In applications where not only P R but also W IPi , STi , and BLi are available from factory floor measurements, the latter can be used for model validation as well. The measure of fidelity (3.34), however, must be modified to avoid small denominators when either W IPi or STi or BLi are very small. Therefore, the following measures for model validation are recommended:

²W IPi ²STi ²BLi

\ |W IPi − W IP i | · 100%, i = 1, . . . , M − 1, Ni c i |, = |STi − ST i = 2, . . . , M, d = |BLi − BLi |, i = 1, . . . , M − 1, =

(3.35) (3.36) (3.37)

c i and BL di are calculated \ where Ni is the i-th buffer capacity and W IP i , ST using the model. When a production system is at the design stage and no measurements of its performance are available, the above model validation process cannot be carried out. In this case, opinion of manufacturing system specialists is the only guide for assessing model adequacy.

92

CHAPTER 3. MATHEMATICAL MODELING

3.8 3.8.1

Steps of Modeling, Analysis, Design, and Improvement Modeling

To summarize, the process of production system modeling consists of the following five steps: • Layout investigation. The goal here is to identify the physical layout of the production system at hand, as exemplified, for instance, by Figures 3.13 and 3.15. • Structural modeling. The goal here is to reduce the identified physical layout to one of the standard types of production systems, as exemplified by Figures 3.14 and 3.17. • Machine parameters identification. The goal here is to determine cycle time τi , average up- and downtime, Tup,i and Tdown,i , and coefficients of variation, CVup,i and CVdown,i of each machine in the system. If the process of structural modeling included aggregation of some of the machines, parameters of the aggregated machines must also be calculated, as described in Subsection 3.3.5. • Buffer parameter identification. The purpose here is to identify the storing capacity of each buffer, as described in Subsection 3.4.2. • Model validation. The purpose is to compare predictions of the model with factory floor measurements, as described in Section 3.7. If it turns out that the errors are too large, all steps must be repeated anew. It must be pointed out that the process of mathematical modeling is, perhaps, the most important stage of production systems analysis, design, and continuous improvement. Engaging in this process, one should keep in mind that, as it was mentioned before, the model should be as simple as possible but not simpler.

3.8.2

Analysis, continuous improvement, and design

Typically, mathematical models are used for analysis, continuous improvement, and design of production systems based on one or more of the following steps: • Evaluation of the model performance measures (e.g., P R, T P , CR, SR, W IP , BL, ST , ts ). • Identification of system bottlenecks, i.e., machines and buffers that impede the system performance in the strongest manner. • Investigation of “what if” scenarios, i.e., prediction of system performance if some of its elements are changed (i.e., improvement/replacement of a machine, increase/re-allocation of buffer capacity, re-assignment of workforce, etc.).

3.9 SIMPLIFICATION

93

• Design of lean buffering, i.e., the smallest buffer capacity, which is necessary and sufficient to ensure the desired throughput of the system. • Based on the above data, determination of the most promising direction for system improvement, taking into account all practical constraints (i.e., financial and personnel resources, space and time availability, etc.). Parts II - IV of this textbook provide tools for carrying out these steps for a wide range of production systems in large volume manufacturing environments.

3.9 3.9.1

Simplification: Transforming Exponential Models into Bernoulli Models Motivation

As it turns out, production systems with Bernoulli machines are easier to analyze than similar systems with exponential machines. Therefore, in some cases it is beneficial to “simplify” an exponential line to a Bernoulli one, if such a simplification does not lead to a substantially lower accuracy. In this section, this simplification, referred to as exp-B (exponential-to-Bernoulli) transformation, is described and its accuracy with respect to P R calculation is evaluated. After the exp-B transformation is performed and the subsequent analysis and improvement of the Bernoulli model is carried out, it is often necessary to return to the exponential description (for instance, to guide the implementation of the improvement measures developed). This is accomplished by the so-called B-exp (Bernoulli-to-exponential) transformation, which is also described below.

3.9.2

Exponential and Bernoulli lines considered

Consider a serial production line with M exponential machines denoted as [τi , expup,i , expdown,i ],

i = 1, . . . , M.

This implies that the i-th machine is capable of producing ci =

1 parts/unit of time, τi

and up- and downtime are distributed exponentially with parameters λi and µi , respectively, i.e., fup,i (t) fdown,i (t)

= =

λi e−λi t , µi e

−µi t

,

t ≥ 0, t ≥ 0.

Thus, the i-th machine is characterized by the triple: (ci , λi , µi ),

i = 1, . . . , M.

94

CHAPTER 3. MATHEMATICAL MODELING

The i-th buffer is characterized by its capacity, Ni , i = 1, . . . , M − 1. Assume that this serial line operates according to the following conventions: (a) Flow model. (b) Machine status is determined independently from each other. (c) Time-dependent failures. (d) Blocked before service. (e) The first machine is never starved and the last machine is never blocked. Along with this system, consider a Bernoulli line with M machines denoted as i = 1, . . . , M. [Bup,i , Bdown,i ], In other words, the i-th machine produces a part during a cycle time with , and the cycle time, τ , of all machines is the same. Denote the probability pBer i buffer capacities of this line as NiBer , i = 1, . . . , M − 1, and assume that the line operates according to the following assumptions: (a) Synchronous model with slot duration τ . (b) The status of the machines is determined at the beginning of each time slot and the state of the buffer at the end of the time slot. (c) Machine status is determined independently from each other. (d) Time-dependent failures. (e) Blocked before service. (f) The first machine is never starved and the last machine is never blocked. Given these two production lines, the purpose of the exp-B transforma, i = 1, . . . , M , NiBer , tion (illustrated in Figure 3.25) is to calculate pBer i i = 1, . . . , M − 1, and τ , so that the throughputs of the two lines, T P exp and T P Ber = (1/τ )P RBer , are sufficiently close to each other. N1

(c1, λ1, µ1)

N M−1

exp−B transformation

Ber

N1

(c M, λM, µM )

Ber

( p , τ) 1

Ber

N M−1

Ber

( pM , τ )

Figure 3.25: Illustration of exp-B transformation The inverse operation, i.e., the B-exp transformation, is illustrated in Figure 3.26. The goal here is to calculate the parameters of the transformed exponential line, i.e., tr tr i = 1, . . . , M ctr i , λi , µi , and Nitr ,

i = 1, . . . , M − 1

so that the throughput of the transformed exponential line, T P tr , is sufficiently close to (1/τ )P RBer . Both of these transformations are described below.

3.9.3

The exp-B transformation

Given the exponential line with machines (ci , λi , µi ) and buffers Ni , introduce , and buffers NiBer defined the Bernoulli line with cycle time τ , machines pBer i

3.9 SIMPLIFICATION Ber

95 B−exp transformation

Ber

N M−1

N1

Ber

Ber

(p , τ) 1

(pM , τ)

tr

tr

tr

N1

tr

N M−1

tr

tr tr (cM , λM, µM )

tr

(c1 , λ1, µ1 )

Figure 3.26: Illustration of B-exp transformation as follows: τ

=

pBer i

=

NiBer

1 cmax

,

(3.38)

ci 1 cmax · λi 1 1 λi + µi

= min

=

ci ei , cmax

i = 1, . . . , M,

´ ³ N Ni i µi , µi+1 + 1, ci+1 ci

i = 1, . . . , M − 1,

(3.39) (3.40)

where cmax ei

= max(ci ), µi = , λi + µi

i = 1, . . . , M, i = 1, . . . , M.

The reason for equations (3.38) and (3.39) are obvious: the cycle time of the Bernoulli machines equals to that of the fastest exponential machine, and the production rate of each exponential machine in isolation equals that of the corresponding Bernoulli machine. The reason behind equation (3.40) is in the following: The buffer in the Bernoulli model can prevent starvation of the downstream machine and blockage of the upstream machine for a number of i represents time slots at most equal to the size of the buffer. In (3.40), cN i+1 the largest time during which the buffer can protect the downstream machine i when the upstream machine is down. Therefore, cN µi is the fraction of average i+1 downtime of the upstream machine that can be accommodated by the buffer. Thus, this quantity can be considered as the equivalent Bernoulli buffer size. Analogously, Ncii µi+1 is the fraction of average downtime of the downstream machine that can be accommodated by the buffer and also can be viewed as the equivalent Bernoulli buffer size. By choosing the worst case, we have the equivalent buffer size in the Bernoulli model given by (3.40). Note that under the blocked before service convention, the buffer size should be greater than or equal to 1, because the machine itself is viewed as a unit of buffer capacity. To account for this fact, we need the 1 in (3.40). In principle, the Bernoulli buffer size should be an integer. However, the theory and calculation formulas developed in Part II do work for fractional buffers as well. Therefore, we allow, according to (3.40), fractional buffer sizes. The accuracy of the exp-B transformation has been assessed numerically by simulating exponential and the corresponding Bernoulli lines, statistically

96

CHAPTER 3. MATHEMATICAL MODELING

evaluating T P exp and T P Ber = cmax P RBer and calculating the error ²T P =

|T P exp − T P Ber | · 100%. T P exp

Numerous systems have been investigated. Typical results are illustrated in Tables 3.2 and 3.3 and Figure 3.27 for three- and ten-machine lines, respectively. As one can see from the tables, for exponential lines, which result in NiBer ≥ 2, the error is quite small (less than 4%); for NiBer < 2, the error is up to 7% - 8%. Similar conclusions follow from Figure 3.27 for ten-machine lines with parameters selected randomly and equiprobably form the following sets: e = [0.7, 0.98], Tdown = [5, 20], N = [5, 40], c = [1, 2].

(3.41)

Specifically, Figure 3.27 illustrates the following three exponential lines: Line 1: e = {0.867, 0.852, 0.925, 0.895, 0.943, 0.897, 0.892, 0.935, 0.903, 0.870}, Tdown = {14.23, 16.89, 18.83, 16.08, 7.65, 11.09, 19.05, 18.76, 11.15, 18.42}, N = {7.026, 17.350, 33.461, 5.345, 9.861, 12.097, 11.955, 26.133, 14.527}, c = {1.950, 1.231, 1.607, 1.486, 1.891, 1.762, 1.457, 1.019, 1.822, 1.445}. Line 2: e

= {0.945, 0.873, 0.911, 0.899, 0.939, 0.926, 0.896, 0.852, 0.932, 0.895}, Tdown = {14.22, 16.89, 18.83, 16.08, 7.65, 11.09, 19.05, 18.76, 11.15, 18.42}, N = {5.535, 31.138, 20.578, 37.614, 21.310, 19.653, 34.618, 23.380, 12.093}, c = {1.672, 1.838, 1.020, 1.681, 1.380, 1.832, 1.503, 1.709, 1.429, 1.305}. Line 3: e = {0.869, 0.869, 0.918, 0.880, 0.904, 0.865, 0.920, 0.888, 0.936, 0.935}, Tdown = {13.91, 12.45, 18.48, 17.33, 14.68, 17.27, 14.90, 10.13, 9.35, 10.12}, N = {26.746, 32.819, 38.490, 23.291, 35.805, 11.054, 39.291, 14.501, 13.832}, c

=

{1.534, 1.727, 1.309, 1.839, 1.568, 1.370, 1.703, 1.547, 1.445, 1.695}.

In this figure, the buffer capacity of the exponential lines is selected as e = αN, N where N is given in the definition of each line and α ∈ [0.1, 10]; the error, ²T P , is illustrated as a function of α. Again, as one can see, the exp-B transformation is quite accurate for buffers that can accommodate at least one largest downtime of the machines.

3.9 SIMPLIFICATION

97

40

30 20

ε

TP

20 10

10 0

0 −10 0

5 α

−10 0

10

5 α

(a) Line 1

(b) Line 2

30

TP

20

ε

ε

TP

30

10 0 −10 0

5 α

10

(c) Line 3 Figure 3.27: Accuracy of exp-B transformation for Lines 1 - 3

10

98

CHAPTER 3. MATHEMATICAL MODELING

Table 3.2: Accuracy of the exp-B transformation for identical machines and buffers (M = 3, λ = 0.01, µ = 0.1) N1 50 50 50 50 50 100 100 100 100 100 200 200 200 200 200 400 400 400 400 400 600 600 600 600 600

N2 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600

T P exp 5.6234±0.0041 5.7491±0.0045 5.8768±0.0046 5.9510±0.0045 5.9641±0.0044 5.8568±0.0041 5.9811±0.0041 6.1092±0.0043 6.1863±0.0040 6.2005±0.0039 6.1323±0.0039 6.2566±0.0038 6.3881±0.004 6.4701±0.0039 6.4864±0.0037 6.3715±0.0036 6.5001±0.0035 6.6392±0.0036 6.7311±0.0033 6.7517±0.0032 6.466±0.0037 6.5998±0.0035 6.7462±0.0034 6.8454±0.0032 6.8683±0.0031

N1Ber 1.35 1.35 1.35 1.35 1.35 1.7 1.7 1.7 1.7 1.7 2.4 2.4 2.4 2.4 2.4 3.8 3.8 3.8 3.8 3.8 5.2 5.2 5.2 5.2 5.2

N2Ber 1.35 1.7 2.4 3.8 5.2 1.35 1.7 2.4 3.8 5.2 1.35 1.7 2.4 3.8 5.2 1.35 1.7 2.4 3.8 5.2 1.35 1.7 2.4 3.8 5.2

T P Ber 6.0895 6.1727 6.2396 6.2652 6.2677 6.2691 6.3565 6.4286 6.4581 6.4612 6.4701 6.564 6.644 6.6794 6.6836 6.6288 6.7324 6.8234 6.8662 6.8719 6.8719 6.7911 6.889 6.9362 6.9428

|T P exp −T P Ber | T P exp

8.3 7.4 6.2 5.3 5.1 7 6.3 5.2 4.4 4.2 5.5 4.9 4 3.2 3 4 3.6 2.8 2 1.8 3.3 2.9 2.1 1.3 1.1

· 100%

3.9 SIMPLIFICATION

99

Table 3.3: Accuracy of the exp-B transformation for non-identical machines and buffers (c1 = 7, λ1 = 0.02, µ1 = 0.08), (c2 = 10, λ2 = 0.03, µ2 = 0.07), (c3 = 9, λ3 = 0.01, µ3 = 0.09) N1 50 50 50 50 50 100 100 100 100 100 200 200 200 200 200 400 400 400 400 400 600 600 600 600 600

N2 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600 50 100 200 400 600

T P exp 4.4971±0.0041 4.4772±0.0044 4.5384±0.0044 4.5577±0.0042 4.5592±0.0041 4.7244±0.0036 4.8001±0.0037 4.8618±0.004 4.8837±0.0036 4.8857±0.0036 5.0632±0.0027 5.1325±0.0027 5.1908±0.0028 5.215±0.0028 5.2176±0.0027 5.3404±0.003 5.3954±0.0029 5.4399±0.0029 5.4602±0.0029 5.4629±0.0029 5.4553±0.0028 5.4978±0.0028 5.5296±0.0028 5.5432±0.0028 5.545±0.0029

N1Ber 1.4 1.4 1.4 1.4 1.4 1.8 1.8 1.8 1.8 1.8 2.6 2.6 2.6 2.6 2.6 4.2 4.2 4.2 4.2 4.2 5.8 5.8 5.8 5.8 5.8

N2Ber 1.4 1.7778 2.5556 4.1111 5.6667 1.3889 1.7778 2.5556 4.1111 5.6667 1.3889 1.7778 2.5556 4.1111 5.6667 1.3889 1.7778 2.5556 4.1111 5.6667 1.3889 1.7778 2.5556 4.1111 5.6667

T P Ber 4.3433 4.791 4.8476 4.8688 4.8708 4.9346 5.0053 5.0609 5.083 5.0853 4.9346 5.2498 5.2993 5.3197 5.3197 5.4098 5.4568 5.4907 5.5043 5.5060 5.5031 5.5359 5.557 5.5647 5.5657

|T P exp −T P Ber | T P exp

7.3 7 6.8 6.8 6.8 4.4 4.3 4.1 4.1 4.1 2.4 2.3 2.1 2 2 1.3 1.1 0.9 0.8 0.8 0.9 0.7 0.5 0.4 0.4

· 100%

100

CHAPTER 3. MATHEMATICAL MODELING

3.9.4

The B-exp transformation

The transformation from Bernoulli to the exponential model is accomplished as follows: Suppose a serial Bernoulli production line is given along with the original exponential line from which it has been deduced. Define the parameters of the new exponential line as follows: µtr i = µi , If

pBer cmax i ci

pBer cmax i ci

(3.42)

< 1, then ctr i = ci ,

If

i = 1, . . . , M.

and

etr i =

pBer cmax i , tr ci

i = 1, . . . , M.

(3.43)

pBer cmax i , tr ci

i = 1, . . . , M.

(3.44)

≥ 1, choose ctr i such that

cmax pBer i < 1, tr ci

etr i =

and

From (3.43) and (3.44), obviously 0 < etr i < 1,

i = 1, . . . , M.

(3.45)

Using (3.42), we have etr i =

µi µtr = tr i tr , λi + µi λi + µi

It follows that λtr i =

i = 1, . . . , M.

tr (1 − etr i )µi . etr i

(3.46)

(3.47)

The equation corresponding to (3.40) can be expressed as Nitr = max

³ N Ber − 1 i

µtr i

ctr i+1 ,

NiBer − 1 tr ´ ci , µtr i+1

i = 1, . . . , M − 1.

(3.48)

Numerical investigations indicate that the accuracy of the B-exp transformation is roughly the same as that of the exp-B transformation. Finally, it should be pointed out that the exp-B and B-exp transformations are reversible in the following sense: Transferring an exponential line to a Bernoulli one using (3.38)-(3.40) and then returning to exponential description using (3.42)-(3.48), results in the original exponential line. PSE Toolbox: The exp-B and B-exp transformations have been implemented as two tools of the toolbox function Modeling. For a description and illustration of these tools, see Section 19.2.

3.9 SIMPLIFICATION

3.9.5

101

Exp-B and B-exp transformations for assembly systems

Exp-B transformaion: This transformation in assembly systems is similar to that in serial lines. Specifically, given the exponential assembly systems with machines (c1i , λ1i , µ1i ), (c2i , λ2i , µ2i ), (c0i , λ0i , µ0i ) and buffers N1i , N2i , N0i (see Figure 1.2), we introduce the Bernoulli assembly system with cycle time τ , Ber Ber Ber Ber Ber machine efficiencies pBer 1i , p2i , p0i , and buffer capacities N1i , N2i , N0i defined as follows: τ pBer ji Ber Nji

Ber NjM j

1

=

cmax

,

(3.49)

cji 1 cmax · λji 1 1 λji + µji

=

= min

= min

=

cji eji , cmax

j = 0, 1, 2, i = 1, . . . , Mj ,

´ ³ N Nji ji µji , µj,i+1 + 1, cj,i+1 cji ³N

jMj

c01

´

µjMj ,

(3.50)

j = 0, 1, 2, i = 1, . . . , Mj − 1,

NjMj µ01 + 1, cjMj

j = 1, 2,

(3.51) (3.52)

where cmax

= max(cji ), µji = , λji + µji

eji

j = 0, 1, 2, i = 1, . . . , Mj , j = 0, 1, 2, i = 1, . . . , Mj .

B-exp transformation: Similar to B-exp transformation in serial lines, the parameters of the transformed exponential assembly system are defined as µtr ji = µji , If

pBer ji cmax cji

pBer ji cmax cji

(3.53)

< 1, then

ctr ji = cji , If

j = 0, 1, 2, i = 1, . . . , Mj .

and etr ji =

pBer ji cmax , ctr ji

j = 0, 1, 2, i = 1, . . . , Mj .

(3.54)

≥ 1, choose ctr ji such that

pBer ji cmax < 1, ctr ji

and etr ji =

pBer ji cmax , ctr ji

j = 0, 1, 2, i = 1, . . . , Mj . (3.55)

From (3.54) and (3.55), 0 < etr ji < 1,

j = 0, 1, 2, i = 1, . . . , Mj .

(3.56)

102

CHAPTER 3. MATHEMATICAL MODELING Using (3.53) we have etr ji =

µtr µji ji = tr , λji + µji λji + µtr ji

j = 0, 1, 2, i = 1, . . . , Mj .

(3.57)

Then, it follows that λtr ji =

tr (1 − etr ji )µji , etr ji

j = 0, 1, 2, i = 1, . . . , Mj .

(3.58)

The buffer capacities of the transformed exponential lines can be expressed as tr Nji

tr NjM j

= max

= max

³ N Ber − 1 ji

µtr ji ³ N Ber − 1 jMj µtr jMj

ctr j,i+1 ,

Ber − 1 tr ´ Nji c , tr µj,i+1 ji

Ber − NjM j ctr , 01 tr µ01

j = 0, 1, 2,

i = 1, . . . , Mj − 1, 1 tr ´ j = 1, 2. cjMj ,

(3.59) (3.60)

PSE Toolbox: The exp-B and B-exp transformations for assembly systems have been implemented as two tools of the toolbox function Modeling. For a description and illustration of these tools, see Section 19.2.

3.10

Case Studies

This section describes modeling of several production systems from the automotive industry. The resulting models are used in Parts II - IV to illustrate methods of analysis, continuous improvement, and design included in this textbook.

3.10.1

Automotive ignition coil processing system

System description and layout: The production system of coils for an automotive ignition unit is shown in Figure 3.28. It consists of 16 operations where the coils are either processed (e.g., Op. 5) or have external parts attached (e.g., Op. 13). Ops. 8 and 12 load the external parts. Ops. 9 and 10 carry out identical operations and are coupled in the sense that one of them being down forces the other down as well; parts are directed to either Op. 9 or 10 automatically. Therefore, Ops. 9 and 10 are aggregated into one machine. The coils are transported within the system on pallets by a conveyor. The empty pallet buffer is implemented by two devices referred to as elevators. The raw material (i.e., an unfinished coil) is loaded at Op. 1, if an empty pallet is available; if not, Op. 1 is starved for pallets. Op. 16 transfers the finished product to the downstream system and releases the pallet, if the subsequent elevator is not full; otherwise, Op. 16 is blocked.

3.10. CASE STUDIES

103 External part 1

Raw material

8

1

2

Trans. Sec. Coil

3

4

5

6

Primary Staker Turnover Turnover Solder

Solder

7

9-10

11

13

Turnover Tower Resistance Trans. 12

14

15

16

Assemble Check Trans. Trans To Epoxy Height Coil Coils Assy

External part 2 Elevator

Elevator

Pallets

Figure 3.28: Layout of the ignition coil processing system The system’s performance during eight consecutive weeks is characterized in Table 3.4. For the first three weeks, the machines cycle time was set to produce 562.53 parts/hour. The measured performance, however, was substantially lower, amounting to an average loss of 16.1%. For the subsequent five weeks, the system was sped up to produce 592.07 parts/hour, while the average throughput remained practically the same (472.6 parts/hr), with average losses of 20.32%. Table 3.4: System performance Week

Week Week Week Week Week Week Week Week

1 2 3 4 5 6 7 8

Nominal throughput (parts/hr) 562.53 562.53 562.53 593.07 593.07 593.07 593.07 593.07

Actual throughput (parts/hr) 464 505 447 501 454 424 480 504

Losses (%) 17.52 10.23 20.54 15.52 23.45 28.51 19.07 15.02

Average throughput (parts/hr)

Average losses (%)

472

16.10

472.6

20.32

The goal of this case study was to identify reasons for these losses and provide suggestions for system improvement. While the description of the analysis and continuous improvement is given in Part II, its modeling is described below. Structural modeling: Neglecting the closed nature of the conveyor and the effect of Ops. 8 and 12 (which are quite fast and reliable and, therefore, do not impede the system performance), the model of the coil processing system can be represented as the serial production line shown in Figure 3.29. This

104

CHAPTER 3. MATHEMATICAL MODELING

structural model is used for the subsequent modeling, analysis, and continuous improvement. 1

2

3

4

5

6

7

9-10

11

13

14

15

16

Figure 3.29: Structural model of ignition coil processing system

Modeling and identification of the machines: Assuming that the machines are exponential, their identification amounts to determining the triple {τi , λi , µi }, for all operations. Based on Table 3.4, the cycle time for Period 1 (the first three weeks) was 6.4 sec/part and for Period 2 (the last five weeks) 6.07 sec/part. The machines’ average up- and downtime were determined from the Weekly Data Report using expression (3.7) and (3.8) (see Table 3.5 and note that Ops. 4 and 13 have zero downtime, and, therefore, are not included in Table 3.5). Based on these data, λi and µi have been calculated as follows: λi

=

µi

=

1

, Tup,i 1 . Tdown,i

The results are given in Table 3.6. Table 3.5: Average up- and downtimes (min) Uptime Downtime Uptime Downtime

Op. 1 227.79 1.837

Op. 2 188.11 1.517

Op. 9-10 13.98 1.571

Op. 3 504.15 1.517

Op. 11 43.07 1.748

Op. 5 1515.5 1.517

Op. 14 74.33 1.517

Op. 6 572.27 16.485

Op. 15 188.11 1.517

Op. 7 1493.2 9.013 Op. 16 356.02 2.149

(a). Period 1 Uptime Downtime Uptime Downtime

Op. 1 141.18 2.15

Op. 2 280.31 1.976

Op. 9-10 16.25 2.05

Op. 3 651.73 3.275

Op. 11 45.11 2.076

Op. 5 438.67 4.879

Op. 14 52.91 1.976

(b). Period 2

Op. 6 450.37 3.632

Op. 15 168.89 2.398

Op. 7 1974 1.976

Op. 16 201 2.854

3.10. CASE STUDIES

105

Table 3.6: Parameters of the machines (1/min) Op. 1 0.0044 0.5444

λi µi

Op. 2 0.0053 0.6592

Op. 9-10 0.0715 0.6365

λi µi

Op. 3 0.0020 0.6592

Op. 11 0.0232 0.5721

Op. 5 0.0007 0.6592

Op. 14 0.0135 0.6592

Op. 6 0.0017 0.0607

Op. 15 0.0053 0.6592

Op. 7 0.0007 0.1110 Op. 16 0.0028 0.4653

(a). Period 1 Op. 1 0.0071 0.4651

λi µi λi µi

Op. 2 0.0036 0.5061

Op. 9-10 0.0615 0.4878

Op. 3 0.0015 0.3053

Op. 11 0.0222 0.4817

Op. 5 0.0023 0.2050

Op. 14 0.0189 0.5061

Op. 6 0.0022 0.2753

Op. 15 0.0059 0.4170

Op. 7 0.0005 0.5061 Op. 16 0.0050 0.3504

(b). Period 2

Modeling and identification of buffers: The buffering capacity of the conveyor between each pair of consecutive operations has been evaluated using the method of Subsection 3.4.2. The results are given in Table 3.7. Table 3.7: Buffer capacity Operation Buffer capacity Operation Buffer capacity

Op. 1 3

Op. 7 4

Op. 2 1

Op. 9-10 1

Op. 3 1 Op. 11 6

Op. 4 4 Op. 13 1

Op. 5 1 Op. 14 1

Op. 6 1 Op. 15 2

Overall system model: Based on the data of Tables 3.6 and 3.7, the exponential model of the coil processing system has been identified as shown in i is the i-th machine efficiency). Figure 3.30 (where, as before, ei = λiµ+µ i The Bernoulli model of this system can be obtained using the exp-B transformation of Section 3.9. The resulting system is shown in Figure 3.31. In addition, Figure 3.31 includes the effect of the conveyor by multiplying p1 by the probability that Op. 1 is not starved for pallets. This probability has been identified experimentally during the steady state of system operation. No blockages of Op. 16 by full elevators have been observed and, therefore, p16 has not

106

CHAPTER 3. MATHEMATICAL MODELING 1

2

3

3

4

1

5

1

6

7

1

4

e i:

0.9920

0.9920

0.9970

1.0

0.9989

Tdown,i:

1.837

1.517

1.517

0.0

1.517

1

2

4

5

9-10

11

0.9728

14

15

16

6

0.9937

0.8990

0.9610

1.0

0.9799

0.9920

0.9940

9.009

1.571

1.748

0.0

1.517

1.517

2.149

11

13

4

16.474

13

1

1

1

1

2

(a). Period 1 3

3

1

1

6

7

1

4

9-10

1

1

4

14

6

15

1

16

1

2

e i:

0.985

0.993

0.995

1.0

0.989

0.992

0.999

0.888

0.956

1.0

0.964

0.986

0.986

Tdown,i:

2.150

1.976

3.275

0.0

4.878

3.632

1.976

2.050

2.076

0.0

1.976

2.398

2.854

(b). Period 2 Figure 3.30: Exponential model of the ignition coil processing system been modified. 1

2 1.2

p:

3

4

1.1

5

1.1

0.9920(1-0.07) 0.9920

6

1.3

0.9970

1.0

7

1.0

9-10

1.0

0.9989

0.9728

1.0

0.9937

11 1.1

0.8990

13 1.4

0.9610

14 1.1

1.0

15 1.1

0.9799

16 1.1

0.9920

0.9940

(a). Period 1 1

2 1.1

p:

0.985(1-0.07)

3 1.0

0.9929

4 1.0

0.9951

5 1.1

1.0

6 1.0

0.9889

7 1.0

0.9921

9-10 1.2

0.9990

11 1.0

0.8880

13 1.3

0.9559

14 1.0

1.0

15 1.0

0.9640

16 1.1

0.9860

0.9859

(b). Period 2 Figure 3.31: Bernoulli model of the ignition coil processing system The validation, analysis, and continuous improvement of this model are described in Part II.

3.10.2

Automotive paint shop production system

System description and layout: The layout of a paint shop production system at an automotive assembly plant is shown in Figure 3.32. It consists of 11 operations in which the car bodies (referred to as jobs) are cleaned (chemically or physically), sealed (against water leaks), painted, and, finally finessed. Operations 5, 6, and 8 consist of two parallel lines (due to capacity considerations). Operation 10 consists of five parallel painting booths (for both capacity reasons and to ensure color variety). The jobs within the system are transported by conveyors on two types of carriers. The transfer from one carrier to another occurs after Op. 3. Thus, carriers of type 1 are used in Ops. 1 - 3 and type 2 in Ops. 4 - 11.

3.10. CASE STUDIES

107

49 20

TO GENERAL

15

Op.11

Empty carrier buffer

ASSEMBLY

42 10

116 40

10 2

18

6 1

10 2

18

6 1

21 2

18

6 1

15 2

18

6 1

10 2

18

6 1

45 19

13

Op.9

Op.10

FROM

20 5

BODY SHOP 71 24

2 43

Op.1

67 22

45

134 32

Op.2

25

Op.3

6 3

6

Op.4

8 4 2

4 2 4 2

Op.5

22

11 14 7

20 5

14

15 4

22

11

Op.6

20 5 20 5

Op.7

18

10 5 15

18

Op.8

116 40

Empty carrier buffer

Figure 3.32: Layout of paint shop system The technological operations are performed while the jobs are moving on their carriers. That is why the parts of the conveyor where the work is being carried out are called operational conveyors, while other parts are referred to as accumulators. The conveyor as a whole has a modular structure in the sense that each operational conveyor can be stopped without stopping other operational conveyors. Workers typically use push-buttons to stop their operational conveyors in order to complete respective operations with the desired quality. Thus, the downtime is typically due to quality issues rather than machine breakdowns. The numbers in the circles of Figure 3.32 indicate the number of jobs within the operational conveyors necessary to ensure continuous production. The numbers in rectangles show the minimal occupancy of accumulators to ensure continuous production and the maximal number of jobs that could be contained within an accumulator. Thus, the difference between these two numbers is the buffering capacity of the accumulator. This system was designed to produce 63 jobs/hour (see the capacity, ci , of each operation given in Table 3.8). In reality, however, the throughput was much lower, averaging 52.1 jobs/hour (see Table 3.9 where the measured average throughput for five consecutive months is shown). The goal of this case study was to determine reasons for the production losses and to provide recommendations for their elimination. These analyses are described in Part II, while the mathematical model used for this purpose is constructed below. Structural modeling: The average production losses in Ops. 1 - 11 due to internal reasons (i.e., excluding blockages and starvations) are shown in Table

108

CHAPTER 3. MATHEMATICAL MODELING

Table 3.8: Capacity of the machines (jobs/hour) Ops. ci

1 63

2 63

3 63

4 63

5 63

6 63

7 63

8 63

9 72

10 100

11 100

Table 3.9: System performance (jobs/hour) Period TP

Month 1 53.5

Month 2 43.81

Month 3 51.27

Month 4 54.28

Month 5 55.89

3.10. Clearly, Ops. 1 and 2 have very low or no losses and, therefore, can be excluded. To accomplish this, we conceptually transfer the common point of the two loops of Figure 3.32 from the output of Op. 3 to its input. This transformation does not lead to reduced accuracy since Op. 3 operates in so-called no-gap mode (i.e., no empty space between consecutive jobs on the operational conveyor is allowed). Therefore, after aggregating the parallel machines of Ops. 5, 6, 8, and 10, we represent the system as shown in Figure 3.33. Table 3.10: Average production losses (jobs/hour) Operation Op. 1 Op. 2 Op. 3 Op. 4 Op. 5 Op. 6 Op. 7 Op. 8 Op. 9 Op. 10 Op. 11

Month 1 0 0.05 2.88 2.77 0.23 0 1.09 1.39 6.18 0.35 0.01

Month 2 0.05 0.45 2.15 2.60 0.01 0 3.13 3.42 7.38 1.40 0.01

Month 3 0.01 0.07 0.64 4.45 0.04 0.001 1.09 1.28 7.01 1.66 0.01

Month 4 0 0.03 2.26 2.00 1.07 0.02 2.68 2.73 6.59 3.09 0.01

Month 5 0 0.00 1.35 2.60 1.64 0.39 2.05 0.41 6.14 3.63 0.01

Modeling and identification of the machines: Since, as it follows from Table 3.10, downtime of each operation is mostly of the same order of magnitude as the cycle time, we adopt the Bernoulli model of machine reliability. The

3.10. CASE STUDIES

109 165 60

FROM PREVIOUS

3 25

SHOP

5

4 6 3

8 4

6

30

7

6 14 7

58

80 20

8 10 5

4

66

9 45 19

10 30 5

13

90

11 108 20

TO NEXT

15

SHOP

Figure 3.33: Structural model of paint shop system parameters pi are calculated as n c −L o i i , pi = min 1, 63

i = 3, . . . , 11,

(3.61)

where ci and Li are the capacity and the losses of the i-th operation given in Tables 3.8 and 3.10, respectively. These parameters are summarized in Table 3.11. Table 3.11: Machine parameters (pi )

Operations Month 1 Month 2 Month 3 Month 4 Month 5

3 0.9543 0.9659 0.9898 0.9641 0.9786

4 0.9560 0.9587 0.9294 0.9683 0.9587

5 0.9963 0.9998 0.9994 0.9830 0.9740

6 1 1 1 0.9997 0.9938

7 0.9827 0.9503 0.9827 0.9575 0.9675

8 0.9779 0.9457 0.9797 0.9567 0.9935

9 1 1 1 1 1

10 1 1 1 1 1

Modeling and identification of the buffers: As indicated above, the buffering capacity of each accumulator is the difference between its maximal and minimal occupancy. The capacity of the buffers after Ops. 6 and 10 is assumed to be the sum of the capacities of the parallel buffers. The buffers within Op. 5 are omitted. The resulting data on buffer capacity are summarized in Table 3.12. Table 3.12: Buffer capacity Operations Ni

3 3

4 4

5 7

6 60

7 5

8 26

9 25

10 88

Overall system model: Based on the above, the Bernoulli model of the automotive paint shop is represented as shown in Figure 3.34 and the parameters

11 1 1 1 1 1

110

CHAPTER 3. MATHEMATICAL MODELING

pi for each of the five months are given in Table 3.11. Operations 9 - 11 are omitted since their efficiency is 1 and, therefore, they do not affect the system performance. The effect of the closed loop of Figure 3.33 is taken into account using the probability Pst that Op. 3 is starved (or, in the original system of Figure 3.32, blocked) by carriers. This probability, evaluated during normal system operation, is given in Table 3.13. In Figure 3.34 this probability is used in the factor (1 − Pst ) multiplying p3 . Thus, a simplified model of the paint shop system is constructed. 3

4 4

3

p (1−Pst ) 3

5

p

6 7

p5

4

7

8

60

p

5

p

6

p

8

7

Figure 3.34: Simplified structural model of paint shop system

Table 3.13: Estimated probability of starvation of Op. 3

Pst

Month 1 0.0981

Month 2 0.1171

Month 3 0.1113

Month 4 0.1046

Month 5 0.0975

The validation of this model and its analysis and continuous improvement are described in Part II.

3.10.3

Automotive ignition module assembly system

The description of this system and its structural modeling are given in Section 3.2. Below, the subsequent modeling steps are carried out. The nominal throughput of the system is 600 parts/hour. The actual performance for six consecutive months is summarized in Table 3.14. As it follows from this table, the average throughput over the six months is 362 parts/hour, i.e., the system operates at 60% of its capacity. Table 3.14: Actual throughput of the system for six months Month T P (parts/hr)

May 337

June 347

July 378

Aug. 340

Sep. 384

Oct. 383

Modeling and identification of the machines: We assume that both the uptime and downtime of the machines are distributed exponentially. Their identification requires to determine the average up- and downtime for each machine,

3.10. CASE STUDIES

111

or their recipricals λi and µi . The data for this identification have been measured during real-time system operation and are summarized in Tables 3.15 and 3.16. This, along with the cycle time of each operation, 6 sec/part, identifies completely the exponential machines that compromise the system. Table 3.15: Machine parameters Operations 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

May Tdown Tup (min) (min) 4.8 38.8 3.0 35.4 4.5 70.5 10.2 68.3 8.9 65.3 2.0 98 1.8 58.2 2.5 47.5 3.9 31.6 2.6 34.5 2.7 31.1 3.3 326.7 3.8 91.2 5.2 98.8 1.8 14.6 2.8 137.2 1.7 55 2.2 107.8

June Tdown Tup (min) (min) 8.4 159.6 1.7 55 5.7 136.8 7.1 57.4 6.3 56.7 6.7 216.6 4.4 142.3 2.4 37.6 7.1 81.7 3.3 33.4 3.4 45.2 0.9 89.1 1.6 38.4 2.5 47.5 2.8 90.5 10.8 529.2 10.3 504.7 1.8 20.7

July Tdown Tup (min) (min) 18.3 286.7 3.7 42.6 12.4 142.6 12.7 66.7 12.7 66.7 5.2 59.8 12.2 231.8 5.7 65.6 7.3 65.7 4.0 53.1 4.1 54.5 16.9 224.5 17.9 144.8 10.2 103.1 16.8 223.2 23.4 269.1 27.7 368 3.4 30.6

Modeling and identification of the buffers: The capacity of the buffers has been identified using the method of Subsection 3.4.2. The resulting capacities are shown in Table 3.17. Overall system model: Based on the data of Tables 3.15 - 3.17, the exponential model of the coil assembly system has been identified as shown in Figure 3.35 (with the up- and downtime data for the month of May). For the subsequent analysis, the exponential model of Figure 3.35 and similar models for five other months have been reduced to Bernoulli models, using the exp-B transformation of Section 3.9. The resulting system is shown in Figure 3.36. In this figure, the effect of the closed nature of circular conveyors, i.e., the starvation of Ops. 1 and 9 and the blockage of Op. 18 are taken into account. Specifically, the average fraction of time when Ops. 1 and 9 were starved for

112

CHAPTER 3. MATHEMATICAL MODELING

Table 3.16: Machine parameters (cont.) August Tdown Tup (min) (min) 4.4 435.6 2.7 267.3 2.0 48 11.4 92.2 10.7 86.6 5.9 190.8 8.2 401.8 2.3 43.7 6.5 47.7 4.0 46 5.1 24.9 9.6 310.4 6.6 125.4 4.5 59.8 6.5 58.5 5.5 73.1 4.1 77.9 3.3 38

Operations 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

5

5

4 7

5 5

September Tdown Tup (min) (min) 3.3 107 3.7 366 1.5 28 2.2 20 4.2 48 3.1 152 4.1 133 0.7 34 4.1 78 3.4 65 2.4 32 14.7 149 1.2 39 2.9 142 1.0 99 2.3 22998 1.0 9999 1.3 64

6 5

7 5

8 8

8

Ave. uptime [m]:

38.8

35.4

70.5

68.3

65.3

98

58.2

47.5

Ave. downtime [m]:

4.8

3.0

4.5

10.2

8.9

2.0

1.8

2.5

9

10 5

Ave. uptime [m]:

31.6

Ave. downtime [m]: 3.9

11 5

October Tdown Tup (min) (min) 9.6 110 10.7 142 2.8 32 7.7 56 11.8 95 5.5 178 6.1 197 2.8 67 2.9 45 3.6 86 1.9 25 1.6 158 2.1 50 2.6 12 1.2 39 1.2 11999 4.9 485 2.3 113

13

12 5

14 5

8

34.5

31.1

326.7

2.6

2.7

3.3

91.2 3.8

15 5

98.8 5.2

16 5

14.6 1.8

17 5

18 6

137.2 55 2.8 1.7

Figure 3.35: Exponential model of the ignition module assembly system

107.8 2.2

3.11. SUMMARY

113 Table 3.17: Buffer capacity

Operations Buffer capacity Operations Buffer capacity

1 5 10 5

2 5 11 5

3 7

4 5

12 8

5 5

13 5

6 5 14 5

7 8 15 5

8 8

9 5

16 5

17 6

pallets and Op. 18 blocked has been identified during real-time operation. The results are summarized in Table 3.18.. To take these losses into account, the efficiencies of Ops. 1, 9 and 18 have been multiplied by a factor (1- average fraction of time when starvation or blockage takes place) (see Figure 3.36). Table 3.18: Average frequency of starvation and blockage Month Average fraction of time when Op. 1 is starved Average fraction of time when Op. 9 is starved Average fraction of time when Op. 18 is blocked

May

June

July

Aug.

Sep.

Oct.

0.257

0.308

0.285

0.28

0.199

0.252

0.134

0.247

0.199

0.099

0.142

0.089

0.236

0.256

0.226

0.189

0.11

0.238

The validation, analysis, and continuous improvement of this model are described in Part IV.

3.11

Summary

• The types of production systems considered in this textbook are serial lines and assembly systems. • The process of mathematical modeling of production systems consists of the following steps: – – – – –

layout investigation structural modeling machine parameter identification buffer parameter identification model validation.

114

CHAPTER 3. MATHEMATICAL MODELING 1

1.1

2

1.1

p : 0.89(1-0.257) 0.92

3

1.1

0.94

4

1.0

0.87

5

1.1

6

1.3

7

1.3

8

1.2

0.88

0.98

0.97

0.95

9

10

11

12

1.1

1.2

p : 0.89(1-0.134) 0.93

1.2

0.92

1.2

13

1.1

0.96

14

1.1

0.95

15

1.2

0.89

16

1.2

0.98

17

1.3

18

0.97 0.98(1-0.236)

0.99

Figure 3.36: Bernoulli model of the ignition module assembly system (based on May data) • A machine model consists of its – – – –

cycle time pmf or pdf of its uptime pmf or pdf of its downtime pmf or pdf of parts quality.

• The specific pmf’s and pdf’s considered in this textbook are: – – – – – – – –

Bernoulli geometric exponential Rayleigh Weibull gamma log-normal general.

• Buffers are modeled by their storing capacity. • A production system can be considered as operating – – – – –

in slotted or continuous time with time-dependent or operation-dependent failures synchronously or asynchronously as a discrete event system or as a flow system with blocked before service or with blocked after service convention.

• Performance measures that characterize the behavior of production systems are: – – – – – –

production rate or throughput, consumption rate, and scrap rate work-in-process finished goods inventory probability of blockages and starvations the level of customer demand satisfaction (or due-time performance) transient properties.

• Mathematical models of production systems described in this chapter are used throughout this textbook for case studies.

3.12. PROBLEMS

3.12

115

Problems

Problem 3.1 A production system manufactures products A and B. Each product consists of two parts: A1 and A2 for product A and B1 and B2 for product B. The processing of A and B require several technological steps. The departments where these steps are carried out are shown in Figure 3.37. The number of each department indicates its order in the technological process. The Department 1 Blanking of A1

Department 2 Power cut of A1 and A2

Department 1’ Blanking of A2

Department II’ Wet cut of B2

Department II Wet cut of B1 Parts A1 & A2 and B1 & B2 are Shipped offsite for heat treatment (This option takes about 48 hours, while the cycle time of the machines in all department is about 30 sec.)

Department 3’ Hard turn of A2

Department III Hard turn of B1 and B2

Department IV Assembly of B

Department 3 Hard turn of A1

Department 4 Assembly of A

Figure 3.37: Problem 3.1 material handling among the departments is carried out by carts, which are pushed by machine operators from one department to another. (a) Construct a structural model of this production system. (b) Describe the data that have to be collected to identify this model. (c) Describe which steps must be taken to collect these data. (d) Describe which steps must be taken to validate this model. Problem 3.2 The layout of a production system for an automotive ignition device is shown in Figure 3.38. It consists of four main operations: Housing Subassembly, Valve Body Assembly, Injector Subassembly, and Injector Final Assembly. In addition, the system contains Shell Assembly, three Welding operations (L.H.W., U.H.W., and Weld), two Overmold operations (O.M.1 and

116

CHAPTER 3. MATHEMATICAL MODELING

O.M.2), two Set Stroke operations (Stroke 1 and Stroke 2), one Leak Test operation (L.T.) and one High Potential operation (Hi Pot). Finally, the system includes five buffers positioned as shown in Figure 3.38 and conveyor buffering among all other operations. Construct a structural model for this system and simplify it to a serial line. L.T.

L.H.W.

U.H.W.

O.M.1

O.M.2

Hi Pot

Buffer

Buffer Buffer Injector Final Asm. Housing Sub.

Buffer

L.C./L.H./L.I.T.

Injector Sub. Valve Body Sub. Buffer

S.V. Stroke 1

Shell Asm.

Stroke 2

Weld

Legend: L.C./L.H./L.I.T.=Load Coil/Load Housing/Load Inlet Tube

L.V.B.S.

L.N.M.S.

L.N.M.S.=Load Non-Magnetic Shell

L.H.W.=Lower Housing Weld

L.T.=Leak Test O.M.=Overmold U.H.W.=Upper Housing Weld

L.V.B.S.=Load Valve Body Shell S.V.=Stroke Verify

Figure 3.38: Problem 3.2 Problem 3.3 Consider the production system of Figure 3.38 and its model as a serial line obtained in Problem 3.2. Assume that the first five machines are of interest and their parameters are as follows: The cycle time of each machine is 3 sec and the breakdown and repair rates are (in units of 1/min): λ λ λ λ λ

(S.A) (H.S.) (L.T.) (L.H.W.) (U.H.W.)

= = = = =

0.1075, 0.3173, 0.0051, 0.0051, 0.0101,

µ µ µ µ µ

(S.A.) (H.S.) (L.T.) (L.H.W.) (U.H.W.)

= = = = =

0.5; 0.6711; 0.5; 0.5; 1.

Assume also that the buffers between these operations have the following capacities: buffer buffer buffer buffer

between between between between

S.A. and H.S. = 125; H.S. and L.T. = 500; L.T. and L.H.W. = 15; L.H.W. and U.H.W. = 15.

3.13. ANNOTATED BIBLIOGRAPHY

117

(a) Construct the Bernoulli model of this five-machine exponential serial line. (b) Using the Simulation function of the PSE Toolbox, investigate the accuracy of the Bernoulli model. Problem 3.4 A serial production line with five exponential machines is defined as follows: λ = [0.0025, 0.0011, 0.0016, 0.0031, 0.0042] (in units of 1/min), µ = [0.025, 0.0333, 0.025, 0.0286, 0.05] (in units of 1/min), c = [1.0714, 1.0714, 0.6667, 0.9375, 1] (in units of parts/min), N = [26, 10, 28, 30]. (a) Construct the Bernoulli model of this five-machine exponential serial line. (b) Using the Simulation function of the PSE Toolbox, investigate the accuracy of the Bernoulli model. Problem 3.5 The layout of a production system for an automotive ignition device is shown in Figure 3.39. It consists of 15 operations, separated by bufferconveyors. Construct a structural model for this system and simplify it to a serial line. Op.2

Op.4

Op.8

Op.12

Op.6

Op.1

Op.3

Op.5

Op.7

Op.14

Op.10

Op.9

Op.11

Op.13

Op.15

Figure 3.39: Problem 3.5

Problem 3.6 The layout of an automotive camshaft production line is shown in Figure 3.40. Construct a structural model for this system and simplify it into two parallel serial lines.

3.13

Annotated Bibliography

Various aspects of production systems modeling can be found in [3.1] J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993.

118

CHAPTER 3. MATHEMATICAL MODELING Sta. 1

Sta. 2

Sta. 3

Qualifier &

CMS

Lobe

Misc Drilling

Broach

Mill

Sta. 4

Grinder

Sta. 5

Lobe Grinder

Sta. 6

Hardener

Sta. 7

Sta. 8

Straighter Polish

Sta. 9

Washer

Sta. 10

Moore Gage

Figure 3.40: Problem 3.6 [3.2] H.T. Papadopoulos, C. Heavey and J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design, Chapman & Hall, London, 1993. [3.3] S.B. Gershwin, Manufacturing Systems Engineering, Prentice Hall, Englewood Cliffs, NJ, 1994. [3.4] T. Altiok, Performance Analysis of Manufacturing Systems, Springer, New York, 1997. A detailed discussion of different types of machine blocking is given in [3.5] H.G. Perros, Queueing Networks with Blocking, Oxford University Press, New York, 1994. The exp-B and B-exp transformations are introduced in [3.6] C.-T. Kuo, Bottlenecks in Production Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1996. The accuracy of parallel machine aggregation (Subsection 3.3.5) has been investigated by F. Xu in the framework of his Research Experience for Undergraduates at the University of Michigan. He also produced the data for Figure 3.27. The case studies reported in Section 3.10 have been carried out by S.-Y. Chiang, D. Jacobs, C.-T. Kuo, J. Li, J.-T. Lim, S. M. Meerkov, F. Top, and L. Zhang. Some of them can be found in [3.7] J.-T. Lim, S.M. Meerkov and F. Top, “Homogeneous, Asymptotically Reliable Serial Production Lines: Theory and a Case Study,” IEEE Transactions on Automatic Control, vol. 35, pp. 524-534, 1990. [3.8] D.A. Jacobs and S.M. Meerkov, “A System-Theoretic Property of Serial Production Lines: Improvability,” International Journal of Systems Science, vol. 26, pp. 755-785, 1995. [3.9] J.-T. Lim and S.M. Meerkov, “On Asymptotically Reliable Serial Production Lines,” Control Engineering Practice, vol. 1, 147-152, 1993.

3.13. ANNOTATED BIBLIOGRAPHY

119

[3.10] C.-T. Kuo, J.-T. Lim and S.M. Meerkov, “Bottlenecks in Serial Production Lines: A System-Theoretic Approach,” Mathematical Problems in Engineering, vol. 2, pp. 233-276, 1996. [3.11] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “DT-Bottlenecks in Serial Production Lines: Theory and Applications,” IEEE Transactions on Robotics and Automation, vol. 16, pp. 567-580, 2000. [3.12] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “Improvability of Assembly Systems II: Improvability Indicator and Case Study,” Mathematical Problems in Engineering, vol. 5, pp. 359-393, 2000. [3.13] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “c-Bottlenecks in Serial Production Lines: Identification and Application,” Mathematical Problems in Engineering, vol. 7, pp. 543-578, 2001. [3.14] J. Li and S.M. Meerkov, “Customer Demand Satisfaction in Production Systems: A Due-Time Performance Approach,” IEEE Transactions on Robotics and Automation, vol. 17, pp. 472-482, 2001.

Chapter 4

Analysis of Bernoulli Lines Motivation: The problem of performance analysis of production systems consists of investigating their performance measures, e.g., production rate (P R), work-in-process (W IPi ), blockages (BLi ) and starvations (STi ), as functions of machine and buffer parameters. In principle, this investigation can be carried out using computer simulations. However, the simulation approach has two drawbacks. First, it is not conducive to analysis of fundamental properties of production systems, e.g., relationships between system parameters and performance measures. Second, simulations require a relatively lengthy and costly process of developing a computer model and its multiple runs for statistical evaluation of the performance measures. In some cases, especially when numerous “what if” scenarios must be analyzed, this approach may become prohibitively expensive and slow. This problem is exacerbated by the exponential explosion of the dimensionality of the system as a function of buffer capacity. Indeed, even in the Bernoulli reliability case (i.e., when the machines are memoryless), a serial line with, say, 11 machines and buffers of capacity 9, has 1010 states, which is overwhelming for simulations. Therefore, a quick, easy and revealing method for production systems analysis, based on formulas, rather than on simulations, is of importance. The purpose of this chapter is to present such a method for serial production lines with Bernoulli machines along with describing system-theoretic properties of these systems. Overview: The analytical approach to calculating P R, W IPi , BLi and STi is based on the mathematical models of production systems discussed in Chapter 3. Due to the complex nature of interactions among the machines, closed-form expressions for their performance measures are all but impossible to derive, except for the case of systems with two machines. Therefore, the approach, developed here, is based on a two-stage procedure: First, analytical formulas for performance analysis of two-machine lines are derived and, second, an aggregation procedure is developed, which reduces longer systems to a set of coupled two-machine lines and recursively evaluates their performance characteristics. This approach, illustrated in Figure 4.1, leads to sufficiently accurate estimates 123

124

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

of performance measures P R, W IPi , BLi and STi . Machine parameters Buffer parameters

Two−machine line analysis

Long line aggregation

PR WIP i BL i ST i

Iterations

Figure 4.1: Block diagram of the analysis procedure In addition, based on the analysis of the aggregation equations, this chapter investigates several system-theoretic properties, which provide qualitative insights into the behavior of serial lines.

4.1 4.1.1

Two-machine Lines Mathematical description

System: The production system considered here is shown in Figure 4.2. The time is slotted with slot duration equal to the cycle time of the machines, and machines m1 and m2 are up during each time slot with probability p1 and p2 , respectively. The buffer is of capacity N < ∞.

p1

N

p2

m1

b

m2

Figure 4.2: Two-machine Bernoulli production line

States of the system: Since the Bernoulli machines are memoryless, the states of the system coincide with the states of the buffer, i.e., the state space consists of N + 1 points: 0, 1, . . . , N . Conventions: The following conventions are used to define the system at hand: (a) Blocked before service. (b) The first machine is never starved; the last machine is never blocked. (c) The status of the machines is determined at the beginning and the state of the buffer at the end of each time slot. (d) Each machine status is determined independently from the other.

4.1. TWO-MACHINE LINES

125

(e) Time-dependent failures.

State transition diagram: Since the buffer occupancy can change in each time slot at most by one part, the state transition diagram is “linear,” as shown in Figure 4.3.

P00

P01

0

P10

P11

Pii

1

i

P

(N−1)(N−1) P (N−1)N

PNN N

N−1

PN(N−1)

Figure 4.3: Transition diagram of two-machine Bernoulli production line

Transition probabilities: In the expressions that follow, the events {mi is up during the time slot n+1} and {mi is down during the time slot n+1} are denoted for the sake of brevity as {mi up} and {mi down}. The buffer occupancy at slot n is denoted as h(n). In these notations, the transition probabilities are: P00 P01 P10

= = =

P [h(n + 1) = 0|h(n) = 0] = P [{m1 down}], P [h(n + 1) = 0|h(n) = 1] = P [{m1 down} ∩ {m2 up}], P [h(n + 1) = 1|h(n) = 0] = P [{m1 up}],

··· ··· ··· ··· ··· Pii = P [h(n + 1) = i|h(n) = i] = P [({m1 up} ∩ {m2 up}) Pi(i+1) P(i+1)i ··· PN N

∪({m1 down} ∩ {m2 down})], i = 1, . . . , N − 1, P [h(n + 1) = i|h(n) = i + 1] = P [{m1 down} ∩ {m2 up}], i = 1, . . . , N − 1, = P [h(n + 1) = i + 1|h(n) = i] = P [{m1 up} ∩ {m2 down}], =

i = 1, . . . , N − 1, ··· ··· ··· ··· = P [h(n + 1) = N |h(n) = N ] = P [({m1 up} ∩ {m2 up}) ∪{m2 down}].

Using the formulas for the probability of the union of mutually exclusive events (2.1) and for the probability of the intersection of independent events (2.3),

126

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

these transition probabilities can be calculated as follows: P00 P01

= =

P10 = ··· Pii = Pi(i+1) P(i+1)i

P [{m1 down}] = 1 − p1 , P [{m1 down} ∩ {m2 up}] = (1 − p1 )p2 , P [{m1 up}] = p1 , ··· ··· ··· ··· P [{m1 up}]P [{m2 up}] + P [{m1 down}]P [{m2 down}]

= =

p1 p2 + (1 − p1 )(1 − p2 ), P [({m1 down}]P [{m2 up})]

i = 1, . . . , N − 1,

=

(1 − p1 )p2 ,

i = 1, . . . , N − 1,

(4.1)

= =

P [({m1 up}]P [{m2 down})] p1 (1 − p2 ), i = 1, . . . , N − 1, ··· ··· ··· ··· ··· PN N = P [{m1 up}]P [{m2 up}] + P [{m2 down}] = p1 p2 + 1 − p2 .

Dynamics of the system: Since the transition probabilities (4.1) are constant, the system under consideration is a Markov chain. Let Pi (n) be the probability of state i, i = 0, 1, . . . , N , at time n. Then, as it is shown in Subsection 2.3.3, the evolution of Pi (n) can be described by the following constrained linear dynamical system: Pi (n + 1)

=

N X

Pij Pj (n),

i = 0, 1, . . . , N,

(4.2)

j=0 N X

Pi (n) =

1.

(4.3)

i=0

Statics of the system: The steady state of the system at hand is described by the balance equations Pi

=

N X

Pij Pj ,

i = 0, 1, . . . , N,

(4.4)

j=0 N X

Pi

= 1.

(4.5)

i=0

Their solution provides a complete characterization of the system behavior. This is accomplished next.

4.1.2

Steady state probabilities

Since the states are communicating and there are “self-loops” (see Figure 4.3), the Markov chain (4.1) is ergodic. Therefore, there exists a unique stationary

4.1. TWO-MACHINE LINES

127

probability mass function. Taking into account (4.1), the balance equations can be re-written as P0

= (1 − p1 )P0 + (1 − p1 )p2 P1 , P1 = p1 P0 + [p1 p2 + (1 − p1 )(1 − p2 )]P1 + (1 − p1 )p2 P2 , ··· ··· ··· ··· ··· Pi

= p1 (1 − p2 )Pi−1 + [p1 p2 + (1 − p1 )(1 − p2 )]Pi +(1 − p1 )p2 Pi+1 , i = 2, . . . , N − 1,

(4.6)

··· ··· ··· ··· ··· PN = p1 (1 − p2 )PN −1 + (p1 p2 + 1 − p2 )PN . These equations can be solved consecutively in terms of P0 . Indeed, from the first equation in (4.6) we have P1 =

p1 P0 . (1 − p1 )p2

It is convenient to re-write this expression as ³ 1 ´³ p (1 − p ) ´ 1 2 P1 = P0 1 − p2 p2 (1 − p1 ) and denote the second factor in the right hand side as α(p1 , p2 ) :=

p1 (1 − p2 ) . p2 (1 − p1 )

(4.7)

Then, from the second equation of (4.6), ³ P2 =

α2 (p1 , p2 ) 1 ´³ p1 (1 − p2 ) ´2 P0 = P0 1 − p2 p2 (1 − p1 ) 1 − p2

and so on, leading to PN =

αN P0 , 1 − p2

where, for the sake of brevity, α(p1 , p2 ) is denoted as α. Thus, Pi =

αi P0 , 1 − p2

i = 1, . . . , N.

(4.8)

To complete the calculation, the expression for P0 must be derived. Using (4.5) and (4.8), we obtain h P0 1 +

α α2 αN i = 1. + + ... + 1 − p2 1 − p2 1 − p2

Thus, P0 =

1 − p2 . 1 − p2 + α + α 2 + . . . + α N

(4.9)

128

CHAPTER 4. ANALYSIS OF BERNOULLI LINES In the special case of identical machines, i.e., when p1 = p2 =: p, α(p1 , p2 ) = 1,

and, therefore, P0 =

1−p , N +1−p

(4.10)

implying that

1 , i = 1, . . . , N. (4.11) N +1−p An illustration of the pmf (4.10), (4.11) is given in Figure 4.4 for p = 0.95 and p = 0.55 with N = 5. Clearly, when p is close to 1 (which is the practical case), Pi =

Pi

u

P0

u

1 , N 0.

i = 1, . . . , N,

0.2 Pi

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

0

1

2

3

4

5

P

i

0 −1

6 i

(a) p1 = p2 = 0.95, N = 5

0

1

2

3

4

5

6 i

(b) p1 = p2 = 0.55, N = 5

Figure 4.4: Stationary pmf of buffer occupancy in two-machine lines with identical Bernoulli machines Intuitively, one would expect that the pmf of the buffer occupancy in the case p1 = p2 would be symmetric in the sense that P0 = PN . The fact that it is not is due to the blocked before service assumption, i.e., m1 is not blocked even if h = N but m2 is up. If this assumption is changed to “m1 is blocked if h = N ” (i.e., irrespective of the status of m2 ), it is possible to show that the buffer occupancy is indeed symmetric, i.e., P0 = PN (see Problem 4.5). We follow, however, the original assumption since it is closer to reality on the factory floor. (Note that in the case of continuous time systems, discussed in Part III, both assumptions lead to the same conclusion – the pdf of buffer occupancy for the case of identical machines is symmetric.) When p1 6= p2 , substituting (4.7) into (4.9) we obtain: P0 =

1 − p2 1 − p2 +

p1 (1−p2 ) p2 (1−p1 ) (1

+ α + α2 + . . . + αN −1 )

.

4.1. TWO-MACHINE LINES

129

Summing up the geometric series in the denominator, this can be re-written as P0 =

1 − p2 1 − p2 +

i.e., P0 =

p1 (1−p2 ) p2 (1−p1 )

·

1−αN 1−α

=

1 1+

p1 (1−p1 )p2

·

1−αN 1−α

,

(1 − p1 )(1 − α) . (1 − p1 )(1 − α) + pp12 (1 − αN )

Substituting (4.7) into the first term in the denominator of the above expression, after simplification we finally obtain: P0 =

(1 − p1 )(1 − α) . 1 − pp12 αN

(4.12)

Thus, equations (4.8), (4.12) describe the steady state pmf of buffer occupancy in two-machine lines with non-identical Bernoulli machines. This pmf is illustrated in Figure 4.5 for the following serial lines: L1 L2 L3 L4

: : : :

p1 p1 p1 p1

= 0.8, p2 = 0.82, N = 5, = 0.82, p2 = 0.8, N = 5, = 0.6, p2 = 0.9, N = 5, = 0.9, p2 = 0.6, N = 5.

(4.13)

Clearly, L1 and L2 are a reverse of each other, in the sense that the first (respectively, second) machine of L1 is the second (respectively, first) machine of L2 , while the buffer remains the same. Similarly, L3 and L4 are also reverse of each other. The pmf’s of Figure 4.5 clearly show that the buffer tends to be empty (respectively, full) if p1 − p2 < 0 (respectively, p1 − p2 > 0), and this phenomenon becomes more pronounced when |p1 − p2 | is large. The functions in the right hand side of (4.10) and (4.12) play an important role in the subsequent analyses. Similar functions appear in all other cases of serial lines (e.g., when the machines are exponential). Therefore, we introduce a special notation:  (1−p )(1−α(p ,p )) 1 1 2   1− pp1 αN (p1 ,p2 ) , if p1 6= p2 , 2 (4.14) Q(p1 , p2 , N ) := P0 =   1−p if p1 = p2 = p. N +1−p , The properties of this function are as follows: Lemma 4.1 Function Q(x, y, N ), where 0 < x < 1, 0 < y < 1, and N ∈ {1, 2, . . .}, takes values on (0, 1) and is • strictly decreasing in x, • strictly increasing in y, • strictly decreasing in N .

130

CHAPTER 4. ANALYSIS OF BERNOULLI LINES P

0.25

0.25 Pi

i

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

0

1

2

3

4

5

0 −1

6 i

0

1

2

(a) L1

4

5

6 i

4

5

6 i

Pi

Pi 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 −1

3

(b) L2

0

1

2

3

4

5

0 −1

6 i

0

1

2

(c) L3

3

(d) L4

Figure 4.5: Stationary pmf of buffer occupancy in two-machine lines with nonidentical Bernoulli machines Proof: See Section 20.1. The properties of Q, as indicated in Section 4.2, ensure the convergence of the aggregation procedure that is used for analysis of serial lines with more than two machines. To conclude this subsection, we re-write the expression for the probability of the buffer being full (i.e., PN ) in a form, which is more convenient for the performance measure formulas described in Subsection 4.1.3. From (4.8) and (4.12) we obtain PN =

(1 − p1 )(1 − α(p1 , p2 ))αN (p1 , p2 ) . (1 − p2 )(1 − pp12 αN (p1 , p2 ))

Dividing both numerator and denominator by αN and replacing 1 α , after simplification, we obtain PN =

1 α(p1 ,p2 ) p2 1 N p1 ( α(p1 ,p2 ) )

1− 1−

Taking into account that, as it follows from (4.7), 1 = α(p2 , p1 ), α(p1 , p2 )

.

p2 (1−p1 ) p1 (1−p2 )

with

(4.15)

4.1. TWO-MACHINE LINES

131

this, finally, can be re-written as PN =

4.1.3

1 − α(p2 , p1 ) . 1 − pp21 αN (p2 , p1 )

(4.16)

Formulas for the performance measures

Using the steady state probabilities derived above and function Q defined by (4.14), the performance measures P R, W IP , BL1 , and ST2 can be expressed as shown below. Production rate: Keeping in mind conventions (a)-(e) listed at the beginning of this section and formula (2.3) for the probability of the intersection of independent events, P R can be represented as PR

= P [{m2 is up at the beginning of a time slot} ∩ {buffer is not empty at the end of the previous time slot}] = P [{m2 up}]P [{buffer not empty}] = p2 (1 − P0 ) = p2 [1 − Q(p1 , p2 , N )].

(4.17)

Alternatively, P R can be expressed as PR

=

P [{m1 is up at the beginning of a time slot} ∩ {buffer is not full at the beginning of this time slot}]. (4.18)

While the probability of the first event in the right hand side of (4.18) is p1 , the probability of the second event is P [{buffer is not full at the beginning of a time slot] = 1 − P [{buffer is full at the end of the previous slot} ∩{m2 is down at the beginning of this time slot}] = 1 − PN (1 − p2 ). Therefore, P R = p1 [1 − (1 − p2 )PN ], where, as it follows from (4.11) and (4.16),  1−α(p ,p ) 2 1  6 p2 ,  1− pp2 αN (p2 ,p1 ) , if p1 = 1 PN =   1 if p1 = p2 = p. N +1−p , This implies that (1 − p2 )PN = Q(p2 , p1 , N ),

132

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

and, therefore, P R = p1 [1 − Q(p2 , p1 , N )]. Thus, P R can be expressed in two equivalent ways: PR

=

p1 [1 − Q(p2 , p1 , N )]

=

p2 [1 − Q(p1 , p2 , N )].

(4.19)

These are important expressions: along with their direct value as P R of twomachine lines, they are the basis of the aggregation procedure for the analysis of M > 2-machine lines (see Section 4.2). Work-in-process: The average value of pmf’s (4.10), (4.11) and (4.8), (4.12), i.e., W IP , is given by W IP =

N X i=1

iPi =

N X iαi Q(p1 , p2 , N ). 1 − p2 i=1

(4.20)

For p1 = p2 = p, this leads to N

W IP =

X 1 N (N + 1) . i= N + 1 − p i=1 2(N + 1 − p)

(4.21)

For p1 6= p2 , after some algebraic manipulations, this can be reduced to W IP =

h 1 − αN (p , p ) i p1 1 2 N − N α (p , p ) . 1 2 p2 − p1 αN (p1 , p2 ) 1 − α(p1 , p2 )

(4.22)

Therefore,    W IP =

 

p1 p2 −p1 αN (p1 ,p2 )

h

1−αN (p1 ,p2 ) 1−α(p1 ,p2 )

i − N αN (p1 , p2 ) ,

N (N +1) 2(N +1−p) ,

if p1 6= p2 , if p1 = p2 = p. (4.23)

Blockages and starvations: As it follows from the definitions of Subsection 3.6.3, BL1

=

P [{m1 up} ∩ {buffer full} ∩ {m2 down}],

ST2

=

P [{m2 up} ∩ {buffer empty}].

Taking into account that the probabilities of the buffer being full, PN , and being empty, P0 , are given by (4.16) and (4.14), respectively, these relationships can be expressed as BL1

=

p1 PN (1 − p2 ) = p1 Q(p2 , p1 , N ),

ST2

=

p2 P0 = p2 Q(p1 , p2 , N ).

(4.24)

4.2. M > 2-MACHINE LINES

133

Clearly, these expressions are in agreement with formulas (4.19) for P R, which now can be understood as = p1 − BL1 = p2 − ST2 .

PR

4.1.4

Asymptotic properties

Expressions (4.19), (4.23) and (4.24) reveal the following properties of the performance measures as N → ∞: Theorem 4.1 In a two-machine line with Bernoulli machines defined by conventions (a)-(e), lim P R

N →∞

lim W IP

N →∞

=

min(p1 , p2 ),  ∞,      p1 (1−p1 ) = p2 −p1 ,      limN →∞ N2 = ∞,

lim BL1

=

lim ST2

=

N →∞

N →∞

0,   0, 

(4.25) if p1 > p2 , if p1 < p2 ,

(4.26)

if p1 = p2 , (4.27)

if p1 ≥ p2 , (4.28)

p2 − p1 ,

if p1 < p2 .

Proof: See Section 20.1. The last expression in (4.26) implies that for N sufficiently large and p1 = p2 , the buffer is, on the average, half full. Theorem 4.1 is illustrated in Figure 4.6 for three serial lines defined by L1 : L2 : L3 :

p1 = p2 = 0.9, p1 = 0.9, p2 = 0.7, p1 = 0.7, p2 = 0.9.

(4.29)

As one can see, P R is monotonically increasing but with a decreasing rate, while W IP increases linearly (if p1 ≥ p2 ). This implies that there is nothing to be gained from having a buffer of capacity larger than 5. The issue of buffer capacity selection is explored in details in Chapter 6.

4.2 4.2.1

M > 2-machine Lines Mathematical description and approach

System: The production system considered here is shown in Figure 4.7. The time is slotted, and each machine mi , i = 1, . . . , M , is up during a time slot with

134

CHAPTER 4. ANALYSIS OF BERNOULLI LINES 10

1

8

0.9

WIP

PR

L1 0.8

0.7

0.6

4

6

N

4

L2

2

L 2 and L 3 2

6

8

0

10

L1 L3 4

2

(a) P R

6

8

10

(b) W IP

0.25

0.25 0.2

L2

L3

ST

BL

2

1

0.2

N

0.15

0.15

0.1

0.1

0.05

L1

0

L3 2

0.05 4

N

6

8

10

(c) BL1

0

L1 L2 2

4

N

6

8

10

(d) ST2

Figure 4.6: Performance measures of two-machine Bernoulli lines as functions of buffer capacity probability pi and down with probability 1 − pi , i = 1, . . . , M . The capacity of buffer i is Ni < ∞, i = 1, . . . , M − 1. p1

N1

p2

N2

pM−1

N M−1

pM

m1

b1

m2

b2

mM−1

bM−1

mM

Figure 4.7: M -machine Bernoulli production line

Conventions: Similar to two-machine lines, the following conventions are used: (a) Blocked before service. (b) Machine m1 is never starved for parts; machine mM is never blocked by subsequent operations. (c) The status of the machines is determined at the beginning, and the state of the buffers at the end of each time slot. (d) Each machine’s status is determined independently from the others. (e) Time-dependent failures.

4.2. M > 2-MACHINE LINES

135

Throughout this book, we refer to (a)-(e) intermittently as either conventions or assumptions. States of the system: The Bernoulli machines are memoryless, and, therefore, the states of the system coincide with the states of the buffers. Since the i-th buffer has Ni + 1 states, the system has (N1 + 1)(N2 + 1) · · · (NM −1 + 1) states. For example, if Ni = 9 for all i and M = 23, the number of states is 1022 , which equals the number of molecules in a cubic centimeter of gas under normal pressure and temperature! Clearly, a direct analysis of such a large system is impossible (and, perhaps, unnecessary as well). Therefore, a simplification is in order. We use for this purpose an aggregation approach. Idea of the aggregation: Consider the M -machine line and aggregate the last two machines, mM −1 , and mM , into a single Bernoulli machine denoted as mbM −1 , where b stands for backward aggregation (see Figure 4.8(a)). The Bernoulli parameter, pbM −1 , of this machine is assigned as the production rate of the aggregated two-machine line, calculated using the first expression of (4.19). Next, aggregate this machine, i.e., mbM −1 , with mM −2 and obtain another aggregated machine, mbM −2 . Continue this process until all the machines are aggregated into mb1 , which completes the backward phase of the aggregation procedure. It turns out that the Bernoulli parameter of mb1 might be quite different from the production rate of the M -machine line under consideration. To remedy this problem, we introduce the forward phase of the aggregation procedure defined as follows: Aggregate the first machine, m1 , with the aggregated version of the rest of the line, i.e., with mb2 . This results in the aggregated machine, denoted as mf2 , where f stands for forward aggregation (see Figure 4.8(b)). The Bernoulli parameter, pf2 , is assigned as the production rate of the aggregated two-machine line, calculated using the second expression of (4.19) with p2 in function Q substituted by pb2 . Next, aggregate mf2 with mb3 , resulting in mf3 and so on until all the machines are aggregated into mfM , which completes the forward phase of the procedure. Again, the Bernoulli parameter of mfM may be quite different from the actual production rate of the M -machine system. To alleviate this discrepancy, we iterate between the backward and forward aggregations. In other words, view the above backward and forward aggregations as the first iteration of the aggregation, i.e., as s = 1. At the second iteration, s = 2, mfM −1 is aggregated with mM to result in mbM −1 for the second iteration, which is then aggregated with mfM −2 and so on until the second iteration of the backward aggregation is complete. Next, the second iteration of the forward aggregation is carried out, followed by the third iteration of the backward aggregation and so on, i.e., s = 3, 4, . . .. We show below that the steady states of this recursive procedure lead to relatively accurate estimates of the performance measures for the M -machine line. We also indicate that the convergence to these steady states is quite rapid,

136

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

p1

N1

p2

N2

N M−3 p M−2 NM−2 p M−1 N M−1 p M

m1

b1

m2

b2

b M−3 m M−2 b M−2 m M−1 b M−1 m M

p1

N1

p2

N2

p bM−1 N M−3 p M−2 N M−2

m1

b1

m2

b2

b M−3

p1

N1

p2

N2

N M−3

m1

b1

m2

b2

b M−3

m M−2 b b M−2 m M−1 b p M−2

b mM−2

pb 1

m 1b

(a). Backward aggregation pb

p1

N1

m1

b1

m b2

pf 2

N2

p3

m 2f

b2

m 3b

2

b

f p M−1

NM−1

f m M−1

b M−1 m M

p

M

f pM

m

f M

(b). Forward aggregation Figure 4.8: Illustration of the aggregation procedure

4.2. M > 2-MACHINE LINES

137

typically requiring less than 10 iterations, which implies that the performance measures are evaluated within a fraction of a second.

4.2.2

Aggregation procedure and its properties

The mathematical representation of the recursive aggregation procedure described above and its properties are given below. Recursive Aggregation Procedure 4.1: pbi (s + 1)

=

pfi (s + 1)

=

pi [1 − Q(pbi+1 (s + 1), pfi (s), Ni )], i = 1, . . . , M − 1, pi [1 − Q(pfi−1 (s + 1), pbi (s + 1), Ni−1 )],

(4.30)

i = 2, . . . , M, s = 0, 1, 2, . . . , with initial conditions pfi (0) = pi ,

i = 1, . . . , M

(4.31)

and boundary conditions pf1 (s) pbM (s)

= =

As before,

p1 , pM ,   

Q(x, y, N ) =

 

s = 0, 1, 2, . . . , s = 0, 1, 2, . . . .

(1−x)(1−α) , N 1− x yα

if x 6= y,

1−x N +1−x ,

if x = y,

(4.32)

(4.33)

where α=

x(1 − y) . y(1 − x)

(4.34)

These equations are solved as follows: With i = M − 1, using the initial condition pfM −1 (0) = pM −1 and the boundary condition pbM (s) = pM , solve the first equation of (4.30) to obtain pbM −1 (1); then solve it with i = M −2 to obtain pbM −2 (1), and so on, until pb1 (1) is obtained. Next, solve the second equation of (4.30) with i = 2 to obtain pf2 (1); then solve it with i = 3 to obtain pf3 (1) and so on, until pfM (1) is obtained. This completes the first iteration of the aggregation procedure. For the second, third, . . ., iterations, this process is repeated anew using pfM −1 (1), pfM −1 (2), . . ., respectively, in the first equation of (4.30).

138

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

Example 4.1 Consider a three-machine serial line with p = [0.9, 0.9, 0.9] and N = [2, 2]. Then, s=1: pb2 (1) pb1 (1) pf2 (1) pf3 (1)

= p2 [1 − Q(p3 , p2 , N2 )] = 0.9[1 − Q(0.9, 0.9, 2)] = 0.8571, = p1 [1 − Q(pb2 (1), p1 , N1 )] = 0.9[1 − Q(0.8571, 0.9, 2)] = 0.8257, = p2 [1 − Q(p1 , pb2 (1), N1 )] = 0.9[1 − Q(0.9, 0.8571, 2)] = 0.8670, = p3 [1 − Q(pf2 (1), p3 , N2 )] = 0.9[1 − Q(0.8670, 0.9, 2)] = 0.8333,

s=2: pb2 (2) = p2 [1 − Q(p3 , pf2 (1), N2 )] = 0.9[1 − Q(0.9, 0.8670, 2)] = 0.8650, pb1 (2) = p1 [1 − Q(pb2 (2), p1 , N1 )] = 0.9[1 − Q(0.8650, 0.9, 2)] = 0.8318, pf2 (2)

= p2 [1 − Q(p1 , pb2 (2), N1 )] = 0.9[1 − Q(0.9, 0.8650, 2)] = 0.8654,

pf3 (2) = p3 [1 − Q(pf2 (2), p3 , N2 )] = 0.9[1 − Q(0.8654, 0.9, 2)] = 0.8321, s=3: pb2 (3) = p2 [1 − Q(p3 , pf2 (2), N2 )] = 0.9[1 − Q(0.9, 0.8654, 2)] = 0.8653, pb1 (3) = p1 [1 − Q(pb2 (3), p1 , N1 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, pf2 (3)

= p2 [1 − Q(p1 , pb2 (3), N1 )] = 0.9[1 − Q(0.9, 0.8653, 2)] = 0.8653,

pf3 (3) = p3 [1 − Q(pf2 (3), p3 , N2 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, s=4: pb2 (4) = p3 [1 − Q(p3 , pf2 (3), N2 )] = 0.9[1 − Q(0.9, 0.8653, 2)] = 0.8653, pb1 (4) = p2 [1 − Q(pb2 (4), p1 , N1 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, pf2 (4)

= p2 [1 − Q(p1 , pb2 (4), N1 )] = 0.9[1 − Q(0.9, 0.8653, 2)] = 0.8653,

pf3 (4) = p3 [1 − Q(pf2 (4), p3 , N2 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, s=5: pb2 (5) = p3 [1 − Q(p3 , pf2 (4), N2 )] = 0.9[1 − Q(0.9, 0.8653, 2)] = 0.8653, pb1 (5) = p2 [1 − Q(pb2 (5), p1 , N1 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, pf2 (5)

= p2 [1 − Q(p1 , pb2 (5), N1 )] = 0.9[1 − Q(0.9, 0.8653, 2)] = 0.8653,

pf3 (5)

= p3 [1 − Q(pf2 (5), p3 , N2 )] = 0.9[1 − Q(0.8653, 0.9, 2)] = 0.8320, ....

All subsequent iterations are carried out similarly. Convergence: Clearly, (4.30) is an (M − 1)-dimensional dynamical system, which iterates pi ’s and Ni ’s and results in two sequences of numbers pb1 (s), . . . , pbM −1 (s), pf2 (s), . . . , pfM (s), s = 1, 2, . . .

4.2. M > 2-MACHINE LINES

139

defined on the interval (0, 1). The properties of these sequences and their physical meaning are described below. Based on (4.30) and Lemma 4.1, the following can be proved: Theorem 4.2 Aggregation procedure (4.30)-(4.33) has the following properties: (i) The sequences, pf2 (s), . . ., pfM (s) and pb1 (s), . . ., pbM −1 (s), s = 1, 2, . . ., are convergent, i.e., the following limits exist: pbi

:=

pfi

:=

lim pbi (s),

s→∞

lim pf (s). s→∞ i

(4.35)

(ii) These limits are unique solutions of the steady state equations corresponding to (4.30), i.e., of pfi pbi

= pi [1 − Q(pfi−1 , pbi , Ni−1 )], =

pi [1 − Q(pbi+1 , pfi , Ni )], pbM = pM . pf1 = p1 ,

2 ≤ i ≤ M, 1 ≤ i ≤ M − 1,

(iii) In addition, these limits satisfy the relationships: pfM

= pb1 = pbi+1 [1 − Q(pfi , pbi+1 , Ni )] = pfi [1 − Q(pbi+1 , pfi , Ni )],

i = 1, . . . , M − 1.

Proof: See Section 20.1. Figure 4.9 illustrates the behavior of pfi (s) and pbi (s), s = 0, 1, 2, . . ., for the following lines: L1 : L2 : L3 :

pi = 0.9, i = 1, . . . , 5; Ni = 3, i = 1, . . . , 4, p = [0.7, 0.75, 0.8, 0.85, 0.9], Ni = 3, i = 1, . . . , 4, p = [0.7, 0.85, 0.9, 0.85, 0.7], Ni = 3, i = 1, . . . , 4,

L4 :

p = [0.9, 0.85, 0.7, 0.85, 0.9], Ni = 3, i = 1, . . . , 4.

Obviously, L1 represents lines with identical machines, while L2 , L3 , and L4 illustrate lines where pi ’s are allocated according to an increasing, inverted bowl, and bowl patterns, respectively. As one can see, pfi and pbi indeed exhibit the properties established in Theorem 4.2. In addition, Figure 4.9 shows that convergence to the limits is quite fast: 2 - 4 iterations of the aggregation procedure for the uniform, ramp, and bowl allocations and about 15 iterations for the inverted bowl.

140

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

0.9

0.95

pf2 pf3

pfi or pbi

0.87

0.85

pf5

0.84 0.83 1

2

0.85

pb3

pf4

0.86

0.82 0

0.9

pb 4

pb2

0.8

pf5

pb4

pf4

pb3 pb2

0.75

i

0.88

pf or pbi

0.89

0.7

pb1

1

0.65 3

Iterations

4

5

6

pf2

pb

0.6 0

1

2

(a) L1

pf3 3

Iterations

4

5

6

4

5

6

(b) L2 0.9

0.9

0.85

pf

4

2

i

0.75 0.7 0.65 0.6 0

pf3 pb3

pb

0.8

pf5

pb1 5

pf

2

0.8

pfi or pbi

i

pf or pb

0.85

4

0.7

15

Iterations

(a) L3

pf2

pb4

pb1

pf

4

b 0.75 p2

pb

10

pf5

20

25

0.65 0

pf3 pb3 1

2

3

Iterations

(b) L4

Figure 4.9: Illustration of the dynamics of pfi (s) and pbi (s)

4.2. M > 2-MACHINE LINES

141

Interpretation: As it follows from statement (iii) of Theorem 4.2, pfi and pbi can be given the following interpretation: From the point of view of each buffer bi , i = 1, . . . , M − 1, the upstream of the line is represented by the “virtual” Bernoulli machine mfi defined by the parameter pfi . Similarly, the downstream is represented by the “virtual” machine mbi+1 defined by pbi+1 . In addition, the whole line can be represented by mb1 or mfM . Thus, the M -machine line can be represented as shown in Figure 4.10. Clearly, all the performance measures of the two-machine lines included in this figure can be calculated using the formulas of Subsection 4.1.3. f

pi

f

mi

b

Ni

pi +1

bi

mib+1

f

b

pM

p1

=

= b m1

f

mM

i = 1 , ..., M −1

Figure 4.10: Equivalent representations of Bernoulli M > 2-machine line through the aggregated machines.

4.2.3

Formulas for the performance measures

Using the equivalent representations of Figure 4.10 and the limits (4.35), estimates of the performance measures of the M > 2-machine line are introduced below. d Production rate: Based on Figure 4.10 and Theorem 4.2, the estimate, P R, of the production rate is defined as d P R

= pb1 = pfM = pbi+1 [1 − Q(pfi , pbi+1 , Ni )] = pfi [1 − Q(pbi+1 , pfi , Ni )], i = 1, . . . , M − 1.

(4.36)

Work-in-process: Using the two-machine representation of the M > 2-machine \ line (Figure 4.10) and expression (4.23), the estimate, W IP i , of the steady state occupancy of buffer i is defined as h i  1−αNi (pfi ,pbi+1 ) pfi Ni f b  − N α (p , p )  i f f f b b b Ni i+1 , i    pi+1 −pi α (pi ,pi+1 ) 1−α(pi ,pi+1 ) f if pi 6= pbi+1 , (4.37) \ W IP i =      Ni (Ni +1) , if pfi = pbi+1 . 2(N +1−pf ) i

i

i = 1, . . . , M − 1.

142

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

Obviously, the estimate of the total W IP is

\ W IP =

M −1 X

\ W IP i .

(4.38)

i=1

Blockages and starvations: Since these probabilities must evaluate blockages and starvations of the real, rather then aggregated, machines, taking into di account expressions (4.24), the estimates of these performance measures, BL c i , are introduced as follows: and ST di BL ci ST

= pi Q(pbi+1 , pfi , Ni ), pi Q(pfi−1 , pbi , Ni−1 ),

=

i = 1, . . . , M − 1,

(4.39)

i = 2, . . . , M.

(4.40)

These expressions confirm the interpretation of pfi and pbi described in the aggregation procedure. Indeed, using the steady states of the aggregation procedure (4.30) and expressions (4.39), (4.40), we obtain pbi pfi

= pi [1 − Q(pbi+1 , pfi , Ni )] di , = pi − BL = pi [1 −

c i. = pi − ST

Residence time: as follows:

(4.41)

Q(pfi−1 , pbi , Ni−1 )] (4.42)

d \ Given P R and W IP , an estimate of RT can be calculated

\ d = W IP [cycle time]. RT d P R

PSE Toolbox: The recursive procedure (4.30) and performance measure estimates (4.36)-(4.40) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Subsection 19.3.1.

4.2. M > 2-MACHINE LINES

4.2.4

143

Asymptotic properties of M > 2-machine lines

Formulas (4.36)-(4.40) can be used to investigate asymptotic properties of Bernoulli lines as N → ∞. This is carried out using the following eight serial lines: L1 : L2 : L3 :

pi = 0.9, i = 1, . . . , 5, Ni = N, i = 1, . . . , 4,

L4 L5 L6 L7 L8

: :

p = [0.9, 0.85, 0.7, 0.85, 0.9], Ni = N, i = 1, . . . , 4, p = [0.7, 0.85, 0.9, 0.85, 0.7], Ni = N, i = 1, . . . , 4,

: : :

p = [0.7, 0.9, 0.7, 0.9, 0.7], Ni = N, i = 1, . . . , 4, p = [0.9, 0.7, 0.9, 0.7, 0.9], Ni = N, i = 1, . . . , 4, p = [0.75, 0.75, 0.95, 0.75, 0.75], Ni = N, i = 1, . . . , 4.

p = [0.9, 0.85, 0.8, 0.75, 0.7], Ni = N, i = 1, . . . , 4, p = [0.7, 0.75, 0.8, 0.85, 0.9], Ni = N, i = 1, . . . , 4, (4.43)

The reasons for selecting these particular lines are as follows: Line 1 illustrates the behavior of systems with identical machines. Lines 2 and 3 represent systems with increasing and decreasing machine efficiency, respectively; clearly, L3 is the reverse of L2 . Lines 4 and 5 illustrate systems with machine efficiency allocated according to a bowl and an inverted bowl patterns, respectively. Lines 6 and 7 exemplify systems with “oscillating” machine efficiency allocation. Finally, Line 8 is selected to illustrate the case of a good machine surrounded by low efficiency ones. Figures 4.11 - 4.18 show the performance measures of these lines as a function of N . Based on this information, the following can be concluded: As N → ∞, d • P R → min pi , i = 1, . . . , 5. \ \ • W IP i , i = 1, . . . , 4, are increasing almost linearly in N , with W IP i ∼ = κi N , κ4 < . . . < κ1 < 1, when pi ’s form a decreasing sequence. \ • W IP i , i = 1, . . . , 4, are also increasing almost linearly but with smaller ∼ κi N , κ4 < . . . < κ1 < 1, when pi ’s are \ coefficients κi , so that W IP i = identical. \ \ \ IP i > W IP i−1 , • W IP i , i = 1, . . . , 4, converges to a finite limit with W i = 2, 3, 4, when pi ’s form an increasing sequence. d In addition, Figures 4.11 P- 4.18 indicate that since P R as a function of N exhibits saturation, while i W IPi tends to infinity (except for the case of increasing machine efficiency), there is no reason for having large N , typically, above 8 - 10. A detailed discussion on selecting N is included in Chapter 6.

4.2.5

Accuracy of the estimates

Clearly, formulas (4.36)-(4.40) are just estimates of the true values of the performance measures. Therefore, an analysis of their accuracy is of importance. In this subsection, such an analysis is carried out using both analytical and numerical tools.

144

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

0.9

8 6

d P R

Wd IP i

0.8

0.7

0.6

d WIP 1 d WIP 2

d WIP 3 d WIP

4

4

2

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL 2

0.2 di BL

d BL 3 d BL 4

0.1

0.3

c ST 5 c ST 4

0.2

c ST 3 c ST

ci ST

0.3

0

4

2

0.1

2

4

N

6

8

0

10

2

4

di (c) BL

N

6

8

10

ci (d) ST

Figure 4.11: Performance of Line L1 (identical machines) 0.7

10

d WIP 1 d WIP

8 d P R

Wd IP i

0.6

0.5

2

d WIP 3 d WIP

6

4

4 2

0.4

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL

0.4

2

d BL 3 d BL

di BL

0.3

4

0.2 0.1

0.5

c ST 5 c ST

0.4

4

c ST 3 c ST

0.3

ci ST

0.5

0

4

2

0.2 0.1

2

4

N

6

di (c) BL

8

10

0

2

4

N

6

8

10

ci (d) ST

Figure 4.12: Performance of Line L2 (decreasing machine efficiency)

4.2. M > 2-MACHINE LINES

145

0.7

4

WIP

1

d P R

Wd IP i

0.6

0.5

0.4

3

WIP2

2

WIP

WIP

3 4

1

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL

0.4

2

d BL 3 d BL

di BL

0.3

4

0.2 0.1

0.5

c ST 5 c ST

0.4

4

c ST 3 c ST

0.3

Sc Ti

0.5

0

4

2

0.2 0.1

2

4

N

6

8

0

10

2

4

di (c) BL

N

6

8

10

ci (d) ST

Figure 4.13: Performance of Line L3 (increasing machine efficiency) 0.7

10

d WIP 1 d WIP

Wd IP i

d P R

8

0.6

2

d WIP 3 d WIP

6

4

4 2

0.5

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL

0.3

2

di BL

d BL 3 d BL

0.2

4

0.1

0.4

c ST 5 c ST

0.3

4

c ST 3 c ST

ci ST

0.4

0

4

0.2

2

0.1

2

4

N

6

di (c) BL

8

10

0

2

4

N

6

8

10

ci (d) ST

Figure 4.14: Performance of Line L4 (bowl machine efficiency allocation)

146

CHAPTER 4. ANALYSIS OF BERNOULLI LINES 0.7

10

d WIP 1 d WIP

d P R

Wd IP i

8

0.6

2

d WIP 3 d WIP

6

4

4 2

0.5

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL 2

0.2 di BL

d BL 3 d BL 4

0.1

0.3

c ST 5 c ST 4

0.2

c ST 3 c ST

ci ST

0.3

0

4

2

0.1

2

4

N

6

8

0

10

2

4

di (c) BL

N

6

8

10

ci (d) ST

Figure 4.15: Performance of Line L5 (inverted bowl machine efficiency allocation) 0.7

10

d WIP 1 d WIP

8 d P R

Wd IP i

0.6

0.5

2

d WIP 3 d WIP

6

4

4 2

0.4

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL

0.3

2

di BL

d BL 3 d BL

0.2

4

0.1

0.4

c ST 5 c ST

0.3

4

c ST 3 c ST

ci ST

0.4

0

4

0.2

2

0.1

2

4

N

6

di (c) BL

8

10

0

2

4

N

6

8

10

ci (d) ST

Figure 4.16: Performance of Line L6 (“oscillating” machine efficiency allocation)

4.2. M > 2-MACHINE LINES

147

0.7

10

d WIP 1 d WIP

8 d P R

Wd IP i

0.6

0.5

2

d WIP 3 d WIP

6

4

4 2

0.4

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL

0.4

2

d BL 3 d BL

di BL

0.3

4

0.2 0.1

0.5

c ST 5 c ST

0.4

4

c ST 3 c ST

0.3

ci ST

0.5

0

4

2

0.2 0.1

2

4

N

6

8

0

10

2

4

di (c) BL

N

6

8

10

ci (d) ST

Figure 4.17: Performance of Line L7 (“oscillating” machine efficiency allocation) 0.8

10

d P R

Wd IP i

0.7 0.6 0.5 0.4

d WIP 1 d WIP

8

2

d WIP 3 d WIP

6

4

4 2

2

4

N

6

8

0

10

2

d (a) P R

N

6

8

10

\ (b) W IP i d BL 1 d BL 2

0.2 di BL

d BL 3 d BL 4

0.1

0.3

c ST 5 c ST 4

0.2

c ST 3 c ST

ci ST

0.3

0

4

2

0.1

2

4

N

6

di (c) BL

8

10

0

2

4

N

6

8

10

ci (d) ST

Figure 4.18: Performance of Line L8 (good machine surrounded by low efficiencies ones)

148

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

Analytical investigation: Consider the joint probability Pi,...,j [hi , . . . , hj ] that buffers i, i + 1, . . ., j, contain hi , hi+1 , . . ., hj parts, respectively. In general, this joint probability is not close to the product of its marginals, i.e., Pi,...,j [hi , . . . , hj ] 6= Pi [hi ]Pi+1,...,j [hi+1 , . . . , hj ], where Pi [hi ] is the probability that the i-th buffer contains hi parts. However, it turns out that for certain values of hi , . . ., hj , related to blockages and starvations, these probabilities are indeed close to each other. Specifically, define δij (b) := δ ij (a) :=

|Pi,...,j [0, b, Ni+2 , . . . , Nj ] − Pi [0]Pi+1,...,j [b, Ni+2 , . . . , Nj ]|, |Pi,...,j [a, Ni+1 , . . . , Nj ] − Pi [a]Pi+1,...,j [Ni+1 , . . . , Nj ]|, 1 ≤ b ≤ Ni+1 , 1 ≤ a ≤ Ni

and

δ := max max{δij (b), δ ij (a)}. i,j

a,b

(4.44)

Extensive numerical simulations show that δ is practically always small. An illustration is given in Table 4.1 for several four-machine lines with Ni = 3, i = 1, 2, 3. Table 4.1: Illustration of δ p1 0.80 0.70 0.70 0.60 0.99

p2 0.80 0.80 0.90 0.99 0.60

p3 0.80 0.70 0.70 0.99 0.60

p4 0.80 0.80 0.90 0.60 0.99

δ 0.0073 0.0233 0.0568 0.1181 0.0083

Thus, we formulate Numerical Fact 4.1 For Bernoulli lines defined by assumptions (a)-(e), δ ¿ 1. Based on this fact, the following can be proved: Theorem 4.3 For Bernoulli lines defined by assumptions (a)-(e), the accuracy of the production rate estimate is characterized by d |P R(p1 , . . . , pM , N1 , . . . , NM −1 ) − P R(p1 , . . . , pM , N1 , . . . , NM −1 )| = O(δ), where δ is defined by (4.44) and O(δ) denotes a quantity of the same order of magnitude as δ. Proof: See Section 20.1.

4.2. M > 2-MACHINE LINES

149

Numerical investigation: The accuracy of the performance measure estimates (4.36)-(4.40) has been investigated numerically using a C++ code, which simulates production lines defined by assumptions (a)-(e). (This code is included in the Simulation function of the PSE Toolbox – see Section 19.9.) Since similar numerical investigations are carried out throughout this textbook, we define them below by a standard procedure. Simulation Procedure 4.1: (1) Select the initial status of each machine up with probability pi and down with probability 1 − pi , i = 1, . . . , M . (2) For each line under consideration, carry out 20 runs of the simulation code. (3) In each run, use the first 20,000 time slots as a warm-up period and the subsequent 400,000 time slots to statistically evaluate P R, W IPi , STi and BLi . This results in 95% confidence intervals of less than 0.001 for P R; 0.02 for W IPi ; and 0.002 for STi and BLi . ¥ The accuracy of the estimates has been evaluated by ²P R ²W IPi ²STi ²BLi

d |P R − P R| · 100%, PR \ |W IP i − W IPi | = · 100%, i = 1, . . . , M − 1, Ni c i − STi |, = |ST i = 2, . . . , M, d = |BLi − BLi |, i = 1, . . . , M − 1. =

(4.45) (4.46) (4.47) (4.48)

The results of this numerical investigation for the set of production lines (4.43) are shown in Figures 4.19 - 4.21. From this information, the following conclusions can be derived: d • In general, P R provides a relatively accurate estimate of P R; in most cases, the error is within 1% and the largest error is about 3%. c i , and BL di is typically lower. \ • The accuracy of W IP i , ST • The highest accuracy of all estimates is for the uniform machine efficiency pattern. • The lowest accuracy is for the inverted bowl and “oscillating” patterns. • Lines, which are inverse of each other, result in identical accuracy of all estimates. In spite of this accuracy limitation, formulas (4.36)-(4.40) provide a useful analytical tool for performance evaluation of Bernoulli lines, especially taking into account that the parameters of the machines are rarely known on the factory floor with accuracy better than 5% - 10%.

5

0.8 0.7 0.6 0.5 0.4 0

5

0.8 0.7 0.6 0.5 0.4 0

PR PR

pi

Line 2 1 0.9 0.8 0.7 0.6

1

2

3

4

Machine i

PR

pi

Line 3 1 0.9 0.8 0.7 0.6

1

2

3

4

Machine i

pi

PR 1

2

3

4

Machine i

2

3

4

Machine i

2

3

4

Machine i

5

0.8 0.7 0.6 0.5 0.4 0

5

0.8 0.7 0.6 0.5 0.4 0

5

0.8 0.7 0.6 0.5 0.4 0

pi pi

PR

pi

Line 7 1 0.9 0.8 0.7 0.6

1

2

3

4

Machine i

PR

pi

Line 8 1 0.9 0.8 0.7 0.6

1

2

3

4

Machine i

5

0.6

5

PR 1

N

d P R PR 10 15 20

5

5

5

5

0.3 0.1 0 0

5

10 15 20

0 0

5

10 15 20

2 1.5 1 0.5 0 0

5

10 15 20

0 0

5

10 15 20

3 2.5 2 1.5 1 0.5 0 0

d P R PR 10 15 20

N

5

10 15 20

3.5 2.5 1.5 0.5 0

d P R PR 10 15 20

N

5

10 15 20

d P R PR 10 15 20

5

10 15 20

5

10 15 20

1 0.5

N

N

1.5 1 0.5

N

N

N

2.5 1.5 0.5 0

N

d P R PR 10 15 20

N

1.5

N

0.7 0.5 0

Line 6 1 0.9 0.8 0.7 0.6

d P R PR 10 15 20

0.8

PR 1

0.6

0.5

N

0.7 0.5 0

5

Line 5 1 0.9 0.8 0.7 0.6

5

d P R PR 10 15 20

0.8

Line 4 1 0.9 0.8 0.7 0.6

5

N

²P R (%)

Machine i

5

²P R (%)

4

²P R (%)

3

²P R (%)

2

d P R PR 10 15 20

²P R (%)

pi

1 0.9 0.8 0.7 0.6 1

²P R (%)

5

1 0.9 0.8 0.7 0.6 0

Line 1

²P R (%)

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

²P R (%)

150

N

2.5 1.5 0.5

N

d Figure 4.19: Accuracy of P R

0

N

4.2. M > 2-MACHINE LINES

2

3

4

Machine i

0 0

5

pi pi

Wd IP 1

2

3

4

Machine i

2

3

4

Machine i

0 0 20

pi pi

Wd IP 1

2

3

4

Machine i

0 0 20

Wd IP

pi

Line 7 1 0.9 0.8 0.7 0.6

1

2

3

4

Machine i

pi

Wd IP 1

2

3

4

Machine i

10

20

Line 8 1 0.9 0.8 0.7 0.6

5

5

10

N

5

10

N

5

20

15

0 0

5

20

15

20

20 10 0 0

10 15 20

10 15 20

10 0 0

N

N

10 0 0

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

10

10 0 0

10 15 20

N

5

10

N

²W IP (%)

²W IP (%)

W IP

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

0 0

5

5

W IP1 W IP2 W IP3 W IP4 15

20 10 0 0

5

10

N

15

20

W IP1 W IP2 W IP3 W IP4 5

10 15 20

N

W IP1 W IP2 W IP3 W IP4 5

10

N

15

20

W IP1 W IP2 W IP3 W IP4 5

10

N

15

20

W IP1 W IP2 W IP3 W IP4 5

10

N

15

20

W IP1 W IP2 W IP3 W IP4 5

2 1

²W IP1 ²W IP2 ²W IP3 ²W IP4

0 0

5

10 15 20

N

\ Figure 4.20: Accuracy of W IP i

10

N

1

0 0

5

10

N

1

5

10

N

1

40 20 0 0 30 20 10 0 0 20 10 0 0 30 20 10 0 0

15

20

15

20

²W I P 1 ²W I P 2 ²W I P 3 ²W I P 4

0.5 0 0

20

²W I P 1 ²W I P 2 ²W I P 3 ²W I P 4

0.5 0 0

15

²W I P 1 ²W I P 2 ²W I P 3 ²W I P 4

0.5

20

W IP1 W IP2 W IP3 W IP4

0 0

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

10

5

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

10

5

Line 6 1 0.9 0.8 0.7 0.6

10

20

Wd IP 1

10

N

d W IP 1 d W IP 2 d W IP 3 d W IP 4

0 0

5

Line 5 1 0.9 0.8 0.7 0.6

5

20

Line 4 1 0.9 0.8 0.7 0.6

5

d W IP 1 d W IP 2 d W IP 3 d W IP 4

Wd IP 1

0 0

10 15 20

N

20

²W IP (%)

5

Line 3 1 0.9 0.8 0.7 0.6

5

15

²W IP (%)

0 0

5

10

N

²W IP (%)

4

10

5

²W IP (%)

3

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

10

0 0

20

²W IP (%)

2

Machine i

15

W IP

pi

Wd IP 1

10

N

W IP1 W IP2 W IP3 W IP4

²W IP (%)

20

Line 2 1 0.9 0.8 0.7 0.6

5

10

W IP

pi

Wd IP

0 0

5

W IP

4

W IP

3

W IP

2

Machine i

10

W IP

1

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

W IP

20

Line 1 1 0.9 0.8 0.7 0.6

151

5

10

15

20

10

15

20

10

15

20

10

15

20

15

20

N

²W IP1 ²W IP2 ²W IP3 ²W IP4 5

N

²W IP1 ²W IP2 ²W IP3 ²W IP4 5

N

²W I P 1 ²W I P 2 ²W I P 3 ²W I P 4 5

N

²W I P 1 ²W I P 2 ²W I P 3 ²W I P 4 5

10

N

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

2

3

4

Machine i

5

2

3

4

Machine i

2

3

4

Machine i

5

2

3

4

Machine i

5

2

3

4

Machine i

0 0

5

5

10

N

d BL 1

2

3

4

Machine i

5

15

5

10

N

15

0.1 5

10

N

15

20

5

10

N

5

10

N

10 15 20

N

BL

²BL ²BL

8 x 10

²BL1 ²BL2 ²BL3 ²BL4

5 0 0

20

15

5

10 15 20

N

15

5

10

N

15

5

10

N

0.3

15

0.1 5

10

N

15

di Figure 4.21: Accuracy of BL

0.1

0 0

10 15 20

N

²BL1 ²BL2 ²BL3 ²BL4 5

10 15 20

N

²BL1 ²BL2 ²BL3 ²BL4

0.02 0.01 0 0

5

10 15 20

N

0.02

²BL1 ²BL2 ²BL3 ²BL4

0.01 0 0

20

20

5

0.03

20

BL1 BL2 BL3 BL4

0.2

²BL1 ²BL2 ²BL3 ²BL4

3 0 0

20

BL1 BL2 BL3 BL4

0.2

6 x 10

20

BL1 BL2 BL3 BL4

0.4

0 0

15

BL1 BL2 BL3 BL4

0.2

0 0

5

−3

0.1

0 0

²BL1 ²BL2 ²BL3 ²BL4

5

0 0

20

BL1 BL2 BL3 BL4

0.4

20

d1 BL d2 BL d3 BL d4 BL

0.2

10

N

0.2

0 0

20

d1 BL d2 BL d3 BL d4 BL

5

0.3

20

10 15 20

N

−3

0.2 0 0

15

BL1 BL2 BL3 BL4

0.4

20

d1 BL d2 BL d3 BL d4 BL

0.3

0 0

15

10

N

0.1 0

5

7 x 10

²BL

pi pi pi pi pi pi pi

10

N

0.5

Line 8 1 0.9 0.8 0.7 0.6

5

d BL 1

15

5

0.2 0

0 0

20

BL1 BL2 BL3 BL4

0.3

20

d1 BL d2 BL d3 BL d4 BL

0.2 0 0

Line 7 1 0.9 0.8 0.7 0.6

10

N

0.4

d BL 1

5

0.1 0 0

15

0 0

15

−3

0.5

20

d1 BL d2 BL d3 BL d4 BL

0.2

Line 6 1 0.9 0.8 0.7 0.6

10

N

0.3

d BL 1

5

0.2 0 0

5

Line 5 1 0.9 0.8 0.7 0.6

0.1

0.4

d BL 1

15

d1 BL d2 BL d3 BL d4 BL

0.2 0 0

Line 4 1 0.9 0.8 0.7 0.6

10

N

0.3

d BL 1

5

10

N

²BL

0 0

5

5

²BL1 ²BL2 ²BL3 ²BL4

2

²BL

4

0 0

20

²BL

3

0.1

²BL

2

Machine i

15

d1 BL d2 BL d3 BL d4 BL

d BL 1

Line 3 1 0.9 0.8 0.7 0.6

10

N

0.5

Line 2 1 0.9 0.8 0.7 0.6

5

0.2

4 x 10

5

10 15 20

N

0.03

²BL

5

BL1 BL2 BL3 BL4

BL

4

BL

3

0 0

BL

2

Machine i

0.1

BL

1

0.2

−3

0.3

BL

d BL

pi

1 0.9 0.8 0.7 0.6

d1 BL d2 BL d3 BL d4 BL

BL

0.3

Line 1

BL

152

²BL1 ²BL2 ²BL3 ²BL4

0.02 0.01 0 0

5

10 15 20

N

4.2. M > 2-MACHINE LINES

2

3

4

Machine i

0 0

5

2

3

4

Machine i

2

3

4

Machine i

2

3

4

Machine i

5

2

3

4

Machine i

0 0

pi

c ST 1

2

3

4

Machine i

10

N

5

10

N

5

15

0.1 5

10

N

15

10

N

5

10

N

²ST2 ²ST3 ²ST4 ²ST5

ST

²ST

²ST 0 0

20

15

5

10 15 20

N

7 x 10

0 0

20

15

6

²ST2 ²ST3 ²ST4 ²ST5

5

10 15 20

N

5

10

N

15

5

10

N

15

0.2 5

10

N

0.3

15

0.1 5

10

N

15

ci Figure 4.22: Accuracy of ST

20

10 15 20

N

²ST2 ²ST3 ²ST4 ²ST5 5

10 15 20

N

0.03

²ST2 ²ST3 ²ST4 ²ST5

0.02 0.01 0 0

5

10 15 20

N

0.02

²ST2 ²ST3 ²ST4 ²ST5

0.01 0 0

20

ST2 ST3 ST4 ST5

0.2

5

0.05 0 0

20

ST2 ST3 ST4 ST5

²ST2 ²ST3 ²ST4 ²ST5

0.1

20

ST2 ST3 ST4 ST5

x 10

0 0

20

ST2 ST3 ST4 ST5

0.4

0 0

20

−3

0.2

0 0

15

ST2 ST3 ST4 ST5

0.4

20

20

5

0.1

0 0

15

−3

0.2

0 0

10

N

7 x 10

ST2 ST3 ST4 ST5

0.3

20

d2 ST d3 ST d4 ST d5 ST

0.2

0 0

15

10

N

0.2 0 0

20

d2 ST d3 ST d4 ST d5 ST

0.3

Line 8 1 0.9 0.8 0.7 0.6

5

5

0.4

20

d2 ST d3 ST d4 ST d5 ST

0.2

5

15

0 0

5

²ST

pi pi pi pi pi

c ST 1

10

N

0.4

Line 7 1 0.9 0.8 0.7 0.6

5

0.2 0 0

15

0.1

0.5

20

d2 ST d3 ST d4 ST d5 ST

0.4

c ST 1

10

N

0.1 0 0

Line 6 1 0.9 0.8 0.7 0.6

5

0.2

5

15

0 0

20

ST2 ST3 ST4 ST5

0.2

0 0

2

−3

0.3

20

d2 ST d3 ST d4 ST d5 ST

0.3

c ST 1

10

N

0.2 0 0

5

Line 5 1 0.9 0.8 0.7 0.6

5

0.4

c ST 1

15

d2 ST d3 ST d4 ST d5 ST

c ST 1

Line 4 1 0.9 0.8 0.7 0.6

10

N

0.5

Line 3 1 0.9 0.8 0.7 0.6

5

15

²ST2 ²ST3 ²ST4 ²ST5

²ST

5

10

N

²ST

4

5

²ST

3

0.1

0 0

20

d2 ST d3 ST d4 ST d5 ST

0.2

0 0

15

0.1

²ST

2

Machine i

10

N

0.2

5

10 15 20

N

0.03

²ST

pi

c ST 1

5

0.3

Line 2 1 0.9 0.8 0.7 0.6

0

ST

0

5

4 x 10

ST2 ST3 ST4 ST5

ST

4

ST

3

ST

2

Machine i

0.1

ST

1

0.2

−3

0.3

ST

c ST

pi

1 0.9 0.8 0.7 0.6

d2 ST d3 ST d4 ST d5 ST

ST

0.3

Line 1

153

²ST2 ²ST3 ²ST4 ²ST5

0.02 0.01 0 0

5

10 15 20

N

154

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

4.3

System-Theoretic Properties

4.3.1

Static laws of production systems

Many engineering systems can be characterized by their static laws. For instance, a static law of mechanical systems is: Sum of all forces acting on a rigid body = 0.

(4.49)

The statics of electric circuits are described by Sum of currents flowing through a node = 0

(4.50)

Sum of all voltage drops and rises in a closed loop = 0.

(4.51)

and by Similarly, the static law of production systems should describe the steady state flow of parts through this system. As it has been shown above, in the case of serial lines with Bernoulli machines, this flow is characterized by the steady states of the recursive procedure (4.30), i.e., by pbi pfi

= pi [1 − Q(pbi+1 , pfi , Ni )], = pi [1 −

i = 1, . . . , M − 1,

Q(pfi−1 , pbi , Ni−1 )],

i = 2, . . . , M.

(4.52)

In the same manner as (4.49)-(4.51) characterize static behavior of mechanical and electrical systems, (4.52) characterizes static properties of the production systems under consideration. Analyzing (4.52), and taking into account the expressions for the performance measures (4.36)-(4.40), we derive below two system-theoretic properties of serial lines with Bernoulli machines – reversibility and monotonicity. One more property – improvability – is analyzed in Chapter 5.

4.3.2

Reversibility

Consider a serial line L, defined by assumptions (a)-(e) of Subsection 4.2.1, and its reverse Lr (see Figure 4.23). Theorem 4.4 The performance measures of a Bernoulli serial line, L, and its reverse, Lr , are related as follows: L

d d P R =P R L d BL i

=

Lr

,

Lr c (M ST −i+1)r ,

i = 1, . . . , M − 1.

Proof: See Section 20.1. The reversibility property of production lines has practical implications. Several of them are mentioned below.

4.3. SYSTEM-THEORETIC PROPERTIES

155

p1

N1

p2

N2

pM−1

NM−1

pM

m1

b1

m2

b2

m M−1

bM−1

mM

pM

NM−1

pM−1

NM−2

p2

N1

p1

r

m1

r

b1

m 2r

r

b2

r

m M−1

r

bM−1

r

mM

Figure 4.23: M -machine production line and its reverse • Some argue that buffers at the end of the line should be larger than those at its beginning, since more work has been put into parts as they progress further along the line and are closer to being complete. The reversibility property says that the same effect can be ensured by reversing this argument. Thus, the argument of “end of the line” or “beginning of the line” is not valid for buffer capacity assignment. • If the machines are identical and only one buffer is available, where should it be placed within the line so that the production rate is maximized? If it is placed anywhere but in the middle of the line (assuming the number of machines is even), as it follows from Theorem 4.4, the buffer will be under utilized either in L or in Lr . Thus, the optimal position is the middle of the line. Improvement due to this positioning might be substantial. For instance, consider a 6-machine Bernoulli serial line when pi = 0.9, i = 1, . . . , 6 and a single buffer of capacity 2 available for placement within this line. If it is placed as either b1 or b5 , the resulting production rate is 0.668, whereas placing it in the middle of the line (i.e., as b3 ) gives P R = 0.694, a 3.8% improvement. • If all machines and buffers are identical and one machine can be improved, which one should it be, so that the production rate of a serial line with an odd number of machines is maximized? Similar to the above, the reversibility property leads to a conclusion that it should be the machine in the middle of the line. • If all machines are identical and the total buffering capacity N ∗ must be allocated among M − 1 buffers, how should their capacity be selected so that the production rate is maximized? The reversibility property tells us that the allocation, whatever it may be, must be symmetric with respect to the middle machine (if the number of machines is odd) or with respect to the two middle machines (if the number of machines is even). So, the only question that remains is whether the buffers should be of equal capacity or not (assuming that N ∗ is divisible by M ). It is easy to show that buffers of equal capacity do not maximize P R. Therefore, due to the reversibility property and the argument of the first bullet above, optimal

156

CHAPTER 4. ANALYSIS OF BERNOULLI LINES allocation must be of an inverted bowl shape, i.e., larger buffers are in the middle of the line. This result is intuitive, since the machines at the beginning and the end of the line experience less perturbations (because the first machine is not starved and the last is not blocked) and, therefore, need less protection than those in the middle of the line. In practical terms, however, the “bowl effect” is not very significant: the difference of P Rs under the uniform and the optimal bowl allocation is typically within 1% (which is often below the accuracy with which parameters of the machines are known). To illustrate this fact, consider a Bernoulli 5machine line with pi = 0.9. Its production rate for the uniform allocation of N ∗ = 20 is 0.8666, while according to the best bowl allocation, it is 0.8667.

4.3.3

Monotonicity

Could P R of a serial line decrease if the machines or buffers are improved? Intuitively, it is clear that this should not happen. A formal statement to this effect follows from (4.52): Theorem 4.5 In Bernoulli serial lines, defined by assumptions (a)-(e) of d d Subsection 4.2.1, P R=P R(p1 , . . . , pM , N1 , . . . , NM −1 ) is • strictly monotonically increasing in Ni , i = 1, . . . , M − 1; • strictly monotonically increasing in pi , i = 1, . . . , M . Proof: See Section 20.1.

4.4 4.4.1

Case Studies Automotive ignition coil processing system

Model validation: The Bernoulli model of this system is obtained in Subsection 3.10.1 and shown in Figure 3.31 for Periods 1 and 2. To validate this model (as well as the model of the next subsection), we use the following procedure: d • Using expressions (4.36)-(4.40), calculate the production rate estimate P R (parts/cycle). • Keeping in mind that the cycle time, τ , is 6.4 sec for Period 1 and 6.07 d d sec for Period 2, convert P R into T P (parts/hour): 3600 d d P R parts/hour. T P = τ • Using the throughput measured on the factory floor, T Pmeas , evaluate the accuracy of the model in terms of the error ²T P =

d |T P − T Pmeas | · 100%. T Pmeas

(4.53)

4.4. CASE STUDIES

157

The results are given in Table 4.2. Clearly, the fidelity of the model for both periods is sufficiently high. In general, it would be desirable to have more data points for the comparison. However, in this application (as well as in many others) the data is quite limited, but the decision nevertheless has to be made. Therefore, we conclude that the model is validated. Table 4.2: Model validation data Data Set Period 1 Period 2

d P R (parts/cycle) 0.8267 0.8143

d T P (parts/hr) 465 482

T Pmeas (parts/hr) 472 472.6

Error (%) 1.48 1.99

Although concrete improvement measures for the system at hand will be developed in Chapters 5 and 6, below we use the model validated to investigate several “what if” scenarios. Effect of starvations by pallets: The model of Figure 3.31 includes the effect of starvations of Op. 1 by empty pallets (i.e., the terms in parenthesis of p1 ). This indicates that the number of pallets in the system is not selected appropriately. It is of interest to know how much the performance would improve if the number of pallets were correct. To answer this question, assume that no starvations of Op. 1 takes place (i.e., eliminate the terms in parenthesis in the expression for p1 ) and re-calculate the production rate and throughput of the system. The results are shown in Table 4.3. Since the improvement is just about 1%, the system modification by adjusting the number of pallets is not an effective way for continuous improvement. Table 4.3: Performance without starvation of Op. 1 for pallets Period 1 Period 2

d T P (parts/hr) 470 488

Improvement (%) 1.08 1.24

Effect of increasing buffer capacity: The buffers in the system of Figure 3.31 are quite small. It is of interest to know how much the performance would improve if the capacity of all buffers were increased. The answer, given in Table 4.4, indicates that increasing the capacity by 50% leads to about 6% - 7% of throughput improvement, while further increases have practically no effect. Thus, increasing (perhaps, only some of the) buffer capacities may be an effective way for system improvement.

158

CHAPTER 4. ANALYSIS OF BERNOULLI LINES Table 4.4: Performance with increased buffer capacity % increment in buffers 50 100 200 300

d T P (parts/hr) 492 498 501 502

Improvement (%) 5.85 7.03 7.75 7.95

(a). Period 1 % increment in buffers 50 100 200 300

d T P (parts/hr) 513 518 522 523

Improvement (%) 6.39 7.45 8.22 8.46

(b). Period 2

Effect of increasing machine efficiency: The worst machine in the model of Figure 3.31 is m9−10 . Assuming that p9−10 is increased by, say, 3%, 5%, or 10%, what would the performance of the system be? The results are shown in Table 4.5. Clearly, increasing the efficiency of machine m9−10 leads to a reasonable improvement in system productivity. Table 4.5: Performance with increased efficiency of m9−10 % increment in p9−10 3 5 10

d T P (parts/hr) 474 480 494

Improvement (%) 2.10 3.40 6.30

(a). Period 1 % increment in p9−10 3 5 10

d T P (parts/hr) 493 499 512

Improvement (%) 2.04 3.32 6.12

(b). Period 2

Returning to the exponential model: As it follows from the exp-B transformation (3.38)-(3.40), the efficiency p9−10 of the Bernoulli machine m9−10 can be increased by either decreasing λ9−10 or increasing µ9−10 given in Table

4.4. CASE STUDIES

159

3.6. Which one of these routes is preferable? In other words, how much should λ9−10 be decreased (i.e., Tup of m9−10 increased) or how much should µ9−10 be increased (i.e., Tdown of m9−10 decreased) so that the desired throughput (see Table 4.5) is obtained? The answer, calculated using (3.39), (3.40) and (4.30)-(4.36), is given in Tables 4.6 and 4.7. Clearly, decreasing Tdown is more effective than increasing Tup . Table 4.6: Performance with increased uptime of m9−10 d To achieve desired T P 474 480 494

Tup 19.60 26.38 137.97

Increment of uptime (%) 40.44 88.97 888.47

(a). Period 1 d To achieve desired T P 493 499 512

Tup 21.84 28.07 84.52

Increment of uptime(%) 34.96 73.47 422.38

(b). Period 2

Table 4.7: Performance with decreased downtime of m9−10 d To achieve desired T P 474 480 494

Tdown 1.18 0.91 0.28

Decrement of downtime (%) 25.11 42.24 82.17

(a). Period 1 d To achieve desired T P 493 499 512

Tdown 1.53 1.19 0.39

Decrement of downtime (%) 25.59 42.08 80.87

(b). Period 2

4.4.2

Automotive paint shop production system

Model validation: The mathematical model of the paint shop system is shown in Figure 3.34 and the machines’ efficiency for five monthly periods is

160

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

given in Table 3.10. To validate this model, we evaluate its production rate, using expressions (4.36)-(4.40), and compare it with that measured on the factory floor (using (4.53)). The results are shown in Table 4.8. As it follows from this table, the model predicts well the system’s performance in all periods except for period 2. This discrepancy is attributed to the fact that during this period a new car model was introduced and, perhaps, some transient phenomena played a substantial role. Thus, omitting period 2, we assume that the model is validated. Table 4.8: Model validation data (jobs/hour)

d T P T Pmeas Error

Time Month 1 54.10 53.5 1.12%

Time Month 2 53.63 43.81 22.41%

Time Month 3 54.81 51.27 6.90%

Time Month 4 54.33 54.28 0.09%

Time Month 5 55.48 55.89 -0.73%

Effect of starvations by carriers: The model of Figure 3.34 and Table 3.13 includes the effect of starvations of Op. 3 by lacking carriers (the terms of p3 in parenthesis). Deleting these terms and re-calculating the production rate, we determine the effect of the starvations. The results are given in Table 4.9. Clearly, elimination of starvations by carriers could yield up to 10% improvement in system production rate. Table 4.9: Performance without starvation of Op. 3 by carriers (jobs/hour)

d T P Improvement

Time Month 1 59.29 9.59%

Time Month 2 59.10 10.20%

Time Month 3 58.54 6.81%

Time Month 4 59.78 10.03%

Time Month 5 60.09 8.31%

Effect of increasing buffer capacity: To investigate this effect, we calculate the production rate of the system with each buffer increased by 50% and by 100%. The results are given in Table 4.10. Clearly, increasing buffer capacity has almost no effect on system production rate. Effect of increasing machine capacity: As it follows from (3.61), machine efficiency depends on both production losses Li and machine capacity ci . While the production losses (due to push button activation) are difficult to eliminate, the speed of the operational conveyors may be modified relatively easily within

4.5. SUMMARY

161

Table 4.10: Performance with increased buffer capacity (jobs/hour) % incre. 50 100

d T P Imp. d T P Imp.

Time Month 1 54.20 0.18% 54.22 0.23%

Time Month 2 53.71 0.15% 53.72 0.17%

Time Month 3 55.19 0.90% 55.33 1.15%

Time Month 4 54.38 0.09% 54.38 0.09%

Time Month 5 55.61 0.23% 55.64 0.29%

±10% of their nominal values given in Table 3.8. Assuming that this would not lead to a substantial change in Li ’s, we calculate the production rate with all operations having the capacity 1.1ci . The results are in Table 4.11. Thus, increasing the speed of operational conveyors by 10% leads to over 11% improvement in system throughput. Table 4.11: Performance with increased machine capacity (jobs/hour)

d T P Improvement

4.5

Time Month 1 60.39 11.42%

Time Month 2 59.93 11.75%

Time Month 3 61.10 11.48%

Time Month 4 60.63 11.60%

Time Month 5 61.78 11.36%

Summary

• For two-machine lines, – the probabilities of buffer occupancy can be expressed in closed form as functions of machine and buffer parameters; – based on these probabilities, closed form expressions for all performance measures are derived. • For M > 2-machine Bernoulli lines, – due to the complexity of Markov chains involved, no closed form expression for the probabilities of buffer occupancy can be derived; – however, a recursive aggregation procedure is developed, the steady states of which lead to closed form estimates of all performance measures; – the accuracy of these estimates is high for the production rate (typically, within 1%) and lower for work-in-process and blockages/starvations;

162

CHAPTER 4. ANALYSIS OF BERNOULLI LINES – the accuracy of the estimates for all performance measures depends on the pattern of machine efficiency allocation, with the lowest accuracy taking place for the inverted bowl and “oscillatory” allocations. • The serial lines under consideration possess the property of reversibility: if the flow of parts is reversed, the production rate remains the same, while the probability of blockage (respectively, starvation) of machine i in the original line becomes the probability of starvation (respectively, blockage) of machine M − i + 1 in the reversed line.

• The serial lines under consideration possess the property of monotonicity: improving any machine efficiency or increasing any buffer capacity always leads to an increased production rate of the system.

4.6

Problems

Problem 4.1 Consider a two-machine Bernoulli production line defined by the conventions of Subsection 4.1.1 with N = 1 (i.e., machine m1 serves as the buffer). (a) Draw the state transition diagram of the ergodic Markov chain that describes this system and determine its transition probabilities. (b) Calculate the stationary probability of each state of this Markov chain and derive the formula for the production rate of this system. (c) Assuming that p1 = p2 =: p, draw the graph of P R as a function of p ∈ [0.5, 0.99] and comment on the qualitative behavior of this graph. Problem 4.2 Consider again a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection 4.1.1. (a) Assume p1 = 0.95 and p2 = 0.6. Calculate and plot P R, W IP , BL1 and ST2 as a function of N for N from 1 to 10. Based on these plots, determine the buffer capacity that is reasonable for this system. (b) Assume p1 = 0.6 and p2 = 0.95. Again calculate and plot P R, W IP , BL1 and ST2 as a function of N for N between 1 and 10. Will your choice of buffer capacity change? Problem 4.3 Consider a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection 4.1.1. Assume that N = 5 and p1 p2 = 0.81. (a) Under this constraint, find p1 and p2 , which maximize P R. (You may use trial and error to accomplish this; alternatively, you may think a little bit, look at the expressions for the performance measure, make an “educated guess” and verify it by calculations.) (b) For these p1 and p2 , calculate P R, W IP , BL1 and ST2 . What can you say about qualitative features of W IP , BL1 and ST2 ?

4.6. PROBLEMS

163

(c) Interpret the results and formulate a conjecture concerning the optimal allocation of pi ’s. Problem 4.4 Consider a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection 4.1.1. Suppose that each machine produces a good part with probability gi and a defective part with probability 1 − gi , i = 1, 2. Assume that quality control devices operate in such a manner that a defective part is removed from the system immediately after the machine that produced this part. (a) Derive expressions for the production rate of good parts and for W IP , BL1 and ST2 in this system. (b) Using any example you wish, check if the reversibility property still holds. Problem 4.5 Consider a two-machine Bernoulli production line defined by all but one of the conventions of Subsection 4.1.1. Specifically, assume that instead of the blocked before service, the following convention is used: Machine m1 is blocked during a time slot if it is up at the beginning of this time slot and the buffer is full at the end of the previous time slot (i.e., the blockage of m1 is independent of the status of m2 ). We refer to this convention as symmetric blocking since both starvation and blocking conventions are of the same nature - they are defined by the state of the buffer and the status of one machine only. Assume for simplicity that the machines are identical, i.e., p1 = p2 =: p. (a) Draw the state transition diagram of the ergodic Markov chain that describes this system and determine the transition probabilities. (b) Derive the expressions for the stationary probabilities of this Markov chain. (c) Plot these stationary probabilities for p = 0.95 and for p = 0.55, assuming that in both cases N = 5; compare the resulting graphs with those of Figure 4.4 and explain what are the differences and why they take place. (d) Derive formulas for P R, W IP , BL1 and ST2 as a function of p and N . (e) Plot P R, W IP , BL1 and ST2 as functions of N for N from 1 to 10 and p = 0.9; compare the resulting graphs with those of Figure 4.6, Line 1, and explain what are the differences and why they take place. Problem 4.6 Consider again the two-machine production line with the symmetric blocking as defined in Problem 4.5. Repeat parts (a)-(e) of Problem 4.2, assuming that p1 6= p2 ; for the questions in which numerical values are required, assume that p1 = 0.9 and p2 = 0.7. Problem 4.7 Investigate if the reversibility property holds for two-machine Bernoulli lines with the symmetric blocking convention defined in Problem 4.5. Problem 4.8 Repeat Problem 4.3 for a two-machine Bernoulli line with the symmetric blocking convention defined in Problem 4.5. Problem 4.9 Consider a 5-machine Bernoulli production line defined by conventions (a)-(e) of Subsection 4.2.1.

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CHAPTER 4. ANALYSIS OF BERNOULLI LINES

(a) Assume pi = 0.9, i = 1, . . . , 5, and all buffers are of equal capacity. Calculate c i as functions of Ni for Ni = 1, 2, 3, 4, and d di and ST \ and plot P R, W IP , BL 5. Based on these results, determine the buffer capacity, which is reasonable for this system. (b) Assume now that pi = 0.7, i = 1, . . . , 5, and buffers are as above. Again c i as functions of Ni . Will your d di and ST \ calculate and plot P R, W IP , BL choice of buffer capacity change? (c) Interpret the results. Formulate your conjecture as to the choice of buffer capacity as a function of machine efficiency. Problem 4.10 Consider a 3-machine Bernoulli production line defined by conventions (a)-(e) of Subsection 4.2.1. Assume Ni = 1, i = 1, 2, and p1 p2 p3 = (0.8)3 . d (a) Under this constraint, find pi , i = 1, 2, 3, which maximize P R. (You may use the trial and error method and the PSE Toolbox to accomplish this.) di and ST c i . What can you say about W \ \ (b) For these pi , calculate W IP , BL IP ? Interpret the results and formulate a conjecture concerning the optimal allocation of pi ’s. Problem 4.11 Consider the production system of Figure 3.38 and its 5machine Bernoulli model constructed in Problem 3.3. (a) (b) (c) (d)

Calculate the production rate of this system. Calculate the average occupancy of each buffer. Calculate the probabilities of blockages and starvations of all machines. Assuming that the Bernoulli buffer capacity is increased by a factor of 2, recalculate the production rate. Does it make practical sense to have all buffer capacities increased? (e) Assume that efficiency of all Bernoulli machines is increased by 10% and again recalculate the production rate. Does it make sense to have all efficiencies increased?

Problem 4.12 Repeat the steps of Problem 4.11 for the Bernoulli model of the production system constructed in Problem 3.4. Problem 4.13 Consider a seven-machine Bernoulli line with machines having identical efficiency. Assume that two buffers with equal capacities are available to be placed in this system. d (a) Where should the buffers be placed so that P R is maximized? (b) Suggest an example illustrating the efficacy of your solution. Problem 4.14 Consider a ten-machine Bernoulli line with machines of identical efficiency and buffers of identical capacity. Assume that the efficiency of two machines can be increased (or new machines with higher efficiency can be purchased).

4.7. ANNOTATED BIBLIOGRAPHY

165

d (a) Which of the machines should be improved (or replaced) so that P R is maximized? (b) Suggest an example illustrating the efficacy of your solution. Problem 4.15 Consider an M > 2-machine Bernoulli line with the symmetric blocking convention defined in Problem 4.5. (a) Develop a recursive aggregation procedure for performance analysis of this line. di and ST c i. d \ (b) Derive formulas for estimates of P R, W IP i , BL Problem 4.16 Using examples, investigate the reversibility property of M > 2-machine Bernoulli lines with the symmetric blocking convention. In particular, investigate if d P R

L

L di BL

Lr

=

d P R

=

Lr c (M ST −i+1)r ,

,

(4.54) i = 1, . . . , M

(4.55) (4.56)

still hold and, in addition, L

r \ W IP ir

=

L

\ NM −i − W IP M −i .

(4.57)

Problem 4.17 Using examples, investigate the monotonicity property of M > 2-machine Bernoulli lines with the symmetric blocking convention.

4.7

Annotated Bibliography

The initial analysis of two-machine Bernoulli lines has been carried out in [4.1] J.-T. Lim, S.M. Meerkov and F. Top, “Homogeneous, Asymtotically Reliable Serial Production Lines: Theory and a Case Study,” IEEE Transactions on Automatic Control, vol. 35, pp. 524-534, 1990. under the assumption that the machines are asymptotically reliable, i.e., their efficiencies are close to 1. The general case has been addressed in [4.2] D.A. Jacobs and S.M. Meerkov, “A System-Theoretic Property of Serial Production Lines: Improvability,” International Journal of System Science, vol. 26, pp. 95-137, 1995. The case of M > 2-machine lines has also been analyzed in [4.1] for the asymptotically reliable machines and then generalized in [4.2]. Numerical investigation of aggregation procedure accuracy has been carried out by J. Huang and A. Khondker in the framework of their course project in

166

CHAPTER 4. ANALYSIS OF BERNOULLI LINES

the class on Production Systems Engineering. The properties of reversibility and monotonicity have been known for a long time. In the framework of serial lines with exponential machines, the reversibility property has been discovered in [4.3] G. Yamazaki and H. Sakasegawa, “Property of Duality in Tandem Queueing Systems,” Annals of the Inst. of Statistical Math., vol. 27, pp. 201212, 1975, while the monotonicity property has been proven to exist in [4.4] J.G. Shanthikumar and D.D. Yao, “Monotonicity and Concavity Properties in Cyclic Queueing Networks with Finite Buffers,” in Queueing Networks with Blocking, eds. H.G. Perros and T. Altiok, North-Holland, pp. 325-345, 1989. For the case of Bernoulli lines, these properties have been investigated in [4.5] D.A. Jacobs and S.M. Meerkov, “Mathematical Theory of Improvability for Production Systems,” Mathematical Problems in Engineering, vol. 1, pp. 95-137, 1995.

Chapter 5

Continuous Improvement of Bernoulli Lines Motivation: It is not uncommon that, due to unscheduled downtime, machining lines in many industries operate at 60%-70% of their capacity. Although assembly systems are typically more efficient (often operating at 80%-90% of their capacity), the losses are still significant. In this situation, continuous improvement is a major tool for production systems management. Typically, continuous improvement projects are developed using common sense, managerial intuition and, in some cases, discrete event simulations. Due to the “soft” nature of these approaches, they often do not result in an actual productivity improvement. The purpose of this chapter is to present analytical methods for designing continuous improvement projects in Bernoulli lines with predictable results. The development is based on the analytical method and recursive equations derived in Chapter 4. Overview: Two approaches to design of continuous improvement projects are developed. They are referred to as constrained and unconstrained improvability. Constrained improvability addresses the issue of improving a production system by re-allocating its limited resources, e.g., buffer capacity or workforce. The main question here is: Can or cannot a production system be improved by utilizing more efficiently its limited resources? If it is possible, the system is called improvable under constraints; otherwise, it is unimprovable. Section 5.1 presents criteria, which allow to determine whether the system is improvable and provides a characterization of unimprovable allocations. Constrained improvability is related to optimality. Indeed, an unimprovable system is, in fact, optimal. We use, however, the term “improvable” to indicate that the goal is not necessarily to render the system optimal but rather to determine whether it can be improved and indicate actions that lead to this improvement. Moreover, given the lack of accurate information on the factory floor, the optimality may not be practically achievable, whereas continuous im167

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CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

provement, being robust with respect to inaccurate information, may. Unconstrained improvability addresses the issue of bottleneck identification and elimination by allocating additional resources (such as additional buffer capacity, machine improvement or replacement, etc.). The concept of bottleneck (BN) is not well understood, and, as a result, it is not unusual that in practice an improvement or replacement of a machine, viewed as the BN, leads to no improvement of the production system as a whole. So, what is a BN? Often, BN is understood as the machine with the smallest production rate in isolation. In other cases, the machine with the largest work-in-process in front of it is viewed as the BN. It is possible to show, however, that neither may be the BN in the sense of being the most impeding for the production rate of the system. This happens because the above intuitive conceptualizations are local in nature and do not take into account the total system properties, such as the location of machines in the production line, capacity of the buffers, types of interactions among the machines and buffers, etc. In Section 5.2, we introduce “system-based” definitions of bottleneck machines (BN-m) and bottleneck buffers (BN-b) in terms of their effect on the production rate of the line. The main practical results of this chapter are the criteria (referred to as indicators of improvability), which allow factory floor personnel to determine if the system is improvable (in the constrained or unconstrained case) and define actions that must be taken to achieve this improvement. In addition, we define the notion of buffering potency and introduce the method of measurement-based management of production systems.

5.1 5.1.1

Constrained Improvability Resource constraints and definitions

Consider a serial production line with M Bernoulli machines defined by parameters pi , i = 1, . . . , M , and M − 1 buffers with capacities Ni , i = 1, . . . , M − 1, which operates according to conventions (a)-(e) of Subsection 4.2.1. Assume that Ni ’s and pi ’s are constrained as follows: M −1 X

Ni

=

N ∗,

(5.1)

=

p∗ ,

(5.2)

i=1 N Y

pi

i=1

where N ∗ and p∗ are positive numbers with p∗ satisfying p∗ < 1. Constraint (5.1) implies that the total buffer capacity cannot exceed N ∗ . Constraint (5.2) can be interpreted as a bound on the machine efficiency or workforce. Indeed, in many systems, assignment of the workforce (both machine operators and skilled trades for repair and maintenance) defines the machine efficiency and, thus, pi ’s.

5.1. CONSTRAINED IMPROVABILITY

169

Therefore, we refer to (5.1) and (5.2) as the buffer capacity (BC) and workforce (WF) constraints, respectively. d d Let, as before, P R=P R(p1 , . . . , pM , N1 , . . . , NM −1 ) denote the production rate of the system, calculated using (4.30)-(4.36). Definition 5.1 A serial production line with Bernoulli machines is: 0 • improvable with respect to BC if there exists a sequence N10 , . . . , NM −1 such that M −1 X Ni0 = N ∗ (5.3) i=1

and 0 d d P R(p1 , . . . , pM , N10 , . . . , NM −1 ) > P R(p1 , . . . , pM , N1 , . . . , NM −1 ); (5.4)

otherwise, it is unimprovable with respect to BC; • improvable with respect to WF if there exists a sequence p01 , . . . , p0M such that M Y p0i = p∗ (5.5) i=1

and d d P R(p01 , . . . , p0M , N1 , . . . , NM −1 ) > P R(p1 , . . . , pM , N1 , . . . , NM −1 ); (5.6) otherwise, it is unimprovable with respect to WF; • improvable with respect to BC and WF simultaneously if there exist se0 0 0 quences N10 , . . . , NM −1 and p1 , . . . , pM such that M −1 X i=1

Ni0 = N ∗ ,

M Y

p0i = p∗

(5.7)

i=1

and 0 d d P R(p01 , . . . , p0M , N10 , . . . , NM −1 ) > P R(p1 , . . . , pM , N1 , . . . , NM −1 ); (5.8)

otherwise, it is unimprovable with respect to BC and WF simultaneously. Conditions for various types of improbability are given next.

5.1.2

Improvability with respect to WF

Necessary and sufficient conditions: Below we provide both theoretical and practical conditions of improvability with respect to WF.

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CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

Theorem 5.1 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to WF if and only if pfi = pbi+1 ,

i = 1, . . . , M − 1,

(5.9)

where pfi and pbi are the steady states (4.35) of the recursive aggregation procedure (4.30). Proof: See Section 20.1. Recall that, as it has been shown in Chapter 4, for each buffer bi , an M machine line can be represented as a two-machine system with virtual machines characterized by pfi and pbi+1 (see Figure 4.10). Thus, Theorem 5.1 implies that the necessary and sufficient condition of unimprovability with respect to WF is that both virtual machines are identical. Using the property (4.21) of two-machine lines with identical machines, we obtain Corollary 5.1 Under condition (5.9), \ W IP i =

Ni (Ni + 1) 2(Ni + 1 − pfi )

,

i = 1, . . . , M − 1.

(5.10)

Proof: See Section 20.1. Since 0 < pfi < 1, expression (5.10) implies that Ni Ni + 1 \ <W IP i < , 2 2

i = 1, . . . , M − 1.

(5.11)

Based on this, we formulate the practical WF-Improvability Indicator 5.1: A Bernoulli line is practically unimprovable with respect to WF if each buffer is, on average, close to being half full. Although this indicator may seem somewhat unexpected, it is, in retrospect, quite natural. Indeed, buffer bi is intended to protect mi from blockage and mi+1 from starvation. From the point of view of mi , buffer bi should be all the time empty; from the point of view of mi+1 it should be full. The compromise is – the buffer is half full (with some correction depending on the machines’ efficiency as indicated in (5.10) by pfi ). Unimprovable allocation of pi : To characterize the unimprovable (i.e., optimal) pi ’s, introduce the notation ∗

d P R := ∗

max Q ∗ p1 ,...,pM ; M i=1 pi =p

d P R(p1 , . . . , pM , N1 , . . . , NM −1 ).

(5.12)

d Clearly, P R is the largest production rate that can be attained in the system ∗ d under constraint (5.2). It turns out that P R can be calculated as follows:

5.1. CONSTRAINED IMPROVABILITY

171

Consider the recursive procedure 1

x(n + 1) = (p∗ ) M

M −1 ³ Y i=1

Ni + x(n) ´ M2 , Ni + 1

x(0) ∈ (0, 1).

(5.13)

PM −1 Theorem 5.2 Assume i=1 Ni−1 ≤ M/2. Then recursive procedure (5.13) is convergent and its steady state, x∗ , is: ∗

d R . x∗ = lim x(n) = P n→∞

(5.14)

Proof: See Section 20.1. Thus, the maximum production rate can be calculated without the knowl∗ d edge of the optimal WF allocation. Moreover, this P R can be used to calculate the optimal allocation of pi ’s, as shown below. QM Theorem 5.3 The sequence p∗1 , . . ., p∗M , i=1 p∗i = p∗ , which renders the serial production line with Bernoulli machines unimprovable with respect to WF, is given by ³ N +1 ´ ∗ 1 d (5.15) p∗1 = ∗ PR , d N1 + P R ´³ N + 1 ´ ³ N ∗ i−1 + 1 i d (5.16) p∗i = ∗ ∗ P R , i = 2, . . . , M − 1, d d Ni−1 + P R Ni + P R ´ ³ N ∗ M −1 + 1 d (5.17) p∗M = ∗ PR . d NM −1 + P R Proof: See Section 20.1. Thus, for any given sequence N1 , . . ., NM −1 and p∗ of (5.2), the optimal allocation of work can be calculated using (5.13)-(5.17). ∗ d Taking into account that P R < 1, expressions (5.15)-(5.17) lead to a “flat” inverted bowl phenomenon: Corollary 5.2 If all buffers are of equal capacity, i.e., Ni =: N , i = 1, . . ., M − 1, then (5.18) p∗1 = p∗M < p∗2 = p∗3 = . . . = p∗M −1 . Although the flat inverted bowl allocation of optimal pi ’s does take place, it does not lead, as it was mentioned before, to a significantly larger P R as compared with the uniform allocation. Indeed, consider a system with five Bernoulli machines, Ni = 2, i = 1, . . . , 4, and p∗ = (0.9)5 = 0.5905. For this system, the optimal pi allocation, according to (5.15)-(5.17) is p∗1 p∗2

= =

p∗5 = 0.8655, p∗3 = p∗4 = 0.9237,

∗

d leading to P R = 0.8110 . If pi ’s were distributed in the uniform manner, i.e., pi = 0.9,

i = 1, . . . , 5,

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CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

d P Runif = 0.8050, which is only 0.74% less than the production rate for the inverted bowl allocation. To reinforce this conclusion, consider one more example: M = 10, p∗ = (0.95)10 = 0.5987, and Ni = 3, i = 1, . . . , 9. Then the optimal allocation (5.15)-(5.17) becomes p∗1 p∗i

= p∗10 = 0.9335, = 0.9542, i = 2, . . . , 9

∗

d and the optimal P R = 0.9132, while the uniform allocation pi = 0.95,

i = 1, . . . , 10 ∗

d d R . results in P Runif = 0.9104, which is only 0.31% less than P Continuous improvement by WF re-allocation: While expressions (5.15)(5.17) can be used for design of new production systems, continuous improvement projects for existing operations can be designed using WF-Improvability Indicator 5.1. This can be carried out as follows: WF-Continuous Improvement Procedure 5.1: (1) By calculations off-line (using the aggregation procedure (4.30) and formula (4.37)) or by measurements on the factory floor, evaluate the average buffer occupancy, W IPi , i = 1, . . ., M − 1. (2) Determine the buffer for which |W IPi − (Ni + 1)/2| is the largest. Assume this is buffer k. (3) If W IPk − (Nk + 1)/2 is positive, re-allocate a sufficiently small amount of work, ²pk , from mk to mk+1 ; if W IPk − (Nk + 1)/2 is negative, re-allocate ²pk+1 from mk+1 to mk (observing, of course, the constraint pk pk+1 = const). (4) Return to step (1). (5) Continue this process until maxi |W IPi − (Ni + 1)/2| is sufficiently small. To illustrate this procedure, consider a four-machine Bernoulli line with d R of this line is 0.8281. Ni = 5 and p = [0.9675, 0.9225, 0.8780, 0.8372]. The P Running WF-Continuous Improvement Procedure 5.1 with ² = 0.01, we arrive at the following WF allocation p = [0.8875, 0.9125, 0.9163, 0.8841]. The resulting d P R = 0.8707, which is a 5.14% improvement.

5.1.3

Improvability with respect to WF and BC simultaneously

Necessary and sufficient conditions: For the case of simultaneous improvability with respect to WF and BC, these conditions can be formulated as follows:

5.1. CONSTRAINED IMPROVABILITY

173

Theorem 5.4 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to WF and BC simultaneously if and only if p1 = pfi = pbi = pM ,

i = 2, . . . , M − 1.

(5.19)

Proof: See Section 20.1. Corollary 5.3 Under condition (5.19), d1 BL di BL

c M, = ST c i , i = 2, . . . , M − 1. = ST

(5.20) (5.21)

In addition, Ni = N, and \ W IP i =

i = 1, . . . , M − 1,

N (N + 1) , 2(N + 1 − p1 )

i = 1, . . . , M − 1.

(5.22) (5.23)

Proof: See Section 20.1. Thus, if a line is unimprovable with respect to WF and BC simultaneously, all internal machines are starved and blocked with equal frequency, the first machine is blocked as frequently as the last machine is starved, all buffers are of equal capacity, have equal steady state occupancy, and this occupancy can be evaluated using (5.23). Clearly, N +1 N \ <W IP i < , 2 2

i = 1, . . . , M − 1.

Unimprovable allocation of pi and Ni : To characterize the unimprovable (i.e., optimal) pi and Ni allocations, introduce d P R

∗∗

:=

d P R(p1 , . . . , pM , N1 , . . . , NM −1 ). max PM −1 N1 , . . . , NM −1 ; i=1 Ni = N ∗ QM p1 , . . . , pM ; i=1 pi = p∗ (5.24)

Theorem 5.5 Let N ∗ be a multiple of M − 1. Then the sequences p∗1 , . . ., ∗ and N1∗ , . . ., NM −1 , which render the serial production line with Bernoulli machines unimprovable with respect to WF and BC simultaneously, are given by p∗M

Ni∗

=

p∗1

=

p∗i

=

N∗ ∗ = Nopt , i = 1, . . . , M − 1, M −1 ³ N∗ + 1 ´ ∗∗ opt d p∗M = ∗∗ P R , ∗ d Nopt + P R ³ N ∗ + 1 ´2 ∗∗ opt d P R , i = 2, . . . , M − 1. ∗∗ ∗ d Nopt + P R

(5.25) (5.26) (5.27)

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CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

Proof: See Section 20.1. ∗∗ ∗ d Thus, based on the capacity Nopt , the value of P R can be calculated using (5.13), and then p∗i ’s can be evaluated from (5.26) and (5.27). This result can be useful at the design stage of serial production lines. Note that the optimal pi allocation is again a flat inverted bowl, while optimal Ni ’s are uniform.

5.1.4

Improvability with respect to BC

Necessary and sufficient conditions: Theorem 5.6 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to BC if and only if the quantity ³ n pf pb o´ min pi min ib , fi i=1,...,M pi pi 0 is maximized over all sequences N10 , . . ., NM −1 such that

(5.28) PM −1 i=1

Ni0 = N ∗ .

Proof: See Section 20.1. Unfortunately, this result is of little practical importance, since its interpretation and utilization are much less obvious than those of (5.9) and (5.19). However, numerical experiments with (5.28) show that in most cases the unimPM −1 ∗ ∗ ∗ provable sequence N1∗ , . . ., NM −1 , i=1 Ni = N , can be determined by a criterion with a clearer interpretation: Numerical Fact 5.1 The production rate ensured by the buffer capacity allocation defined by Theorem 5.6 is almost always the same as the production rate defined by the allocation that minimizes max

i=2,...,M −1

\ \ |W IP i−1 − (Ni − W IP i )|

0 over all sequences N10 , . . ., NM −1 such that

PM −1 i=1

(5.29)

Ni0 = N ∗ .

This implies that a production line is practically unimprovable with respect to BC if the occupancy of each buffer bi−1 is as close to the availability of buffer bi as possible. In other words, bi−1 and bi offer practically equal protection of machine mi against starvations and blockages, respectively. This is illustrated \ in Figure 5.1 where the occupancy of bi−1 (i.e., W IP i−1 ) and the availability \ IP i ) are indicated by shaded areas. Based on the above, we of bi (i.e., Ni − W formulate the practical BC-Improvability Indicator 5.1: A Bernoulli line is practically unimprovable with respect to BC if the average occupancy of each buffer is as close as possible to the average availability of its downstream buffer.

5.1. CONSTRAINED IMPROVABILITY

b i−1

mi

175

bi

Figure 5.1: Illustration of Numerical Fact 5.1 Continuous improvement by BC re-allocation: improvability indicator can be used in the following

Similar to WF, the above

BC-Continuous Improvement Procedure 5.1: (1) By calculations off-line (using (4.30) and (4.37)) or by measurements on the factory floor, evaluate the average occupancy of each buffer, W IPi , i = 1, . . ., M − 1. (2) Determine the buffer for which |W IPi −(Ni+1 −W IPi+1 )|, i = 1, . . . , M −2, is the largest. Assume this is buffer k. (3) If W IPk − (Nk+1 − W IPk+1 ) is positive, transfer a unit of capacity from bk to bk+1 ; if W IPk − (Nk+1 − W IPk+1 ) is negative, re-allocate a unit of capacity from bk+1 to bk . (4) Return to step (1). (5) Continue this process until arriving at a limit cycle and choose the buffer capacity allocation on the limit cycle, which maximizes P R. As an illustration, consider an 11-machine serial line with pi = 0.8, i 6= 6 and p6 = 0.6. Assume that the total buffer capacity is 24 and determine the unimprovable buffer capacity allocation. Based on the above procedure and \ calculating W IP i using (4.30), (4.37), the following result is obtained: N1 N2 N5 N6 N7

= = = = =

1, N3 = N4 = 2, 5, 4, N8 = N9 = N10 = 2.

d This allocation ensures P R = 0.5843, which is almost the maximum possible in this system. Note that if, following Goldratt’s Theory of Constraints, all available buffer capacity were placed in front of the “bottleneck” machine m6 , i.e., Ni N5

= =

1, 15,

i 6= 5,

d the resulting P R = 0.427, which is 27% lower than that of the unimprovable allocation. If, keeping in mind the property of reversibility (see Section 4.3),

176

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

the “bottleneck” machine is protected on both sides, i.e., the allocation is Ni N5

= =

1, i 6= 5, 6, N6 = 8,

d the resulting P R = 0.491, which is still 16% lower than that of the unimprovable allocation obtained using BC-Continuous Improvement Procedure 5.1. As a final note, it should be pointed out that in some cases, the unimprovable allocation may not appear in the limit cycle, but a few steps before the limit cycle is reached. However, the production rate obtained using BC-Continuous Improvement Procedure 5.1 is typically very close to the production rate ensured by the unimprovable allocation. PSE Toolbox: The unimprovable allocations of WF and BC, as well as WFand BC-Continuous Improvement Procedures, are implemented in the Continuous Improvement function of the toolbox. For a description and illustration of these tools, see Subsections 19.4.1–19.4.4.

5.2 5.2.1

Unconstrained Improvability Definitions

Bottleneck machine: Consider a serial production line with M Bernoulli machines defined by parameters pi , i = 1, . . . , M , and M − 1 buffers with capacities Ni , i = 1, . . . , M − 1. Assume that the line operates according to conventions (a)-(e) of Subsection 4.2.1. Let, as before, P R, denote the production rate of the system, i.e., P R = P R(p1 , . . . , pM , N1 , . . . , NM −1 ). Definition 5.2 Machine mi , i ∈ {1, . . . , M }, is the bottleneck machine (BN-m) of a Bernoulli line if ∂P R ∂P R > , ∂pi ∂pj

∀j 6= i.

(5.30)

Due to the monotonicity properties of P R with respect to pi ’s (see Section 4.3), both derivatives in (5.30) are positive. Thus, Definition 5.2 implies that mi is the BN-m if its infinitesimal improvement leads to the largest increase of the production rate, as compared with a similar improvement of any other machine in the system. A machine with the smallest pi is not necessarily the BN-m in the sense of Definition 5.2. Indeed, consider the production lines shown in Figure 5.2, where the numbers in the circles and the rectangles are pi and Ni , respectively, and the row of numbers under the machines represent the estimates of partial R derivatives ∂P ∂pi evaluated by numerical simulations. Clearly, the bottleneck

5.2. UNCONSTRAINED IMPROVABILITY

177

machines are m2 (in Figure 5.2(a)) and m4 (in Figure 5.2(b)), none of which corresponds to the worst machine (i.e., the machine with the smallest pi ). In fact, m2 in Figure 5.2(a) is the best machine in the system. BN−m

∆PR

m1

b1

0.8

2

0.369

∆p i

m2 0.83

b2

m3

b3

m4

2

0.77

3

0.8

0.452

0.443

0.022

(a). The best machine is the bottleneck. BN−m

∆PR ∆p i WIP i

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

b6

m7

b7

m8

0.9

6

0.7

6

0.8

1

0.7

1

0.75

4

0.6

6

0.7

2

0.85

0.05

0.06 5.59

0.28 5.39

0.38 0.87

0.31 0.69

0.17 1.68

0.06 1.18

0.05 0.66

(b). The worst machine is not the bottleneck. Figure 5.2: Examples of bottleneck machines in Bernoulli lines Similarly, a machine with the largest work-in-process in front of it is not necessarily the bottleneck. An example is given in Figure 5.2(b), where m2 has the largest W IP to be processed, while the BN-m is m4 . While Definition 5.2 provides a formal characterization of the BN-m, its practical application is not straightforward because the partial derivatives involved could be neither measured on the factory floor nor calculated analytically with any acceptable accuracy. Indeed, while P R itself is often measured on the factory floor, its sensitivity to pi ’s is not and could hardly be expected to be measured, since it would require increasing efficiency of each machine and evaluating the resulting increase in P R. Such a procedure is hardly possible in most practical situations. Although the analytical calculations, described in Chapd ter 4, lead to acceptable estimates P R, no analytical methods for evaluating ∂P R/∂pi are available. Therefore, to make Definition 5.2 practical, it has to be reformulated in terms of quantities, which are either available through measurements on the factory floor or through analytical calculations or both. It is shown in Subsections 5.2.2 and 5.2.3 that this can be accomplished using the probabilities of blockages and starvations, BLi and STi , defined in Chapter 3 and evaluated in Chapter 4. The above arguments notwithstanding, the machine with the smallest pi is, in fact, the bottleneck in the sense of (5.30) if the production line is WFunimprovable. This follows from the following Theorem 5.7 In WF-unimprovable Bernoulli lines defined by assumptions

178

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

(a)-(e) of Subsection 4.2.1, d ∂P R pi = const, ∂pi

i = 1, . . . , M.

(5.31)

Proof: See Section 20.1. Thus, the BN-m in WF-unimprovable lines can be identified quite easily. Bottleneck buffer: While the term “bottleneck machine” is widely used in practice, the term “bottleneck buffer” is not. This happens, perhaps, because thinking locally, one usually pays more attention to the efficiency of the machines, as part-producing devices, and less to buffers, as “shock absorbers” of perturbations. Nevertheless, the notion of a bottleneck buffer could and, moreover, must be introduced in order to explore all means of system improvements. Definition 5.3 Buffer bi , i ∈ {1, . . . , M −1}, is the bottleneck buffer (BN-b) of a Bernoulli line if P R(p1 , . . . , pM , N1 , . . . , Ni + 1, . . . , NM −1 ) > P R(p1 , . . . , pM , N1 , . . . , Nj + 1, . . . , NM −1 ),

∀j 6= i. (5.32)

In other words, BN-b is the buffer, which leads to the largest increase of the P R if its capacity is increased by 1, as compared with increasing any other buffer in the system. The buffer with the smallest capacity is not necessarily the BN-b. An example is shown in Figure 5.3, where the numbers under each buffer correspond to the P R of the system obtained by simulations when the capacity of this buffer is increased by one. Clearly, the BN-b is b1 while the smallest buffer is b3 . BN−b

PR(N i+1)

m1

b1

0.8

3

0.769

m2 0.85

b2

m3

b3

m4

3

0.85

2

0.9

0.766

0.763

Figure 5.3: Example of bottleneck buffer in a Bernoulli line To identify the BN-b using Definition 5.3, one would have to experiment with the system by increasing each buffer and measuring the resulting production rate, which is hardly possible in practice. It turns out that this is also unnecessary: as it is shown below, BLi and STi can be used to identify not only the BN-m but also the BN-b.

5.2. UNCONSTRAINED IMPROVABILITY

5.2.2

179

Identification of bottlenecks in two-machine lines

Theorem 5.8 For a two-machine Bernoulli line defined by assumptions (a)-(e) of Subsection 4.1.1, the inequality ∂P R ∂P R > ∂p1 ∂p2

(respectively,

∂P R ∂P R < ) ∂p1 ∂p2

(5.33)

takes place if and only if BL1 < ST2

(respectively, BL1 > ST2 ).

Proof: See Section 20.1. There are three benefits offered by this theorem. First, it provides a relationship between the “non-measurable” and “non-calculable” partial derivatives of P R and “measurable” and “calculable” probabilities of blockages and starvations. Second, it offers a possibility of identifying the BN-m without even knowing parameters of the machines and buffer, but just by measuring ST2 and BL1 . Third, it offers a simple graphical way of representing the BN-m. To illustrate this, consider the production line of Figure 5.4, where the two rows of numbers under the machines represent STi and BLi . Place an appropriate inequality sign between ST2 and BL1 and turn the inequality into an arrow by adding a line within the sign of the inequality. According to Theorem 5.8, the machine, to which the arrow is pointed, is the BN-m. As it turns out, this procedure can be extended to M > 2-machine lines as well. BN−m

ST i BL i

m1

b1

m2

0.9

2

0.8

0 0.1215

>

0.0215 0

Figure 5.4: Arrow-based method of bottleneck identification Note that for the case of M = 2, • the problem of BN-b does not arise; • the machine with the smallest pi is the BN-m since, as it follows from the results of Section 4.1, ST2 > BL1 , if and only if p1 < p2 .

5.2.3

Identification of bottlenecks in M > 2-machine lines

Bottleneck Indicator: As it was discussed in Chapter 4, STi and BLi in lines ci with M > 2 machines cannot be calculated exactly, and only estimates, ST

180

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

di , are available. Therefore, the rules for BN identification in M > 2and BL machine lines are formulated either in terms of STi and BLi , which may be di , which c i and BL available from factory floor measurements, or in terms of ST may be calculated using (4.30) and (4.39), (4.40)). Note that the application of the former requires no knowledge of the machine and buffer parameters. Consider the production lines shown in Figures 5.5 and 5.6 with two rows of numbers under each machine, the first one indicating STi and the second BLi . Place arrows directed from one machine to another in the same manner as in Subsection 5.2.2, i.e., according to the following rule BN−m BN−b

ST

i

BL i

m1

b1

m2

b2

m3

b3

m4

b4

0.9

6

0.7

6

0.8

1

0.7

1

m5

b5

m6

b6

m7

b7

m8

0.75

4

0.6

6

0.7

2

0.85

0

0

0

0.09

0.23

0.1

0.2

0.36

0.4

0.2

0.3

0.14

0.03

0

0.01

0

Figure 5.5: Illustration of a Bernoulli line with a single bottleneck machine PBN−m BN−b

ST

i

BL i

BN−m

m1

b1

m2

b2

m3

b3

m4

b4

0.9

2

0.5

2

0.9

2

0.9

2

m5

b5

m6

b6

m7

b7

m8

0.9

2

0.9

2

0.6

2

0.9

0

0

0.39

0.37

0.33

0.27

0.11

0.41

0.01

0.03

0.05

0.1

0.17

0

Si

0.7845

0.41 0

0.4682

Figure 5.6: Illustration of a Bernoulli line with multiple bottleneck machines Arrow Assignment Rule 5.1: If BLi > STi+1 , assign the arrow pointing form mi to mi+1 . If BLi < STi+1 , assign the arrow pointing from mi+1 to mi . Bottleneck Indicator 5.1: In a Bernoulli line with M > 2 machines, • if there is a single machine with no emanating arrows, it is the BN-m; • if there are multiple machines with no emanating arrows, the one with the largest severity is the Primary BN-m (PBN-m), where the severity of each (local) BN-m is defined by Si S1

= =

|STi+1 − BLi | + |STi − BLi−1 |, |ST2 − BL1 |,

SM

=

|STM − BLM −1 |;

i = 2, . . . , M − 1, (5.34)

• the BN-b is the buffer immediately upstream of the BN-m (or PBN-m) if it is more often starved than blocked, or immediately downstream of the BN-m (or PBN-m) if it is more often blocked than starved.

5.2. UNCONSTRAINED IMPROVABILITY

181

Thus, according to this indicator, m4 and b4 are the bottlenecks in Figure 5.5, and m2 and b2 are the PBN-m and BN-b in Figure 5.6. Bottleneck Indicator 5.1 is justified below using both numerical and analytical approaches. Numerical justification: It is carried out by calculations and simulations. di , identifying BNc i and BL The calculation approach consists of calculating ST m and BN-b using Bottleneck Indicator 5.1, and then verifying the conclusions d d R(Ni + 1) and Definitions 5.2 and using the calculated quantities ∆P R/∆pi , P 5.3. The simulation approach is carried out analogously but using STi , BLi , and ∆P R/∆pi , P R(Ni + 1) evaluated numerically based on Simulation Procedure 4.1. In both calculation and simulation approaches, ∆pi was selected as 0.03 d P R R (or ∆P and ∆∆p ∆pi ) was evaluated as i d d d P R(p1 , . . . , pi + ∆pi ; . . . , pM ) − P R(p1 , . . . , pi , . . . , pM ) ∆P R = . ∆pi ∆pi In the majority of cases analyzed, Bottleneck Indicator 5.1 identified BNs correctly. Typical examples are shown in Figures 5.7 (single BN case) and 5.8 (multiple BN case). Some counterexamples, however, have also been discovered. Two of them are shown in Figures 5.9 and 5.10 for single and multiple BN cases, respectively. To investigate the “frequency” of the counterexamples, the following statistical experiment was carried out: 5000 five-machine serial lines were constructed by selecting pi ’s and Ni ’s randomly and equiprobably from the sets pi Ni

∈ {0.75, 0.80, 0.85, 0.90.0.95}, ∈ {1, 2, 3},

respectively. For each of these lines, BNs were identified using the calculation and simulation approaches, and the percent of correct and incorrect BN identifications were evaluated. The results are shown in Figures 5.11 and 5.13 for calculation and simulation approaches, respectively. Figure 5.11(a) indicates that among the 5000 lines, investigated by calculations, about 80% had a single BN-m. Within those, Bottleneck Indicator 5.1 identified the BN-m and BN-b correctly in over 95% and 87% of cases, respectively (Figures 5.11(b) and (c)). The lower accuracy of BN-b identification is, perhaps, due to the fact that, unlike pi ’s, buffer capacity cannot be increased infinitesimally. This conjecture is supported by Figure 5.11(d), which shows that the BN-b, identified by Definition 5.3, is one of the buffers around the BN-m in over 91% of cases. Thus, Figures 5.11(b)-(d) indicate that Bottleneck Indicator 5.1 is a sufficiently reliable tool for BN-m and BN-b identification. Note that, as it is shown in Figure 5.12(a), the BN-m is the worst machine of the system in only 62% of cases analyzed; thus, assuming that the worst machine is the bottleneck leads to a much lower frequency of correct BN identification.

182

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

BN−m BN−b

^ ST

i

^ BL i

m1

b1

m2

b2

m3

b3

m4

b4

0.9

6

0.7

6

0.8

1

0.7

1

m7

b7

m8

0.75

0.6

6

0.7

2

0.85

0.09

0.23

0.11

0.20

0.36

0.41

0.21

0.31

0.14

0.04

0.00

0.01

0

0

0.01

0.26

0.42

0.34

0.18

0

^

0.4927

0.4929

0.5407

0.5240

0.4980

0.4928

0 0.4927

0

0

0

0.09

0.23

0.1

0.2

0.36

0.4

0.2

0.3

0.14

0.03

0

0.01

0

0.05

0.06

0.28

0.38

0.31

0.17

0.06

0.05

i

∆pi

b6

0

PR(N i+1)

∆ PR

m6

4

0

∆p i

BL i

b5

0

^ ∆PR

ST

m5

0.4963

PR(N i+1)

0.4965

0.5308

0.5267

0.4965

0.5007

0.4965

(a) BN−b BN−m m1 0.9

ST

∆p i

6

0.8

b3 3

m4

b4

m5

b5

m6

b6

m7

b7

m8

0.7

3

0.75

4

0.6

6

0.7

2

0.85

0.01

0.04

0.01

0.11

0.27

0.32

0.12

0.22

0.11

0.13

0.01

0.01

0

0

0.01

0.02

0.09

0.15

0.64

0

0

0.5784

PR(N i+1)

∆PR

6

0.7

m3

0.01

∆p i

0.5785

0.5835

0.5818

0.5790

0.5796

0.803

0

0

0.01

0.01

0.05

0.02

0.11

0.27

0.32

0.12

0.22

0.12

0.13

0.01

0.01

0

0.03

0.02

0.06

0

i

BL i

b2

0

∆PR

ST

m2

0

i

BL i

b1

0.01

PR(N i+1)

0.5762

0.5770

0.12

0.5775

0.61

0.09

0.5807

0.5811

0.5777

0.5769

(b) Figure 5.7: Examples of bottleneck identification using Bottleneck Indicator 5.1; single bottleneck case

5.2. UNCONSTRAINED IMPROVABILITY

183

PBN−m BN−b

^ ST

^S

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

b6

m7

b7

0.9

2

0.5

2

0.9

2

0.9

2

0.9

2

0.9

2

0.6

2

0.40

0.40

0.38

0.31

0.10

0.41

0.41

0

0.01

0.01

0.04

0.15

0

0

0.8025

^ +1) PR(N i ST

i

BL i

0.4481

0.95

0.05

∆p i

0.9

0

i

^ ∆PR

m8

0

i

^ BL i

BN−m

m1

0.06

0.4969

0.4966

0.01

0

0.4945

0.4943

0

0.4943

0

0.4942

0

0.39

0.37

0.33

0.27

0.11

0.41

0.01

0.03

0.05

0.10

0.17

0

0.7845 0.11

∆p i

0.4942

0

Si ∆PR

0.11 0.4919

0.4913

0.41 0

0.4682

0.76

PR(N i +1)

0

0.09

0.09

0.4908

0.4917

0.09 0.4916

0.13 0.4892

0.09 0.4868

(a) BN−m

^ ST

i

^ BL i ^S

b1

m2

b2

m3

b3

0.7

2

0.9

2

0.8

2

m4

b4

m5

b5

m6

b6

m7

b7

m8

2

0.9

2

0.7

2

0.7

2

0.9

0.9

0

0.11

0.04

0.06

0.02

0.01

0.10

0.30

0.10

0.22

0.17

0.26

0.28

0.09

0.01

0

0.50

0.07

0.2772

0.0096

i

^ ∆PR

0.01

∆p i

^ PR(N

PBN−m BN−b

m1

0.5991

+1)

i

0.01

0.02

0.03 0.6004

0.5998

0.5992

0.09

0.51 0.6237

0.6034

0.6017

0

0.12

0.07

0.1

0.06

0.03

0.1

0.31

BL i

0.11

0.2

0.15

0.23

0.26

0.09

0.01

0

Si

0.0157

0.38

0.07

ST

i

∆PR ∆p i PR(Ni +1)

0.13

0.2552 0.08

0.5954

0.1 0.5956

0.08 0.5954

0.09 0.5968

0.46 0.5971

0.6142

0.5933

(b) Figure 5.8: Examples of bottleneck identification using Bottleneck Indicator 5.1; multiple bottlenecks case

184

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

m1

b1

m2

b2

0.8

2

0.83

4

m3

b3

m4

0.77

6

0.8

ST i

0

0.0739

0.0458

0.0817

BL i

0.0817

0.0442

0.0063

0

∆PR ∆ pi

0.362

0.384

0.381

0.072

ST i

0

0.0711

0.0471

0.0839

BL i

0.0846

0.0435

0.0048

0

∆PR ∆ pi

0.4

0.385

0.35

0.145

Figure 5.9: Counterexample for Bottleneck Indicator 5.1, single bottleneck case

^ ST

i

^

BL i

^S

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

b6

m7

b7

0.9

2

0.7

2

0.9

2

0.9

2

0.7

2

0.9

2

0.7

2

0 0.28

∆p i

ST

i

BL i

∆PR ∆ pi

0.01

0.07

0.02

0.16

0.07

0.28

0.07

0.16

0.22

0.05

0.14

0.01

0

0.02

0.12

0

0.01

0.29

0.09

0.3089

0.08

0.06

0.27

0.3378

0.12

0.39

0.2

0.14

0.09

0.04

0.19

0.09

0.3

0.17

0.22

0.06

0.13

0.01

0

0.3053

0.3394

Si

0.9

0.14

0.3378

i

^ ∆PR

m8

0.1

0.08

0.37

0.31

0.05

0.3332

0.1

0.27

0.06

Figure 5.10: Counterexample for Bottleneck Indicator 5.1, multiple bottlenecks case

5.2. UNCONSTRAINED IMPROVABILITY

185

4.70%

19.47%

Correct single BN-m identified Incorrect single BN-m identified

Single BN-m Multiple BN-m 80.53%

95.30%

(a) Single vs. multiple BN-m

12.46%

Correct BN-b identified

(b) Accuracy of BN-m identification in single BN-m case

BN-b is around BN-m

8.65%

BN-b is not around BN-m

Incorrect BN-b identified 87.54%

91.35%

(c) Accuracy of BN-b identification in single BN-m case

Correct PBN-m identified

27.91%

(d) BN-b is one of the buffers surrounding the BN-m in single BN-m case PBN-m is within a set of local BN-m's

5.74%

PBN-m is not within a set of local BN-m's

Incorrect PBN-m identified 72.09% 94.26%

(e) Accuracy of PBN-m identification

(f) PBN-m is within the set of local BN-m’s

BN-b is around PBN-m identified

Correct BN-b identified 46.66%

53.34%

Incorrect BN-b identified

(g) Accuracy of BN-b identification in multiple BN-m case

43.66% 56.34%

BN-b is not around PBN-m identified

(h) BN-b is one of the buffers surrounding PBN-m

Figure 5.11: Accuracy of Bottleneck Indicator 5.1 using calculation data

186

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

Worst machine is PBN-m

Worst machine is BN-m

37.89%

Worst machine is not BN-m 62.11%

(a) BN-m vs. the worst machine in the line for single BN-m case

42.29% 57.71%

Worst machine is not PBN-m

(b) PBN-m vs. the worst machine in the line for multiple BN-m case

Figure 5.12: Frequency of worst machine being BN-m using calculation data The frequency of correct PBN identification using Bottleneck Indicator 5.1 for multiple bottlenecks case is illustrated in Figure 5.11(e). Although it is relatively low (about 72%), the true PBN is within the set of local BN-m’s in over 94% of cases (Figure 5.11(f)). The frequency of correct BN-b identification is illustrated in Figures 5.11(g) and (h)). Clearly, accuracy of PBN-m identification is lower than that of just a bottleneck. This is, perhaps, due to the fact that the bottleneck severity (5.34) has been defined in an ad-hoc manner and better definitions might be possible. Nevertheless, we conclude that Bottleneck Indicator 5.1 is a useful tool for bottleneck identification, especially taking into account that in only 42% of cases analyzed the worst machine was indeed the PBN (see Figure 5.12(b)). Similar results were obtained using the simulation approach. These results are illustrated in Figures 5.13 and 5.14. Thus, the conclusion from this investigation is that Bottleneck Indicator 5.1 provides a sufficiently accurate method for identifying BNs using either meac i and BL di . We also remark sured quantities STi and BLi or calculated ones ST that, in our experience, even when Bottleneck Indicator 5.1 leads to an incorrect BN identification, the true BN has just a slightly higher ∆P R/∆pi and P R(Ni + 1) than those identified by the indicator. Based on the above, we formulate BN-Continuous Improvement Procedure 5.1: (1) By off-line calculations (using (4.30), (4.39), and (4.40)) or by measurements on the factory floor, evaluate the probabilities (or frequencies) of blockages and starvations of each machine. (2) Using Bottleneck Indicator 5.1, identify the BN-m (or PBN-m) and BN-b. (3) Take actions to increase the efficiency, pi , of this machine (for instance, by improved preventative maintenance, assigning additional work force and the like). (4) If the above, for one reason or another, is impossible, increase the BN-b or both buffers around the BN-m (or PBN-m). (5) Return to step (1).

5.2. UNCONSTRAINED IMPROVABILITY

187

8.80%

15.93%

Correct single BN-m identified Incorrect single BN-m identified

single BN-m Multiple BN-m 84.07%

91.20%

(a) Single vs. multiple BN-m

(b) Accuracy of BN-m identification in single BN-m case

7.39%

12.19% Correct BN-b identified Incorrect BN-b identified

BN-b is around BN-m BN-b is not around BN-m

87.81%

92.61%

(c) Accuracy of BN-b identification in single BN-m case

(d) BN-b is one of the buffers surrounding the BN-m in single BN-m case

2.82% 34.73%

65.27%

PBN-m is within a set of local BN-m's PBN-m is not within a set of local BN-m's

Correct PBN-m identified Incorrect PBN-m identified

97.18%

(e) Accuracy of PBN-m identification

53.14%

46.86%

Correct BN-b identified Incorrect BN-b identified

(g) Accuracy of BN-b identification in multiple BN-m case

(f) PBN-m is within the set of local BN-m’s

BN-b is around PBN-m identified 47.07%

52.93%

BN-b is not around PBN-m identified

(h) BN-b is one of the buffers surrounding PBN-m

Figure 5.13: Accuracy of Bottleneck Indicator 5.1 using simulation data

188

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

39.51% 60.49%

Worst machine is BN-m Worst machine is not BN-m

(a) BN-m vs. the worst machine in the line for single BN-m case

36.92% 63.08%

Worst machine is PBN-m Worst machine is not PBN-m

(b) PBN-m vs. the worst machine in the line for multiple BN-m case

Figure 5.14: Frequency of worst machine being BN-m using simulation data Our industrial experience, gained through numerous case studies, indicates that BN-Continuous Improvement Procedure 5.1 (and its generalization for production lines with exponential and general models of machine reliability - see Parts III and IV) is one of the most efficient ways of managing production systems. Analytical justification: An analytical justification is available only for the case of a single bottleneck machine. It is based on the following Hypothesis 5.1 Inequalities BLj−1 > STj ,

j = 2, . . . , M

and BLj < STj+1 ,

j = 1, . . . , M − 1

imply, respectively, that

and

²j1 := Pj−1 (0) ≈ Q(pfj−1 , pbj , Nj−1 ) ¿ 1,

(5.35)

²j2 := (1 − pbj+1 )Pj (Nj ) ≈ Q(pbj+1 , pfj , Nj ) ¿ 1.

(5.36)

The following lemma states that Hypothesis 5.1 indeed holds, at least for Nj sufficiently large: Lemma 5.1 In a Bernoulli line defined by assumptions (a)-(e), for any 0 < ²0 ¿ 1, there exists N ∗ such that if Nj > N ∗ , j = 1, . . . , M − 1, then ² = max(²j1 , ²j2 ) < ²0 . Proof: See Section 20.1. Theorem 5.9 Under Hypothesis 5.1, the BN-m is downstream of mj if c j+1 and upstream of mj if BL dj−1 < ST c j. d BLj > ST Proof: See Section 20.1. Thus, this theorem confirms Bottleneck Indicator 5.1 as far as a single BN-m is concerned.

5.2. UNCONSTRAINED IMPROVABILITY

189

PSE Toolbox: The method of BN-m and BN-b identification, using both calculated and measured data, is implemented in the Bottleneck Identification function of the toolbox. For a description of these tools, see Subsections 19.5.1 and 19.5.3.

5.2.4

Potency of buffering

As it has been shown above, the worst machine often is not the BN of the system. Why does this happen? Clearly, this is because of an inappropriate buffer capacity allocation. To formalize the notion of buffering quality, introduce Definition 5.4 The buffering of a production system is • weakly potent if the BN-m is the worst machine in the system (i.e., the machine with the smallest efficiency); otherwise, it is not potent; • potent if it is weakly potent and its production rate is sufficiently close to the BN-m efficiency (e.g., within 5% of the BN machine efficiency in isolation); • strongly potent if it is potentPand the system has the smallest possible total M −1 buffer capacity (i.e., N ∗ = i=1 Ni is the smallest possible to ensure the desired production rate). For serial lines with Bernoulli machines, one can determine if the buffering is weakly potent or not using the method of BN-m identification described in this section. To determine if it is potent, the method of P R calculation, described in Chapter 4, can be used. To investigate the notion of strong potency, methods that allow one to calculate the smallest total buffers capacity, which is necessary to ensure the desired production rate, must be available; these methods are described in Chapter 6. For serial lines with exponential and general models of machine reliability, similar techniques are discussed in Part III, and for assembly systems in Part IV. Along with its practical utility, the notion of buffering potency has conceptual significance. Indeed, production systems consist of two distinct entities: machines and buffers. The quality of the machines is characterized by their efficiency. In practice, machine efficiency is often monitored, and continuous improvement efforts are largely centered on its modification. In contrast, the quality of buffering is rarely monitored and even more rarely viewed as a resource for continuous improvement. The quantification provided by Definition 5.4 brings buffering to the same level of monitoring potential as that for machine efficiency. To illustrate the importance of buffering potency, consider the automotive ignition module assembly system described in Sections 3.2 and 3.10. Its throughput losses due to machines and due to MHS are analyzed in Table 5.1. Here, the losses due to machines are evaluated as the difference between the nominal throughput (600 parts/h) and the isolation throughput of the worst machine; the

190

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

losses due to MHS are evaluated as the difference between the isolation throughput of the worst machine and the actual throughput of the system. As follows from these data, out of roughly 240 parts/h lost, 80 parts/h are attributed to the machines and 160 parts/h to MHS. Clearly, this MHS is non-potent. (Chapter 16 shows that the worst machine is not the bottleneck of the system.) Interestingly, this 1:2 ratio has been observed in other production systems as well. Thus, ensuring potency of MHS is an important resource of production system improvements. A detailed analysis of the automotive ignition module assembly system is described in Chapter 16. Table 5.1: Losses analysis in automotive ignition module assembly system Month Isolation T P of the slowest machine (parts/h) Losses due to machine (parts/h) T P of the system (parts/h) Losses due to MHS (parts/h)

5.2.5

May

June

July

Aug.

Sept.

Oct.

522

534

468

498

540

492

78

66

132

102

60

108

337

347

378

340

384

383

185

187

90

158

156

109

Designing continuous improvement projects

Based on the methods developed in this chapter, the design of continuous improvement (CI) projects can be carried out following the procedure illustrated in Figure 5.15. As it is shown in this figure, after modeling the production system at hand and model validation (as described in Chapter 3), methods of constrained and/or unconstrained improvability (Chapter 5) can be used to determine possible avenues for system improvement, and the most efficient one is identified (using the performance evaluation techniques of Chapter 4). In this manner, rigorous improvement projects with quantitatively predicted results can be designed. Following their implementation and evaluation on the factory floor, the process must be repeated anew, in a never ending quest for improvement.

5.3

Measurement-based Management of Production Systems

The process of designing continuous improvement projects described above requires mathematical modeling of the production systems at hand. In particular,

5.3. MEASUREMENT-BASED MANAGEMENT

Modeling Validation

Improvement under constraints

Unconstrained improvability (BN identification)

Model−based analysis of the efficacy of the suggested CI projects

Selection of CI project to be implemented

Implementation

Evaluation of the efficacy of CI project by factory floor measurements

Figure 5.15: Procedure for designing continuous improvement projects

191

192

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

it requires a relatively detailed block-diagram of the system and identification of machine and buffer parameters. This information may be difficult to obtain on the factory floor and, more importantly, maintain on a daily basis. Therefore, a simpler method is desirable to exercise daily managerial duties. The bottleneck identification technique of Section 5.2 leads to such a simpler method. It is illustrated in Figure 5.16 and is referred to as Measurement-based Management (MBM) of production systems. Simplified Block− diagram

Measurement of blockages and starvations

Arrow−based Identification of BNs

Managerial decisions

Figure 5.16: Procedure for Measurement-based Management MBM consists of the following: • Simplified block-diagram. Every manager must maintain a simplified blockdiagram of his/her production system. This diagram should include all major operations and their interconnections but may omit conveyors, buffers and other elements of material handling. For example, an automotive assembly plant manager may have the simplified block-diagram as shown in Figure 5.17. For a paint shop manager, the simplified blockdiagram may be as the one of Figure 5.18. Body shop

Paint shop

General assembly

Figure 5.17: Simplified block diagram of automotive assembly plant

5.3. MEASUREMENT-BASED MANAGEMENT Apply primer

Seal

Clean

193

Apply paint

Finesse

Figure 5.18: Simplified block diagram of automotive paint shop • Measurements of blockages and starvation for each block of the simplified block-diagram. These measurements must be carried out on a continuous basis, either automatically or manually – whichever method is available. Then, estimates of the frequencies of blockages and starvations can be computed as BLi

=

STi

=

Time of blockages of operation i , Total time of observation Time of starvations of operation i . Total time of observation

In practice, these calculations must be carried out on a daily or weekly basis, depending on the scale of the system. For instance, a plant manager may have these data on a weekly basis, while a paint shop manager on a daily basis or even for each shift separately. • Identification of bottlenecks. Using the measured data and the method of Subsection 5.2.3, the bottleneck block must be identified. This is illustrated in Figures 5.19 and 5.20 for the assembly plant and paint shop, respectively. BN Body shop

Paint shop

General assembly

ST i :

0

xxxx

xxxx

BLi :

xxxx

xxxx

0

Figure 5.19: Illustration of bottleneck identification for automotive assembly plant • Managerial decisions. Using the information derived above, managers should develop and implement actions, which would improve the performance of the bottleneck block. Similar to a physician, who cannot treat a patient without taking vital signs, no production system should be “treated” without measuring its “vital signs.” It is shown in this chapter that: The “vital signs” that characterize a production system as a whole are blockages and starvations.

194

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES BN Clean

Seal

Apply primer

Apply paint

Finesse

ST i :

0

xxxx

xxxx

xxxx

xxxx

BL i :

xxxx

xxxx

xxxx

xxxx

0

Figure 5.20: Illustration of bottleneck identification for automotive paint shop

5.4

Case Studies

5.4.1

Automotive ignition coil processing system

The mathematical model of this system is constructed in Subsection 3.10.1 and validated in Subsection 4.4.1. Below, we analyze the possibilities of its performance improvement using the methods developed in this chapter. Based on the model for Period 1 and expressions (4.39), (4.40), we calculate the probabilities of blockages and starvations of all machines in the system. The results are shown in Figure 5.21. Arranging the arrows according to Arrow Assignment Rule 5.1, and using Bottleneck Indicator 5.1, we find that the BNm and BN-b are m9−10 and b9−10 . Increasing the capacity of the BN-b by 1 d d leads to P R = 0.8475 and T P = 476.7 parts/hour. In addition, increasing the BN−m BN−b m1

^

ST

^

i

0

BLi 0.0956

b1

m2

b2

m3

b3

m

4

b4

m5

b5

m

6

b

6

m7

b7

m 9−10 b 9−10 m 11

b 11 m13 b 13

m14

0.1464

0.0570

0.0475

0.0382

0.0200

0.0193

0.0406

0.0333

0.1255

0.1556

0.1205

0.1305

0.1405

0.1525

0.1283

0.1322

0.0344

0.0073

0.0206

0.0077

b 14 m 15

0.1624 0.0034

b 15 m 16

0.1675 0

Figure 5.21: Bottleneck identification in coil processing system (Period 1) d d efficiencies of m9−10 by 10% leads to P R = 0.8976 and T P = 505 parts/hour. After these improvements, the bottleneck shifts to m6 and m1 with m6 being the primary bottleneck machine and b5 the bottleneck buffer (Figure 5.22). d d Increasing the capacity of b5 by 1, we obtain P R = 0.9092 and T P = 511.4 parts/hour. Thus, these two steps of improvement result in significant recovery of losses in the ignition coil processing system. A similar conclusion is obtained using the model for Period 2. Using the B-exp transformation of Subsection 3.9.4, we obtain the transformed machine average uptime and buffer capacity as follows: tr = 123.11 min, Tup,9−10

N5tr = 17,

N7tr = 10.

5.5. SUMMARY

195 BN−b PBN−m

m1

^

STi

0

^

BLi 0.0218

b1

m2

b2

m

b3

3

m

4

b4

m5

b5

m

6

b

6

m7

b7

m 9−10b 9−10m 11

b 11 m13 b 13

m14

0.0714

0.0676

0.0640

0.0581

0.0425

0.0427

0.0681

0.0387

0.0471

0.0784

0.0307

0.0366

0.0436

0.0551

0.0299

0.0270

0.0446

0.0117

0.0225

S i 0.0458

b 14 m 15

0.0084

b 15 m 16

0.0876 0.0040

0.0933 0

0.0506

Figure 5.22: Bottleneck identification in improved coil processing system (Period 1)

5.4.2

Automotive paint shop production system

The mathematical model of this system is constructed in Subsection 3.10.2 and validated in Subsection 4.4.2. Below, we analyze its performance improvement. Based on the model for Month 1 and expressions (4.39) and (4.40), we calculate the probabilities of blockages and starvations of all machines. The results are shown in Figure 5.23. Using Bottleneck Indicator 5.1, we find that the BN-m is m3 . BN−m m3 b 3

^

STi

0

^

BLi 0.0020

m4

b4

m5

b5

m6

b6

m7

b7

m8

0.0973

0.1376

0.1412

0.1240

0.1192

0

0

0

0

0

Figure 5.23: Bottleneck identification for paint shop system (Month 1) The main reason for m3 to be the bottleneck is starvation by empty carriers. Assuming the empty carriers are always available so that the starvation probad d R = 0.9411 and T P = 59.29 jobs/hour. bility Pst can be eliminated, we obtain P Again, since all buffers are large, machine m3 is the system bottleneck due to its relatively low reliability (see Figure 5.24). Increasing the efficiency of m3 by d d 4% leads to P R = 0.9558 and T P = 60.22 jobs/hour, and machine m4 becomes the new bottleneck (Figure 5.25). Thus, these improvements result in almost complete recovery of losses in the paint shop system. Similar conclusions are obtained using the models for Months 2 - 5.

5.5

Summary

• Production systems can be improved in a constrained or unconstrained scenario. In the constrained case, the system is improvable if its work

196

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES BN−m m3 b 3

^

STi

0

^

BLi 0.0132

m4

b4

m5

b5

m6

b6

m7

b7

m8

0.0149

0.0552

0.0558

0.0414

0.0368

0

0

0

0.0002

0

Figure 5.24: Bottleneck identification for paint shop system without starvation by empty carriers (Month 1) m3

STi

0

BLi 0.0367

b3

BN−m m4 b 4

m5

b5

m6

b6

m7

b7

m8

0.0002

0.0405

0.0441

0.0262

0.0221

0

0

0

0.0007

0

Figure 5.25: Bottleneck identification for improved paint shop system (Month 1)

• •

•

•

• •

force (WF) and/or buffer capacity (BC) can be re-allocated among various operations so that the production rate is increased. In the unconstrained case, improvement involves identifying the machine and/or buffer that impedes the system performance in the strongest manner, followed by improving either this machine, or this buffer, or both. Production lines with Bernoulli machines are unimprovable with respect to WF re-allocation if each buffer is, on average, close to being half full. Production lines with Bernoulli machines are unimprovable with respect to WF and BC re-allocation simultaneously if all buffers are of equal capacity and, on average, are close to being half full. Production lines with Bernoulli machines are unimprovable with respect to BC re-allocation if the average occupancy of buffer i is close to the average availability of buffer i + 1, i = 1, . . . , M − 2. A machine is the bottleneck machine (BN-m) of a Bernoulli line if increasing its efficiency has the largest effect on the production rate of the line. A buffer is the bottleneck buffer (BN-b) of a Bernoulli line if increasing its capacity has the largest effect on the production rate of the line. Both BN-m and BN-b can be identified during normal system operation by measuring machine blockages and starvations and using Bottleneck Indicator 5.1.

5.6. PROBLEMS

197

• The buffering of a production line is potent if the BN-m is indeed the machine with the smallest efficiency. • Identifying the BN-m can be used as a basis for Measurement-based Management of production systems.

5.6

Problems

Problem 5.1 Consider a five-machine line with Bernoulli machines. Assume N1 = N4 = 1, N2 = N3 = 3 and the product of all pi ’s of the machines is (0.9)5 . (a) Design an unimprovable (i.e., optimal) system with respect to WF. (b) Does the inverted bowl phenomenon take place? Explain why it does or does not. Problem 5.2 Consider the same system as in Problem 5.1. (a) Using WF-Continuous Improvement Procedure 5.1, obtain an unimprovable system. (b) Does it coincide with the one obtained in Problem 5.1? Explain why it does or does not. d (c) Calculate P R of the design obtained and compare it with that ensured by the design of Problem 5.1. Problem 5.3 Consider again a five-machine production line where the sum of Ni ’s is 8 and the product of pi ’s is (0.9)5 . (a) Design a system that is unimprovable with respect to both WF and BC simultaneously. (b) Compare it with the designs obtained in Problems 5.1 and 5.2 and comment on the differences in structures and production rates of each design. Problem 5.4 Consider a four-machine Bernoulli line, where the sum of Ni ’s is 5. (a) Assume each pi is 0.8. Using BC-Continuous Improvement Procedure 5.1, find the unimprovable allocation of Ni ’s. Comment on the shape of this allocation as a function of i. (b) Assume now that p1 = p3 = 0.7 and p2 = p4 = 0.9. Again, using BCContinuous Improvement Procedure 5.1, determine the unimprovable allocation of Ni ’s and compare it with the one obtained in (a). Comment on the reason for the differences. Problem 5.5 Consider the Bernoulli model of the production system analyzed in Problem 3.3.

198

CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES

(a) Determine if this system is WF-improvable. If so, calculate the unimprovable work allocation and the resulting production rate. (b) Determine if this system is BC-improvable. If so, calculate the unimprovable buffer capacity allocation and the resulting production rate. (c) Calculate the simultaneously unimprovable WF and BC allocations and the resulting production rate. (d) Compare the production rates obtained among themselves and with that obtained in Problem 4.9 for the original system. (e) Which, if any, of the above improvement projects would you recommend for implementation? Problem 5.6 Consider the Bernoulli model constructed in Problem 3.4 and analyzed in Problem 4.11. Repeat steps (a)-(e) of Problem 5.5. Problem 5.7 Consider a five-machine serial line with Bernoulli machines. Assume Ni = 2, i = 1, . . . , 4, and pi ’s are as follows: [0.9, 0.7, 0.9, 0.9, 0.7]. (1) Determine the BN-m and BN-b. (2) Change the buffer capacities around the BN-m so that the bottleneck moves to another machine. What is the new BN-m? Why did the bottleneck move? (3) What is the BN-m when all buffers are infinite? Problem 5.8 Consider the production system analyzed in Problem 5.5. (a) Identify its BN-m (or PBN-m) and BN-b and determine if the buffering is potent. (b) Based on this information, design the best, from your point of view, continuous improvement project that results in 10% increase of the production rate of the system (as compared with the original one). (c) Using the B-exp transformation of Chapter 3, return to the exponential description and formulate measures, which would have to be carried out in order to implement this continuous improvement project. Problem 5.9 Repeat steps (a)-(c) of Problem 5.8 for the system considered in Problem 5.6. Problem 5.10 Derive a WF-Improvability Indicator for Bernoulli lines with the symmetric blocking convention. Problem 5.11 Derive a BC-Improvability Indicator for Bernoulli lines with the symmetric blocking convention. Problem 5.12 Derive a Bottleneck Indicator for identifying BN-m and BNb in Bernoulli lines with the symmetric blocking convention.

5.7. ANNOTATED BIBLIOGRAPHY

5.7

199

Annotated Bibliography

The notion of improvability of serial lines with Bernoulli machines has been introduced in [5.1] D.A. Jacobs and S.M. Meerkov, “A System-theoretic Property of Serial Production Lines: Improvability,” International Journal of System Science, vol. 26, pp. 95-137, 1995. A detailed study of improvability under constraints has been carried out in [5.2] D.A. Jacobs and S.M. Meerkov, “Mathematical Theory of Improvability for Production Systems,” Mathematical Problems in Engineering, vol. 1, pp. 95-137, 1995. The bowl phenomenon was discovered in [5.3] F.S. Hillier and R.W. Bolling, “The Effect of Some Design Factors on the Efficiency of Production Lines with Variable Operation Times,” Journal of Industrial Engineering, vol. 27 , pp. 351-358, 1966. The Theory of Constraints is described in a “manufacturing novel:” [5.4] E.M. Goldratt and J. Cox, The Goal, North River Press Inc, Croton-onHudson, NY, 1984, see also [5.5] E.M. Goldratt and R.E. Fox, The Race, North River Press Inc, Crotonon-Hudson, NY, 1986. These are simple and useful readings; they provide a good foundation for qualitative understanding of production systems. A drawback is that they are based just on common sense, without analytical investigations. Is it possible to design, say, Boeing 777, just using common sense? Equally it is impossible to design and manage a production system using only common sense; quantitative methods are necessary. The definitions of bottleneck machine (5.30) and bottleneck buffer (5.32) have been introduced and investigated in [5.6] C.-T. Kuo, J.-T. Lim and S. M. Meerkov, “Bottlenecks in Serial Production Lines: A System-theoretic Approach,” Mathematical Problems in Engineering, vol. 2, pp. 233-276, 1996. More details can be found in [5.7] C.-T. Kuo, Bottlenecks in Production Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1996.

Chapter 6

Design of Lean Bernoulli Lines Motivation: Designers of manufacturing systems select machines for production lines based on their technological characteristics. Material handling devices are selected based on the nature of parts produced, available space, type of production, etc. Their capacity, in their function as in-process buffers, is typically selected as small as possible, i.e., lean. But how lean can lean buffers be? In other words, what is the smallest buffer capacity, which is necessary and sufficient to ensure the desired production rate of a system? This is the question addressed in this chapter. Overview: Closed formulas for lean buffering in systems with identical Bernoulli machines are derived. These formulas are exact for two- and threemachine lines and approximate for longer lines. For the case of non-identical Bernoulli machines, both closed-form expressions and recursive approaches are developed.

6.1

Parametrization and Problem Formulation

Consider a serial production line with Bernoulli machines defined by conventions (a)-(e) of Subsection 4.2.1. Introduce the following notions: Line efficiency (E) – production rate of the line, P R, in units of the largest possible production rate of the system. As it is clear from Chapter 4, the largest production rate is obtained when all buffers are infinite. Denote this production rate as P R∞ . Then, the line efficiency can be expressed as E=

PR PR . = P R∞ min{p1 , . . . , pM }

Obviously, 0 < E < 1. 201

(6.1)

202

CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES Lean buffer capacity (LBC) – the sequence N1,E , . . . , NM −1,E

(6.2)

PM −1 such that the desired line efficiency E is achieved while i=1 Ni,E is minimized. In other words, LBC is the buffer capacity, which is necessary and sufficient to obtain the desired P R, quantified by E. The problem addressed in this chapter is to develop analytical methods for calculating LBC as a function of machine efficiencies pi , i = 1, . . . , M , line efficiency E, and the number of machines in the system M . For the case of identical machines, i.e., pi =: p, i = 1, . . . , M , this is carried out in Section 6.2, while the case of non-identical machines, pi 6= pj , i, j = 1, . . . , M , is addressed in Section 6.3.

6.2

Lean Buffering in Bernoulli Lines with Identical Machines

In this section, we assume that all machines have identical efficiency, pi =: p,

i = 1, . . . , M,

(6.3)

and, in addition, all buffers are of identical capacity, Ni =: N,

i = 1, . . . , M − 1.

(6.4)

Assumption (6.4) is introduced in order to obtain a compact representation of the results. It should be pointed out that a more efficient buffer capacity allocation in systems satisfying (6.3) is the inverted bowl pattern (see Chapter 5). However, this leads to just a small improvement of the production rate in comparison to the uniform allocation (6.4) (typically, within 1%) and, therefore, is not considered here.

6.2.1

Two-machine lines

In the case of two identical machines, expression (4.19) for the production rate becomes P R = p[1 − Q(p, N )], (6.5) where, as it follows from (4.14), Q(p, N ) =

1−p . N +1−p

(6.6)

Thus, using (6.1), we obtain P R = P R∞ E = p[1 − Q(p, NE )],

(6.7)

6.2. LEAN BUFFERING FOR IDENTICAL MACHINES

203

where NE is the buffer capacity necessary and sufficient to ensure E. Taking into account that P R∞ = p, (6.6) and (6.7) result in the following equation: E =1−

1−p . NE + 1 − p

(6.8)

Solving for NE and taking into account that NE is an integer, we obtain Theorem 6.1 The lean buffer capacity in Bernoulli lines defined by assumptions (a)-(e) of Subsection 4.2.1 with M = 2 and p1 = p2 =: p is given by ' & E(1 − p) , (6.9) NE (M = 2) = 1−E where dxe denotes the smallest integer not less than or equal to x. Note that according to (6.9), NE cannot be less than 1. Under the blocked before service assumption, buffering NE = 1 implies that the machine itself stores a part being processed and no additional buffering between the machines is required. This can be interpreted as Just-in-Time (JIT) operation. Figures 6.1(a) and 6.2(a) illustrate the behavior of the lean buffer capacity as a function of machine efficiency p and line efficiency E, respectively. From these figures and expression (6.9), we observe:

(a) M = 2

(b) M = 3

(c) M = 10

Figure 6.1: Lean buffering as a function of machine efficiency

• For each E, LBC is a monotonically decreasing function of p, with a practically constant slope. • For each p, LBC is a monotonically increasing function of E, exhibiting a hyperbolic behavior in 1 − E. • JIT operation is acceptable only if p’s are sufficiently large. For instance, if the desired line efficiency is 0.85, JIT can be used only if p > 0.83, while for E = 0.95, p must be larger than 0.95. • In a practical range of p’s, e.g., 0.7 < p < 0.98, relatively small buffers are required to achieve a large E. For instance, N0.95 = 6 if p = 0.7; if p = 0.9, N0.95 = 2.

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

(a) M = 2

(b) M = 3

(c) M = 10

Figure 6.2: Lean buffering as a function of line efficiency

6.2.2

Three-machine lines

For a three-machine line, using the aggregation procedure (4.30), the following can be derived: Theorem 6.2 The lean buffer capacity in Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with M = 3 and p1 = p2 = p3 =: p is given by  © √E ª  ln 1− 1−E  (6.10) NE (M = 3) =   © (1−p)√E ª  . √   ln 1−p E

Proof: See Section 20.1. The behavior of this NE is illustrated in Figures 6.1(b) and 6.2(b). Obviously, for most values of p, the lean buffer capacity is increased, as compared with the case of M = 2, and the range of p’s where JIT is possible is decreased. For instance, if p = 0.95, JIT is acceptable for E < 0.91, while for p = 0.85 it is acceptable for E < 0.77.

6.2.3

M > 3-machine lines

Theorem 6.3 The lean buffer capacity in Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with M > 3 and pi =: p, i = 1, . . . , M , is given by  © 1−E−Q(pf , pb , N ª M −2 ) M −1 M −2 ln (1−E)(1−Q(p f b   M −2 , pM −1 , NM −2 )) (6.11) NE (M > 3) =  © , (1−p)(1−Q(pfM −2 , pbM −1 , NM −2 )) ª   ln   1−p(1−Q(pf , pb ,N )) M −2

M −1

M −2

where pfM −2 and pbM −1 are the steady states of aggregation procedure (4.30) and function Q is defined by (4.14). Proof: See Section 20.1.

6.2. LEAN BUFFERING FOR IDENTICAL MACHINES

205

Although a closed-form expression for Q cannot been derived (since N ’s are unknown and, therefore, pfi and pbi cannot be calculated), its estimate can be given as follows: ) ( µ ¶ 1 M −1 − p M −3 M/4 M −3 M/4 M −2 1 1 E b = 1−E 2 [1+( M −1 ) ] + E 2 [1+( M −1 ) ] −E ( M −1 ) exp − Q . (1 − E)(1/E)2E (6.12) Thus, an estimate of LBC for M > 3 is defined as  © b ª 1−E−Q ln (1−E)(1− b Q) bE (M > 3) =  , (6.13) N  © (1−p)(1−Q) b ª   ln b 1−p(1−Q)

b is given in (6.12). where Q The accuracy of estimate (6.12), (6.13) has been evaluated numerically by calculating the exact value of NE (using aggregation procedure (4.30) to evaluate bE as follows: Q) and comparing it with N ∆E =

bE − NE N · 100%. NE

(6.14)

The values of ∆E have been calculated for p ∈ {0.85, 0.9, 0.95}, M ∈ {5, 10, 15, 20, 25, 30}, and E ∈ {0.85, 0.9, 0.95}. It turned out that ∆E = 0 for all combinations of these parameters except when [p = 0.85, M = 5, E = 0.85], where it is 50%. (Such a large error is due to the integer nature of NE and bE .) Thus, we conclude that in most cases N bE provides a sufficiently accurate N estimate of NE . bE for M = 10 is illustrated in Figures 6.1(c) and 6.2(c). The behavior of N Clearly, the buffer capacity is increased as compared to M = 3, and JIT operation becomes unacceptable for all values of p and E considered. Using (6.12), (6.13), the behavior of lean buffering as a function of M can be investigated. This is illustrated in Figure 6.3. Interestingly, and to a certain bE is constant for all M ≥ 10. This implies that the lean degree unexpectedly, N buffering, appropriate for lines with 10 machines, is also appropriate for lines with any larger number of machines. Based on this observation, the following can be formulated: Rule-of-thumb for selecting lean buffering: In Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with M ≥ 10, the capacity of the lean buffering can be selected as shown in Table 6.1. Interestingly, the diagonal elements of this “matrix” are all identical, implying that buffer capacity 3 is necessary and sufficient to ensure line efficiency E if p = E. The lower triangle of this matrix (p > E) has all elements smaller than 3 and the upper (p < E) – larger than 3.

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

(a) p = 0.85

(b) p = 0.90

(c) p = 0.95

bE as a function of the number of machines in the Figure 6.3: Lean buffering N system

Table 6.1: Rule-of-thumb for selecting lean buffer capacity in Bernoulli lines with M ≥ 10 p = 0.85 p = 0.90 p = 0.95

E = 0.85 3 2 2

E = 0.90 4 3 2

E = 0.95 7 5 3

6.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

6.3

207

Lean Buffering in Serial Lines with Nonidentical Bernoulli Machines

6.3.1

Two-machine lines

In the case of two-machine lines with non-identical machines, as it follows from (6.1) and (4.19), the equation that defines NE becomes " # (1 − p1 )(1 − α) , (6.15) P R = P R∞ E = p2 [1 − Q(p1 , p2 , NE )] = p2 1 − 1 − pp12 αNE where P R∞ = min(p1 , p2 ) and α=

(6.16)

p1 (1 − p2 ) . p2 (1 − p1 )

(6.17)

Solving (6.15) for NE , we obtain: Theorem 6.4 The lean buffer capacity in Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with M = 2 is given by ¸¾   ½ · p2 p1 −P R∞ E   ln p1 p2 −P R∞ E , (6.18) NE (p1 , p2 ) =    ln α     where P R∞ and α are defined by (6.16) and (6.17), respectively.

5

4

4

5

p2 = 0.85 p2 = 0.75

4

3

N

3

6

p2 = 0.95

E

6

5

E

6

N

N

E

Figure 6.4 illustrates the behavior of NE as a function of p1 for various values of p2 and E, while Figure 6.5 shows NE as a function of E for various p1 and p2 . From these figures, we conclude:

3

2

2

2

1

1

1

0

0

0

0.2

0.4

0.6

0.8

p1

(a) E = 0.85

1

0

0.2

0.4

0.6

0.8

p1

(b) E = 0.90

1

0

0

0.2

0.4

0.6

0.8

1

p1

(c) E = 0.95

Figure 6.4: Lean buffering in two-machine lines as a function of the first machine efficiency

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

10

10

8

8

6

6

10

p2 = 0.95 p2 = 0.85

8

E

N

E

6

N

N

E

p2 = 0.75

4

4

4

2

2

2

0 0.8

0 0.8

0.85

0.9 E

0.95

(a) p1 = 0.75

0.85

0.9 E

0.95

(b) p1 = 0.85

0 0.8

0.85

0.9 E

0.95

(c) p1 = 0.95

Figure 6.5: Lean buffering in two-machine lines as a function of line efficiency

• For p2 sufficiently large, JIT operation is acceptable for all values of p1 and E. • For small p2 , JIT is acceptable only when p1 is sufficiently large. For instance, if p2 = 0.75, JIT represents LBC only if p1 > 0.88 for E = 0.9 and p1 > 0.94 for E > 0.95. • The maximum of NE tends to take place when p1 = p2 . Intuitively, it is expected that the lean buffering in a line {p1 , p2 } is the same as in the reversed line, i.e., {p2 , p1 }. It turns out that this is indeed true as stated below. Theorem 6.5 Lean buffer capacity has the property of reversibility, i.e., NE (p1 , p2 ) = NE (p2 , p1 ).

(6.19)

Proof: See Section 20.1.

6.3.2

M > 2-machine lines

Exact formulas for LBC in the case of M > 2 are all but impossible to derive. Therefore, we limit our attention to estimates of Ni,E . These estimates are obtained based both on closed formulas (6.9)-(6.12), (6.18) and on recursive calculations. Each of these approaches is described below. Closed formula approaches: The following four methods have been investigated: I. Local pair-wise approach. Consider every pair of consecutive machines, mi and mi+1 , i = 1, . . . , M − 1, and select LBC using formula (6.18). This results in the sequence of buffer capacities denoted as I I , . . . , NM N1,E −1,E .

6.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

209

II. Global pair-wise approach. It is based on applying formula (6.18) to all possible pairs of machines (not necessarily consecutive) and then selecting the capacity of each buffer equal to the largest buffer obtained by this procedure. Clearly, this results in buffers of equal capacity, which is denoted as NEII . III. Local upper bound approach. Consider all pairs of consecutive machines, mi and mi+1 , i = 1, . . . , M − 1, substitute each of them by a two-machine line with identical machines defined by pbi := min{pi , pi+1 }, i = 1, . . . , M − 1, and select LBC using formula (6.9) with p = pbi . This results in the sequence of buffer capacities III III N1,E , . . . , NM −1,E .

IV. Global upper bound approach. Instead of the original line, consider a line with all identical machines specified by pb := min{p1 , p2 , . . . , pM } and select the buffer capacity, denoted as NEIV , using expressions (6.12) and (6.13). Due to the monotonicity of P R with respect to machine efficiency and buffer capacity (see Chapter 4), this approach provides an upper bound of LBC: Ni,E ≤ NEIV , i = 1, . . . , M − 1. If the desired line efficiency for two-machine lines, involved in approaches I III, were selected as E, the resulting efficiency of the M -machine line would be certainly less than E. To avoid this, the efficiency, E 0 , of each of the two-machine lines is calculated as follows: For a given M -machine line, find the buffer capacity using approach IV. Then consider a two-machine line with identical machines, where each machine is defined by pb = min{p1 , . . . , pM }, and the buffer with the capacity as found above. Finally, calculate the production rate and the efficiency of this two-machine line and use it as E 0 in approaches I - III. To analyze the performance of approaches I - IV, we consider 100, 000 lines formed by selecting M and pi randomly and equiprobably from the sets M ∈ {4, 5, . . . , 30},

(6.20)

0.70 ≤ p ≤ 0.97.

(6.21)

The desired efficiency for each of these lines is also selected randomly and equiprobably from the set 0.80 ≤ E ≤ 0.98. (6.22)

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

For each k-th line, thus formed, we calculate the vector of buffer capacities   j N1,k   j  N2,k  (6.23) Njk =   , k = 1, . . . , 100, 000, j = I, II, III, IV, ...   j NM −1,k j using the four approaches introduced above. The subscripts of Ni,k represent i-th buffer, i = 1, . . . , Mk − 1, of the k-th line, k = 1, 2, . . . , 100, 000; the superscript j = I, II, III, IV represents the approach used for this calculation. In addition, we calculate the production rate, P Rkj , and the efficiency, Ekj , using expressions (4.30)-(4.36) and (6.1), respectively. The efficacy of approaches I - IV is characterized by the following two metrics:

1. The average buffer capacity per machine among all systems analyzed: j = Nave

K 1 X j Nk K

(6.24)

k=1

where K = 100, 000 and Nkj =

M k −1 X 1 Nj . Mk − 1 i=1 i,k

2. The percent of systems whose Ekj turns out to be less than the desired efficiency Ek : K 1 X ∆j = Sg(Ek − Ekj ) · 100%, (6.25) K k=1

where K = 100, 000 and ( Sg(x) =

1, x > 0 0, x ≤ 0.

The results are given in Table 6.2. Clearly, approach I leads to the smallest average buffer capacity but, unfortunately, almost always results in line efficiency less than desired. Thus, a “local” selection of LBC (i.e., based on the two machines surrounding the buffer) is, practically always, unacceptable. Approaches II and III provide line efficiency less than desired in only a small fraction of cases and result in average buffer capacity 2 - 3 times larger than approach I. Approach IV, as expected, always guarantees the desired performance but requires the largest buffering. To further differentiate between the four approaches, consider their performance as a function of M . To accomplish this, we formed 1000 lines for each M ∈ {4, 6, 8, 10, 15, 20, 25, 30, 80} by selecting pi ’s and E’s randomly and

6.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

211

Table 6.2: Performance characteristics of approaches I - IV Approach j Nave ∆j

I 2.0 97.3

II 6.2 0.1

III 5.3 0.1

IV 7.2 0.0

equiprobably from sets (6.21) and (6.22), respectively. For each of these lines, we calculated buffer capacities using approaches I - IV and evaluated the performance metrics (6.24) and (6.25). The results are shown in Table 6.3. Examining these data, we conclude the following: • The local pair-wise approach, in most cases, leads to a lower line efficiency than desired. • The global pair-wise approach results in good performance from the point of view of both Nave and ∆. For M ≤ 10, it outperforms approach III from the point of view of Nave . However, it is quite sensitive to M : Nave increases substantially with M . • The local upper bound approach is less sensitive to M and outperforms approach II for M > 10. • The global upper bound approach substantially overestimates the LBC. Based on the above, it is recommended to use the global pair-wise approach in systems with M ≤ 10 and local upper bound approach in systems with M > 10. Table 6.3: Effect of M on the performance of approaches I - IV M I Nave II Nave III Nave IV Nave ∆I ∆II ∆III ∆IV

4 1.7 3.0 4.2 5.0 88.7 3.1 0.0 0.0

6 1.9 4.2 5.0 6.3 92.8 0.3 0.1 0.0

8 2.0 4.8 5.2 6.7 95.6 0.1 0.1 0.0

10 2.0 5.2 5.2 6.8 97.1 0.0 0.1 0.0

15 2.0 5.9 5.2 7.0 98.4 0.0 0.1 0.0

20 2.1 6.4 5.4 7.4 98.0 0.0 0.0 0.0

25 2.0 6.5 5.3 7.3 99.2 0.0 0.0 0.0

30 2.1 7.1 5.7 7.9 99.3 0.0 0.0 0.0

80 2.1 7.6 5.6 7.9 100.0 0.0 0.3 0.0

Recursive approaches: The following two recursive methods have been investigated: V. Full search approach. Start from all buffers of capacity 1. Increase the capacity of the first buffer by 1 and, using the aggregation procedure (4.30), calculate the production rate of the system. Return the first buffer capacity to its initial value, increase the second buffer capacity

212

CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES by 1, and calculate the resulting production rate. Repeat the same procedure for all buffers, determine the buffer that leads to the largest production rate, and permanently increase its capacity by 1. Repeat the same procedure until the desired line efficiency is reached. This results in the sequence of buffer capacities V V , . . . , NM N1,E −1,E .

VI. Bottleneck-based approach. Consider a production line with buffer capacity calculated according to approach I but rounded down in formula (6.18) rather than up. Although being relatively small, this buffering often leads, as it follows from Tables 6.2 and 6.3, to line efficiency less than desired. Therefore, to improve the line efficiency, increase the buffering according to the following procedure: Using Bottleneck Indicator 5.1, identify the bottleneck machine (or, when applicable, primary bottleneck machine) and increase the capacity of both buffers surrounding this machine by 1. Repeat this procedure until the desired line efficiency is reached. This results in the sequence of buffer capacities denoted as VI VI , . . . , NM N1,E −1,E .

Clearly, approach V gives a smaller buffer capacity than approach VI. However, the latter might require a shorter computation time. Therefore, in order to compare V and VI, the computation time should be taken into account. This additional performance metric is defined as the total computer time necessary to carry out the calculation: tj = tjend − tjstart ,

(6.26)

where tjstart and tjend , j ∈ {I, . . . , V I}, are the times (in seconds) of the beginning and the end of the computation. Based on metrics (6.24), (6.25), and (6.26), we compared approaches I-VI using the production systems generated by selecting pi ’s randomly and equiprobably from set (6.21). The results are shown in Tables 6.4-6.6. Specifically, Tables 6.4 and 6.5 present the results obtained using 5000 randomly generated 5- and 10-machine lines, respectively, while Table 6.6 is based on the analysis of 2000 randomly generated 15-machine lines. Examining these data, we conclude the following: • Full search approach, as expected, results in the smallest buffer capacity and the longest calculation time. • Approaches II - IV, being based on closed-form expressions, are much faster than V (up to two orders of magnitude for long lines) but lead to average buffering 2 - 3 times larger than that of V. • Approach VI provides a good tradeoff between the calculation time and buffer capacity. In long lines, it is about 20 times faster than V and results

6.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

213

in average buffering almost the same as V (about 10% difference). Also, it is about 6 times slower than I - IV but gives buffering 2 - 3 times smaller than II - IV.

Table 6.4: Performance characteristics of approaches I - VI in five-machine lines

Approaches j Nave ∆j tj

(a) I 1.5 27.8 8

E = 0.80 II III 2.2 2.4 0.0 0.0 8 8

IV 2.9 0.2 8

V 1.4 0.0 59

VI 1.5 0.0 28

Approaches j Nave ∆j tj

(b) I 1.7 36.0 8

E = 0.85 II III 2.7 3.0 0.0 0.0 8 8

IV 3.6 0.2 8

V 1.7 0.0 91

VI 1.8 0.0 38

Approaches j Nave ∆j tj

(c) I 2.2 33.8 6

E = 0.90 II III 3.6 4.2 0.0 0.0 6 6

IV 5.0 0.0 6

V 2.0 0.0 91

VI 2.3 0.0 27

Approaches j Nave ∆j tj

(d) I 3.2 25.5 6

E = 0.95 II III 6.0 7.9 0.0 0.0 6 6

IV 9.5 0.0 6

V 2.8 0.0 150

VI 3.3 0.0 29

Thus, if a recursive approach is to be used, the bottleneck-based one is recommended. Illustrative examples: To illustrate particular cases of lean buffering designed using approaches I - VI, we provide several examples. Consider the four production lines with machines specified in Table 6.7 along with the desired line efficiency. The estimates of LBC for each of these lines, calculated using approaches I - VI, are shown in Tables 6.8 - 6.11, along with resulting line efficiency. These examples clearly indicate that: • Approach VI is almost as good as the full search approach V. • Approach II in most cases outperforms approach III (since M = 5); however, it still leads to buffer capacity 2 - 3 times larger than approach V.

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

Table 6.5: Performance characteristics of approaches I - VI in 10-machine lines

Approaches j Nave ∆j tj

Approaches j Nave ∆j tj

(a) E I 1.8 66.7 15

∈ [0.80, 0.89] II III IV 3.5 3.2 4.1 0.0 0.0 0.0 15 15 15

(b) E ∈ [0.89, 0.98] I II III IV 3.2 8.0 8.2 10.6 48.5 0.0 0.0 0.0 26 26 26 26

V 1.8 0.0 757

VI 1.9 0.0 76

V 2.7 0.0 2339

VI 3.2 0.0 124

Table 6.6: Performance characteristics of approaches I - VI in 15-machine lines

Approaches j Nave ∆j tj

(a) E ∈ [0.80, 0.89] I II III IV 1.8 3.8 3.3 4.2 81.0 0.0 0.0 0.0 18 18 18 18

V 1.8 0.0 2452

VI 2.0 0.0 107

Approaches j Nave ∆j tj

(b) E ∈ [0.89, 0.98] I II III IV 3.2 9.0 8.2 10.9 57.6 0.0 0.0 0.0 24 24 24 24

V 2.6 0.0 5837

VI 3.2 0.0 138

Table 6.7: Machine parameters and desired line efficiencies Line 1 2 3 4

m1 0.78 0.79 0.72 0.77

m2 0.88 0.84 0.85 0.87

m3 0.75 0.85 0.74 0.90

m4 0.91 0.94 0.82 0.90

m5 0.83 0.76 0.84 0.72

E 0.80 0.85 0.90 0.95

6.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

215

Table 6.8: LBC estimates and resulting efficiency for Line 1 (desired E = 0.80) Buffer Desired NiI NiII NiIII NiIV NiV NiV I

bj1

bj2

bj3

bj4

1 2 3 3 1 1

1 2 3 3 2 2

1 2 3 3 2 2

1 2 2 3 1 1

Ej 0.80 0.71 0.90 0.96 0.96 0.83 0.83

Table 6.9: LBC estimates and resulting efficiency for Line 2 (desired E = 0.85) Buffer Desired NiI NiII NiIII NiIV NiV NiV I

bj1

bj2

bj3

bj4

2 3 3 3 1 1

2 3 2 3 2 2

1 3 2 3 2 2

1 3 3 3 1 1

Ej 0.85 0.84 0.99 0.96 0.99 0.85 0.85

Table 6.10: LBC estimates and resulting efficiency for Line 3 (desired E = 0.90) Buffer Desired NiI NiII NiIII NiIV NiV NiV I

bj1

bj2

bj3

bj4

2 4 5 5 2 2

2 4 5 5 3 3

3 4 5 5 2 3

3 4 4 5 2 2

Ej 0.90 0.90 0.97 0.99 0.99 0.90 0.92

Table 6.11: LBC estimates and resulting efficiency for Line 4 (desired E = 0.95) Buffer Desired NiI NiII NiIII NiIV NiV NiV I

bj1

bj2

bj3

bj4

3 5 9 10 2 2

3 5 5 10 2 2

4 5 4 10 2 3

2 5 10 10 2 2

Ej 0.95 0.98 1.00 1.00 1.00 0.96 0.97

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

PSE Toolbox: The six methods of lean buffering calculation for Bernoulli lines are implemented in the Lean Buffer Design function of the toolbox. For a description and illustration of these tools, see Subsections 19.6.1 and 19.6.2.

6.4 6.4.1

Case Studies Automotive ignition coil processing system

The Bernoulli model of this system is given in Figure 3.31 (for Periods 1 and 2). Below, its lean buffering is evaluated using the model for Period 2 (since it corresponds to system conditions during and after this case study). Rule-of-thumb approach: The rule-of-thumb for selecting lean buffering in Bernoulli lines with ten or more identical machines is given in Table 6.1. Although the system at hand consists of non-identical machines, we use this rule to obtain a quick (but, clearly, not tight) upper bound for the lean buffering. To accomplish this, “convert” the coil processing system into a virtual production line with identical Bernoulli machines defined by pb = min{p1 , . . . , p13 } = 0.8825, where pi , i = 1, . . . , 13, is the efficiency of the i-th machine in Figure 3.31(b). Since the cycle time in the original line is selected to ensure the nominal throughput of 593.07 parts/hour, the maximum throughput of the virtual line is 593.07 × 0.8825 = 523.38 parts/hour. Assume that the desired line efficiency E = 0.9 (i.e., the desired throughput is 523.38 × 0.9 = 471.05 parts/hour). Then, taking into account that pb is close to 0.9 and using Table 6.1, we infer that the lean buffer capacity in the virtual line is i = 1, . . . , 12. (6.27) NiBer = 3, Due to the monotonicity of P R with respect to machine and buffer parameters (see Chapter 4), we conclude that (6.27) is an upper bound of lean buffering for the original Bernoulli line. Clearly, this buffering is 2 - 3 times larger than that in the model of Figure 3.31(b). (Note that this configuration gives throughput of 521.35 parts/hour.) Formula-based approach: Since the coil processing line contains more than ten machines, the local upper bound approach (i.e., Approach III of Subsection 6.3.2) is recommended for selecting lean buffering. Using this approach and the machine parameters of Figure 3.31(b), we obtain the following vector of lean buffering estimates for E = 0.9, i.e., 471.05 parts/hour: N Ber = [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]. This buffering ensures a throughput of 517.29 parts/hour, which is much higher than the desired one, i.e., the buffering is not lean.

6.4. CASE STUDIES

217

Recursive approach: Applying the bottleneck-based approach (i.e., approach VI of Subsection 6.3.2) with E = 0.9, we obtain a tighter estimate: N Ber = [1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1]. Finally, using the full search approach, we arrive at a still tighter estimate: N Ber = [1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1], and this buffering leads to the line efficiency E = 0.923, i.e., the throughput of 483.13 parts/hour.

6.4.2

Automotive paint shop production system

The Bernoulli model of this system is given in Tables 3.11 and 3.12 (for months 1-5). To evaluate lean buffering, we average machine parameters over all five monthly periods and obtain the machine parameter values given in Table 6.12. Assuming that empty carriers are managed appropriately so that m3 is never starved (see Chapter 5) and neglecting machines 9 - 11 (since they operate perfectly), we use the model of Table 6.12 to characterize lean buffering for the paint shop. Table 6.12: Five-month averages of machine parameters p3 0.9705

p4 0.9542

p5 0.9905

p6 0.9987

p7 0.9681

p8 0.9707

Rule-of-thumb approach: Convert the system into a virtual line with identical Bernoulli machines defined by pb = min{p3 , . . . , p8 } = 0.954, where pi , i = 3, . . . , 8, is given in Table 6.12. This implies that the maximum throughput is 63 × 0.954 = 60.116 jobs/hour. Assuming that the desired efficiency is E = 0.95 (which ensures the production of 60.116 × 0.95 = 57.110 jobs/hour) we obtain the lean buffering for the virtual system given by Ni = 3,

i = 3, . . . , 7.

This results in throughput of 59.76 jobs/hour. Clearly, this buffering is far below that of Table 3.12, implying that the paint shop system is far from being lean. Recursive approach: Applying the bottleneck-based approach (i.e., approach VI of Subsection 6.3.2) with E = 0.95, we obtain a tighter estimate of lean buffering: N = [2, 2, 1, 1, 1], which leads to throughput of 57.56 jobs/hour. This results in over 90% reduction in buffer capacities.

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CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

6.5

Summary

• Lean buffer capacity (LBC) is the smallest buffering necessary and sufficient to ensure the desired efficiency of a production line. Thus, LBC is the “just-right” rather than “just-in-time” buffering. • Lean buffer capacity in serial lines with identical Bernoulli machines can be evaluated using closed-form expressions (6.9)-(6.13) or the rule-of-thumb given in Table 6.1. • In the case of non-identical machines, a closed formula is available only for two-machine lines (6.18). • For longer lines, estimates of lean buffer capacity can be obtained using either closed-form expressions or recursive calculations. • If closed formulas are used, the global pair-wise approach and the local upper bound approach are recommended when M ≤ 10 and M > 10, respectively. • If recursive calculations are used, the bottleneck-based approach is recommended.

6.6

Problems

Problem 6.1 Consider a Bernoulli serial line with two identical machines. Assume that it operates under the assumption of symmetric blocking described in Problem 4.5. (a) (b) (c) (d)

Derive the formula for lean buffer capacity in such a system. Plot the lean buffer capacity as a function of machine efficiency. Plot the lean buffer capacity as a function of line efficiency. Does the symmetric blocking require larger or smaller lean buffering than that defined by the blocked before service assumption (see Subsection 6.2.1)?

Problem 6.2 Consider the two-machine Bernoulli serial line with defective parts produced as described in Problem 4.4. (a) (b) (c) (d)

Derive the formula for lean buffer capacity in such a system. Plot the lean buffer capacity as a function of machine efficiency. Plot the lean buffer capacity as a function of line efficiency. Compare the resulting graphs with those obtained in Subsection 6.3.1 and comment on the effect of the defectives on the lean buffer capacity.

Problem 6.3 Investigate the precision of the rule-of-thumb given in Table 6.1. Specifically, consider the Bernoulli serial line with 15 identical machines having p = 0.87. Assume that the desired line efficiency is 0.93. (a) Using expressions (6.12) and (6.13), calculate NE .

6.7. ANNOTATED BIBLIOGRAPHY

219

(b) Using the rule-of-thumb, determine an estimate of the lean buffer capacity for this system. (c) Compare the two results and comment on the applicability of the rule-ofthumb for p and E, which are not included in Table 6.1. Problem 6.4 Consider the ignition device production line of Problem 3.3. (a) Using any approach of Section 6.3, design the lean buffering for this system with E = 0.90. (b) Using the same approach as in (a), design the lean buffering for this system with E = 0.97. (c) Compare the two results and recommend which line efficiency should be used for practical implementation. Problem 6.5 Consider the five-machine serial line of Problem 3.4. Assume that the desired line efficiency is 0.97. Using the Bernoulli model of this line: (a) Determine the estimate of the lean buffer capacity using the global upper bound approach of Subsection 6.3.2. (b) Determine the estimate of the lean buffer capacity using the bottleneckbased approach. (c) Compare the two results and state which of these approaches is preferable (from all points of view you find important). Problem 6.6 Consider a production line with p = [0.88, 0.95, 0.9, 0.92]. Select the lean buffering capacity for E = 0.95 using the following approaches: (a) (b) (c) (d)

Global pair-wise approach. Local upper bound approach. Bottleneck-based recursive approach. Full search recursive approach.

Compare the results and comment on the advantages and disadvantages of each method.

6.7

Annotated Bibliography

The quantitative notion of lean buffering has been introduced and analyzed in [6.1] E. Enginarlar, J. Li and S.M. Meerkov, “How Lean Can Lean Buffers Be?” IIE Transactions, vol. 37, pp. 333-342, 2005. The lean buffering in Bernoulli lines, has been investigated in [6.2] A.B. Hu and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Bernoulli Machines,” Mathematical Problems in Engineering, vol. 2006, Article ID 17105, 2006.

220

CHAPTER 6. DESIGN OF LEAN BERNOULLI LINES

It is the basis for the material of this chapter. Additional results on lean buffering (for the exponential and non-exponential reliability models) can be found in [6.3] E. Enginarlar, J. Li and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Non-exponential Machines,” OR Spectrum, vol. 27, pp. 195-219, 2005. [6.4] S.-Y. Chiang, A.B. Hu and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Non-identical Exponential Machines,” IEEE Transactions on Automation Science and Engineering, vol. 5, pp. 298-306, 2008.

Chapter 7

Closed Bernoulli Lines Motivation: Production lines in large volume manufacturing environments often have parts transported from one operation to another on carriers (sometimes referred to as pallets, skids, etc.). Examples of such lines are the automotive ignition processing system and paint shop production system introduced in Chapter 3. Since in these situations the number of parts in the system is bounded by the number of available carriers, these lines are called closed with respect to carriers (or just closed). Such a line with M machines is shown in Figure 7.1, where the empty carrier buffer, b0 , has capacity N0 and the number of carriers in the system is S.

Figure 7.1: Closed serial production line Since in a closed line the first machine can be starved for carriers and the last blocked by b0 , the production rate of the closed line, P Rcl , is, at best, equal to that of the corresponding open line, P Ro . If, however, either N0 or S or both are chosen inappropriately, the closed nature of the line impedes system performance and, as a result, P Rcl can be substantially lower than P Ro . An illustration is given in Figure 7.2, where P Rcl is shown as a function of S for various values of N0 ; P Ro is also indicated by the broken lines. An interpretation of these graphs is as follows: For the system of Figure 7.2(a), the empty carrier buffer capacity N0 = 2 is too small, since P Rcl < P Ro for any S. With N0 = 4, there is a single value of S (specifically, S = 4) that 221

222

CHAPTER 7. CLOSED BERNOULLI LINES

guarantees P Rcl = P Ro . When N0 = 6, the equality P Rcl = P Ro holds on the set S = {4, 5, 6}. Finally, when N0 = 10, the set of “non-impeding” S’s becomes even larger (S = {4, 5, 6, 7, 8, 9, 10}). Clearly, the drop of P Rcl for small and for large values of S is due to starvation of m1 for carriers and blockage of m2 by b0 , respectively, and, as Figure 7.2 shows, P Rcl is practically (however, not exactly) symmetric in S. A similar interpretation can be given for Figure 7.2(b) as well. p =p =0.8, N =3 1

2

p =0.9, N =3

1

i

0.8 0.75

0.8

cl

0.65

0.6

PR

cl

0.7 PR

i

1

N0=2

0.6

0.4

N =4

N =2

0

N =6

0.55 0.5 2

0

0

0.2

N =8 0

N =10 0

4

6

S

8

10

(a) Two-machine line

12

N =14 0

0

5

10

S

15

20

25

(b) Five-machine line

Figure 7.2: Production rate of closed lines as a function of the number of carriers Given the above, a question arises: How should N0 and S be selected so that, on one hand, the closed nature of the line does not impede the open line performance and, on the other hand, N0 and S are sufficiently small so that the closed line is “lean?” This is the question addressed in this chapter. Note that in Chapter 3, we avoided the closed nature of the case studies by assuming that the probabilities of starvation of the first machine and the blockage of the last machine were known from factory floor measurements. These probabilities were used to convert a closed line into an open one by modifying p1 and pM accordingly. In the current chapter, we do not use this simplified approach and develop methods for analysis, continuous improvement, and design directly applicable to closed lines.

Overview: In Section 7.1, the model of the system under consideration and the problems addressed are formulated. In Section 7.2, a method for performance analysis is developed, monotonicity and reversibility properties of P Rcl are analyzed, and conditions are established under which the closed nature of the line does not impede the open line performance. Improvability of closed lines with respect to S and N0 is investigated in Section 7.3, and a method for bottleneck identification is developed in Section 7.4. Section 7.5 addresses the issue of (N0 , S) leanness and in Section 7.6 a case study is described.

7.1. SYSTEM MODEL AND PROBLEM FORMULATION

7.1 7.1.1

223

System Model and Problem Formulation Model

Consider the production system shown in Figure 7.1. Assume that it operates according to assumptions (a)-(e) of Subsection 4.2.1 and, in addition, (f) The parts are transported within the system on carriers; the total number of carries is S and the capacity of the empty carrier buffer, b0 , is N0 . (g) The parts are placed on carriers at the input of machine m1 , and m1 is starved for carriers when b0 is empty. (h) The parts are removed from carriers at the output of machine mM , and mM is blocked when b0 is full and m1 is down or blocked. Due to the blocked before service convention (see assumptions (a) and (h)), the part being processed by mi is viewed as if it were already in bi (or b0 , in the case of mM ). That is why Ni ≥ 1, i = 0, 1, . . . , M − 1.

7.1.2

Problems addressed

In the framework of the model defined by assumptions (a)-(h), this chapter addresses the problems listed below. Performance analysis problem: Given the machine and in-process buffer parameters as well as N0 and S, calculate the production rate, work-in-process, and probability of blockages/starvations of the machines, i.e., evaluate

P Rcl W IPicl BLcl i STicl

= = = =

P Rcl (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S), W IPicl (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S), i = 1, . . . , M − 1, cl BLi (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S), i = 1, . . . , M, cl STi (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S), i = 1, . . . , M.

This is carried out in Section 7.2 along with the study of monotonicity and reversibility properties of P Rcl . In addition, Section 7.2 discusses the issues of performance impediment in closed lines defined as follows: Definition 7.1 A pair (N0 , S) is unimpeding if

P Rcl (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S) = P Ro (p1 , . . . , pM , N1 , . . . , NM −1 ); (7.1) otherwise, (N0 , S) is called impeding.

224

CHAPTER 7. CLOSED BERNOULLI LINES

Improvability problem: Improvability of open lines with Bernoulli machines has been analyzed in Chapter 5. Here, we address improvability issues for closed lines. Specifically, assuming that the machine and in-process buffer parameters are fixed, i.e., pi , i = 1, . . . , M , and Ni , i = 1, . . . , M − 1, are given, consider the function P Rcl (N0 , S). Definition 7.2 A closed line is: • S + -improvable if P Rcl (N0 , S + 1) > P Rcl (N0 , S); • S − -improvable if P Rcl (N0 , S − 1) > P Rcl (N0 , S); • S-unimprovable if P Rcl (N0 , S ∗ ) cannot be increased for any other S, i.e., P Rcl (N0 , S ∗ ) ≥ P Rcl (N0 , S),

∀S 6= S ∗ .

Note that S ∗ may be non-unique, as illustrated in Figure 7.2. Clearly, the unimprovable S ∗ is a function of N0 , and is denoted throughout this chapter as S ∗ = S ∗ (N0 ). Definition 7.3 A closed line is N0 -improvable if P Rcl (N0 + 1, S ∗ (N0 + 1)) > P Rcl (N0 , S ∗ (N0 )); otherwise, it is unimprovable and the pair (N0∗ , S ∗ (N0∗ )) is called (N0 , S)-unimprovable. Criteria for S + -, S − -, and N0 -improvability are given in Section 7.3. Bottleneck identification problem: The bottleneck (BN) machine of a closed line is defined in the same manner as that of an open line (see Definition 5.2), i.e., mi , i ∈ {1, . . . , M }, is the BN if ¯ ¯ ¯ ¯ ¯ ∂P Rcl ¯ ¯ ∂P Rcl ¯ ¯>¯ ¯ ¯ ∀j 6= i. (7.2) ¯ ∂pi ¯ ¯ ∂pj ¯ , Here, the absolute values of the derivatives are used since it is not a priori clear that P Rcl is monotonically increasing with respect to pi ’s. Unfortunately, Bottleneck Indicator 5.1 is not applicable to closed lines. Indeed, while in open lines there always exists at least one machine with no emanating arrows (because ST1 = BLM = 0), in closed lines there may be none. This is illustrated in Figure 7.3, where, in addition to the usual arrows, the arrow between m1 and mM is assigned according to the same rule: if ST1 > BLM , the arrow is directed to the left; ST1 < BLM , it is directed to the right. In this situation, which machine is the BN? The answer to this question is provided in Section 7.4.

7.2. PERFORMANCE ANALYSIS OF CLOSED LINES

225

Figure 7.3: Towards bottleneck identification in closed serial lines The problem of (N0 , S) leanness: unimpeding pair, introduce:

Recalling that (N0∗ , S ∗ (N0∗ )) denotes an

Definition 7.4 An unimpeding pair (N0∗ , S ∗ (N0∗ )) is lean if N0∗ and S ∗ (N0∗ ) are the smallest among all possible unimpeding pairs. The issue of leanness is discussed in Section 7.5.

7.2 7.2.1

Performance Analysis, Monotonicity, Reversibility, and Unimpeding Closed Lines Two-machine lines

Theorem 7.1 The performance characteristics of a closed Bernoulli line defined by assumptions (a)-(h) of Subsection 7.1.1 with M = 2 can be evaluated as follows: P Rcl = p2 [1 − Qcl (p1 , p2 , N1 , N0 , S)],

W IP1cl =

´  ³PS iα(p ,p )i p1 Sα(p1 ,p2 )S 1 2  −  i=1 1−p 1−p 2 2     ·Qcl (p1 , p2 , N1 , N0 , S),  1      PN1 iα(p1 ,p2 )i cl    Q2 (p1 , p2 , N1 , N0 , S), i=1 1−p2

(7.3)

if S ≤ min(N1 , N0 ), if N1 < S ≤ N0 ,

(7.4)   PN0 iα(p1 ,p2 )i cl   S − i=1 1−p2 Q2 (p2 , p1 , N1 , N0 , S), if N0 < S ≤ N1 ,       PN1 +N2 −S    iα(p1 , p2 )i N2 + i=1   S− cl ·Q3 (p1 , p2 , N1 , N0 , S), if S > max(N1 , N0 ),

226

CHAPTER 7. CLOSED BERNOULLI LINES

BLcl 1 = ST1cl = BLcl 2 = ST2cl =

 if S > N1  p1 Qcl (p2 , p1 , N1 , N0 , S), p1 (1 − p2 )Qcl (p2 , p1 , N1 , N0 , S), if S = N1  0, otherwise ½ cl p1 Q (p2 , p1 , N1 , N0 , S), if S ≤ N1 , 0, otherwise,  cl if S > N0  p2 Q (p1 , p2 , N1 , N0 , S), p2 (1 − p1 )Qcl (p1 , p2 , N1 , N0 , S), if S = N0  0, otherwise, ½ cl p2 Q (p1 , p2 , N1 , N0 , S), if S ≤ N0 , 0, otherwise,

or S = N1 ≥ N0 , < N0 , (7.5) (7.6) or S = N0 ≥ N1 , < N1 , (7.7) (7.8)

where p1 (1 − p2 ) , (7.9) p2 (1 − p1 )  cl Q (p1 , p2 , N1 , N0 , S), if S ≤ min(N0 , N1 ),    1cl Q3 (p1 , p2 , N1 , N0 , S), if S > max(N0 , N1 ), cl Q (p1 , p2 , N1 , N0 , S) = (7.10) cl Q  2 (p1 , p2 , N1 , N0 , S), if S > min(N1 , N0 )   and S ≤ max(N1 , N0 ),  1−p  S+1−2p , if p1 = p2 = p, (1−p )(1−α) Qcl (7.11) (p , p , N , N , S) = 1 1 2 1 0 1 p1 (1−p1 ) S , if p1 6= p2 ,  α(p1 , p2 ) =

( Qcl 2 (p1 , p2 , N1 , N0 , S) = ( Qcl 3 (p1 , p2 , N1 , N0 , S) =

1− p

2 (1−p2 )

α

1−p min(N1 ,N0 )+1−p , (1−p1 )(1−α) p 1− p1 αmin(N1 ,N0 )

if p1 = p2 = p, if p1 6= p2 ,

(7.12)

2

1−p N1 +N0 −S+1 , (1−p1 )(1−α) , 1−αN1 +N0 −S+1

if p1 = p2 = p, if p1 = 6 p2 .

(7.13)

Proof : See Section 20.1. Theorem 7.2 Function P Rcl (p1 , p2 , N1 , N0 , S) is • strictly increasing in p1 and p2 ; • non-strictly increasing in N1 and N0 ; • nonmonotonic concave in S. Proof : See Section 20.1. This theorem is of practical importance. First, it states that, similar to open lines, increasing pi ’s always leads to an increased production rate in closed lines as well. Second, it states that, unlike open lines, increasing buffer capacity does not always lead to improved performance. Finally, it reaffirms the evidence of Figure 7.2 that P R is nonmonotonic concave in S. Theorem 7.3 The pair (N0 , S) is unimpeding if and only if N1 < S ≤ N0 .

(7.14)

7.2. PERFORMANCE ANALYSIS OF CLOSED LINES

227

Proof : See Section 20.1. Along with characterizing the unimpeding values of N0 and S, this theorem has another important implication. It states that, in fact, unimpeding N0 and S are independent of the machine efficiencies p1 and p2 : as long as (7.14) is observed, the closed nature of the line does not impede the open line behavior, i.e., P Rcl (p1 , p2 , N1 , N0 , S) = P Ro (p1 , p2 , N1 ), no matter what p1 and p2 are. Thus, changing pi ’s cannot change an unimpeding pair (N0 , S) into an impeding one, and vise versa. Next, we describe a property of asymptotic equivalence of closed and open lines. To accomplish this, consider the closed two-machine line of Figure 7.4(a) and the open two-machine line of Figure 7.4(b). Note that the efficiencies of the machines in these two lines are the same, while Neo in Figure 7.4(b) is not yet determined. Is it possible to select Neo so that the production rates of these lines are the same or, at least, almost the same? Referring to the system of Figure 7.4(b) as the equivalent open line, the above question amounts to determining Neo such that P Rcl (p1 , p2 , N1 , N0 , S) ≈ P Reo (p1 , p2 , Neo ),

(7.15)

where P Reo denotes the production rate of the equivalent open line. Let W IP1eo , eo BLeo 1 and ST2 denote the average buffer occupancy and the probabilities of blockage of m1 and starvation of m2 in the equivalent open line. Then:

(a) Closed line

(b) Equivalent open line

Figure 7.4: Closed and equivalent open two-machine lines Theorem 7.4 Assume that the machines are asymptotically reliable, i.e., pi = 1 − ²ki ,

i = 1, 2,

where 0 < ² ¿ 1 and ki > 0 is independent of ². Then, if  if S ≤ min(N1 , N0 ),  S − 1, if min(N1 , N0 ) < S ≤ max(N1 , N0 ), min(N1 , N0 ), Neo =  N1 + N0 − S + 1, if S > max(N1 , N0 ),

(7.16)

(7.17)

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CHAPTER 7. CLOSED BERNOULLI LINES

the performance characteristics of the closed two-machine line defined by assumptions (a)-(h) and the equivalent open two-machine line defined by assumptions (a)-(e) are related as follows: P Rcl = P Reo (p1 , p2 , Neo ) + O(²2 ), (7.18) ½ eo ) + W IP (p , p , N ) + O(²), N ≤ N , max (0, S − N 0 1 2 eo 1 0 1 (7.19) W IP1cl = S − max (0, S − N1 ) − W IP1eo (p2 , p1 , Neo ) + O(²), N1 > N0 ,  eo 2  BL1 (p1 , p2 , Neo ) + O(² ), S ≥ N1 or S = N1 ≥ N0 , cl 2 O(² ), S = N1 < N0 (7.20) BL1 =  0, otherwise, ½ 2 S ≤ N1 , BLeo 1 (p1 , p2 , Neo ) + O(² ), ST1cl = (7.21) 0, S > N1 ,  eo 2  ST2 (p1 , p2 , Neo ) + O(² ), S ≥ N0 or S = N0 ≥ N1 , cl O(²2 ), S = N0 < N 1 (7.22) BL2 =  0, otherwise, ½ ST2eo (p1 , p2 , Neo ) + O(²2 ) S ≤ N0 , ST2cl = (7.23) 0, S > N0 ,

where O(²) and O(²2 ) denote terms of order of magnitude ² and ²2 , respectively. Proof : See Section 20.1. Thus, for closed lines with pi ’s close to 1, all the performance characteristics can be evaluated using the open line expressions with buffer capacity defined in (7.17).

7.2.2

M > 2-machine lines

Similar to M = 2, the behavior of closed lines with M > 2 can be described by ergodic Markov chains. However, unlike the two-machine case, closed-form expressions for the performance measures are all but impossible to derive. Therefore, we resort to a more restrictive statement: Theorem 7.5 Assume that the Bernoulli line defined by assumptions (a)(h) of Subsection 7.1.1 satisfies the condition: M −1 X

Ni < S ≤ N0 .

(7.24)

i=1

Then the pair (N0 , S) is unimpeding and, therefore, all performance characteristics of this line coincide with those of the corresponding open line, i.e., P Rcl (p1 , . . . , pM , N1 , . . . , N0 , S) W IPicl (p1 , . . . , pM , N1 , . . . , N0 , S)

= =

BLcl i (p1 , . . . , pM , N1 , . . . , N0 , S) STicl (p1 , . . . , pM , N1 , . . . , N0 , S)

= =

P Ro (p1 , . . . , pM , N1 , . . . , NM −1 ), W IPio (p1 , . . . , pM , N1 , . . . , NM −1 ), i = 1, . . . , M − 1, (7.25) BLoi (p1 , . . . , pM , N1 , . . . , NM −1 ), o STi (p1 , . . . , pM , N1 , . . . , NM −1 ), i = 1, . . . , M.

7.2. PERFORMANCE ANALYSIS OF CLOSED LINES

229

Proof : See Section 20.1. Theorem 7.5 implies, in particular, that under condition (7.24), the unimpeding pair (N0 , S) is independent of pi ’s. We show below that a similar property holds even when (7.24) does not take place. Since we show this by simulations, the notion of unimpediment is modified to account for the fact that P Rcl and P Ro are determined with finite accuracy and, therefore, equality (7.1) may hold only approximately. Definition 7.5 A pair (N0 , S) is practically unimpeding if |P Rcl (N0 , S) − P Ro | ≤ δ ¿ 1. P Ro Numerical Fact 7.1 In closed Bernoulli lines defined by assumptions (a)(h) of Subsection 7.1.1, a practically unimpeding pair (N0 , S) remains practically unimpeding for all values of pi , i = 1, . . . , M , as long as N0 , N1 , . . . , NM −1 remain the same. Justification: The justification was carried out using the C++ codes, which simulate the open and closed lines at hand, and Simulation Procedure 4.1. In these simulations, the carriers were initially placed in the empty carrier buffer, with the excess carriers (if any) placed randomly and equiprobably in the ino o process buffers. As a result, P Rcl , STicl , BLcl i , P Ro , STi , BLi have been evaluated with 95% confidence intervals 0.001 for production rates and 0.002 for blockages and starvations. We constructed 30,000 closed lines by selecting M , pi ’s, and Ni ’s randomly and equiprobably from the following sets: M pi Ni

∈ {3, 5, 10}, ∈ [0.7, 0.95], i = 1, . . . , M, ∈ {1, 2, 3, 4, 5}, i = 1, . . . , M − 1

N0

∈

{3, 4, . . . ,

M −1 X

Ni − 1}.

(7.26) (7.27) (7.28) (7.29)

i=1

Note that on the set (7.29), condition (7.24) does not hold. In addition, δ was selected as 0.01. For each of these lines, using Simulation Procedure 4.1, we evaluated P Rcl and P Ro and selected randomly and equiprobably a pair (N0∗ ,S ∗ (N0∗ )) that was practically unimpeding in the sense of Definition 7.5. To verify whether this pair remained practically unimpeding with other values of pi ’s, for each of the 30,000 lines mentioned above, we constructed 10 additional lines with Ni , i = 0, 1, . . . , M − 1, being the same but with new pi ’s selected randomly and equiprobably from the set (7.27). Again, using Simulation Procedure 4.1, we evaluated P Rcl and P Ro for each of the new lines, thus constructed, and checked whether Definition 7.5 holds. The results are shown in Figure 7.5. As one can see, (N0∗ , S ∗ (N0∗ )) remains unimpeding in almost 100% of the cases analyzed. Thus, we conclude that Numerical Fact 7.1 indeed takes place.

230

CHAPTER 7. CLOSED BERNOULLI LINES

Figure 7.5: Accuracy of Numerical Fact 7.1 In conclusion of this section, guided by Theorem 7.2, the following systemtheoretic properties can be established: Theorem 7.6 Function P Rcl (p1 , . . . , pM , N1 , . . . , NM −1 , N0 , S) is • strictly increasing in pi , i = 1, . . . , M ; • non-strictly increasing in Ni , i = 0, . . . , M − 1; • nonmonotonic concave in S. Thus, based on the first of the above statements, the sign of the absolute values in (7.2) can be removed. Theorem 7.7 Closed Bernoulli lines defined by assumptions (a)-(h) of Subsection 7.1.1 are reversible in the sense that Lr L = P Rcl , P Rcl

where L and Lr are a closed line and its reverse, respectively. PSE Toolbox: The performance analysis of closed lines is one of the tools in the Performance Analysis function of the toolbox. Since no closed formulas for performance measures of closed lines are available (for M > 2), it is based on simulations. See Subsection 19.3.4 for details.

7.3

Improvability

The definitions of S- and N0 -improvability are given in Subsection 7.1.2. Below we provide methods for identifying whether a line is improvable in an appropri∗ ∗ and Smax the smallest and ate sense or not. Throughout, we denote as Smin ∗ ∗ (see Figure 7.2). largest unimprovable S. Clearly, in some systems Smin = Smax

7.3.1

Two-machine lines

∗ ∗ Theorem 7.8 For S ∈ / [Smin , Smax ], a closed Bernoulli line defined by assumptions (a)-(h) of Subsection 7.1.1 with M = 2 is

7.3. IMPROVABILITY OF CLOSED LINES

231

• S + -improvable if 2 X

STi >

2 X

i=1

i=1

2 X

2 X

BLi ;

• S − -improvable if STi <

i=1

BLi .

i=1

Proof : See Section 20.1. ∗ ∗ , Smax ], increasing or decreasing S leads to a limit cycle, i.e., For S ∈ [Smin “oscillations” between S and S − 1 or S + 1. In this case, the best S (i.e., the one resulting in the largest P Rcl ) must be selected from the limit cycle. It is convenient to introduce the notation: I=

M X

STi −

i=1

M X

BLi ,

(7.30)

i=1

and refer to I as the S-improvability indicator. Thus, positive (respectively, negative) I’s imply S + - (respectively, S − -) improvability. In addition to its direct value as a tool for S-improvability identification, the utility of this theorem (and the subsequent similar statement for M > 2) is in the fact that S-improvability can be identified without knowing the machine and buffer parameters but just by measuring the frequency of blockages and starvations of the machines during normal system operation. Finally, we formulate: Theorem 7.9 If a system is S-unimprovable and ST1 or BL2 is non-zero, then the system is N0 -improvable. Proof : Follows directly from Theorem 7.8 and the definition of N0 -improvability.

7.3.2

M > 2-machine lines

An extension of Theorem 7.8 for the case of M > 2 is all but impossible to derive analytically. However, based on simulations, we conclude that it takes place for any M . Specifically, ∗ ∗ , Smax ], a closed Bernoulli line defined Numerical Fact 7.2 For S ∈ / [Smin by assumptions (a)-(h) of Subsection 7.1.1 with M > 2 is

• S + -improvable if M X

STi >

i=1

M X

BLi ;

i=1

• S − -improvable if M X i=1

STi <

M X i=1

BLi .

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CHAPTER 7. CLOSED BERNOULLI LINES

Thus, I of (7.30) is still the indicator of improvability: S + if I is positive and S − if I is negative. Justification: We constructed 300,000 closed lines by selecting M , pi ’s, and Ni ’s randomly and equiprobably from the following sets: M

∈

{3, 5, 10},

(7.31)

pi Ni

∈ ∈

[0.7, 0.95], i = 1, . . . , M, {1, 2, 3, 4, 5}, i = 1, . . . , M − 1

(7.32) (7.33)

N0

∈

{1, 2, . . . , 15}.

(7.34)

For each of these lines, using Simulation Procedure 4.1, we evaluated P Rcl (S) PM −1 for all S ∈ {1, 2, . . . , i=0 Ni − 1} and determined Sopt for which P Rcl (S) is maximized. Also, for each of these lines, using Simulation Procedure 4.1 and Numerical Fact 7.2, we obtained Sunimp , i.e., the value of S at which the improvability indicator I changes its sign. Then we compared the values of P Rcl (Sopt ) and P Rcl (Sunimp ). The results are shown in Figure 7.6. As one can see, the two production rates are within 1% of each other in almost 100% of the cases analyzed. Thus we conclude that Numerical Fact 7.2 indeed defines the conditions of S-improvability.

Figure 7.6: Accuracy of Numerical Fact 7.2 Based on this, we formulate: S-Continuous Improvement Procedure 7.1: (1) (2) (3) (4) (5)

Evaluate STi and BLi for all machines in the system. PM PM Calculate the S-improvability indicator I = i=1 STi − i=1 BLi . If I > 0, increase S by one; if I < 0, decrease S by one. Return to (1) and continue until a limit cycle is reached. Select the S from the limit cycle, which gives the largest P Rcl ; this S is unimprovable and is denoted as S ∗ (N0 ). Clearly, if for the above S ∗ (N0 ), P Rcl (N0 , S ∗ (N0 )) < P Rcl (N0 + 1, S ∗ (N0 + 1)),

7.3. IMPROVABILITY OF CLOSED LINES

233

the pair (N0 , S ∗ (N0 )) is impeding and, therefore, is improvable with respect to N0 . This improvement can be carried out using N0 -Continuous Improvement Procedure 7.1: (1) For a given N0 , carry out S-Continuous Improvement Procedure 7.1 and determine S ∗ (N0 ) and S ∗ (N0 + 1). (2) If |P Rcl (N0 , S ∗ (N0 )) − P Rcl (N0 + 1, S ∗ (N0 + 1))| > δP Rcl (N0 , S ∗ (N0 )), δ ¿ 1, increase N0 by 1, return to (1). (3) If |P Rcl (N0 , S ∗ (N0 )) − P Rcl (N0 + 1, S ∗ (N0 + 1))| ≤ δP Rcl (N0 , S ∗ (N0 )), δ ¿ 1, the system is unimprovable with respect to N0 ; this N0 and the resulting S is an unimpeding pair and is denoted as (N0∗ , S ∗ (N0∗ )). The accuracy of this procedure in identifying unimpeding pairs (N0∗ , S ∗ (N0∗ )) is illustrated in Figure 7.7.

Figure 7.7: Accuracy of continuous improvement with respect to N0 Below, two examples illustrating S- and N0 -improvement procedures are given. In the first example, the system of Figure 7.8 is considered and SContinuous Improvement Procedure 7.1 is carried out starting from S = 2 and S = 21. The results are given in Tables 7.1 and 7.2, respectively. In both cases, the unimprovable number of carriers is 10. In the second example, N0 -Continuous Improvement Procedure 7.1 (with δ = 0.01) is applied to the system of Figure 7.9. As a result, an unimprovable pair (N0∗ , S ∗ (N0∗ )) is obtained with N0∗ = 5 and S ∗ (N0∗ ) = 11 (Table 7.3).

Figure 7.8: Example of S-Continuous Improvement Procedure 7.1

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CHAPTER 7. CLOSED BERNOULLI LINES

Table 7.1: Example of S-Continuous Improvement Procedure 7.1 (starting from S = 2) Step # 1 2 3 4 5 6 7 8 9 10

S 2 3 4 5 6 7 8 9 10 9

I 2.3126 1.4071 0.7802 0.4940 0.3302 0.2572 0.1922 0.0841 -0.0577 0.0841

PR 0.4189 0.5877 0.7006 0.7447 0.7623 0.7664 0.7673 0.7673 0.7674 0.7673

Table 7.2: Example of S-Continuous Improvement Procedure 7.1 (starting from S = 21) Step # 0 1 2 3 4 5 6 7 8 9 10 11 12 13

S 21 20 19 18 17 16 15 14 13 12 11 10 9 10

I −0.4005 −0.1886 −0.1111 −0.0936 −0.0927 −0.0940 −0.0916 −0.0934 −0.0917 −0.0921 −0.0888 −0.0577 0.0841 -0.0577

PR 0.7596 0.7665 0.7671 0.7672 0.7673 0.7673 0.7672 0.7672 0.7666 0.7672 0.7672 0.7674 0.7673 0.7674

Figure 7.9: Example of N0 -Continuous Improvement Procedure 7.1

7.3. IMPROVABILITY OF CLOSED LINES

235

Table 7.3: Example of N0 -Continuous Improvement Procedure 7.1, P Ro = 0.6665 Step # 0 1 2 3 4 5 6 7 8 9 10 11 12

N0 2 2 2 2 2 2 3 3 4 4 4 5 5

S 5 6 7 8 9 10 10 11 10 11 12 11 12

I 0.7551 0.5114 0.3380 0.2045 0.0693 -0.1017 0.0367 -0.1200 0.1345 0.0094 -0.1337 0.1055 -0.0081

PR 0.5823 0.6116 0.6287 0.6386 0.6425 0.6439 0.6531 0.6530 0.6580 0.6597 0.6575 0.6627 0.6618

PSE Toolbox: S-Continuous Improvement Procedure 7.1 is implemented as one of the tools in the Continuous Improvement function of the toolbox (see Subsection 19.4.5 for details).

7.3.3

Comparisons

The existing literature offers an interesting formula for selecting S in closed lines with machines having random processing time and with blocked after service (BAS) convention (which implies that even if the downstream buffer is full, a machine can process a part). This formula is: ' &P M −1 BAS i=0 Ni b , (7.35) S=M+ 2 where NiBAS , i = 0, 1, . . . , M − 1, is the i-th buffer capacity under the BAS convention and dxe denotes the smallest integer not less than or equal to x. The blocked before service (BBS) convention, used in this chapter, implies that the machine itself is a unit of buffer capacity; therefore, NiBAS = Ni − 1,

(7.36)

where Ni is the i-th buffer capacity under the BBS convention. Thus, formula (7.35) for systems under the BBS convention becomes ' & &P PM −1 ' M −1 M + (N − 1) i i=0 i=0 Ni = . (7.37) Sb = M + 2 2 To investigate the relationship between S ∗ obtained by S-Continuous Improvement Procedure 7.1 and the Sb provided in expression (7.37), we use the

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CHAPTER 7. CLOSED BERNOULLI LINES

examples of the previous subsection. The results are as follows: In the first example, S ∗ = 10, while Sb = 15. This results in P R(S ∗ ) = 0.7674 and b = 0.7672. In the second example, the results are summarized in TaP R(S) ble 7.4. As one can see, both approaches lead to similar results with S ∗ being b somewhat smaller than S. Table 7.4: Comparison of the S-Continuous Improvement Procedure and equation (7.37) N0 2 3 4 5

7.4 7.4.1

Sopt 10 10 12 12

P R(Sopt ) 0.6439 0.6531 0.6600 0.6630

S∗ 10 10 11 11

P R(S ∗ ) 0.6439 0.6531 0.6597 0.6627

Sb 11 12 12 13

b P R(S) 0.6438 0.6526 0.6600 0.6629

Bottleneck Identification Two-machine lines

Theorem 7.10 In closed Bernoulli lines defined by assumptions (a)-(h) of Subsection 7.1.1 with M = 2, machine m1 (respectively, machine m2 ) is the bottleneck (BN) if and only if ST1 + BL1 < ST2 + BL2 ,

(respectively, ST1 + BL1 > ST2 + BL2 ). (7.38)

Proof : Follows the proof of Theorem 5.8 in Section 20.1 for open lines. This theorem can be interpreted as follows: Refer to ST1 +BL1 as the virtual blockage of m1 and to ST2 + BL2 as the virtual starvation of m2 , i.e., BL1,v ST2,v

= =

ST1 + BL1 , ST2 + BL2 ,

and assume that the virtual starvation of m1 and virtual blockage of m2 are 0, i.e., ST1,v = BL2,v = 0.

(7.39)

Assign arrows between m1 and m2 according to the same rule as in the case of open lines but using virtual blockages and starvations of m1 and m2 . Then the machine with no emanating arrows is the BN of the closed line. This is illustrated in Figure 7.10. Thus, using virtual, rather than real, blockages and starvations allows us to extend the open line BN identification technique for two-machine lines to closed ones. As it is shown below, this can be done for M > 2 as well.

7.4. BOTTLENECK IDENTIFICATION IN CLOSED LINES

237

Figure 7.10: BN identification in two-machine closed lines

7.4.2

M > 2-machine lines

Consider an M > 2-machine closed line and assume that STi and BLi , i = 1, . . . , M , are identified during normal system operation. Similar to the case M = 2, introduce the virtual blockages and starvations of the machines as follows: BL1,v ST1,v BLi,v STi,v BLM,v STM,v

:= ST1 + BL1 , := := := := :=

0, BLi , i = 2, . . . , M − 1, STi , i = 2, . . . , M − 1, 0, STM + BLM .

Using STi,v and BLi,v , assign arrows between mi and mi+1 according to the same rule as in the open lines, i.e., an arrow is directed from mi to mi+1 if BLi,v > STi+1,v and from mi+1 to mi if BLi,v < STi+1,v . Since ST1,v = BLM,v = 0, there is at least one machine with no emanating arrow (see Figure 7.11). Bottleneck Indicator 7.1: In closed Bernoulli lines defined by assumptions (a)-(h), • if there is a single machine with no emanating arrows, it is the BN-m;

238

CHAPTER 7. CLOSED BERNOULLI LINES • if there are multiple machines with no emanating arrows, the one with the largest severity is the Primary BN (PBN), where the severity of each (local) BN is defined by Si,v S1,v SM,v

= max(|STi+1,v − BLi,v |, |STi − BLi−1,v |), = max(|ST2,v − BL1,v |, |ST1,v − BLM,v |),

i = 2, . . . , M − 1, (7.40)

= max(|STM,v − BLM −1,v |, |ST1,v − BLM,v |).

Figure 7.11: BN identification in five-machine closed line

Justification: This justification has been carried out as follows: A total of 1,000,000 closed lines have been generated with parameters selected Prandomly M −1 and equiprobably from sets (7.31)-(7.34) and S ∈ {M, M + 1, . . . , i=0 Ni }. Each of these lines has been analyzed using Simulation Procedure 4.1. Specifically, the probabilities of blockages and starvations of all machines have been estimated and, in addition, partial derivatives of the production rate with respect to pi ’s have been evaluated (with ∆pi = 0.01). The probabilities of blockages and starvations have been used to identify the BN by Bottleneck Indicator 7.1, and the partial derivatives have been used to identify the BN by (7.2). If the BNs identified by both method were the same, we conclude that Bottleneck Indicator 7.1 holds; otherwise, we conclude that it does not. The results obtained using this approach are summarized in Figure 7.12. Among the 1,000,000 lines analyzed, 87.59% had a single machine with no emanating arrows, and the BN machine was identified by Bottleneck Indicator 7.1

7.5. LEANNESS OF CLOSED LINES

239

correctly in 92.76% of these cases. For the 12.41% of systems with more than one machine having no emanating arrows, Bottleneck Indicator 7.1 identified correctly the PBN in 71.1% of cases, while the PBN was indeed in the set of local BN’s in 97.20% of cases. These results are similar to those obtained in Chapter 5 for BN identification in open lines. Thus, we conclude that Bottleneck Indicator 7.1 provides a sufficiently accurate tool for bottleneck identification in closed lines.

Figure 7.12: Accuracy of Bottleneck Indicator 7.1

PSE Toolbox: Bottleneck Indicator 7.1 is implemented as one of the tools in the Bottleneck Identification function of the toolbox (see Subsection 19.5.4 for details).

7.5

Leanness

In this section, we discuss the selection of the smallest N0 , i.e., N0lean , and the corresponding smallest S, i.e., Slean (N0lean ), which result in P Rcl = P Ro . Such a pair, as defined in Subsection 7.1.2, is called lean. For two-machine lines, the lean pair (N0 , S) can be obtained immediately from Theorem 7.3 Slean (N0lean ) = N0lean = N1 + 1.

(7.41)

240

CHAPTER 7. CLOSED BERNOULLI LINES For M > 2, the lean (N0lean , Slean ) can be obtained using the following:

Lean (N0 , S)-Design Procedure 7.1: (1) Using N0 -Continuous Improvement Procedure 7.1, determine an unimpeding pair (N0∗ , S ∗ (N0∗ )). (2) Decrease N0∗ by 1 and determine S ∗ (N0∗ − 1). (3) If |P Rcl (N0∗ − 1, S ∗ (N0∗ − 1)) − P Rcl (N0∗ , S ∗ (N0∗ ))| < δP R(N0∗ , S ∗ (N0∗ )), δ ¿ 1, return to step (2). (4) If |P Rcl (N0∗ − 1, S ∗ (N0∗ − 1)) − P Rcl (N0∗ , S ∗ (N0∗ ))| > δP R(N0∗ , S ∗ (N0∗ )), δ ¿ 1, then N0∗ = N0lean . As an example, this procedure (with δ = 0.01) is applied to the system of Figure 7.13, and the results are given in Table 7.5. Clearly, it leads to the reduction of N0 from 13 to 4 and S ∗ from 13 to 11, practically without losses in the production rate.

Figure 7.13: Example of Lean (N0 , S)-Design Procedure 7.1

Table 7.5: Example of Lean (N0 , S)-Design Procedure 7.1 Step # 0 1 2 3 4 5 6 7 8 10 11

N0 13 12 11 10 9 8 7 6 5 4 3

S∗ 13 13 13 13 13 13 12 12 12 11 11

P R(S ∗ ) 0.7001 0.7001 0.7000 0.7000 0.7001 0.7000 0.6994 0.6994 0.6989 0.6982 0.6910

7.6. CASE STUDY

7.6

241

Case Study

7.6.1

Modeling and model validation of closed automotive paint shop production system

An open line model for the paint shop system is developed in Subsection 3.10.2. Here, we model the system as a closed Bernoulli line (see Figure 7.14). The machine and buffer parameters are given in Tables 3.10 and 3.11, respectively. The minimum number of carriers to ensure continuous operation is 401, while the total number of carries available in the system is 409. Therefore, the effective number of carriers, S, is 8.

Figure 7.14: Closed line model of the automotive paint shop system The open line model, where p3 is modified to p3 (1 − Pst ) to account for the starvation of Op.3 for carriers, is validated in Subsection 4.4.2. To validate the closed line model, we evaluate the performance of the system in Figure 7.14 using Simulation Procedure 4.1 and compare it to that measured on the factory floor. The results are summarized in Table 7.6 for the throughput and in Table 7.7 for the starvations of Op.3. Clearly, the estimated and measured quantities are close to each other, except for Month 2, where, as mentioned in Subsection 3.10.2, a new product has been introduced. Therefore, we conclude that the closed line model is validated. Table 7.6: Closed line model validation data: T P Est. T P Actual T P Error

Month 1 52.59 53.50 −1.70%

Month 2 51.54 43.81 17.64%

Month 3 52.77 51.27 2.93%

Month 4 51.92 54.28 −4.35%

Month 5 52.64 55.89 −5.81%

Table 7.7: Closed line model validation data: Pst Est. Pst Actual Pst

Month 1 0.1191 0.0981

Month 2 0.1476 0.1171

Month 3 0.1505 0.1113

Month 4 0.1413 0.1046

Month 5 0.1428 0.0975

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CHAPTER 7. CLOSED BERNOULLI LINES

7.6.2

S-improvability

As it has been shown in Subsection 4.4.2, the throughput of the paint shop system can be improved by about 10% if starvations due to lack of carriers are eliminated (see Table 4.9). Therefore, we carry out S-Continuous Improvement Procedure 7.1 in order to determine the smallest number of carriers, which ensure the desired performance. The results, based on Month 1 data, are shown in Table 7.8. Clearly, with S = 16 the system attains the open line throughput and, thus, is unimpeded by its closed nature. The same property holds for all other months, since, as it is stated in Numerical Fact 7.2, the unimpeding pair (N0 , S) is independent of machine parameters. Thus, the performance of the paint shop with S = 16 is as shown in Table 4.9, which is about a 10% improvement over the performance ensured by S = 8. Table 7.8: S-Continuous Improvement Procedure 7.1 for Month 1 Step # 0 1 2 3 4 5 6 7 8

7.6.3

S 8 9 10 11 12 13 14 15 16

I 1.3534 0.7217 0.5216 0.4297 0.3926 0.3780 0.3735 0.3711 0.3707

TP 52.59 57.00 58.36 58.93 59.16 59.24 59.27 59.28 59.29

Lean system design

It has been shown in Subsection 6.4.2 that the lean in-process buffering that ensures paint shop efficiency E = 0.95 is N = [2, 2, 1, 1, 1, 1, 1, 1].

(7.42)

Given the above, we design below the lean (N0 , S) pair, which renders the closed system lean. To accomplish that, we carry out Lean (N0 , S)-Design Procedure 7.1 with initial pair N0 = 11 and S = 11. The results for all five months are shown in Table 7.9. As one can see, the pair (2,10) is lean, implying that if the in-process buffering is reduced to (7.42), the empty buffer capacity N0 = 2 and S = 10 would be sufficient to ensure the line efficiency E = 0.95.

7.7

Summary

• The performance of a closed line can be impeded, in comparison to the corresponding open line, if the number of carriers, S, and the capacity of the empty carrier buffer, N0 , are not selected correctly.

7.8. PROBLEMS

243

Table 7.9: Lean (N0 , S) and the resulting throughput Lean (N0 , S) T P (N0 , S)

Month 1 (2,10) 57.52

Month 2 (2,10) 55.88

Month 3 (2,10) 57.71

Month 4 (2,10) 56.29

Month 5 (2,10) 57.41

• A method for calculating performance measures in two-machine closed Bernoulli lines is derived, and a more restrictive result for longer lines is obtained. • A criterion of improvability PM to the number of carriers PMis estabPM with respect lished. Specifically, if i=1 STi > i=1 BLi , (respectively, i=1 STi < PM + − i=1 BLi ), the system is S - (respectively, S -) improvable, i.e., P Rcl can be increased by adding (respectively, removing) a carrier. • A criterion of improvability with respect to the capacity of the empty carrier buffer is derived, and a corresponding continuous improvement procedure is proposed. • A method for identifying bottleneck machines in closed lines is suggested. Specifically, it is shown that bottlenecks in closed lines can be identified based on the same procedure as that for open lines but using the so-called virtual, rather than real, probabilities of blockages and starvations. • A procedure for calculating the lean empty carrier buffer capacity and the lean number of carriers is proposed.

7.8

Problems

Problem 7.1 For a closed two-machine Bernoulli line with p1 = 0.9, p2 = 0.95, N1 = 3, N0 = 2, and S = 3, (a) Determine the equivalent open line. (b) Calculate P Rcl using the equivalent and the original line. (c) Determine the difference between these two results. Problem 7.2 For the system of Problem 7.1, determine if it is (a) S + -improvable or S − -improvable, (b) N0 -improvable, and (c) Calculate the unimprovable pair (N0 , S). (d) Compare the resulting S with Sb provided by formula (7.37). (e) Interpret the results. Problem 7.3 Consider a closed two-machine line with p1 = p2 = 0.9. Determine the smallest N1 , N0 , and S so that the line efficiency is 0.95.

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CHAPTER 7. CLOSED BERNOULLI LINES

Problem 7.4 Consider a closed two-machine line with p1 = 0.9, p2 = 0.91, N1 = 3, N0 = 2, and S = 5. (a) Identify the BN machine. (b) Can this machine be the BN of the open line with the same p1 , p2 , and N1 ? (c) Verify your answer to (a) using the equivalent open line representation. Problem 7.5 Develop a method for performance analysis of two-machine closed Bernoulli lines with the symmetric blocking convention. Problem 7.6 Investigate the property of asymptotic equivalence of twomachine closed Bernoulli lines with the symmetric blocking convention. Problem 7.7 Derive criteria for S + - and S − -improvability for two-machine closed Bernoulli lines with the symmetric blocking convention. Problem 7.8 Derive a criterion for BN-m identification in two-machine closed Bernoulli lines with the symmetric blocking convention.

7.9

Annotated Bibliography

Closed queueing systems have been studied in numerous publications. A few examples are: [7.1] R. Suri and G.W. Diehl, “A Variable Buffer-Size Model and Its Use in Analyzing Closed Queueing Networks with Blocking,” Management Science, vol. 32, pp. 206-224, 1986. [7.2] X.-G. Liu, L. Zhuang and J.A. Buzacott, “A Decomposition Method for Throughput Analysis in Cyclic Queues with Production Blocking,” in Queueing Networks with Finite Capacity, R.O. Onvural and I.F. Akyildiz (Editors), Elsevier, Amsterdam, 1993. [7.3] Y. Dallery, Z. Liu and D. Towsley, “Properties of Fork/Join Queueing Networks with Blocking under Various Operating Mechanisms,” IEEE Transaction on Robotics and Automation, vol.13, pp. 503-518, 1997. Formula (7.35) has been proposed in [7.4] D.S. Kim, D.M. Kulkarny and F. Lin, “An Upper Bound for Carriers in a Three-Workstation Closed Serial Production System Operating under Production Blocking,” IEEE Transactions on Automatic Control, vol. 47, pp. 1134-1138, 2002. The material of this chapter is based on [7.5] J.-T. Lim and S.M. Meerkov, “On Asymptotically Reliable Closed Serial Production Lines,” Control Engineering Practice, vol. 1, pp. 147-152, 1993.

7.9. ANNOTATED BIBLIOGRAPHY

245

[7.6] S. Biller, S.P. Marin, S.M. Meerkov and L. Zhang, “Closed Bernoulli Lines: Analysis, Continuous Improvement, and Leanness,” IEEE Transactions on Automation Science and Engineering, vol. 5, 2008. Additional results (for exponential lines) can be found in [7.7] S. Biller, S.P. Marin, S.M. Meerkov and L. Zhang, “Closed Production Lines with Arbitrary Models of Machine Reliability,” Proceedings of IEEE Conference on Automation Science and Engineering, Washington DC, August 2008.

Chapter 8

Product Quality in Bernoulli Lines Motivation: Along with productivity, product quality is of central importance in manufacturing. While all previous chapters address mostly productivity, this chapter is devoted to quality. Specifically, we study Bernoulli serial lines with machines of the following types: • perfect quality machines, i.e., producing parts with no defects; • non-perfect quality machines obeying the Bernoulli quality model, i.e., producing a good (non-defective) part with probability g and a defective with probability 1 − g; • non-perfect quality machines with quality-quantity coupling (QQC), i.e., having their quality parameters g as a monotonically decreasing function of their efficiency, p. • inspection machines, i.e., inspecting parts quality, perhaps along with carrying out some technological operations. Such a production line is shown in Figure 8.1, where the perfect, non-perfect, and QQC machines are represented by white, shaded, and double-shaded circles, respectively, while the inspection machines are black ones; the arrows under the inspection machines indicate scrapping defective parts. For such systems, this chapter presents methods for performance analysis, continuous improvement, and design.

Figure 8.1: Serial production line with perfect quality, non-perfect quality, QQC, and inspection machines

247

248

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

Overview: Section 8.1 is devoted to production lines with constant g’s, i.e., without QQC machines. Section 8.2 considers systems with QQC. We show that such systems have fundamentally different monotonicity properties from all other systems considered in this textbook; this leads to different ways of bottlenecks identification and elimination. Finally, Section 8.3 addresses serial lines with rework in which defective parts are repaired and returned to the main line for reprocessing.

8.1 8.1.1

Bernoulli Lines with Non-perfect Quality Machines Model and problem formulation

Model: Consider the production line shown in Figure 8.2. Suppose that it operates according to assumptions (a)-(e) of Subsection 4.2.1 along with:

Figure 8.2: Serial production line with perfect quality, non-perfect quality, and inspection machines (f) The machines are of three types: perfect quality machines mi , i ∈ Ip ; non-perfect quality machines mi , i ∈ Inp ; and inspection machines mi , i ∈ Iinsp . Here Ip , Inp , and Iinsp represent a partitioning of {1, . . . , M } into the sets of indices, which indicate the positions of perfect, non-perfect, and inspection machines, respectively. Both perfect and non-perfect quality machines are referred to as producing machines. (g) The non-perfect machines obey the Bernoulli quality model, i.e., each part produced by a non-perfect quality machine mi , i ∈ Inp , is good with probability gi and defective with probability 1 − gi , independently from all other parts produced. Parameter gi is referred to as the quality of mi , i ∈ Inp . For convenience, the expression gi , i ∈ {1, . . . , M }, is used to / Inp . indicate the quality parameters of all machines, where gi = 1 for i ∈ (h) For each inspection machine mi , i ∈ Iinsp , “producing a part” implies that the part quality is identified. The defects are assumed to be identified perfectly, i.e., no defects are missed and no perfect parts are identified as defective. Defective parts are discarded (i.e., scrapped), while nondefective parts are placed in the downstream buffer or, in the case of mM , shipped to the customer. Model (a)-(h) can represent production lines without inspection machines as well. In this case, the last machine, mM , can be viewed, conceptually, as performing, along with its technological operation, a quality inspection and separation of good and defective parts produced.

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

249

Problems: In the framework of the above model, this section considers the following three problems: Performance analysis: Given the machine and buffer parameters, pi , gi , and Ni , evaluate • the production rate (P R), i.e., the average number of non-defective parts produced by the last machine per cycle time; • the scrap rate (SRi ), i.e., the average number of defective parts scrapped by inspection machine mi , i ∈ Iinsp , per cycle time; • the consumption rate (CR), i.e., the average number of raw parts consumed by the first machine per cycle time; • the average number of parts in each buffer, i.e., work-in-process (W IPi ); • the probabilities of blockages (BLi ) and starvations (STi ) of each machine; • the monotonicity properties of P R, SR, and CR with respect to machine and buffer parameters; • the reversibility property of P R. A solution of this problem is given in Subsection 8.1.2. Bottlenecks: Production lines with non-perfect quality machines have bottlenecks of two types: those with the largest effect on P R through their efficiency, pi , i ∈ {1, . . . , M }, and those with the largest effect on P R through their quality, gi , i ∈ Inp . Accordingly, we introduce: Definition 8.1 Machine mi , i ∈ {1, . . . , M }, is the production rate bottleneck (PR-BN) of a line defined by assumptions (a)-(h) if ¯ ¯ ¯ ¯ ¯ ∂P R ¯ ¯ ∂P R ¯ ¯ ¯>¯ ¯ (8.1) ¯ ∂pi ¯ ¯ ∂pj ¯ , ∀ j 6= i. This definition is similar to Definition 5.2 for production lines with perfect quality machines. The only difference is that absolute values of partial derivatives are used in (8.1), because it is not a priori clear that P R in systems defined by assumptions (a)-(h) is monotonic with respect to pi ’s. Definition 8.2 Machine mi , i ∈ {1, . . . , M }, is the quality bottleneck (QBN) of a line defined by assumptions (a)-(h) if ¯ ¯ ¯ ¯ ¯ ∂P R ¯ ¯ ∂P R ¯ ¯ ¯>¯ ¯ (8.2) ¯ ∂gi ¯ ¯ ∂gj ¯ , ∀ j 6= i. Methods of PR- and Q-BNs identification are presented in Subsection 8.1.3. Design: In general, design of quality inspection systems consists of two problems: (1) selecting the number of inspection machines, and (2) placing them within the production line. The first of these problems, which is based on both

250

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

economic and technological considerations, is beyond the scope of this textbook. Addressed here is the second one: Given the number of inspection machines and the parameters pi , gi and Ni of all machines and buffers, position the inspection machines so that: • no defective parts are shipped to the customer; • P R is maximized. Clearly, if the last producing machine is of non-perfect quality and only one inspection machine is available, the solution of this problem is trivial: the inspection machine should be placed at the end of the line. However, when the last machine produces no defectives or two or more inspection machines are available, the design problem is non-trivial. This problem is considered in Subsection 8.1.4.

8.1.2

Performance analysis

Production lines with a single inspection machine mM : When the only inspection machine is the last one, mM , the performance analysis can be carried out using the recursive aggregation procedure developed in Chapter 4 for serial lines producing no defectives, i.e., defined by assumptions (a)-(e) of Subsection 4.2.1. The only difference is that the expression for P R must be modified to account for scrap. Specifically, consider aggregation procedure (4.30) and its steady states (4.35). Then the performance measures of a line defined by assumptions (a)-(h) with the only inspection machine being mM are given by: d P R cM SR

=

pfM qM ,

(8.3)

=

(8.4)

d CR

=

pfM (1 − qM ), pb1 = pfM pbi (1 − Q(pfi−1 , pbi , Ni−1 ))

=

= pfi (1 − Q(pbi+1 , pfi , Ni )),

(8.5)

where qM , referred to as the quality buy rate of the system, is given by qM =

Y

gi ,

(8.6)

i∈Inp

c i , and BL di remain the same as in (4.37)\ while the expressions for W IP i , ST (4.40). The accuracy of these estimates is the same as that of (4.36)-(4.40). In d particular, the accuracy of P R is typically within 1%. Clearly, expressions (8.3)-(8.6) can be used for performance evaluation of Bernoulli lines without inspection machines as well. In this case, (8.3) and (8.4) represent the production rate of non-defective and defective parts, respectively.

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

251

Production lines with multiple inspection machines or a single inspection machine other than mM : Aggregation procedure: For the case of multiple inspection machines or a single inspection machine mi , i 6= M , aggregation procedure (4.30) is modified as follows: pbi (s + 1) pfi (s + 1) pfi (s + 1)

= = =

pi [1 − Q(pbi+1 (s + 1), pfi (s), Ni )], i = 1, . . . , M − 1, pi [1 − Q(pfi−1 (s + 1), pbi (s + 1), Ni−1 )], i ∈ / Iinsp , pi [1 − Q(pfi−1 (s + 1), pbi (s + 1), Ni−1 )]qi , i ∈ Iinsp , s = 0, 1, 2, . . . ,

(8.7)

with initial conditions, boundary conditions, and function Q defined by (4.31)(4.34), respectively, and with qi given by Y

qi =

gl ,

(8.8)

i−k
where i − k is the index of the upstream inspection machine, mi−k , which is the closest to the inspection machine mi . The quantity qi , i ∈ Iinsp , is referred to as the quality buy rate of inspection machine mi , i ∈ Iinsp . Theorem 8.1 Sequences pf2 (s), . . . , pfM (s) and pb1 (s), . . . , pbM −1 (s), s = 1, 2, . . . , defined by recursive procedure (8.7), are convergent and the following limits exist: pfi := lims→∞ pfi (s),

pbi := lims→∞ pbi (s).

(8.9)

These limits are unique solutions of the steady state equations corresponding to (8.7), i.e., of pbi pfi pfi

= pi [1 − Q(pbi+1 , pfi , Ni )], i = 1, . . . , M − 1, = pi [1 − Q(pfi−1 , pbi , Ni−1 )], i ∈ / Iinsp , f b = pi [1 − Q(pi−1 , pi , Ni−1 )]qi , i ∈ Iinsp .

Proof: Similar to the proof of Theorem 4.2. Performance measure estimates: Using pfi and pbi , the estimates of the performance measures for the production line at hand are defined as follows: d P R ci SR d CR

= pfM , =

pbi [1 − Q(pfi−1 , pbi , Ni−1 )](1 − qi ) pfi−1 [1 − Q(pbi , pfi−1 , Ni−1 )](1 − qi ),

=

pb1 ,

=

(8.10) i ∈ Iinsp ,

(8.11) (8.12)

c i and BL di remain the same as in the previous \ and the expressions for W IP i , ST case, i.e., given by (4.37)-(4.40).

252

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES Also, as in the previous case, it can be shown that d=P d c CR R + SR,

(8.13)

where c = SR

X

c i. SR

(8.14)

i∈Iinsp

Accuracy of the estimates: The accuracy of these estimates has been investigated numerically using a C++ code, which simulates production lines defined by assumptions (a)-(h) of Subsection 8.1.1, and Simulation Procedure 4.1 (evaluating not only P R, BLi and STi but also SRi and CR). Using this procedure, a total of 100,000 lines with M = 10 were investigated with the parameters pi ’s, gi ’s, Ni ’s, and the number of inspection machines, |Iinsp |, selected randomly and equiprobably from the sets pi gi

∈ ∈ ∈

Ni |Iinsp | ∈

[0.7, 0.95], [0.7, 1), {1, 2, 3, 4, 5},

(8.15) (8.16) (8.17)

{1, 2, 3}.

(8.18)

The positions of the inspection machines were selected randomly and equiprobably from the sets {2, . . . , 10} (without replacement). Each remaining (producing) machine was chosen to be perfect or non-perfect with probability 1/2. For each non-perfect machine, the quality parameter was selected randomly and equiprobably from the set (8.16), and the efficiency of each machine was selected randomly and equiprobably from set (8.15). The performance measures of the systems, thus constructed, have been calculated analytically using expressions (8.10)-(8.12), (4.37)-(4.40) and numerically using Simulation Procedure 4.1. The accuracy of the analytical estimates, as compared with those obtained by simulations, has been evaluated by ²P R

=

²SR

=

²CR

=

²W IP

=

d |P R − P R| · 100%, PR c i| |SRi − SR max · 100%, i∈Iinsp SRi d |CR − CR| · 100%, CR M −1 X \ 1 |W IPi − W IP i | M −1

i=1 M

²ST

=

Ni

1 X c i |, |STi − ST M − 1 i=2

· 100%,

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

²BL

=

253

M −1 X 1 di |. |BLi − BL M − 1 i=1

The results are as follows: Among the 100,000 lines analyzed, the averages of ²P R , ²SR and ²CR are less than 1%, the average of ²W IP is less than 3%, and the averages of ²ST and ²BL are less than 0.005. Therefore, we conclude that aggregation procedure (8.7) results in an acceptable accuracy of performance estimates for the serial lines under consideration. d Monotonicity properties: It has been shown in Subsection 4.3.3 that P R in production lines defined by assumptions (a)-(e) is monotonically increasing in d R, as pi and Ni . A question arises: What are the monotonicity properties of P c and CR, d in production lines defined by assumptions (a)-(h)? well as those of SR The answer is as follows. Theorem 8.2 In Bernoulli lines defined by assumptions (a)-(h) of Subsection 8.1.1, d c and CR d are monotonically increasing in pi , i = 1, . . . , M , and • P R, SR, Ni , i = 1, . . . , M − 1; d • P R is monotonically increasing in gi , i ∈ Inp ; c is monotonically decreasing in gi , i ∈ Inp ; • SR d is monotonically decreasing in gi , i ∈ Inp , unless the only inspection • CR d is independent of gi . machine is the last one, in which case CR Proof: See Section 20.1. Based on this theorem, the signs of absolute values in Definitions 8.1 and 8.2 can be removed. Reversibility property: Consider a production line with perfect and nonperfect quality machines but without inspection machines (i.e., with the last machine separating good and defective parts). It is possible to show that such a line is reversible in the sense that L

d d R P R =P

Lr

,

(8.19)

where L and Lr are the original line and its reverse. PSE Toolbox: The performance analysis of Bernoulli lines with non-perfect quality and inspection machines is one of the tools in the Product Quality function of the toolbox (see Subsection 19.7.1 for details).

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CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

8.1.3

Bottlenecks

Bottlenecks in production lines with a single inspection machine mM : Production rate bottlenecks: Consider two production lines − L1 defined by assumptions (a)-(h) of Subsection 8.1.1 and L2 defined by assumptions (a)-(e) of Subsection 4.2.1 and having all machines of the same efficiency and all buffers of the same capacity as the corresponding machines and buffers in L1. Let d d P R(1) and P R(2) be the production rates of L1 and L2, respectively, obtained using aggregation procedure (4.30) and (8.3) (with qM = 1 in the case of L2). Theorem 8.3 Under the above assumptions, d d ∂P R(2) ∂P R(2) > , ∂pi ∂pj

∀ j 6= i

⇐⇒

d d ∂P R(1) ∂P R(1) > , ∂pi ∂pj

∀ j 6= i,(8.20)

i.e., mi is the PR-BN of L1 if and only if it is the bottleneck of L2. Proof: See Section 20.1. Thus, based on this theorem, PR-BN of a serial line with one inspection machine mM can be identified using Bottleneck Indicator 5.1. Note that since the PR-BN is independent of gi ’s, it follows from Theorem 8.3 that the PR- and Q-BNs are decoupled. Thus, the PR-BN remains the same no matter how gi ’s are changed. Example 8.1: Consider the production line shown in Figure 8.3. In this and the subsequent figures, the first number above each machine represents its efficiency and the second its quality (for non-perfect quality machines); the number above each buffer is its capacity. Based on the data shown under the machines and using Bottleneck Indicator 5.1, we conclude that the primary bottleneck in this system is m4 . Note that it is not the worst machine, as far as pi ’s are concerned, and it remains PR-PBN no matter how gi ’s, i ∈ Inp , are changed.

Figure 8.3: Example 8.1 Quality bottlenecks: The following theorem identifies Q-BNs.

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

255

Theorem 8.4 In Bernoulli lines defined by assumptions (a)-(h) of Subsection 8.1.1 with the only inspection machine mM , the inequality d d ∂P R ∂P R > , ∂gi ∂gj

∀ j 6= i,

(8.21)

takes place if and only if gi = min gl ,

(8.22)

l∈Inp

i.e., the Q-BN is the machine with the worst quality; if min gl is achieved at more than one machine, each is a Q-BN. Proof : Follows immediately from (8.3), which implies that d d P R ∂P R = , ∂gi gi

i ∈ Inp .

(8.23)

¥ Thus, in the production line of Figure 8.3, m9 is the Q-BN and remains the Q-BN, no matter how p9 or any other pi ’s are changed. Bottlenecks in production lines with multiple inspection machines or a single inspection machine other than mM : Unfortunately, none of the above properties of PR- and Q-BNs hold for lines with multiple inspection machines or with one inspection machine other than the last one. Therefore, bottlenecks are identified directly based on Definitions 8.1 and 8.2, using finite differences instead of derivatives in (8.1) and (8.2), i.e., mi is the PR-BN if ∆P Rj (p1 , . . . , pM ) ∆P Ri (p1 , . . . , pM ) > , ∆pi ∆pj

∀ j 6= i,

(8.24)

∀ j 6= i,

(8.25)

and mi is the Q-BN if ∆P Rj (g1 , . . . , gM ) ∆P Ri (g1 , . . . , gM ) > , ∆gi ∆gj where ∆P Ri (p1 , . . . , pM ) ∆P Ri (g1 , . . . , gM )

= =

P R(p1 , . . . , pi + ∆pi , . . . , pM ) − P R(p1 , . . . , pi , . . . , pM ), P R(g1 , . . . , gi + ∆gi , . . . , gM ) − P R(g1 , . . . , gi , . . . , gM ),

and ∆pi ¿ 1,

∆gi ¿ 1.

(8.26)

Since neither ∆P Ri (p1 , . . . , pM ) nor ∆P Ri (g1 , . . . , gM ) can be evaluated on the factory floor during normal system operation, we replace them by i

d ∆P R (p1 , . . . , pM )

=

d P R(p1 , . . . , pi + ∆pi , . . . , pM )

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CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

i

d ∆P R (g1 , . . . , gM )

=

d −P R(p1 , . . . , pi , . . . , pM ),

(8.27)

d P R(g1 , . . . , gi + ∆gi , . . . , gM ) d −P R(g1 , . . . , gi , . . . , gM ),

(8.28)

where the right hand sides of (8.26) and (8.28) are calculated using recursive procedure (8.7). A question arises whether (8.23) and (8.25) imply and are implied by i

d ∆P R (p1 , . . . , pM ) ∆pi

j

>

d ∆P R (p1 , . . . , pM ) , ∆pj

>

d ∆P R (g1 , . . . , gM ) , ∆gj

i

d ∆P R (g1 , . . . , gM ) ∆gi

∀ j 6= i,

(8.29)

∀ j 6= i.

(8.30)

j

d This question arises because, although P R and P R are close to each other, their partial derivatives, in general, may be quite different. The answer is given below. Numerical Fact 8.1 For ∆pi = 0.01 and ∆gi = 0.01, (8.29) and (8.30) practically always imply and are implied by (8.24) and (8.25), respectively. Justification: The justification has been carried out using Simulation Procedure 4.1 and the 100,000 production lines constructed as mentioned in Subsection 8.1.2. As a result, it has been determined that (8.29) implies and is implied by (8.24) in 96.2% and (8.30) implies and is implied by (8.25) in 95.8% of cases considered. Thus, we conclude that ∆P Ri ∆pi

∆P Ri ∆gi

d ∆P R ∆pi

i

and

d ∆P R ∆gi

i

can be used instead of

and for PR- and Q-BNs identification. Example 8.2: Consider Line 1 shown in Figure 8.4(a). Based on (8.28) and (8.30), we determine that m2 is both PR- and Q-BN. Thus, the most effective way of productivity improvement is by increasing either p2 or g2 . Increasing p2 from 0.8 to 0.9, shifts PR-BN to m3 and Q-BN to m4 (see Figure 8.4(b)); note that the Q-BN is not the worst quality machine in the system. If, instead of increasing p2 , the value of g2 is increased from 0.82 to 0.9, the bottlenecks, shown in Figure 8.4(c), are m2 (PR-BN) and m4 (Q-BN). Note that although the line is perfectly symmetric and m2 and m4 are identical, their effect on the production rate is different. The performance characteristics of Lines 1-3 are summarized in Table 8.1. It quantifies the advantages of increasing gi ’s, rather than increasing pi ’s, from the point of view of both the production and scrap rates. Local bottlenecks: As described above, identification of bottlenecks in the general case requires evaluating all partial derivatives involved in Definitions 8.1 and 8.2. Below, we introduce the notions of local bottlenecks and show that they can be identified using the same techniques as in the case of a single inspection machine mM . In addition, we show that one of the local bottlenecks is, practically always, the global one.

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

(a) Line 1

(b) Line 2

(c) Line 3 Figure 8.4: Example 8.2

d c of Lines 1-3 in Example 8.2 Table 8.1: P R and SR d c P R SR Line 1 0.5494 0.1951 Line 2 0.5886 0.2090 Line 3 0.5967 0.1400

257

258

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

Consider the production line defined by assumptions (a)-(h) of Subsection 8.1.1 and its segmentations shown in Figure 8.5. Note that the segments in

(a) Q-segments

(b) PR-segments Figure 8.5: Segmentations Figure 8.5(a) are not overlapping and consist of producing machines located between each pair of consecutive inspection machines (or the producing machines before the first inspection machine); they are referred to as Q-segments. In contrast, the segments of Figure 8.5(b) are overlapping and consist of a pair of consecutive inspection machines and all the producing machines between them (or the producing machines before the first and after the last inspection machine); they are referred to as PR-segments. l if

Definition 8.3 Machine mli is the local PR-BN (LPR-BN) of PR-segment ∂P R ∂P R > , ∂pli ∂plj

∀ j 6= i, i, j ∈ l-th PR-segment.

(8.31)

Definition 8.4 Machine mli is the local Q-BN (LQ-BN) of Q-segment l if ∂P R ∂P R > , ∂gil ∂gjl

∀ j 6= i, i, j ∈ l-th Q-segment.

(8.32)

Theorem 8.5 The LQ-BN of each Q-segment is the machine with the smallest gi . One of the LQ-BNs is the Q-BN of the line. Proof: See Section 20.1. Example 8.3: Consider the production line shown in Figure 8.6. According d R and P R to Theorem 8.5, its LQ-BNs are m2 and m7 . The sensitivities of P to g2 and g7 are also shown in Figure 8.6 (for ∆gi = 0.01). Thus, m7 is the Q-BN. Note that it is not the worst quality machine in the system. Note also that based on LQ-BNs, two, rather than six, sensitivities must be evaluated to identify Q-BN. Numerical Fact 8.2 The LPR-BNs can be practically always identified using Bottleneck Indicator 5.1 applied to each P R-segment, under the assumption

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

259

Figure 8.6: Example 8.3 that its first machine is not starved and the last is not blocked. One of the LPR-BNs is, practically always, the PR-BN of the line. Justification: The justification has been carried out using the 100,000 production lines constructed as mentioned in Subsection 8.1.2. Both simulation and calculation approaches have been used. In the simulation approach, STi and BLi were estimated using Simulation Procedure 4.1, LPR-BNs have been identified using Bottleneck Indicator 5.1, and PR-BN has been identified using di were calculated using aggregac i and BL (8.24). In the calculation approach, ST d tion procedure (8.7), LP R-BNs have been identified using Bottleneck Indicator d 5.1, and P R-BN has been identified using (8.29). The results are as follows: • In the simulation approach, LPR-BNs are identified correctly in 91.1% of all segments, and the PR-BN is one of the LPR-BNs in 93.9% of cases. d • In the calculation approach, LP R-BNs are identified correctly in 90.3% of d d all segments, and the P R-BN is one of the LP R-BNs in 94.5% of cases. Based on these data, we conclude that Numerical Fact 8.2 can be used for PR-BN identification. Example 8.4: Consider the production line shown in Figure 8.7 along with its PR-segments 1-3. The LPR-BNs, identified using Numerical Fact 8.2 by d R both simulations and calculations, are m1 , m6 and m8 . The sensitivities of P and P R to pi ’s are also shown in Figure 8.7(a) (for ∆pi = 0.01). Thus, m1 is the PR-BN, and it can be identified, using local bottlenecks, based on three, rather than ten, sensitivities.

8.1.4

Design

Motivating examples: The problem of design considered here consists of placing inspection machines so that no defective parts are shipped to the customer and the production rate is maximized. As it is illustrated below, the solution of this problem is quite sensitive to the parameters of the machines involved.

260

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

(a) Production line

(b) PR-segment 1

(c) PR-segment 2

(d) PR-segment 3

Figure 8.7: Example 8.4 Example 8.5: Consider the production line with no inspection machines shown in Figure 8.8(a). Its bottleneck, identified using Bottleneck Indicator 5.1, is m3 . Consider also the inspection module, shown in Figure 8.8(b), with the inspection machine of efficiency pinsp and buffer of capacity 3. The problem d is to position the inspection module so that P R is maximized. To accomplish this, we carry out the full search, i.e., we place the inspection module in all possible positions, Iinsp = {i}, i = 2, 3, 4, 5, and, using aggregation procedure d (8.7), calculate P R as a function of pinsp for g1 = 0.98, 0.90 and 0.80. The results are shown in Figure 8.9. From these results, the following conclusions can be made: • For g1 = 0.98: If pinsp is low, the optimal position of minsp is at the end of the line, i.e., m5 ; if pinsp is mid-range, it is immediately after the nonperfect machine, i.e., m2 ; if pinsp is high, the optimal position of minsp is immediately before the BN, i.e., m3 . • For g1 = 0.90: The situation is quite similar but with different ranges of pinsp leading to various optimal positions of the inspection machine. • For g1 = 0.80: For all values of pinsp , the optimal position of the inspection machine is immediately after the non-perfect machine, i.e., m2 . Example 8.6: Consider the production line of Figure 8.10 with m2 as its BN. Placing the inspection module in all possible positions (i.e., Iinsp = {i},

8.1. LINES WITH NON-PERFECT QUALITY MACHINES

(a) Original production line and its BN

261

(b) Inspection module

Figure 8.8: Example 8.5 0.7

0.7

0.7

0.65

0.65

0.65

0.6

0.6

d PR

d PR

0.55

0.55

0.5

0.5

0.45 0.6

0.7

0.8 p

0.9

insp

(a) g1 = 0.98

1

0.45 0.6

Iinsp={2}

d PR

0.55

Iinsp={4}

0.5

insp

I

={5}

insp

0.7

0.8 p

0.6

={3}

I

0.9

1

0.45 0.6

0.7

insp

(b) g1 = 0.90

0.8 p

0.9

insp

(c) g1 = 0.80

d Figure 8.9: P R as a function of pinsp and Iinsp for Example 8.5 i = 4, 5, 6, 7) and using aggregation procedure (8.7), we obtain the results shown in Figure 8.11. Although they are quite similar to those of Figure 8.9, there is an important difference: for large pinsp , the optimal position of the inspection module is immediately after the non-perfect quality machine, no matter what the value of g3 is.

(a) Original production line and its BN

(b) Inspection module

Figure 8.10: Example 8.6 Based on the above observations and taking into account that in most cases inspection machines are of high efficiency, we limit below our attention to the case when pinsp is sufficiently close to 1 and formulate an empirical rule for a single inspection machine positioning. Empirical design rule: Let mlast denote the last non-perfect quality machine in the line defined by assumptions (a)-(h) with |Iinsp | = 0, i.e., the line

1

262

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

0.7

0.7

0.7

0.65

0.65

0.65

0.6

0.6

d PR

d PR

0.55

0.55

0.5

0.5

0.45 0.6

0.7

0.8 p

0.9

insp

(a) g3 = 0.98

1

0.45 0.6

I

={4}

I

={5}

I

={6}

I

={7}

d PR

insp insp

0.5

insp insp

0.7

0.8 p

0.9

insp

(b) g3 = 0.90

0.6

0.55

1

0.45 0.6

0.7

0.8 p

0.9

insp

(c) g3 = 0.80

d Figure 8.11: P R as a function of pinsp and Iinsp for Example 8.6 with perfect and non-perfect quality machines but without an inspection system. Refer to such a line as the original line. The position of a single inspection d machine, which results in the maximum P R, is referred to as optimal. Let BN denote a bottleneck of the original line identified using Bottleneck Indicator 5.1. Numerical Fact 8.3 There exists a p∗ such that for all pinsp ≥ p∗ , the optimal position of the inspection module is, practically always, either immediately after mlast or around a BN located downstream of mlast . This fact provides an empirical rule: If there are no BNs of the original line downstream of mlast , the optimal position of an efficient inspection machine is immediately after mlast . If there are BNs downstream of mlast , there are several candidates that must be explored before the optimal position of minsp is determined − immediately after mlast and immediately after or before each BN located downstream of mlast . d Justification: The justification is carried out by calculating P R for each possible position of the inspection module and determining whether there exists a p∗ such that for all pinsp ≥ p∗ , the stated fact takes place. Specifically, 100,000 of five- and 100,000 of ten-machine lines defined by assumptions (a)-(h) with |Iinsp | = 0 have been investigated. The parameters pi , gi , and Ni have been selected randomly and equiprobably from sets (8.15)-(8.17). The position of the last non-perfect quality machine, mlast , was selected randomly and equiprobably from the set {1, 2, . . . , M − 1}. Each machine upstream of mlast was chosen to be perfect and non-perfect with probability 1/2. For each of these lines, BNs have been identified using Bottleneck Indicator 5.1. It turned out that all BNs were upstream of mlast in 42.7% of cases. In the remaining 57.3% of cases, one or more BNs were downstream of mlast . The inspection module consisted of a buffer and an inspection machine. The buffer capacity was selected as 2 or 3 with probability 1/2. The production rate of each line, thus constructed, with the inspection module placed in all possible positions, was evaluated using aggregation procedure (8.7) for pinsp ∈ [0.6, 1). The results of this analysis are as follows: Numerical Fact 8.3 holds in 99.7% of cases when all BNs of the original line are upstream of mlast and in 97.2% of

1

8.2. BERNOULLI LINES WITH QQC MACHINES

263

cases when one or more BNs are downstream of mlast . Thus, we conclude that Numerical Fact 8.3 practically always takes place. Numerical Fact 8.3 allows for the optimal placement of a single, sufficiently efficient inspection machine without evaluating the efficacy of all possible placements. No results of this nature are available for placing multiple inspection machines. The recommendation that can be given at this time is to carry out a full search of all possible placements, evaluating the efficacy of each one using aggregation procedure (8.7).

8.2 8.2.1

Bernoulli Lines with Quality-Quantity Coupling Machines Model and problem formulation

Model: We consider here the same model as in the previous section, i.e., defined by conventions (a)-(h) of Subsection 8.1.1, along with an additional assumption: (i) The non-perfect quality machines are of two types: with and without quality-quantity coupling (QQC). In the former case, gi = gi (pi ), i ∈ IQQC ⊆ Inp , where gi (pi ) is a differentiable monotonically decreasing function. Here IQQC is a set of indices representing the positions of the QQC machines. Also, to simplify the presentation, we assume that there is no inspection machines, and the last producing machine, mM , separates good and defective parts. Such a system is illustrated in Figure 8.12.

Figure 8.12: Serial production line with perfect quality, non-perfect quality, and QQC machines

Problems: Clearly, when pi ’s are fixed, the QQC plays no role, and therefore d c and CR d in systems defined by assumptions (a)-(i) the calculation of P R, SR, can be carried out using the method developed in the previous section. Also, the PR-BNs and Q-BNs can be defined as in Definitions 8.1 and 8.2. However, the monotonicity and PR-BNs might have different properties. Therefore, the following problems are addressed in this section: d Monotonicity: Investigate the monotonicity properties of P R with respect to pi , i ∈ IQQC . Bottlenecks: Introduce the following

264

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES Definition 8.5 In a serial line defined by assumptions (a)-(i), machine mi

is: • p+ -BN if

¯ ¯ ¯ ¯ ¯ ∂P R ¯ ¯ ∂P R ¯ ¯ ¯>¯ ¯ ¯ ∂pi ¯ ¯ ∂pj ¯ ,

∀ j 6= i,

(8.33)

and ∂P R > 0; ∂pi • p− -BN if (8.33) holds and ∂P R < 0. ∂pi Clearly, if the BN is a p+ -BN, the largest improvement of P R is obtained by increasing the efficiency of the BN machine; if the BN is a p− -BN, decreasing its efficiency leads to the largest increase of P R. The second problem addressed in this section is to provide methods for p+ and p− -BN identification.

8.2.2

Monotonicity properties

Two-machine lines: Theorem 8.6 Consider a Bernoulli line defined by assumptions (a)-(i) of Subsection 8.2.1 with M = 2 and N ≥ 2. Then, there exist p∗i and p∗∗ i , 0 < p∗i ≤ p∗∗ i < 1, i ∈ IQQC , such that P R, as a function of pi , is monotonically increasing for pi ∈ (0, p∗i ) and monotonically decreasing for pi ∈ (p∗∗ i , 1). Proof : See Section 20.1. Example 8.7: Consider the production line shown in Figure 8.13. Assume g1 (p1 ) = 1 − 0.4p1 .

(8.34)

The production rate of this system, calculated using (8.3), is illustrated in Figure 8.14. Clearly for every p2 , P R is a non-monotonic concave function of p1 , i.e., in this case, p∗1 = p∗∗ 1 .

Figure 8.13: Example 8.7

8.2. BERNOULLI LINES WITH QQC MACHINES 0.55 0.5 PR 0.45

265

p2 = 0.9 p2 = 0.8 p2 = 0.7

0.4 0.35 0.5

0.6

0.7

p1

0.8

0.9

1

Figure 8.14: P R as a function of p1 and p2 in Example 8.7 M > 2-machine lines: Due to the complexity of recursive procedure (4.30), an analytical extension of Theorem 8.6 for M > 2 is all but impossible to derive. However, based on numerical calculations, the follows can be established: Numerical Fact 8.4 For a Bernoulli line defined by assumptions (a)-(i) of Subsection 8.2.1 with M > 2 and Ni ≥ 2, i = 1, . . . , M − 1, there exist d 0 < p∗i ≤ p∗∗ i < 1, i ∈ IQQC , such that P R, as a function of pi , is monotonically increasing for pi ∈ (0, p∗ ) and monotonically decreasing for pi ∈ (p∗∗ i , 1). Justification: We constructed 500,000 lines by selecting M , pi ’s and Ni ’s randomly and equiprobably from the following sets, respectively: M pi Ni

∈ {3, 5, 10}, ∈ [0.7, 0.95], ∈ {2, 3, 4, 5}.

(8.35) (8.36) (8.37)

Each machine was chosen to be perfect quality, non-perfect without QQC or non-perfect with QQC with probability 1/3. For non-perfect machines without QQC, the machine quality parameter gi was selected randomly and equiprobably from the set gi ∈ [0.7, 1).

(8.38)

For each QQC machine, function gi (pi ) was selected randomly and equiprobably from the nine QQC functions shown in Figure 8.15. The production rate was evaluated using recursive procedure (4.30). Among all the lines, thus constructed, no counter example to Numerical Fact 8.4 has been found. Example 8.8: Consider the production line shown in Figure 8.16. Assume that g3 (p3 ) =

1 0.6p3 + 1

(8.39)

The production rate of the system as a function of p3 for various values of p4 is illustrated in Figure 8.17.

266

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

(a) Linear quality

(b) Concave quality

(c) Convex quality

Figure 8.15: QQC functions for the justification of Numerical Fact 8.4

0.92

m1

2 0.83, 0.95 3 p3, g3(p3) 3

b1

m2

b2

m3

b3

p4

m4

2 0.85, 0.90

b4

m5

Figure 8.16: Example 8.8

d Figure 8.17: P R as a function of p3 of Example 8.8

8.2. BERNOULLI LINES WITH QQC MACHINES

8.2.3

267

Bottlenecks

Two-machine lines: Given the machine and buffer parameters, the p+ - and p− -BNs of two-machine lines defined by assumptions (a)-(i) of Subsection 8.2.1 R ∂P R can be identified directly using expressions for ∂P ∂p1 and ∂p2 derived in the proof of Theorem 8.6 (see Section 20.1). Specifically, these expressions are ¶ µ ∂P R ∂Q(p1 , p2 , N1 ) dg(p1 ) = p2 (1 − Q(p1 , p2 , N1 )) − g1 (p1 ) g2 (p2 ), (8.40) ∂p1 dp1 ∂p1 ¶ µ ∂P R ∂Q(p2 , p1 , N1 ) dg(p2 ) = p1 (1 − Q(p2 , p1 , N1 )) − g2 (p2 ) g1 (p1 ), (8.41) ∂p2 dp2 ∂p2 where  

y(1−x)(αN (x,y)−1)+(y−x)N αN (x,y) ∂Q(x, y, N ) , (y−xαN (x,y))2 (1−x) =  − N (N +1) 2 , ∂x 2p(N +1−p)

if x 6= y, if x = y = p.

(8.42)

Based on these expressions, the properties of the bottlenecks are illustrated below. Example 8.9: Consider the production line shown in Figure 8.18 with gi (pi ) =

1 , 0.6p3i + 1

pi ∈ (0, 1), i = 1, 2.

Its bottlenecks, identified using (8.40) and (8.41), are shown in Figure 8.19(a). Note that if both machines were of perfect quality, m1 would be p+ -BN in the lower triangle of Figure 8.19(a) and m2 in the upper one. Thus, having QQC machines preserves the same bottlenecks for pi ’s suffciently small and reverses them for larger pi ’s.

Figure 8.18: Example 8.9 A similar situation takes place when the quality functions, gi (pi ), are not identical. This is illustrated in Figure 8.19(b) for g1 (p1 )

=

g2 (p2 )

=

1 , 0.6p31 + 1 1 , 0.4p32 + 1

p1 ∈ (0, 1), p2 ∈ (0, 1).

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CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

(a) Identical QQC machines

(b) Non-identical QQC machines

(c) QQC and non-QQC machines

Figure 8.19: Bottlenecks in Example 8.9 Finally, the bottlenecks in systems with only one QQC machine are illustrated in Figure 8.19(c) for g1 (p1 )

=

g2 (p2 )

=

1 , 0.6p31 + 1 0.9.

p1 ∈ (0, 1),

Thus, when both machines are QQC, the bottleneck can be eliminated by increasing its efficiency if pi ’s are sufficiently small and decreasing its efficiency if pi ’s are large. When only one machine is QQC, if the non-QQC machine is the bottleneck, it is always p+ -BN, while if the QQC machine is the bottleneck, it is p+ -BN for small pi ’s and p− -BN for large ones. M > 2-machine lines: Unfortunately, the partial derivatives involved in Definition 8.5 cannot be calculated using closed-form expressions for M > 2. Therefore, the BNs are identified using finite difference estimates instead of the partial derivatives in (8.32), i.e., mi is the p+ -BN if ¯ ¯ ¯ ¯ j ¯ ¯ ¯ ∆P Ri (p1 , . . . , pM ) ¯ ¯ > ¯ ∆P R (p1 , . . . , pM ) ¯ , ∀ j 6= i, ¯ (8.43) ¯ ¯ ¯ ¯ ∆pi ∆pj ∆P R > 0, (8.44) ∆pi where ∆P Ri (p1 , . . . , pM ) = P R(p1 , . . . , pi + ∆pi , . . . , pM ) − P R(p1 , . . . , pi , . . . , pM ), (8.45) ∆pi ¿ 1,

(8.46)

and p− -BN if the inequality in (8.44) is reversed. Since ∆P Ri (p1 , . . . , pM ) cannot be evaluated on the factory floor during normal system operation, we replace it by i

d ∆P R (p1 , . . . , pM )

8.3. BERNOULLI LINES WITH REWORK

269

d d =P R(p1 , . . . , pi + ∆pi , . . . , pM ) − P R(p1 , . . . , pi , . . . , pM ),(8.47) where the right hand side of (8.47) is calculated using recursive procedure (4.30). Thus, inequalities (8.43) and (8.44) become ¯ ¯ ¯ ¯ i j ¯ ∆P ¯ ∆P R (p1 , . . . , pM ) ¯¯ R (p1 , . . . , pM ) ¯¯ ¯ d ¯ d (8.48) ¯ > ¯ ¯ , ∀ j 6= i, ¯ ¯ ¯ ¯ ¯ ∆pi ∆pj d ∆P R ∆pi

>

0.

(8.49)

For production lines defined by assumptions (a)-(h), i.e., lines with no QQC machines, the equivalence of (8.43) and (8.48) has been justified in the previous section. A similar result is obtained for systems considered here. Numerical Fact 8.5 For ∆pi = 0.01, (8.43) and (8.44) practically always imply and are implied by (8.48) and (8.49), respectively. Justification: The justification has been carried out using the 500,000 prod duction lines constructed as mentioned in Subsection 8.2.1. For each line, P R was calculated using aggregation procedure (4.30) and P R was evaluated using Simulation Procedure 4.1. As a result, it has been determined that (8.43) implies and is implied by (8.48), and (8.44) implies and is implied by (8.49) in 93.7% and 95.8% of cases considered, respectively. Thus, we conclude that (8.48) and (8.49) can be used instead of (8.43) and (8.44) for BN identification. Example 8.10: Consider the production lines shown in Figures 8.20(a) with g3 (p3 ) = g4 (p4 ) =

1 , 0.34p33 + 1 1 . 0.43p34 + 1

(8.50) (8.51)

Using simulations and calculations to identify its BN (with ∆pi = 0.01), we determine that m4 is the p− -BN. Decreasing its efficiency by 0.05, we observe that m2 becomes the p+ -BN (Figure 8.20(b)). With its efficiency increased by 0.05, the bottleneck shifts to m3 , which is the p+ -BN (Figure 8.20(c)). Increasing now its efficiency by 0.1, we determine that it remains the bottleneck, however, now it is p− -BN (Figure 8.20(d)). Decreasing its efficiency by 0.05, results in m1 as the p+ -BN (Figure 8.20(e)). This example illustrates that in serial lines with QQC machines the process of bottleneck identification and elimination is more involved than that for other serial lines.

8.3

Bernoulli Lines with Rework

In some production systems, it is more economical to have defective parts repaired and reworked, rather than scrapped. In these cases, the system has a

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CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

(a)

(b)

(c)

(d)

(e) Figure 8.20: Bottlenecks in Example 8.10

8.3. BERNOULLI LINES WITH REWORK

271

structure illustrated in Figure 8.21. Here machines m1 , . . . , mM constitute the main line, mr1 , . . . , mrMr represent the repair line, and mk and mj are the split and merge machines, respectively; clearly, the split machine carries out the inspection operation. The probability that a good part emerges at the output of mk , denoted as q, is referred to as the quality buy rate. Systems of this type are the topic of study in this section.

Figure 8.21: Serial production line with rework

8.3.1

Model and problem formulation

Model: We consider the same model as in the previous section, i.e., defined by conventions (a)-(i) of Subsection 8.2.1, along with the following additional assumptions: (j) The merge machine mj is starved if both bj−1 and brMr are empty. The split machine mk is blocked by the main line if bk is full and mk+1 does not take a part during this time slot; it is blocked by the rework loop if br0 is full and mr1 does not take a part during this time slot. (k) Parts from the repair line have a higher priority than those from the main line. In other words, mj does not take a part from bj−1 unless brMr is empty. To simplify the presentation, we assume that mk is the only inspection machine in the system (i.e., Iinsp = {k}) and no QQC machines are present (i.e., |IQQC | = 0). Clearly, in the line defined by assumptions (a)-(k), the quality buy rate q depends on the quality parameters of the machines gi , i ∈ Inp . Indeed, introduce gI =

j−1 Y i=1

gi ,

gII =

k Y

gi ,

i=j

gIII =

M Y

gi = 1,

i=k+1

gIV =

rM Yr

gi .

i=r1

Then, q can be expressed as q = gII [ qgI + (1 − q)gIV ] ,

(8.52)

i.e., q=

gII gIV . 1 + (gIV − gI )gII

(8.53)

272

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

Problems: lems:

In the framework of model (a)-(k), we consider the following prob-

Performance analysis: Given the machine and buffer parameters, evaluate the production rate of the line with rework (P Rlwr ) and the probabilities of lwr ) of each machine in the system. Since blockages (BLlwr i ) and starvations (STi mj has two upstream buffers, bj−1 and brMr , and mk has two downstream buffers, bk+1 and br0 , their starvations and blockages are denoted as follows: STjlwr 1

=

P [mj is starved by bj−1 ],

STjlwr 2 BLlwr k1 BLlwr k2

=

P [mj is starved by brMr ],

=

P [mj is blocked by bk+1 ],

=

P [mj is blocked by br0 ].

In addition, investigate system-theoretic properties of Bernoulli lines with rework. A solution to this problem is given in Subsection 8.3.2. Bottlenecks: In the same manner as in Section 8.1, it is possible to show that P Rlwk is a monotonically increasing function of pi ’s. Therefore, the bottleneck machine can be defined as in Definition 5.2: mi is the bottleneck (BN) if ∂P Rlwr ∂P Rlwr > , ∂pi ∂pl

∀l 6= i.

(8.54)

However, due to the split and merge operations, Bottleneck Indicator 5.1 is not directly applicable. An extension of this indicator to lines with rework is described in Section 8.3.3.

8.3.2

Performance analysis

Approach: Due to the complexity of Markov chains involved in their description, a direct analysis of Bernoulli lines with rework does not seem feasible. Therefore, we use a simplification technique based on overlapping decomposition, which represents the line with rework as four overlapping serial lines and four virtual serial lines shown in Figure 8.22(a) and (b), respectively. The overlapping lines include the overlapping machines mj and mk , each belonging to three lines. The virtual lines do not contain overlapping machines; instead, they include six virtual machines, m1j , m2j , m4j and m2k , m3k , m4k , the efficiencies of which are selected so as to represent the effect of the rest of the system on a particular virtual line. Calculating appropriately the efficiencies, p1j , p2j , p4j and p2k , p3k , p4k , of the virtual machines, allows us to analyze the performance of the original line with rework. To calculate the efficiencies of these virtual machines, we use the recursive d aggregation procedure (4.30), the performance measures P R, given by (4.36), d and ST c , given by (4.39) and (4.40). In addition, we need the probability and BL that in a serial line with M machines the first buffer is full while the second

8.3. BERNOULLI LINES WITH REWORK

273

(a) Overlapping serial lines

(b) Virtual serial lines Figure 8.22: Overlapping decomposition of serial production lines with rework

274

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

machine is either blocked or down and the probability that the last buffer is b 1 and st b M , respectively. As it follows empty; we denote these probabilities as bl from (4.39) and (4.40), they can be evaluated as b 1 = BL d1 /p1 = Q(pb2 , p1 , N1 ), bl

(8.55)

c M /pM = Q(pf b M = ST st M −1 , pM , NM −1 ).

(8.56)

and

Recursive aggregation procedure for overlapping decomposition: Consider the virtual lines of Figure 8.22(b) and introduce the following notations: P Rl BLli

= =

Production rate of virtual line l, l = 1, 2, 3, 4, P [machine mi in virtual line l is blocked], l = 1, 2, 3, 4,

STil

=

P [machine mi in virtual line l is starved], l = 1, 2, 3, 4,

blj stj1 blk1

= = =

P [bj is full and mj+1 is either down or blocked], P [bj−1 is empty], stj2 = P [brMr is empty], P [bk is full and mk+1 is either down or blocked],

blk2 stk

= =

P [br0 is full and mr1 is either down or blocked], P [bk−1 is empty].

The estimates of these probabilities, which allow us to evaluate the parameters p1j , p2j , p4j and p2k , p3k , p4k of the virtual machines m1j , m2j , m4j and m2k , m3k , m4k , can be calculated using a recursive procedure described below. Recursive Procedure 8.1: b j (0), and bl b k (0) randomly b k (0), bl Step 0: Select the initial conditions st 1 and equiprobably from the interval (0,1). Step 1: Consider virtual line 4 of Figure 8.22(b) and update the efficiencies of machines m4j and m4k as follows: p4k (n + 1)

=

p4j (n + 1)

=

b k (n)], b k (n)][1 − bl pk (1 − q)[1 − st 1 b pj [1 − blj (n)], n = 0, 1, . . . .

Then perform recursive procedure (4.30) on virtual line 4 and, using (4.36), 4 4 d di (n + 1), ST c i (n + (4,39), (4,40), (8.55), and (8.56), calculate P R4 (n + 1), BL b k (n + 1), and st b j2 (n + 1). 1), bl 2 Step 2: Consider virtual line 3 of Figure 8.22(b) and update the efficiency of machine m3k as follows: p3k (n + 1)

=

b k (n + 1)], b k (n)][1 − bl pk q[1 − st 2

8.3. BERNOULLI LINES WITH REWORK

275

n = 0, 1, . . . . Then perform recursive procedure (4.30) on virtual line 3 and, using (4.36), 3 3 d di (n + 1), ST c i (n + 1), (4,39), (4,40), and (8.55), calculate P R3 (n + 1), BL b k (n + 1). and bl 1

Step 3: Consider virtual line 1 of Figure 8.22(b) and update the efficiency of machine m1j as follows: p1j (n + 1)

=

b j (n)]st b j2 (n + 1), pj [1 − bl n = 0, 1, . . . .

Then perform recursive procedure (4.30) on virtual line 1 and, using (4.36), 1 1 d di (n + 1), ST c i (n + 1), (4,39), (4,40), and (8.56), calculate P R1 (n + 1), BL b j1 (n + 1). and st Step 4: Consider virtual line 2 of Figure 8.22(b) and update the efficiencies of machines m2j and m2k as follows: p2j (n + 1)

=

b j1 (n + 1)st b j2 (n + 1)], pj [1 − st

p2k (n + 1)

=

b k (n + 1)][1 − bl b k (n + 1)], pk [1 − bl 1 2 n = 0, 1, . . . .

Then perform recursive procedure (4.30) on virtual line 2 and using (4.36), 2 2 d d (n + 1), ST c (n + (4,39), (4,40), (8.55), and (8.56), calculate P R2 (n + 1), BL i i b j (n + 1), and st b k (n + 1). 1), bl Step 5: If the stopping rule 4 X

d d |P Rl (n + 1) − P Rl (n)| < ε,

ε ¿ 1,

(8.57)

l=1

is satisfied, the procedure is terminated; otherwise return to Step 1. Denote the limits of this procedure as d d P Rl := lim P Rl (n), n→∞

l = 1, 2, 3, 4,

b j := lim bl b j (n), bl n→∞

b j1 := lim st b j1 (n), st n→∞

b j2 := lim st b j2 (n), st n→∞

(8.58)

b k := lim st b k (n), st n→∞

b k := lim bl b k (n), bl 1 1 n→∞

b k (n). b k := lim bl bl 2 2 n→∞

Unfortunately, the existence of, and convergence to, these limits cannot be b k (n), n = 0, 1, . . . ). proved analytically (due to a non-monotonic behavior of bl 2 Therefore, it has been investigated numerically.

276

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES Numerical Fact 8.6 Recursive Procedure 8.1 is convergent.

Justification: A total of 5,000,000 lines have been generated by selecting j, k, M , Mr , pi ’s, Ni ’s, and q randomly and equiprobably from the sets: j

∈ {2, 3, 4, 5},

(8.59)

k−j M −k

∈ {2, 3, 4, 5}, ∈ {1, 2, 3, 4},

(8.60) (8.61)

Mr

∈ {1, 2, 3, 4},

(8.62)

∈ ∈

(8.63) (8.64)

pi pi Ni q

[0.7, 0.95], i = 1, . . . , M, [0.1, 0.5], i = r1, . . . , rMr ,

∈ {1, 2, 3, 4, 5}, ∈ [0.7, 0.95].

i ∈ Ib ,

(8.65) (8.66)

Note that the efficiencies of the machines in the rework loop are selected lower than those of the main line because, in practice, the capacity of the repair part of the system is typically smaller than that of the main line. For each of the lines, we ran Recursive Procedure 8.1 (with ε = 10−10 in (8.59)) and observed the convergence taking place within a second using a standard laptop. Thus, we conclude that this procedure can be used for analysis of production lines with rework defined by assumptions (a)-(k). Concluding this paragraph, we point out the following relationships among the production rates of the virtual lines 1-4: Theorem 8.7 The production rates of the virtual lines are related as follows: d P R1 d P R3 d P R4

= = =

d P R3 , d q P R2 , d (1 − q)P R2 .

Proof : See Section 20.1.

Performance measure estimates and their accuracy: its (8.58) of Recursive Procedure 8.1, introduce:

Based on the lim-

d d R3 , P Rlwr = P lwr ci ST lwr cj ST 1 lwr ck ST

=

l c i, ST

b j1 pj , = st b k pk , = st

(8.67) lwr l di = BL di , i BL lwr c j = st b j2 pj , ST 2 lwr b k pk , dk = bl BL 1 1

6= j, k; l = 1, 2, 3, 4, lwr dj BL lwr dk BL 2

(8.68)

b j pj , = bl

(8.69)

b k pk . = bl 2

(8.70)

The accuracy of these estimates has been investigated using a C++ code that simulates the system at hand and Simulation Procedure 4.1. Specifically,

8.3. BERNOULLI LINES WITH REWORK

277

for the system parameters M = 10, j = 4, k = 7, and Mr = 2, we constructed 100,000 lines with pi ’s, Ni ’s and q’s selected randomly and equiprobably from sets (8.63)-(8.66). The following metrics were used to evaluate the accuracy of the estimates: ²P R

=

²ST

=

d |P Rlwr − P Rlwr | · 100%, P Rlwr X lwr 1 c i |, |STilwr − ST M + Mr

(8.71) (8.72)

i∈S1

²BL

S1 = {1, . . . , j − 1, j1 , j2 , j + 1, . . . , rMr }, X lwr 1 di |, = |BLlwr − BL i M + Mr

(8.73)

i∈S2

S2 = {1, . . . , k − 1, k1 , k2 , k + 1, . . . , rMr }. Among the 100,000 lines studied, the average of ²P R was 3.97%, with very few extreme cases resulting in ²P R up to 20.4%. The averages of ²ST and ²BL were both less than 0.01. Therefore, we conclude that Recursive Procedure 8.1 and (8.67)-(8.70) provide an effective tool for performance evaluation of serial lines with rework defined by assumptions (a)-(k). System-theoretic properties of lines with rework: Reversibility: As it is shown in Subsection 4.3.2, serial production lines with no rework observe the property of reversibility: the production rates of a serial line and its reverse are the same. However, as we show below, reversibility does not hold for lines with rework. Indeed, consider the serial line and its reverse shown in Figure 8.23(a) and (b), respectively. Using Recursive Procedure 8.1, we determine that the production rate of the reverse line is not the same as that of the original one; this conclusion is also supported by simulations (see the data of Figure 8.23). Thus, reversibility is violated. The lack of reversibility constitutes a fundamental difference between the usual (i.e., open) Bernoulli lines and those with rework. In addition, comparing the data of Figure 8.23, we observe that placing more efficient machines toward the end of the line yields a higher production rate than placing them upstream. This is also qualitatively different from serial lines with no rework where the position of a machine does not indicate its importance for the performance of the system. The reasons for the loss of reversibility in Bernoulli lines with rework can be explained by the asymmetric routing of jobs at the merge and the split machines. Indeed, at the input of the merge machine, the repaired jobs have higher priority than the new ones. Also, after the split machine, the routing of the jobs is based on the quality buy rate. Finally, the merge machine is starved when both bj−1 and brMr are empty, while, due to the blocked before service convention, the split machine does not produce a part when it is blocked either by bk or by br0 . As a result, the flow of jobs in Bernoulli lines with rework is

278

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

0.9

3

0.9

3

0.9

3

0.9

m1

m b1 m2 b2 m3 b3 4

3

0.9

3

0.9

3

0.7

m b4 m5 b5 m6 b6 7 0.2

3

3

0.2

3

0.7

3

0.7

3

0.7

b7 m8 b8 m9 b9 m10 q = 0.80

3

br2 mr2 br1 mr1 br0

d (a) Production System 1: P Rlwr = 0.4779, P Rlwr = 0.4657 0.7

3

m1r

b1

r

0.7

3

m2r

b2

r

0.7

3

m3r

b3

r

0.7 m4r

3 r

b4 3

0.9

3

m5r

b5

m6r b6r m7

3

0.2

0.2

r r br2 mr2

0.9

3

0.9 r

r

3

3

0.9

3

b7r

m8r

b8

r

0.9

3

0.9

m9r

b9

r

r m10

q = 0.80

r r r br1 mr1 br0

d (b) Reverse of Production System 1: P Rlwr = 0.5330, P Rlwr = 0.4903 Figure 8.23: Production system with rework and its reverse no longer reversible, and the downstream machines have a larger effect on the production rate than do the upstream ones. Note that the above reasons do not hold if the split and the merge machines are mM and m1 , respectively. Thus, lines with rework, where the repaired parts are processed by all machines in the system, are reversible. Monotonicity properties: It is also possible to show that the production rate of Bernoulli lines with rework is monotonically increasing as a function of all machine and buffer parameters, including gi ’s, and, thus, the quality buy rate q.

8.3.3

Bottleneck identification

Approach: The method of the previous subsection reduces a line with rework to four usual serial lines, i.e., the virtual lines of Figure 8.22(b). Their BNs can be identified using BLli and STil , l = 1, 2, 3, 4, evaluated according to Recursive Procedure 8.1. This, however, would identify not the BN of the line with rework but the BNs of the serial lines, where the rest of the system is represented by virtual machines m1j , m2j , m4j and m2k , m3k , m4k . In other words, each of these BNs would be a machine with the strongest effect on the overall production rate from the point of view of an individual virtual line. Although these bottlenecks may be of interest in some applications, our goal is to identify the true BN of the line with rework in the sense of (8.54). Therefore, rather than using the virtual lines, at the first stage of the identification procedure, we determine BNs of the four overlapping lines of Figure 8.22(a); we refer to them as local bottlenecks (LBNs). Then, at the second stage, we identify the overall bottleneck of the line with rework, which is referred to as the global bottleneck (GBN). We describe below how LBNs can be identi-

8.3. BERNOULLI LINES WITH REWORK

279

fied (using either calculated or measured data) and show that one of them is, practically always, the GBN.

Local bottlenecks identification: Consider the line with rework of Figure 8.22(a). Represent its overlapping lines as shown in Figure 8.24. Clearly, each overlapping machine enters three of them. Assume that these lines are in isolation, i.e., the first machines are not starved and the last ones are not blocked. The probabilities of blockages and starvations of all other machines can be either calculated or measured during normal system operation. Specifically, when the parameters of the machines and buffers are known, lwr lwr di can be calculated using Recursive Procedure 8.1 and expresc ST i and BL sions (8.68)-(8.70). Based on these data, the BN of each overlapping line can be identified using Bottleneck Indicator 5.1. When the parameters of the machines and buffers or the quality buy rate are and BLlwr can be evaluated during normal system operanot available, ST lwr i i tion, keeping in mind that mj and mk can be starved and blocked, respectively, by two buffers. Based on these data, again the four LBNs can be identified using Bottleneck Indicator 5.1. Clearly, the same can be carried out when a simulation model is available.

Figure 8.24: LBNs identification Thus, based on either Recursive Procedure 8.1 and expressions (8.68)-(8.70) or factory floor measurements or simulations, one can identify four local bottlenecks of a Bernoulli line with rework.

280

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

Global bottleneck identification The GBN identification is based on the following Numerical Fact 8.7 In Bernoulli lines with rework defined by assumptions (a)-(k) of Subsection 8.3.1, the GBN is, practically always, one of the four LBNs. Although a justification of this fact (including the quantification of the term “practically always”) is given later on (along with other numerical facts formulated below), its application is clear: To identify the GBN, one must test the effect of each LBN on the production rate of the system; the LBN with the largest effect is the GBN. While this process is somewhat involved, it can be facilitated by the following: Numerical Fact 8.8 For Bernoulli lines with rework defined by assumptions (a)-(k) of Subsection 8.3.1, (α) if an overlapping machine is the LBN in three overlapping lines, it is practically always the GBN; (β) if an overlapping machine is the LBN in only one overlapping line, it is practically never the GBN. Numerical Fact 8.9 In a Bernoulli line with rework defined by assumptions (a)-(k) of Subsection 8.3.1 and with quality buy rate q ∗ , (α) if GBN is a non-overlapping machine of line 1, then this machine is practically always the GBN for all q > q ∗ ; (β) if GBN is a non-overlapping machine of line 3, then this machine is practically always the GBN for all q > q ∗ ; (γ) if GBN is a non-overlapping machine of line 4, then this machine is practically always the GBN for all q < q ∗ ; (δ) if GBN is a non-overlapping machine of line 2 or line 4, then the GBN is practically always in either line 2 or line 4 for all q < q ∗ . Justification: The justification of Numerical Facts 8.7-8.9 has been carried out as follows: A total of 100,000 lines have been generated with M = 10, j = 4, k = 7, and Mr = 2 and parameters of machines and buffers selected randomly and equiprobably from sets (8.63)-(8.66). For the calculation-based approach, each of these lines has been analyzed using Recursive Procedure 8.1 and the LBNs have been identified by calculating blockages and starvations using (8.68)-(8.70). Then, the partial derivatives have been estimated using d d ∆P Rlwr ∂P Rlwr ≈ , ∂pi ∆pi with ∆pi = 0.001. For the measurement-based approach, each of these lines has been analyzed using Simulation Procedure 4.1 and the partial derivatives have been estimated via simulation using ∆P Rlwr ∂P Rlwr ≈ , ∂pi ∆pi

8.3. BERNOULLI LINES WITH REWORK

281

with ∆pi = 0.01. Based on this procedure, the following results, quantifying the term “practically always,” have been obtained: • Numerical Fact 8.7 holds in 97.7% of cases using calculations and 93.3% of cases using measurements. • Numerical Fact 8.8: – Claim (α) holds in 88.1% of cases using calculations and 86.9% of cases using measurements. – Claim (β) holds in 95.3% of cases using calculations and 94.9% of cases using measurements. • Numerical Fact 8.9: – Claim (α) holds in 91.3% of cases using calculations and 92.2% of cases using measurements. – Claim (β) holds in 94.6% of cases using calculations and 94.0% of cases using measurements. – Claim (γ) holds in 96.7% of cases using calculations and 95.4% of cases using measurements. – Claim (δ) holds in 97.0% of cases using calculations and 97.2% of cases using measurements. Based on these data, we conclude that Numerical Facts 8.7-8.9 indeed take place. Example 8.11: Consider the line with rework shown in Figure 8.25 along with corresponding overlapping lines 1-4. Their local bottlenecks, identified both by calculation- and measurement-based approaches (using simulations), are also indicated. Since m4 is the LBN in only one of the overlapping lines, according to Numerical Fact 8.8(β) it is not the GBN. Thus, the candidates are m7 and mr1 . Increasing their efficiencies by 0.01, we determine that mr1 is the GBN. According to Numerical Fact 8.9(γ), this machine remains the GBN for all quality buy rate less than 0.8. Note that in this case (as well as in all other cases considered below), the two-stage procedure identifies correctly the GBN (verified by increasing each pi , i ∈ Im , by 0.01 and evaluating the resulting effect on the P R). Assume now that the quality buy rate in this system increases to 0.85. Then, as it follows from Figure 8.26 and Numerical Fact 8.8(β), the GBN is either m7 or m6 . Increasing their efficiencies by 0.01 leads to the conclusion that m6 is the GBN. Note that it is not the worst machine of the main line. Further, Figures 8.27 and 8.28 show that the GBNs are m8 and m1 , if the quality buy rates are 0.9 and 0.95, respectively. This example indicates that bottlenecks may be shifting not only because of changes in the machine and buffer parameters but also due to changes in the

282

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

0.76 3 0.83 4

m1

0.86 3 0.88

3 0.93 2 0.87 3 0.95 3 0.79 3 0.82 2 0.90

m b1 m2 b2 m3 b3 4

b4 m5

m b5 m6 b6 7

GBN 4 0.25 3 0.25 3

b7 m8 b8

m9 b9 m10

q = 0.80

br2 mr2 br1 mr1 br0

PRlwr = 0.29, ∆ PRlwr = 0.55 ∆ ∆p7 ∆pr1 ∆PRlwr ∆PRlwr = 0.19, = 0.53 ∆p7 ∆pr1 LBN

m1 b1 m2 b2 lwr ST i lwr BLi

LBN

m3 b3 m4

m4 b

m5 b

m6 b

m7

0 0.135

0.029 0.182

0.006 0.230

0.006 0

lwr ST i lwr BLi

0 0.094

0.018 0.134

0.018 0.071

0.031 0

STilwr 0 0.164 BLlwr i

0.036 0.202

0.016 0.250

0.012 0

STilwr 0 BLlwr i 0.126

0.025 0.163

0.026 0.099

0.056 0

(a) LBN identification of overlapping line 1

(b) LBN identification of overlapping line 2

LBN

m7 lwr ST i lwr BLi

LBN

b7 m8 b8 m9 b

m7 b mr1 b m6 b m4 r1 r0 r2

m10

0 0.025

0.155 0.012

0.188 0.009

0.275 0

lwr ST i lwr BLi

0 0.121

0.078 0.023

0.094 0.000

0.031 0

STilwr 0 0.038 BLlwr i

0.181 0.016

0.216 0.010

0.304 0

STilwr 0 0.113 BLlwr i

0.086 0.020

0.101 0.000

0.694 0

(c) LBN identification of overlapping line 3

(d) LBN identification of overlapping line 4

Figure 8.25: GBN identification of Example 8.11 for q = 0.8

8.3. BERNOULLI LINES WITH REWORK

0.76 3 0.83 4

m1

283

GBN 3 0.93 2 0.87 3 0.95 3 0.79 3 0.82 2 0.90

0.86 3 0.88

m b1 m2 b2 m3 b3 4

b4 m5

m b5 m6 b6 7

b7 m8 b8

m9 b9 m10

q = 0.85 4 0.25 3 0.25 3

br2 mr2 br1 mr1 br0

∆ PRlwr = 0.30, ∆ PRlwr = 0.24 ∆p6 ∆p7 ∆PRlwr ∆PRlwr = 0.19, = 0.09 ∆p6 ∆p7 LBN

m1 b1 m2 b2 lwr ST i

0

LBN

m3 b3 m4

lwr BLi 0.074

0.051 0.099

0.022 0.154

0.014 0

STilwr 0 0.098 BLlwr i

0.053 0.120

0.032 0.168

0.024 0

(a) LBN identification of overlapping line 1

m4 b lwr ST i

m7

m7

m6 b

lwr BLi 0.062

0.029 0.098

0.028 0.037

0.052 0

STilwr 0 BLlwr i 0.082

0.036 0.118

0.034 0.058

0.070 0

(b) LBN identification of overlapping line 2

LBN

lwr ST i lwr BLi

0

m5 b

LBN

b7 m8 b8 m9 b

m7 b mr1 b m6 b m4 r1 r0 r2

m10

0 0.046

0.084 0.023

0.121 0.013

0.212 0

lwr ST i lwr BLi

0 0.047

0.123 0.011

0.129 0.000

0.750 0

STilwr 0 0.053 BLlwr i

0.110 0.022

0.148 0.012

0.239 0

STilwr 0 0.048 BLlwr i

0.127 0.010

0.134 0.000

0.748 0

(c) LBN identification of overlapping line 3

(d) LBN identification of overlapping line 4

Figure 8.26: GBN identification of Example 8.11 for q = 0.85

284

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

0.76 3 0.83 4

m1

GBN 3 0.93 2 0.87 3 0.95 3 0.79 3 0.82 2 0.90

0.86 3 0.88

m b1 m2 b2 m3 b3 4

b4 m5

m b5 m6 b6 7

b7 m8 b8

m9 b9 m10

q = 0.90 4 0.25 3 0.25 3

br2 mr2 br1 mr1 br0

∆ PRlwr = 0.18, ∆ PRlwr = 0.22 ∆p6 ∆p8 ∆PRlwr ∆PRlwr = 0.17, = 0.23 ∆p6 ∆p8 LBN

m1 b1 m2 b2 lwr ST i

0

LBN

m3 b3 m4

lwr BLi 0.041

0.074 0.069

0.056 0.089

0.038 0

STilwr 0 0.060 BLlwr i

0.067 0.066

0.056 0.108

0.045 0

(a) LBN identification of overlapping line 1

m4 b lwr ST i

m7

m6 b

m7

lwr BLi 0.049

0.042 0.092

0.034 0.037

0.055 0

STilwr 0 BLlwr i 0.063

0.054 0.103

0.046 0.048

0.078 0

(b) LBN identification of overlapping line 2

LBN

lwr ST i lwr BLi

0

m5 b

LBN

b7 m8 b8 m9 b

m7 b mr1 b m6 b m4 r1 r0 r2

m10

0 0.089

0.039 0.033

0.086 0.015

0.180 0

lwr ST i lwr BLi

0 0.013

0.169 0.003

0.170 0.000

0.795 0

STilwr 0 0.085 BLlwr i

0.063 0.029

0.108 0.014

0.209 0

STilwr 0 0.013 BLlwr i

0.171 0.003

0.172 0.000

0.796 0

(c) LBN identification of overlapping line 3

(d) LBN identification of overlapping line 4

Figure 8.27: GBN identification of Example 8.11 for q = 0.9

8.3. BERNOULLI LINES WITH REWORK

GBN 0.76 3 0.83 4

m1

0.86 3 0.88

m b1 m2 b2 m3 b3 4

285

3 0.93 2 0.87 3 0.95 3 0.79 3 0.82 2 0.90

b4 m5

m b5 m6 b6 7

b7 m8 b8

m9 b9 m10

q = 0.95 4 0.25 3 0.25 3

br2 mr2 br1 mr1 br0

PRlwr = 0.46, ∆ PRlwr = 0.02, ∆ PRlwr = 0.03 ∆ ∆p1 ∆p6 ∆p8 ∆PRlwr ∆PRlwr ∆PRlwr = 0.33, = 0.10, = 0.16 ∆p1 ∆p6 ∆p8 LBN

LBN

m1 b1 m2 b2 lwr ST i

0

m3 b3 m4

lwr BLi 0.028

0.087 0.012

0.101 0.031

0.094 0

STilwr 0 0.041 BLlwr i

0.077 0.036

0.079 0.066

0.074 0

(a) LBN identification of overlapping line 1

m4 b lwr ST i

m7

m7

m6 b

lwr BLi 0.042

0.084 0.084

0.065 0.037

0.071 0

STilwr 0 BLlwr i 0.055

0.078 0.101

0.063 0.054

0.085 0

(b) LBN identification of overlapping line 2

LBN

lwr ST i lwr BLi

0

m5 b

LBN

b7 m8 b8 m9 b

m7 b mr1 b m6 b m4 r1 r0 r2

m10

0 0.116

0.021 0.038

0.073 0.017

0.168 0

lwr ST i lwr BLi

0 0.001

0.211 0.000

0.211 0.000

0.841 0

STilwr 0 0.112 BLlwr i

0.039 0.033

0.088 0.016

0.181 0

STilwr 0 0.002 BLlwr i

0.212 0.000

0.212 0.000

0.840 0

(c) LBN identification of overlapping line 3

(d) LBN identification of overlapping line 4

Figure 8.28: GBN identification of Example 8.11 for q = 0.95

286

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES

quality buy rates. Thus, BNs must be tracked when the product quality is fluctuating. Example 8.12 Consider the system shown in Figure 8.29. Assume first that the quality buy rate is 1, i.e., the rework loop is not activated. Then, since the machine efficiencies are allocated symmetrically and the buffers are identical, machines m2 and m9 are the bottlenecks, and they have an identical effect on the system’s production rate. 0.90 3 0.75 3

m1

0.90 3 0.90

m b1 m2 b2 m3 b3 4

3 0.85 3 0.85 3

b4 m5

GBN 0.90 3 0.90 3 0.75 3 0.90

m b5 m6 b6 7

b7 m8 b8 m9

b9 m10

q = 0.95 3 0.25 3 0.25 3

br2 mr2 br1 mr1 br0

∆ PRlwr = 0.18, ∆ PRlwr = 0.12, ∆ PRlwr = 0.23 ∆p2 ∆p6 ∆p9 ∆PRlwr ∆PRlwr ∆PRlwr = 0.28, = 0.13, = 0.31 ∆p2 ∆p6 ∆p9 LBN

m1 b1 m2 b2 lwr ST i

0

LBN

m3 b3 m4

lwr BLi 0.170

0.004 0.016

0.123 0.055

0.067 0

STilwr 0 0.190 BLlwr i

0.003 0.037

0.110 0.086

0.072 0

(a) LBN identification of overlapping line 1

m4 b lwr ST i

m7

b7 m8 b8 m9 b

m6 b

m7

lwr BLi 0.073

0.033 0.051

0.048 0.036

0.068 0

STilwr 0 BLlwr i 0.087

0.044 0.060

0.057 0.047

0.084 0

(b) LBN identification of overlapping line 2

LBN

lwr ST i lwr BLi

0

m5 b

LBN

m7 b mr1 b m6 b m4 r1 r0 r2

m10

0 0.067

0.055 0.123

0.016 0.004

0.170 0

lwr ST i lwr BLi

0 0.001

0.212 0.000

0.212 0.000

0.858 0

STilwr 0 0.071 BLlwr i

0.085 0.110

0.037 0.002

0.190 0

STilwr 0 0.001 BLlwr i

0.213 0.000

0.213 0.000

0.859 0

(c) LBN identification of overlapping line 3

(d) LBN identification of overlapping line 4

Figure 8.29: GBN identification of Example 8.12 On the other hand, when q < 1, for instance, q = 0.95, the data of Figure 8.29 show that the GBN is unique: machine m9 . This difference between lines

8.4. SUMMARY

287

with and without the rework loop is due to the lack of reversibility in the former and the fact that downstream machines have a larger effect on the production rate than those upstream.

8.4

Summary

• All performance measures in Bernoulli lines with non-perfect quality machines and inspection machines can be evaluated using a recursive procedure, which is a generalization of a similar procedure for lines with only perfect quality machines. • Monotonicity and reversibility properties of Bernoulli lines with non-perfect quality machines are similar to those with only perfect quality machines. • Production lines with non-perfect quality and inspection machines have two types of bottlenecks: production rate bottlenecks (PR-BN) and quality bottlenecks (Q-BN). • In a special case where the only inspection machine is the last one, PR-BN can be identified using Bottleneck Indicator 5.1 and Q-BN is the machine with the worst quality. Also, PR- and Q-BN are decoupled in the sense that elimination of one does not affect the position of the other. • In general, PR-BN cannot be identified using Bottleneck Indicator 5.1, Q-BN is not necessarily the worst quality machine of the system, and PRand Q-BNs are coupled, i.e., elimination of one affects the position of the other. • Although in the general case, neither PR- nor Q-BNs can be identified using Bottleneck Indicator 5.1, the notion of local bottlenecks alleviates some of the difficulties in their identification. • The optimal position of a single sufficiently efficient inspection machine is either immediately after the last non-perfect quality machine or around the PR-BN located downstream of this machine. • The production rate of good parts in Bernoulli lines with quality-quantity coupling (QQC) machines is a non-monotonic (concave) function of machine efficiency. • Therefore, a BN can be eliminated either by increasing machine efficiency (for p+ -BN) or decreasing machine efficiency (for p− -BN). • Performance of Bernoulli lines with rework can be analyzed using an overlapping decomposition approach. • Monotonicity properties of lines with rework are similar to those without rework. • However, serial lines with rework do not possess the property of reversibility. • In lines with rework, more efficient machines must be placed towards the end of the line to ensure the largest throughput.

288

CHAPTER 8. PRODUCT QUALITY IN BERNOULLI LINES • Local bottlenecks of lines with rework can be identified using Bottleneck Indicator 5.1. • One of the local bottlenecks in lines with rework is, practically always, the global bottleneck.

8.5

Problems

Problem 8.1 Consider the production line shown in Figure 8.30. 0.9, 0.9 m1

2

b1

0.9, 0.9 m2

2

b2

0.9, 0.9 m3

2

b3

0.97 m4

Figure 8.30: Production line of Problem 8.1 d d SR, c and W \ (a) Evaluate P R, CR, IP i . (b) Identify PR- and Q-BNs. (c) Design the best, from your point of view, continuous improvement project d that would result in 10% of P R improvement. Justify why this project is the best. Problem 8.2 Consider again the production line shown in Figure 8.30. Assume that an additional inspection module, consisting of a buffer with capacity 2 and an inspection machine with efficiency 0.97, can be placed within the system. (a) Determine the position of the new inspection module, which leads to the d largest P R. (b) Would you recommend including this inspection machine in the system? (c) Identify PR- and Q-BNs of the new system. Problem 8.3 For the production system, which includes perfect and nonperfect quality machines as well as inspection machines, develop a method for performance analysis under the symmetric blocking convention. d Problem 8.4 Prove that P R in Bernoulli lines with QQC machines is monotonically increasing with respect to buffer capacities. Problem 8.5 Consider a two-machine Bernoulli line with QQC machines defined by g1 (p1 )

=

g2 (p2 )

=

Assume that the buffer capacity is 2.

1 − 0.5p1 , 1 . 3 p2 + 1

(8.74)

8.6. ANNOTATED BIBLIOGRAPHY

289

(a) Calculate and plot P R as a function of p2 for several values of p1 . (b) Assume that p1 = p2 = 0.95. Which machine is the PR-BN and what is the type of the bottleneck (i.e., p+ or p− )? (c) Assume now that p1 = p2 = 0.6. Which machine is the PR-BN and what is the type of the bottleneck? Problem 8.6 Investigate the monotonicity properties of Bernoulli lines with QQC machines under the symmetric blocking convention. Problem 8.7 Consider a Bernoulli production line with rework and QQC machines. (a) Develop a method for performance analysis of such a line. (b) Analyze the monotonicity properties of this system. Problem 8.8 Consider a Bernoulli line with rework and assume that m1 and mM are the merge and split machines, respectively. (a) Develop a method for performance analysis of such a line. (b) Develop a method for BN identification in such a system. Problem 8.9 Develop methods for performance analysis and BN identification in Bernoulli lines with rework under the symmetric blocking convention.

8.6

Annotated Bibliography

The Bernoulli lines with quality control devices have been studied in [8.1] D. Jacobs and S.M. Meerkov, “Asymptotically Reliable Serial Production Lines with a Quality Control System,” Computers and Mathematics with Applications, vol. 22, pp. 85–90, 1991. [8.2] M.-S. Han, J.-T. Lim and D.-J. Park, “Performance Analysis of Serial Production Lines with Quality Inspection Machines,” International Journal of Systems Science, vol. 29, no. 9, pp. 939–951, 1998. [8.3] S.-Y. Chiang, “Bernoulli Serial Production Lines with Quality Control Devices: Theory and Application,” Mathematical Problems in Engineering, vol. 2006, Article ID 81950, 2006. A different approach has been developed in [8.4] J. Li and N. Huang, “Quality Evaluation in Flexible Manufacturing Systems: A Markovian Approach,” Mathematical Problems in Engineering, vol. 2007, Article ID 57128, 2007. [8.5] J. Kim and S.B. Gershwin, “Integrated Quality and Quantity Modeling of a Production Line,” OR Spectrum, vol. 27, pp. 287–314, 2005.

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[8.6] M. Colledani and T. Tolio, “Impact of Statistical Process Control (SPC) on the Performance of Production Systems,” Proceedings of 5th International Conference on the Analysis of Manufacturing Systems, pp. 77–92, Zakynthos Island, Greece, 2005. The material of Section 8.1 is based on [8.7] S.M. Meerkov and L. Zhang, “Product Quality Inspection in Bernoulli Lines: Analysis, Bottlenecks, and Design,” Control Group Report No. CGR-07-10, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007. The lines with QQC machines were investigated in [8.8] J.H. Owen and D.E. Blumenfeld, “Effects of Operating Speed on Production Quality and Throughput,” International Journal of Production Research, 2008. The material of Section 8.2 is based on [8.9] S.M. Meerkov and L. Zhang, “Bernoulli Production Lines with QualityQuantity Coupling Machines: Monotonicity Properties and Bottlenecks,” Control Group Report No. CGR-07-11, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007. The idea of overlapping decomposition of lines with rework has been proposed in [8.10] J. Li, “Performance Analysis of Production Systems with Rework Loops,” IIE Transactions, vol. 36, pp. 755–765, 2004. Lines with rework have been analyzed also in [8.10] J. Li, “Throughput Analysis in Automotive Paint Shops: A Case Study,” IEEE Transactions on Automation Science and Engineering, vol. 1, no. 1, pp. 90–98, 2004. [8.11] J. Li, “Overlapping Decomposition: A System-Theoretic Method for Modeling and Analysis of Complex Manufacturing Systems,” IEEE Transactions on Automation Science and Engineering, vol. 2, pp. 40–53, 2005. [8.12] J. Li, D.E. Blumenfeld and S.P. Marin, “Manufacturing System Design to Improve Quality Buy Rate: An Automotive Paint Shop Application Study,” IEEE Transactions on Automation Science and Engineering, vol. 4, pp. 75–79, 2007. [8.13] A. Korugan and O.F. Hancer, “On the Quality Information Feedback and the Rework Loop,” Proceedings of 6th International Conference on the Analysis of Manufacturing Systems, pp. 7–14, Lunteren, Holland, 2007.

8.6. ANNOTATED BIBLIOGRAPHY

291

[8.14] D. Borgh, M. Colledani, F. Simone and T. Tolio, “Integrated Analysis of Production Logistics and Quality Performance in Transfer Lines with Rework,” Proceedings of 6th International Conference on the Analysis of Manufacturing Systems, pp. 15–20, Lunteren, Holland, 2007. The material of Section 8.3 is based on [8.15] S. Biller, J. Li, S.P. Marin, S.M. Meerkov and L. Zhang, “Bottlenecks in Bernoulli Serial Lines with Rework,” Control Group Report No. CGR-0706, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007.

Chapter 9

Customer Demand Satisfaction in Bernoulli Lines Motivation: While the previous chapters are devoted to internal performance measures, such as P R, W IP , etc., the current one addresses the issue of customer demand satisfaction. This issue arises because, even if P R is sufficiently large, the number of parts produced per fixed interval of time may vary dramatically and, thus, customer demand may not be satisfied during some time intervals. Figure 9.1(a) illustrates this situation. It shows the number of parts produced by a serial line with two Bernoulli machines per epoch, where each epoch consists of 40 cycles. These data, obtained by simulations, indicate that the number of parts produced per epoch in the steady state of system operation can be as high as 35 and as low as 18, although the P R of this system is 0.65 (i.e., the line efficiency E = 0.92) and 26 parts are produced on the average per epoch. Thus, if the shipment size per epoch (which we will call the shipping period) is, say, 25 parts, the demand is satisfied in less than 78% of the 100 epochs shown in Figure 9.1(a). We refer to this performance measure, formalized rigorously in Section 9.1, as the Due-Time Performance (DT P ). It is intuitively clear that high DT P can be obtained by “filtering out” production randomness. This filtering can be accomplished in two ways – by filtering in time or filtering in space. Filtering in time refers to increasing the shipping period. Indeed, if the shipping period in the above example is increased from 40 to 45 cycles, the demand of 25 parts is satisfied in 94% of the 100 epochs (see Figure 9.1(b)). However, increasing the shipping period may be unacceptable for the customer. Indeed, in the modern manufacturing environment, customers prefer short shipping periods. For instance, in the automotive industry, car and truck assembly plants are often supplied by engine and seat plants on a four-hour delivery schedule. Similarly, engine plants receive their components, say, ignition and injection equipment, from the second tier suppli293

294

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

p=[0.7, 0.9], N=1, PR=0.65

32

30

30

28

28

Parts

34

32

26

26

24

24

22

22

20

20

18 0

20

40

60

Epoch (40 cycles)

p=[0.7, 0.9], N=1, PR=0.65

36

34

80

18 0

100

20

40

(b) 36

p=[0.7, 0.9], N=1, PR=0.65, N FGB=2

34 32 30 28 26 24 22 20 18 0

20

40

60

Epoch (40 cycles)

80

60

Epoch (45 cycles)

(a)

Parts

Parts

36

100

(c) Figure 9.1: Illustration of throughput variability

80

100

9.1. MODELING AND PARAMETRIZATION

295

ers on 6 - 8 hours shipping schedules, etc. In this situation, filtering in time must be complemented by filtering in space. Filtering in space refers to the effect of the finished goods buffer (FGB), which, on one hand, blocks the last machine of the system when it is full but, on the other hand, accumulates parts between shipping periods. The efficacy of the FGB in filtering out production randomness is illustrated in Figures 9.1(c) and 9.2. In Figure 9.1(c) the same system as in Figure 9.1(a) is considered but with a finished goods buffer of capacity NF GB = 2. Clearly, the variability of the number of parts available for shipment at the end of the epoch is decreased, and in 86% of epochs the demand of 25 parts per 40 cycle shipping period is satisfied. In Figure 9.2, DT P in a two-machine line with the demand of 27 parts per shipping period of 40 cycles is plotted for two cases – with and without FGB – as a function of the in-process buffer capacity N . While the details of this figure and the method of its calculation become clear in Section 9.2, it shows that adding even a relatively small FGB leads to a substantial increase of DT P . Note that such an increase is not attainable by any size of the in-process buffer and, thus, FGB plays a role that cannot be accomplished by in-process buffering.

p=[0.7, 0.9], T=40, D=27 1

N

=2

FGB

0.9

DTP

0.8 0.7

N

=0

FGB

0.6 0.5 0.4 1

2

3

4

5

N

6

7

8

9

10

Figure 9.2: Illustration of filtering effect of the finished goods buffer The arguments described above motivate the topic considered in this chapter – analysis and design of serial lines with FGBs that lead to the desired level of customer demand satisfaction. Overview: A method for calculating DT P in serial lines with Bernoulli machines is derived. The issue of lean FGB is discussed. Effects of demand randomness on DT P are analyzed.

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9.1

Modeling and Parametrization

9.1.1

Production-inventory-customer (PIC) model

The manufacturing system considered in this chapter is shown in Figure 9.3. It consists of three subsystems: the production subsystem (PS), the inventory subsystem (IS), and the customer subsystem (CS). Each of them and their interactions are defined as follows: Inventory Subsystem

Production Subsystem

p1

N1

p2

pM−1

N M−1

pM

Customer Subsystem

D,T N FGB

Figure 9.3: Production-inventory-customer system Production subsystem: (i) The production subsystem consists of a serial production line with M Bernoulli machines and M − 1 buffers defined by parameters 0 < pi < 1 and 1 ≤ Ni < ∞, respectively. The PS operates according to assumptions (a)-(e) of Subsection 4.2.1. Inventory subsystem: (ii) The inventory subsystem consists of a finished goods buffer (FGB) with capacity 1 ≤ NF GB < ∞. Parts produced by PS are immediately transferred to the FGB. Interaction between PS and IS: (iii) The production subsystem is blocked during a time slot (i.e., mM is blocked) if FGB is full at the beginning of this slot and the customer subsystem does not take parts from the buffer during this time slot (blocked before service convention). Customer subsystem: (iv) The customer requires D parts to be shipped during each shipping period. The duration of the shipping period is T cycles of time, where, as in all previous chapters, the cycle is the time necessary to process a part. To avoid triviality, it is assumed that D ≤ T · P R,

(9.1)

where P R is the production rate of the PS in isolation, i.e., when neither IS nor CS is present

9.1. MODELING AND PARAMETRIZATION

297

Interactions among PS, IS, and CS: (v) At the beginning of each shipping period i, parts are removed from the FGB in the amount of min{H(i − 1), D}, where H(i − 1) is the number of parts in the FGB at the end of the previous, (i − 1)-th, shipping period. If H(i − 1) ≥ D, the shipment is complete; if H(i − 1) < D, the balance of the shipment, i.e., D − H(i − 1) parts, is to be produced by the PS during the i-th shipping period. The parts produced are immediately removed from the FGB and prepared for shipment, until the shipment is complete, i.e., D parts are available. If the shipment is complete before the end of the shipping period, the PS continues operating, but with the parts accumulating in the FGB, either until the end of the shipping period or until mM is blocked, whichever occurs first. If the shipment is not complete by the end of the shipping period, an incomplete shipment is sent to the customer. No backlog is allowed. The assumption that demand D is constant is introduced to simplify the presentation; in Section 9.4, the results are generalized for the case of a random demand. The “no backlog” assumption is also introduced for simplification; we keep this assumption throughout this chapter.

9.1.2

Due-time performance measure

Let tˆ(i) be the random variable representing the number of parts produced by the PS during the i-th shipping period in the steady state of PIC operation. Then, the due-time performance measure (DT P ) can be defined as follows: DT P = P [tˆi + H(i − 1) ≥ D],

(9.2)

where, as before, H(i − 1) is the number of parts in the FGB at the end of the (i − 1)-th shipping period. Note that the steady state mentioned above exists since, in the time scale T , the PIC system defined by assumptions (i)-(v) is an ergodic Markov chain. Given the production-inventory-customer system defined by (i)-(v), the goal of this chapter is to develop methods for DT P analysis and synthesis.

9.1.3

Parametrization

The above PIC system is characterized by the following parameters: p1 , . . ., pM , N1 , . . ., NM −1 , NDT P , D, and T . None of them independently characterizes the “regime” of PIC operation. Indeed, the same D may represent either high or low demand (depending on the value of P R); similarly, the same NDT P may represent either high or low level of filtering in space. Therefore, the original parameters must be normalized so that the regime of PIC operation becomes explicit. This situation is similar to that in fluid mechanics, where the Reynolds number and other dimensionless parameters are introduced to quantify flow regimes. Below, two dimensionless parameters are introduced in order to quantify PICs regimes.

298

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES Relative FGB capacity (ν) – the value of NF GB in units of D, i.e., ν=

NF GB . D

(9.3)

Clearly, ν characterizes the operation regimes from the point of view of filtering in space: ν ¿ 1 implies that shipments are practically just-in-time (even if NF GB is large), while ν À 1 means that the system operates in a regime with a high level of space filtering (even if NF GB is small). Load factor (L) – the value of D in units of the average production during T , i.e., D . (9.4) L= T · PR Due to (9.1), L ≤ 1. When L is close to 1, the PS operates in a heavy load regime. Often, manufacturing managers view large L as desirable. In Japanese industry, however, small L’s are preferred. The benefits of small L will become apparent in Section 9.2. The parameters ν and L, along with T , are used in this chapter to characterize the fundamental properties of DT P .

9.2

Analysis of DT P

9.2.1

Evaluating DT P in PIC systems with one-machine PS

Let t(i) be the number of parts produced by PS during shipping period i if IS were of infinite capacity. Using t(i), introduce the notations: k = 1, 2, . . . , NF GB − 1, j = 0, 1, . . . , NF GB , rk,j = P [t(i) = D + k − j], rbNF GB ,j = P [t(i) ≥ D + NF GB − j], j = 0, 1, . . . , NF GB . (9.5) Note that the probabilities rk,j and rbNF GB ,j can be easily calculated since they refer to the system without an FGB. Indeed, µ ¶ T pD+k−j (1 − p)T −(D+k−j) , (9.6) rk,j = D+k−j rbN,j =

T X k=D+N −j

µ

T k

¶ pk (1 − p)T −k ,

(9.7)

where N := NF GB . Using the above notations, introduce matrix R and vector Z0 as follows:   r1,1 − r1,0 − 1 r1,2 − r1,0 ··· r1,N − r1,0   r2,1 − r2,0 r2,2 − r2,0 − 1 · · · r2,N − r2,0     , (9.8) ··· ··· ··· ··· R=   rN −1,1 − rN −1,0 rN −1,2 − rN −1,0 · · · rN −1,N − rN −1,0  rbN,1 − rbN,0 rbN,2 − rbN,0 · · · rbN,N − rbN,0 − 1

9.2. ANALYSIS OF DT P

299  r1,0  r2,0   Z0 =   ··· . rbN,0 

(9.9)

Note that due to the uniqueness of the stationary probability distribution defined by assumptions (i)-(v), matrix R is nonsingular. Theorem 9.1 Under assumptions (i)-(v) with M = 1, DT P =

NX F GB

T X

µ zk

k=0 j=D−k

T j

¶ pj (1 − p)T −j ,

(9.10)

where zk = P [H(i−1) = k], k = 0, 1, . . . , NF GB , and vector Z = [z1 , z2 , · · · , zNF GB ]T is calculated according to Z = −R−1 Z0 . (9.11) Proof: See Section 20.1. In terms of the parameterizations of Section 9.1.3, DT P can be represented as DT P = DT P (p, T, L, ν). The behavior of this function, calculated using (9.10), is illustrated in Figures 9.4 and 9.5 as a function of L and ν, respectively. From these figures, we conclude:

p=0.9, T=52 1

ν=0.5 ν=0.33 ν=0.25

DTP

0.8 0.6 0.4 0.2 0

0.9

L

1

1.1

Figure 9.4: DT P as a function of L

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CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

p=0.9, T=50 1

DTP

0.9

L=0.9556 L=0.9778 L=1

0.8 0.7 0.2

0.4

ν

0.6

0.8

Figure 9.5: DT P as a function of ν • DT P is monotonically decreasing as a function of L. For small ν’s, DT P can be much smaller than 1 even if L < 1. (The values of L > 1 are included in Figure 9.4 for the sake of completeness.) For L = 1, DT P is quite small for any ν. Thus, having L away from 1 (say, L < 0.9 in Figure 9.4), allows the manufacturer to ensure a high probability of customer demand satisfaction, even if ν is small. Perhaps, that is why the Japanese manufacturers operate their production system with a relatively small load factor. • DT P is monotonically increasing as a function of ν. For L < 1, ν does not have to be too large to ensure DT P close to 1. Indeed, in the system of Figure 9.5, starting from ν = 0.25, DT P is close to 1 even if L is as large as 0.9778. For L = 1, however, this is not true: DT P = 0.97 even if ν = 0.7. Thus, again, L should be away from 1 to ensure DT P ≈ 1 with a relatively small finished goods buffer. PSE Toolbox: The DT P calculation for PIC systems with one machine is implemented as one of the tools in the Customer Demand Satisfaction function of the toolbox (see Section 19.8 for details).

9.2.2

Evaluating DT P in PIC systems with M -machine PS

It is possible to give a method for an approximate evaluation of DT P in the M -machine case. However, since this method is quite involved, we provide here only a lower bound on DT P , which is, on one hand, easy to calculate and, on the other hand, informative enough for practical utilization. Let DT PM be the DT P for an M -machine line with FGB of capacity NF GB , and let P R be the production rate of this line when no FGB is present. Let DT P1 be the DT P measure of a one-machine line with p = P R and FGB of

9.2. ANALYSIS OF DT P

301

capacity NF GB . Then, Numerical Fact 9.1 Under assumptions (i)-(v), the following inequality holds: DT PM ≥ DT P1 .

(9.12)

This fact has been justified by simulating numerous PIC systems. In these studies, the left hand side of the inequality was evaluated numerically and the right hand side was calculated using (9.10). In every system analyzed, (9.12) took place. An illustration is given in Figure 9.6 for three-, four, and fivemachine lines. Example 1

Example 2 1

1 3

DTP

0.9 p=[0.75, 0.75, 0.75]

0.85

0.75

0.85

T=10, D=6 0

0.5

ν

p=[0.7, 0.8, 0.75]

0.9

N=[3, 2], PR=0.6472

N=[2, 2], PR=0.6247 0.8

DTP1

0.95

1

DTP

DTP

DTP3

DTP

0.95

1

0.8

1.5

T=40, D=24

0

0.1

Example 3

0.3

0.4

1 DTP

0.95

0.9

DTP1

0.9

p=[0.8, 0.75, 0.7, 0.8]

0.85

N=[2, 3, 2], PR=0.6213

0.8

0.25

ν

0.5

DTP

0.85

0.75

1

0.8 p=[0.8, 0.75, 0.75, 0.7, 0.8] N=[2, 2, 3, 3], PR=0.61

0.75 0.7

T=20, D=12 0

DTP5

0.95

4

DTP

DTP

ν

Example 4

1

0.75

0.2

0.65

T=30, D=18 0

0.1

0.2

ν

0.3

0.4

0.5

Figure 9.6: Illustration of DT PM and its lower bound DT P1 Inequality 9.12 takes place, perhaps, due to the fact that if one-machine line is blocked by FGB, no production takes place; however, if the last machine in the M -machine line is blocked, other machines may be working and filling up in-process buffers. Numerical Fact 9.1 can be used for design of finished goods buffers in serial production lines. Indeed, since one is interested in having DT P as high as possible, if the lower bound meets the requirement, the real DT P surely does as well.

302

9.3 9.3.1

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

Design of Lean FGB for Desired DT P Lean FGB

Lean FGB capacity (NDT P ) – the smallest FGB capacity that ensures the desired DT P . In other words, NDT P is the capacity of FGB, which is necessary and sufficient to achieve DT Pdesired . Lean relative FGB capacity (νDT P ) – NDT P in units of D, i.e., νDT P =

NDT P . D

(9.13)

For the case of PIC systems with PS consisting of one Bernoulli machine, a “brute-force” approach to calculating NDT P is as follows: For given p, T , and D, using Theorem 9.1, calculate DT P assuming NF GB = 1, i.e., DT P (NF GB = 1). If it is larger than DT Pdesired , conclude that the lean FGB capacity is 1. (Note that, since the system operates under the blocked before service assumption, NF GB = 1 implies that physically no FGB is present, and the machine itself serves as an FGB of capacity 1.) Otherwise, assume that NF GB = 2, calculate DT P (NF GB = 2), and again compare it with DT Pdesired . Continue this process until arriving at the smallest finished goods buffer capacity (denoted as NDT P ), for which DT P (NDT P ) is equal to or larger than DT Pdesired . Then, evaluate νDT P using (9.13). Thus, νDT P as a function of p, T , and D (or L = TDp ) is calculated. Below, we analyze the properties of νDT P and provide a method for FGB capacity management.

9.3.2

Conservation of filtering

The behavior of νDT P (T ) for various p and L and DT P = 0.99 is shown in Figure 9.7. The curves in Figure 9.7 suggest that νDT P and T may be related in a hyperbolic manner, i.e., L=0.88 L=0.94 L=0.98

L=0.88 L=0.94 L=0.98

1.5

0.99

1

1

ν

ν

0.99

1.5

0.5

0

0.5

10

20

30 T

40

50

60

0

10

(a) p = 0.85

20

30

T

40

50

60

(b) p = 0.9

Figure 9.7: Behavior of νDT P as a function of T T · νbDT P = const,

(9.14)

9.4. ANALYSIS OF DT P FOR RANDOM DEMAND

303

where νbDT P is an approximate value of νDT P and the constant in the right hand side is defined by p and L. Since T and νDT P represent the amount of production randomness filtering in time and space, respectively, (9.14) can be viewed as a conservation law: the product of filtering in time and space is constant. This implies, in particular, that changing T by a factor of k would require changing νDT P by a factor of 1/k in order to maintain the same DT P . To determine the constant in the right side of (9.14), calculate νDT P , using the method of Subsection 9.3.1, for T = 1 and given values of p and L, i.e., determine νDT P (T = 1) := FDT P (p, L). Then, use the value of the right hand side of this expression as the constant in (9.14). This results in the conservation law T · νbDT P = FDT P (p, L).

(9.15)

Clearly, this relationship is exact for T = 1. For other T ’s, the accuracy of (9.15) is evaluated by calculating νDT P (using the method of Subsection 9.3.1) for various values of T and comparing it with νbDT P obtained from (9.15), i.e., with FDT P (p, L) . νbDT P = T The results are illustrated in Figure 9.8. For most values of T , the approximation is relatively precise, and we can conclude that the conservation law (9.15) provides a sufficiently accurate tool for managing lean FGB capacity in response to changing shipping periods T . Thus, T and νDT P trade-off against each other one-to-one.

9.4 9.4.1

Analysis of DT P for Random Demand Random demand modeling

So far, it has been assumed that the demand D per shipping period T is constant. Although, strictly speaking, this is the case in many manufacturing systems with long-term contractual relationships between supplier and customer, in reality D may have some variability. In this section, we model this variability by assuming that D is a random variable. More precisely, instead of assumption (iv) of Section 9.1, assume the following: (iv0 ) At the end of each shipping period i, D(i) parts must be shipped to the customer, where D(i), i = 1, 2 . . ., is a sequence of independent, identically distributed random variables defined by the probability mass function P [D = Dj ] = Pj , j = 1, . . . , J. Thus, the customer is defined by the triplet (D, P, T ), where D = (D1 , . . . , DJ ), P = (P1 , . . . , PJ ), and T is the shipping period. It is assumed that D ≤ T · P R,

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CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

p = 0.85

p = 0.9 0.8

1.5

ν0.99

0.4

ν

L = 0.98

0.99

0.6 1

0.5

0

0.2

10

20

1

40

50

DTP

20

40

60 T

80

100

20

40

60 T

80

100

20

40

60 T

80

100

0.5 0.4

DTP

0.6

0.3

0.4

0.2

0.2 0

0

60

ν0.99

ν0.99

T

νDTP=FDTP(T) ν^ =F (p,L)/T

0.8

L = 0.94

30

0.1 10

20

30

T

40

50

0

60

0.6 0.6

0.5 0.4

0.4

ν0.99

L = 0.88

ν0.99

0.5

0.3

0.3

0.2

0.2

0.1

0.1 0

10

20

30

T

40

50

60

0

Figure 9.8: Accuracy of the conservation law

9.4. ANALYSIS OF DT P FOR RANDOM DEMAND

305

where D is the expected value of D(i) and P R is the production rate of the PS. Below, a method for calculating DT P for this random demand is provided.

9.4.2

Evaluating DT P for random demand

Introduce the following quantities: rk,l,j rbNF GB ,l,j

= P [t(i) = D(i) + k − l], k = 1, . . . , NF GB − 1, l = 0, 1, . . . , NF GB , D(i) = Dj , j = 1, · · · , J, = P [t(i) ≥ D(i) + NF GB − l],

l = 0, 1, . . . , NF GB , D(i) = Dj , j = 1, . . . , J.

These quantities can be calculated as follows: µ ¶ T rk,l,j = pDj +k−l (1 − p)T −(Dj +k−l) , Dj + k − l

(9.16)

and T X

rbNF GB ,l,j =

µ

n=Dj +NF GB −l

T n

¶ pn (1 − p)T −n .

(9.17)

Introduce matrix R1 and vector Z01 defined by 1

R =  PJ j=1 (r1,1,j − r1,0,j )Pj − 1 P J  j=1 (r2,1,j − r2,0,j )Pj   ... PJ rN,1,j − r bN,0,j )Pj j=1 (b

PJ (r − r1,0,j )Pj PJ j=1 1,2,j j=1 (r2,2,j − r2,0,j )Pj − 1 ... PJ rN,2,j − r bN,0,j )Pj j=1 (b

... ... ... ...

PJ (r1,N,j − r1,0,j )Pj Pj=1 J j=1 (r2,N,j − r2,0,j )Pj ... PJ rN,N,j − r bN,0,j )Pj − 1 j=1 (b

  , 

(9.18)

 PJ j=1 r1,0,j Pj  PJ  j=1 r2,0,j Pj Z01 =   ··· PJ bN,0,j Pj j=1 r

   . 

(9.19)

Theorem 9.2 Under assumptions (i)-(iii),(iv0 ),(v) with M = 1, DT P =

NX F GB

J X

T X

k=0 j=1 n=Dj −k

µ zk

T n

¶ pn (1 − p)T −n Pj

(9.20)

where zk = P [H(i−1) = k], k = 0, 1, . . . , NF GB , and vector Z 1 = [z1 , z2 , . . . , zNF GB ]T is calculated according to (9.21) Z 1 = −R1 Z01 .

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CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

Proof: See Section 20.1. Again, Theorem 9.2 can be used for DT P evaluation in production-inventorycustomer systems with more than one Bernoulli machine, since it is possible to show that (9.22) DT PM ≥ DT P1 , where, as before, DT PM is the due-time performance of an M -machine PIC system and DT P1 is the due-time performance of the PIC system with onemachine PS, both under random demand. PSE Toolbox: The DT P calculation for PIC systems with one machine is implemented as one of the tools in the Customer Demand Satisfaction function of the toolbox (see Section 19.8 for details).

9.4.3

DT P degradation as a function of demand variability

Theorem 9.2 can be used to analyze the level of DT P degradation as a function of the coefficient of variation of the demand (CVdemand ). To accomplish this, consider the set of uniform demand pmf’s shown in Table 9.1, with CVdemand varying from 0.2 to 0.8. The PIC systems under consideration have a PS consisting of a single Bernoulli machine with p = 0.68, or 0.7, or 0.72. The shipping period T equals 15 time slots. Finally, the loads addressed are either low (L = 0.925), or medium (L = 0.95), or high (L = 0.98). The resulting behavior of DT P as a function of CVdemand is shown in Figure 9.9. From this figure we can conclude: Table 9.1: Uniform pmf’s considered CV 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Di 8, 12 7, 13 6, 14 5, 15 4, 16 3, 17 2, 18

Pj 0.5 0.5 0.5 0.5 0.5 0.5 0.5

• DT P is a monotonically decreasing function of the coefficient of variation of the demand. In other words, increasing CVdemand inevitably leads to a lower DT P . • This degradation of DT P is more significant in PIC systems with heavy loads and small FGBs. Indeed, while for L = 0.925 and N = 50, the DT P drops by about 5% when CVdemand increases from 0.2 to 0.8, it drops by almost 50% when L = 0.98 and N = 5.

9.4. ANALYSIS OF DT P FOR RANDOM DEMAND

1

p=0.72, T=15, Daverage=10, L=0.9259

0.9

0.85

0.85 0.8

0.6 0.55 0.5 0.2

DTP

N=5 N=10 N=15 N=20 N=25 N=30 N=35 N=40 N=45 N=50 0.3

0.75 0.7 0.65 0.6 0.55

0.4

CV

0.5

0.6

0.7

0.5 0.2

0.8

N=5 N=10 N=15 N=20 N=25 N=30 N=35 N=40 N=45 N=50 0.3

0.4

demand

(a) L = 0.9259 1

0.5

CVdemand

0.6

0.7

(b) L = 0.9524 p=0.68, T=15, Daverage=10, L=0.9804

0.95 0.9 0.85 0.8 DTP

DTP

0.8

0.65

=10, L=0.9524

average

0.95

0.9

0.7

p=0.7, T=15, D

1

0.95

0.75

307

0.75 0.7 0.65 0.6 0.55 0.5 0.2

N=5 N=10 N=15 N=20 N=25 N=30 N=35 N=40 N=45 N=50 0.3

0.4

0.5

CVdemand

0.6

0.7

0.8

(c) L = 0.9804 Figure 9.9: DT P degradation as a function of the CV of the demand

0.8

308

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES • Although it is not illustrated in Figure 9.9, the effect of changing T is similar to that of changing N : smaller T ’s lead to a larger DT P degradation than larger ones.

Thus, to combat demand variability, manufacturers must operate with either light loads or large filtering capabilities (by having either sufficiently large N , or T , or both). The amount of filtering, necessary and sufficient to ensure an appropriate DT P , can be calculated using Theorem 9.2.

9.4.4

Dependence of DT P on the shape of demand pmf ’s

Theorem 9.2 can be used to investigate whether DT P depends on the shape of the demand pmf. In this subsection, we show by examples that, in fact, DT P is practically independent of the shape of the demand pmf and depends mainly on its coefficient of variation. To accomplish this, consider pmf’s of the demand shown in Figure 9.10. For each value of CVdemand (0.1, 0.25, 0.4 and 0.8), there are three pmf’s having uniform, triangular, and ramp shapes. For each of these pmf’s and PIC systems discussed in the previous subsection, the values of DT P are calculated using Theorem 9.2. The results are shown in Figure 9.11. This figure shows that DT P is indeed practically independent of pmf shape and is defined by its coefficient of variation.

9.5

Case Study

In this case study, we consider the automotive ignition coil processing system introduced and modeled in Subsection 3.10.1. The customer (which is the downstream operation) requires 114 coils to be delivered every 15 minutes. The problem is to determine the lean FGB capacity that ensures DT P = 0.999. Using the techniques developed in this chapter, we solve this problem as follows: Since the cycle time of the system is 6.4 sec, the 15 min shipping period is equivalent to 141 cycles. The demand, as indicated above, is 114 coils per shipping period. Taking into account that the production rate of the system is 0.8267 parts/cycle and using Theorem 9.1 and Numerical Fact 9.1, we determine that DT P ≥ 0.9988

for N = 22

and DT P ≥ 0.9991 for N = 23. 23 = 0.202). With these parameters, Thus, we select N0.999 = 23 (i.e., ν0.999 = 114 the right hand side of the conservation law becomes

ν0.999 · T = (0.202)(141) = 28.48.

9.5. CASE STUDY

309

0.2

0.12 0.1

Uniform Triangle Ramp

0.08 j

0.1

P

j

0.15

P

0.14

Uniform Triangle Ramp

0.06 0.04

0.05

0.02

0 0

10

D

20

0 0

30

(a) CV = 0.1 0.07 0.06 0.05

10

D

20

30

(b) CV = 0.25 0.5

Uniform Triangle Ramp

Uniform Triangle Ramp

0.4 0.3 P

P

j

j

0.04 0.03

0.2

0.02 0.1

0.01 0 0

10

D

20

(c) CV = 0.4

30

0 0

10

D

20

(d) CV = 0.8

Figure 9.10: Illustration of the shape of the demand pmf’s

30

310

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

1

1

0.95

0.95 0.9 0.85 DTP

DTP

0.9 0.85

0.75

0.8

Uniform Triangle Ramp

0.75 0.7 0

0.8

10

20 N

30

Uniform Triangle Ramp

0.7 0.65 0

40

(a) CV = 0.1

10

20 N

30

40

(b) CV = 0.25 1

1

0.9

0.9

DTP

DTP

0.8 0.8 0.7

Uniform Triangle Ramp

0.6 10

20 N

(c) CV = 0.4

30

0.7 Uniform Triangle Ramp

0.6 0.5 40

10

20 N

30

40

(d) CV = 0.8

Figure 9.11: Illustration of the dependence of DT P on the shape of the demand pmf’s

9.6. SUMMARY

311

Assume now that the customer changed the required shipping period to 10 min (i.e., 94 cycles). How many parts can the manufacturer ship to the customer so that load factor, L=

114 D = = 0.978, d (141)(0.8267) T · PR

remains the same, and what is the lean FGB capacity for DT P = 0.999? To determine the new shipment size, we write: L=

D1 d T1 · P R

=

D1 = 0.978, (94)(0.8267)

i.e., D1 = 76 parts/shipping period. d Since L and P R remain the same from the conservation law (9.15), we determine that νb0.999 =

28.48 = 0.303 94

and, finally, b0.999 = (0.303)(76) = 23, N i.e., the lean FGB remains the same.

9.6

Summary

• The level of customer demand satisfaction in PIC systems can be quantified by DT P , which can be evaluated using Theorems 9.1 and 9.2 and Numerical Fact 9.1. • The regimes of PIC operation can be parametrized in terms of two dimensionless parameters: the load factor, L, and relative FGB capacity, ν. • To ensure a desired DT P , PIC systems must have a sufficient amount of filtering in time and space, quantified by T and ν, respectively. • PIC systems with lean FGBs satisfy a conservation law: the product of filtering in time and space is practically constant. • Demand variability leads to deterioration of DT P , which is more pronounced under heavy loads and small filtering capabilities. • DT P for random demand is practically independent of the shape of the demand pmf and is defined mainly by its coefficient of variation.

312

CHAPTER 9. DEMAND SATISFACTION IN BERNOULLI LINES

9.7

Problems

Problem 9.1 Consider the PIC system defined by assumptions (i)-(v) of Subsection 9.1.1 with M = 1, p = 0.9, T = 40, NF GB = 6. (a) Calculate and plot DT P as a function of the load factor L. (b) Select the value of the shipment size D, which you would offer to the customer. Problem 9.2 Consider the PIC system defined by assumptions (i)-(v) of Subsection 9.1.1 with M = 1, p = 0.9, T = 40, D = 35. (a) Calculate and plot DT P as a function of the finished good buffer capacity NF GB . (b) Select NF GB , which you recommend for this system. Problem 9.3 Consider the PIC system defined by assumptions (i)-(v) of Subsection 9.1.1. Assume that M = 1, D = 16, T = 20, and the required DT P = 0.99. (a) For p = 0.9, determine the capacity of the finished goods buffer, which ensures the required DT P . (b) Repeat the same calculation for p = 0.8. (c) Compare the two results and recommend ways to keep the finished goods buffer small. Problem 9.4 Consider the PIC system defined by assumptions (i)-(v) of Subsection 9.1.1 and assume that the Production Subsystem is that of Problem 6.4 with the buffering calculated in part (b) of this problem. Using the Bernoulli models of this system and assuming D = 35, T = 40, calculate an estimate of DT P for (a) NF GB = 6, (b) NF GB = 12, (c) what are the pluses and minuses of each of the above choices of NF GB ? Problem 9.5 Consider the PIC system with M = 1, p = 0.7, T = 15 and with (1) constant demand, D = 10, (2) random demand with a uniform pmf defined on {7, 8, 9, 10, 11, 12, 13}, (3) random demand with a triangle pmf, which takes values on {6, 7, 8, 9, 10, 11, 12, 13, 14} with the probabilities {0.04, 0.08, 0.12, 0.16, 0.2, 0.16, 0.12, 0.08, 0.04}, respectively. Calculate: (a) (b) (c) (d)

The load, L, of the system. The CV of the random demands. The DT P as a function of NF GB ; plot this function. Summarize and interpret the results.

9.8. ANNOTATED BIBLIOGRAPHY

9.8

313

Annotated Bibliography

The production variability over a fixed interval of time was analyzed (for exponential machines) in [9.1] S.B. Gershwin, “Variance of Output of a Tandem Production System,” in Queuing Networks with Finite Capacity, ed. R. O. Onvaral and I. F. Akyildi, Elsevier Science Publishers, Amsterdam, 1993. For PIC systems without a finished goods buffer, this issue has been addressed in [9.2] D.A. Jacobs and S. M. Meerkov, “System-Theoretic Analysis of DueTime Performance in Production Systems,” Mathematical Problems in Engineering, vol. 1, pp. 225-243, 1995. The calculation of DT P for Bernoulli lines has been reported in [9.3] J. Li and S.M. Meerkov, “Production Variability in Manufacturing Systems: Bernoulli Reliability Case,” Annals of Operations Research, vol. 93, pp. 299-324, 2000 and [9.4] J. Li and S.M. Meerkov, “Customer Demand Satisfaction in Production Systems: A Due-Time Performance Approach,” IEEE Transactions on Robotics and Automation, vol. 17, pp. 472-482, 2001. More details are available in [9.5] J. Li, Production Variability in Manufacturing Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 2000. The conservation of filtering for exponential machines was analyzed in [9.6] J. Li, E. Enginarlar and S.M. Meerkov, “Conservation of Filtering in Manufacturing Systems with Unreliable Machines and Finished Goods Buffers,” Mathematical Problems in Engineering, vol. 2006, pp. 1-12, 2006.

Chapter 10

Transient Behavior of Bernoulli Lines Motivation: All previous chapters address the steady state properties of Bernoulli lines. The current one is intended to investigate their transient behavior, i.e., to study how fast they reach their steady states. The transient properties of production systems are of practical importance. Indeed, if the steady state is reached after a relatively long period of time, the system may suffer substantial production losses. For instance, if the cycle time of a production system is 1 minute and the plant shift is 500 minutes, the system may lose more than 10% of its production due to transients, if at the beginning of the shift all buffers were empty (see Section 10.5 below). Transient properties are also important in systems that operate with the socalled “floats”. Typically, systems with floats are used when it is desired that the production rate of the line be equal or, at least, close to that of the best machine. This is accomplished by extending the operating time of all other machines in order to build-up floats, which would prevent starvations of the best machine during the overall system operation. Thus, in this case, the system operates in a transient regime, having buffers initially full. In this situation, the main technical problem is to select the minimum float that leads to the maximum throughput. Thus, analysis of transients in production lines is of importance from the practical point of view. Clearly, it is important from the theoretic perspective as well. This is carried out in this chapter. Overview: First, we show that the transients of Bernoulli lines can be characterized in terms of the following metrics: • the second largest eigenvalue of the transition matrix associated with the Markov chain describing the system at hand; • pre-exponential factors associated with the second largest eigenvalue in analytical expressions for P R and W IP ; 315

316

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES • settling time of P R and W IP ; • production losses due to transients.

Then, each of these performance metrics is analyzed as a function of machine and buffer parameters and the number of machines in the system. Finally, we study production losses as a function of the initial buffer occupancy and determine the smallest inventory that leads to no losses due to transients.

10.1

Problem Formulation

10.1.1

Mathematical description

Consider a serial production line defined by assumptions (a)-(e) of Subsection 4.2.1. Assume that it consists of two identical machines, i.e., M = 2 and p1 = p2 =: p. Then the transition probabilities, which describe the line, are given in (4.1). These probabilities can be arranged in a transition matrix as follows: 

1−p  p    0  A=   0   ..  . 0

p(1 − p) p2 + (1 − p)2

0 p(1 − p)

··· 0

p(1 − p)

p2 + (1 − p)2 .. . .. . ···

p(1 − p) .. .

0 ··· .. . .. .

p(1 − p) 0

p2 + (1 − p)2 p(1 − p)

0 .. . 0

0 0 0 .. .

      .     

p(1 − p) p2 + 1 − p (10.1)

Let xi (n), i = 0, 1, 2, . . . , N , n = 0, 1, . . ., be the probability that there are i parts in the buffer at the end of time slot n. Then the evolution of vector x(n) = [x0 (n), . . . , xN (n)]T can be described by the following constrained linear equation: x(n + 1) N X

=

Ax(n),

xi (n) =

1.

(10.2)

i=0

Given this description, P R and W IP can be viewed as the outputs of the state-space model (10.2), defined as follows: P R(n)

=

p

N X

xi (n),

(10.3)

i=1

W IP (n)

=

N X i=1

ixi (n).

(10.4)

10.1. PROBLEM FORMULATION

317

Together, (10.2), (10.3), and (10.4) represent a state space description: x(n + 1) = y(n) =

Ax(n), Cx(n),

kx(n)k1 = 1,

(10.5)

where A is given in (10.1), · C

=

0 p p ··· p 0 1 2 ··· N

¸ ,

(10.6)

and kx(n)k1 =

N X

xi (n).

(10.7)

i=0

This description, along with its generalization for M > 2, is the basis for transient analysis of Bernoulli lines described in this chapter.

10.1.2

Second largest eigenvalue problem

Since A, defined by (10.1), is a transition matrix of an ergodic Markov chain, it has a unique largest eigenvalue λ0 = 1. Assume that the second largest eigenvalue (SLE), λ1 , is real and simple while all other eigenvalues are distinct. Numerical calculations suggest that this assumption holds for matrix (10.1) regardless of the values of p and N . Arrange all eigenvalues of A as follows: 1 = λ0 > λ1 > |λ2 | ≥ · · · ≥ |λN |.

(10.8)

Due to (10.8), there exists a nonsingular matrix Q ∈ R(N +1)×(N +1) such that the substitution x ˜(n) = Qx(n), (10.9) transforms (10.5) - (10.7) into x ˜(n + 1) = y(n) =

˜x(n), A˜ C˜ x ˜(n),

(10.10)

where A˜ = C˜

=

QAQ−1 = diag[1 λ1 · · · λN ], · ¸ C˜10 C˜11 · · · C˜1N CQ−1 = . C˜20 C˜21 · · · C˜2N

(10.11) (10.12)

For any admissible initial condition x(0), i.e., an initial condition such that PN i=0 xi (0) = 1, the solution of (10.10) is x ˜(n) = diag[1

x(0), λn1 · · · λnN ]˜

(10.13)

318

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

where x ˜(0) = Qx(0).

(10.14)

This implies that states x ˜(n) and, due to (10.9), x(n), converge to their steady states as exponential functions of time with bases λi . Hence, the duration of transients is defined by the largest of |λi |, i 6= 0, i.e., by the SLE. In the framework of a Bernoulli line with two identical machines, the SLE is a function of the machine efficiency p and the buffer capacity N . For M > 2, it depends also on M , i.e., λ1 = λ1 (p, N, M ). The first problem addressed in this chapter is to analyze the behavior of λ1 as a function of all three arguments. The results of this analysis should reveal which values of system parameters lead to inherently long and which to inherently short transients of buffer occupancy. This is carried out in Section 10.2.

10.1.3

Pre-exponential factor problem

As it follows from (10.10) - (10.14), the dynamics of the output y(n) are described by the equations: P R(n) W IP (n)

C˜10 + C˜11 x ˜1 (0)λn1 + . . . + C˜1N x ˜N (0)λnN , n = C˜20 + C˜21 x ˜1 (0)λ1 + · · · + C˜2N x ˜N (0)λnN ,

=

(10.15) (10.16)

where it is taken into account that x ˜0 (0) = 1 (since the first row of Q is the left eigenvector of A given by [1, 1, . . . , 1]). Clearly, lim P R(n)

=

C˜10 = P Rss ,

(10.17)

lim W IP (n)

=

C˜20 = W IPss ,

(10.18)

n→∞ n→∞

where P Rss and W IPss are the steady state values of the production rate and work-in-process, respectively, which have been analyzed in details in Chapter 4. Thus, " # C˜1N C˜11 n n x ˜1 (0)λ1 + . . . + x ˜N (0)λN , (10.19) P R(n) = P Rss 1 + C˜10 C˜10 " # ˜2N C C˜21 x ˜1 (0)λn1 + . . . + x ˜N (0)λnN . (10.20) W IP (n) = W IPss 1 + C˜20 C˜20 These equations indicate that the transients of PR and WIP are characterized not only by the eigenvalues of A but also by the pre-exponential factors ˜ij C ˜j (0), i = 1, 2, j = 1, 2, . . . , N . Since the initial conditions x ˜j (0) enter ˜i0 x C (10.19) and (10.20) in a similar manner and since λ1 is the SLE, the most ˜11 ˜21 C important pre-exponential factors are C ˜10 and C ˜20 . Denote them as C ¯ ¯ ¯ ¯ ¯ C˜ ¯ ¯ C˜ ¯ ¯ 11 ¯ ¯ 21 ¯ (10.21) Ψ11 = ¯ ¯ , Ψ21 = ¯ ¯. ¯ C˜10 ¯ ¯ C˜20 ¯

10.1. PROBLEM FORMULATION

319

Coefficients Ψ11 and Ψ21 describe to what extent the SLE affects the transients of the outputs. In addition, the relationship between these coefficients shows which of the outputs, PR or WIP, has faster transients: if Ψ11 < Ψ21 , PR converges to its steady state value faster than WIP ; if the inequality is reversed, the opposite takes place. The second problem considered in this chapter is to analyze the behavior of the pre-exponential factors Ψ11 = Ψ11 (p, N, M ) and Ψ21 = Ψ21 (p, N, M ). This is carried out in Section 10.3.

10.1.4

Settling time problem

In Control Theory, the transients of feedback systems are often characterized by the settling time, ts , which is the time necessary for the output to reach and remain within ±5% of its steady state value, provided that the input is a unit step function and the initial conditions are zero. A similar notion can be used to characterize the transients of production lines as well. Indeed, the zero initial condition could be interpreted as having all buffers initially empty, i.e., xi (0) = [1, 0, . . . , 0]T ,

i = 1, . . . , M − 1,

(10.22)

where xi (0) is the vector of probabilities of initial occupancy of buffer i. The step input, in the framework of production systems, is incorporated in the fact that matrix A has an eigenvalue equal to 1 and, therefore, the outputs converge to non-zero steady state values, P Rss and W IPss . Thus, the production system defined by conventions (a)-(e) of Subsection 4.2.1 can be characterized by two settling times − with respect to PR and WIP − denoted as tsP R and tsW IP , respectively. The third problem considered in this chapter is to analyze settling times tsP R and tsW IP as functions of p, N , and M . This is carried out in Section 10.4.

10.1.5

Production losses problem

The three problems formulated above are mathematical in nature. The problem of production losses, although based on the results of the previous three, is clearly practical. It addresses the question of how much production is lost due to transients if the initial buffer occupancy is zero, and what is the smallest initial buffer occupancy necessary to guarantee no production losses due to transients. To formulate this question precisely, introduce the notion of production losses during period T : LT (x1 (0), . . . , xM −1 (0)) =

T X £

¤ P Rss − P R(n; x1 (0), . . . , xM −1 (0)) , (10.23)

n=0

where xi (0) is the vector of probability of initial occupancy of buffer i, i.e., xi (0) = [xi0 (0), . . ., xiNi (0)]T , i = 1, 2, . . . , M − 1. Of particular interest are

320

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

the initial conditions corresponding to all buffers being empty and all buffers having nonzero identical occupancy h(0). The corresponding production losses h(0) are denoted as L0T (p, N, M ) and LT (p, N, M ), respectively. A more telling metric of production losses is the percent of loss defined by L0T (p, N, M ) · 100%, T · P Rss

h(0)

LT

(p, N, M ) · 100%. T · P Rss (10.24) The fourth problem considered in this chapter is to analyze the properties of Λ0T (p, N, M ) and to determine the smallest initial buffer occupancy h∗ (0) such h∗ (0) that ΛT (p, N, M ) = 0. This problem is discussed in Section 10.5. Λ0T (p, N, M ) =

h(0)

ΛT

(p, N, M ) =

10.2

Analysis of the Second Largest Eigenvalue

10.2.1

Two-machine lines

For N = 1 and 2, the characteristic polynomials of matrix A defined by (10.1) (with (λ − 1) being factored out) are: ¢ ¡ N =1: det(λI − A) = (λ − 1) λ + 2 p − p2 − 1 , £ ¢ ¤ ¡ N =2: det(λI − A) = (λ − 1) λ2 + −2 + 4 p − 3 p2 λ + (1 − p)4 , leading to the following expressions for the SLE as functions of p: λ1 (p, N = 1, M = 2)

=

(1 − p)2 ,

(10.25) p 2 2 2 + 3p − 4p + p 4(1 − p) + p . (10.26) 2 2

λ1 (p, N = 2, M = 2)

=

Characteristic polynomials for N ≥ 3 can also be calculated. For instance, ¡ ¢ N = 3 : det(λI − A) = (λ − 1)[λ3 + −3 − 5 p2 + 6 p λ2 + ¡ ¢ −12 p + 20 p2 + 3 + 6 p4 − 16 p3 λ − (1 − p)6 ], ¡ ¢ N = 4 : det(λI − A) = (λ − 1)[λ4 + −4 − 7 p2 + 8 p λ3 + ¡ ¢ −24 p + 6 + 42 p2 − 36 p3 + 15 p4 λ2 + ¡ ¢ 24 p + 40 p5 − 80 p4 − 4 − 63 p2 + 92 p3 − 10 p6 λ + (1 − p)8 ]. However, since their roots, as well as those for N ≥ 5, cannot be found analytically, we calculate the SLE numerically for various values of p ∈ (0, 1). The results, along with (10.25) and (10.26), are shown in Figure 10.1. As it follows from this figure, the behavior of the SLE as a function of p is qualitatively different for N = 1 and N ≥ 2: monotonically decreasing and non-monotonic convex, respectively. In other words, increasing machine efficiency speeds up the transients if N = 1 and may slow them down if N ≥ 2.

10.2. SECOND LARGEST EIGENVALUE

321

1 0.8

λ

1

0.6 0.4 0.2 0 0

N=1 N=2 N=3 N=5 N = 10 0.2

0.4

p

0.6

0.8

1

Figure 10.1: Second largest eigenvalue as a function of p in two-machine lines The explanation of this phenomenon is as follows: Due to the blocked before service assumption, N = 1 implies that each machine serves as a buffer capable of storing one part and no additional buffering is present. In other words, N = 1 represents just-in-time (JIT) operation. If a buffer is present (i.e., N ≥ 2), the states evolve slowly when the machines operate almost “synchronously”, i.e., are up or down almost simultaneously. Roughly, this synchronism can be characterized by the probability that both machines are up or down simultaneously. Since this probability is p2 + (1 − p)2 , the states move slowly when p is close to either 0 or 1. As one can see from Figure 10.1, for each N ≥ 2, there exists a unique p∗ , which to the smallest SLE, i.e., p∗ (N ) = arg min λ1 (p, N, M = 2). p

∗

The behavior of p as a function of N is shown in Figure 10.2. Interestingly, for all N ≥ 2, p∗ belongs to a relatively narrow int 0.6). This can be explained by the fact that the probability of one machine being up and the other being down is 2p(1 − p), which reaches its maximum at p = 0.5. To illustrate the duration of the transients for p = p∗ and for p’s close to 1, Figure 10.3(a) shows x5 (n) for a system with N = 5 and the initial condition x(0) = [1, 0, 0, 0, 0, 0]T . Clearly, large p’s lead to transients that are an order of magnitude slower than those defined by p∗ . The graphs of Figure 10.1 indicate that the SLE is a monotonically increasing function of N . More explicitly, the behavior of SLE as a function of N is illustrated in Figure 10.4. It shows, in particular, that for N = 10, the SLE is close to 1 for any p. Thus, large buffers lead to very long transients. This is illustrated in Figure 10.3(b), which shows x3 (n) for three systems − with N = 3, 5 and 10 − and with p = 0.9 and initially empty buffers. Clearly, N = 10 results in transients that are an order of magnitude longer than those for N = 3. The above analysis leads to:

322

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

0.62 0.6

p

*

0.58 0.56 0.54 0.52 2

4

6 N

8

10

Figure 10.2: Machine efficiency, which results in the fastest transients of buffer occupancy as a function of buffer capacity 0.35 0.3

0.25

0.25 x (n)

0.15 0.1

200

400

n

600

800

0.15 0.1

p=p*=0.5455 p=0.95 p=0.99

0.05 0 0

0.2

3

5

x (n)

0.2

N=3 N=5 N=10

0.05 0 0

1000

100

(a) N = 5

200

n

300

400

500

(b) p = 0.9

Figure 10.3: Transients of buffer occupancy for M = 2 1 0.8

λ

1

0.6 0.4

p = 0.3 p = 0.6 p = 0.9

0.2 0

2

4

N

6

8

10

Figure 10.4: Second largest eigenvalue as a function of buffer capacity

10.3. PRE-EXPONENTIAL FACTORS

323

Numerical Fact 10.1 In Bernoulli lines with identical machines defined by conventions (a)-(e) of Subsection 4.2.1 with M = 2, function λ1 (p, N ), N ≥ 2, is monotonically increasing in N and non-monotonic convex in p.

10.2.2

Three-machine lines

For M = 3, the transition matrix of a system defined with identical machines can be explicitly written for N  1 − p p(1 − p) 0 2  0 p(1 − p) (1 − p) A(N = 1) =   p 1−p p2 0 p(1 − p) p2

by assumptions (a)-(e) = 1 and N = 2:  0 p2 (1 − p)   , (10.27) p(1 − p)  (1 − p) + p3

A(N = 2) =                

1−p 0 0 p 0 0 0 0 0

(1 − p)p (1 − p)2 0 p2 (1 − p)p 0 0 0 0

0 (1 − p)p (1 − p)2 0 p2

0 (1 − p)p 0 (1 − p)2 p2

(1 − p)p 0 0 0

0 (1 − p)p 0 0

0 (1 − p)p2 (1 − p)2 p (1 − p)2 p (1 − p)3 + p3 (1 − p)p2 (1 − p)p2 (1 − p)2 p 0

0 0 (1 − p)p2 0 (1 − p)2 p p3 + (1 − p)2 0 (1 − p)p2 (1 − p)p

0 0 0 0 (1 − p)p 0 1−p p2 0

0 0 0 0 (1 − p)p2 (1 − p)2 p (1 − p)p p3 + (1 − p)2 (1 − p)p2

0 0 0 0 0 (1 − p)p2 0 (1 − p)p p3 + (1 − p)

        .       

(10.28)

It can be verified analytically that λ = 1 − p is an eigenvalue of (10.27) and numerically that it is the SLE. Thus, λ1 (p, N = 1, M = 3) = 1 − p.

(10.29)

The SLE of (10.28) can be evaluated numerically. These SLEs are illustrated in Figure 10.5 (along with their counterparts for M = 2). From this figure, we conclude that the SLEs for M = 3 and M = 2 are qualitatively similar as functions of p. In addition, λ1 (p, N, M = 3) > λ1 (p, N, M = 2), N ∈ {1, 2}, implying that transients for M = 3 are longer than those for M = 2. Since no results on the behavior of SLE for N ≥ 3 could be derived at this time, based on the above observations, we conjecture that: In serial production lines defined by conventions (a)-(e) of Subsection 4.2.1, function λ1 (p, N, M ) is monotonically increasing in N and M and non-monotonic convex in p.

10.3

Analysis of the Pre-exponential Factors

The effect of the pre-exponential factors on the transients of outputs, PR and WIP, is shown in expressions (10.19) and (10.20). In this section, we analyze their behavior as functions of system parameters.

324

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES 1 0.8

λ

1

0.6 0.4 0.2 0 0

N = 1, M = 2 N = 1, M = 3 N = 2, M = 2 N = 2, M = 3 0.2

0.4

p

0.6

0.8

1

Figure 10.5: Second largest eigenvalue as a function of machine efficiency in two- and three-machine lines Theorem 10.1 In Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1, p Ψ11 (p, N = 1, M = 2) = Ψ21 (p, N = 1, M = 2) = 1 + (1 − p)2 . (10.30) Proof: See Section 20.1. 1.8

Ψ11

1.6

Ψ

1.4

21

21

0.8

Ψ ,Ψ

1

11

1.2

0.6 0.4 0.2 0

0.2

0.4

p

0.6

0.8

1

Figure 10.6: Behavior of pre-exponential factors Ψ11 and Ψ21 as a function of machine efficiency in two-machine lines with N = 1 Thus, if N = 1, the SLE affects both outputs, PR and WIP, identically. In addition, since Ψ11 (p, N = 1, M = 2), as a function of p, behaves as shown in Figure 10.6, JIT leads not only to a monotonically decreasing SLE but also to a monotonically decreasing pre-exponential factor, implying that the transients of PR and WIP are somewhat faster in a system with p close to 1. For N ≥ 2, the situation is qualitatively different.

10.3. PRE-EXPONENTIAL FACTORS

325

Numerical Fact 10.2 In Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with N ≥ 2, Ψ11 (p, N, M = 2) < Ψ21 (p, N, M = 2)

(10.31)

and, in addition, Ψ11 (p, N, M = 2) → 0

1.8

as

21

1

11

1.2 0.8

Ψ ,Ψ

21 11

Ψ ,Ψ

0.8 0.6

0.6

0.4

0.4

0.2

0.2

0

Ψ21

1.4

1.2 1

Ψ11

1.6

Ψ21

1.4

(10.32)

1.8

Ψ11

1.6

p → 1.

0.2

0.4

p

0.6

0.8

1

0

0.2

(a) N = 2

0.4

p

0.6

0.8

1

(b) N = 3

1.8

Ψ11

1.6

Ψ21

1.4 21

0.8

Ψ ,Ψ

1

11

1.2

0.6 0.4 0.2 0

0.2

0.4

p

0.6

0.8

1

(c) N = 5 Figure 10.7: Behavior of pre-exponential factors Ψ11 and Ψ21 as a function of machine efficiency in two-machine lines with N ≥ 2 Justification: This numerical fact is justified by calculating Ψ11 and Ψ21 using the symbolic manipulation function of MATLAB. The analytical expressions for these functions are too long to be included here. However, their behavior for N = 2, 3, and 5 is shown in Figure 10.7. From this figure and (10.15), (10.16), we conclude: • Since both pre-exponential factors are monotonically decreasing in p, the effect of the SLE on the transients of PR and WIP is decreasing when p is increasing. • Since Ψ11 (p, N, M = 2) < Ψ21 (p, N, M = 2), the SLE affects WIP more than PR, i.e., transients of PR are faster than those of WIP.

326

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES • Since for most p’s, Ψ11 (p, N, M = 2) < 1, the transients of PR are faster than those of the states (i.e., buffer occupancy). • Finally, since Ψ11 (p, N, M = 2) → 0 as p → 1, the effect of SLE on the duration of transients of PR becomes negligible when p is close to 1.

Unfortunately, no general statement concerning the pre-exponential factors for M ≥ 3 could be derived. However, based on the above observations and numerical evidence, we conjecture that: Numerical Fact 10.2 holds for M ≥ 3 as well.

10.4

Analysis of the Settling Time

To analyze the settling times, tsP R and tsW IP , one has to know the behavior of PR and WIP as a function of n. Therefore, in this section, we first analyze the trajectories of P R(n) and W IP (n) and then utilize them to evaluate the settling time.

10.4.1

Behavior of P R(n) and W IP (n)

Two-machine lines: The trajectories of P R(n) and W IP (n) can be analyzed numerically by solving (10.5)-(10.7). In addition, analytical approximations, based on the SLE, can be constructed. This is carried out below. Approximation of P R(n): Consider the serial production line defined by conventions (a)-(e) of Subsection 4.2.1 with M = 2. Assume that at the initial time the buffer is empty. Then, P R(0)

= P [a part is produced during time slot 1] = 0,

(10.33)

P R(1)

= P [a part is produced during time slot 2] = P [machine 1 is up during time slot 1] ·P [machine 2 is up during time slot 2] = p2 .

(10.34)

For time slots n ≥ 2, introduce the following approximation: £ ¤ d P R(n) = P Rss 1 − βλ1n−1 ,

(10.35)

where λ1 is the SLE and P Rss is the steady state production rate, i.e., P Rss

=

Np . N +1−p

(10.36)

d Selecting β so that P R(1) = P R(1), we obtain β =1−

p(N + 1 − p) , N

0 < p < 1.

(10.37)

10.4. SETTLING TIME

327

Note that β tends to 0 as p tends to 1, which makes (10.35)-(10.37) in agreement with conclusion (δ) of Section 10.3: when p is close to 1 the effect of λ1 on PR diminishes to 0. The accuracy of approximation (10.35)-(10.37) is illustrated in Figure 10.8 by comparing it with P R(n), obtained by solving (10.5)-(10.7) numerically. This accuracy is quantified in Table 10.1 by ∆P R

d |P R(n) − P R(n)| · 100%. n=1,2,... P R(n)

=

(10.38)

max

Expressions (10.35)-(10.37) are utilized in Subsection 10.4.2 to evaluate tsP R . N =3 0.8

0.8

0.6

0.6 PR

1

PR

p = 0.7

N =5

1

0.4

0.4

0.2

0.2

0 0

20

n

30

40

1

1

0.8

0.8

0.6

0.6

10

20

n

30

40

50

PR

0.4

0.4

0.2

0.2

0 0

10

20

n

30

40

0 0

50

1

0.8

0.8

0.6

0.6

PR ^ PR 10

20

10

20

n

30

40

50

30

40

50

PR

1

PR

p = 0.9

0 0

50

PR

p = 0.8

10

0.4

0.4

0.2

0.2

0 0

10

20

n

30

40

50

0 0

n

d Figure 10.8: Comparison of P R(n) and P R(n) for M = 2 Approximation of W IP (n): Consider the serial production line defined by conventions (a)-(e) of Subsection 4.2.1 with M = 2 and the buffer initially

328

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

d Table 10.1: Accuracy of approximation P R(n) N N N N

=2 =3 =4 =5

p = 0.6 1.72 6.45 10.40 13.64

p = 0.7 0.65 3.98 6.91 9.27

p = 0.8 0.15 2.10 3.99 5.55

p = 0.9 0.01 0.82 1.72 2.49

empty. Then, W IP (0)

=

0,

(10.39)

W IP (1)

= P [a part is present in the buffer at the end of time slot 1] = p. (10.40)

For time slots n ≥ 2, the following approximation is introduced: £ ¤ \ W IP (n) = W IPss 1 − γλn−1 , 1

(10.41)

where, as before, λ1 is the SLE, W IPss is the steady state work-in-process, i.e., W IPss

=

N (N + 1) , 2(N + 1 − p)

(10.42)

\ and γ is selected so that W IP (1) = W IP (1), i.e., γ =1−

2p(N + 1 − p) . N (N + 1)

(10.43)

Note that γ tends to 0 as p tends to 1 only for N = 1, which makes (10.43) in agreement with Numerical Fact 10.2 of Section 10.3. The accuracy of approximation (10.41)-(10.43) is illustrated in Figure 10.9, and quantified in Table 10.2 by ∆W IP

=

\ |W IP (n) − W IP (n)| · 100%. n=1,2,... W IP (n) max

(10.44)

Expressions (10.39)-(10.43) are used in Subsection 10.4.2 for tsW IP evaluation. Comparison of P R(n) and W IP (n): To compare the transients of PR and d \ IP (n)/W IPss for WIP, Figure 10.10 shows the graphs of P R(n)/P Rss and W various p’s and N ’s. This figure also supports the conclusions of Section 10.3. Specifically, for N = 1, the transients of PR and WIP are identical. As N becomes larger, the difference becomes more pronounced. For large N , the transients of PR are orders of magnitude faster than those of WIP.

10.4. SETTLING TIME

329

N =3

p = 0.7

N =5

3

3

2.5

2.5 2 WIP

WIP

2 1.5 1

1

0.5

0.5

0 0

p = 0.8

10

20

n

30

40

0 0

50

3

3

2.5

2.5

1.5

40

n

60

80

1

1 0.5 10

20

n

30

40

0 0

50

3

3

2.5

2.5

WIP ^ WIP 20

40

20

40

n

60

80

100

60

80

100

2 WIP

2 1.5

1.5

1

1

0.5

0.5

0 0

10

20

n

30

40

50

0 0

n

\ Figure 10.9: Comparison of W IP (n) and W IP (n) for M = 2

\ Table 10.2: Accuracy of approximation W IP (n) N N N N

100

1.5

0.5 0 0

WIP

20

2 WIP

WIP

2

p = 0.9

1.5

=2 =3 =4 =5

p = 0.6 0.71 2.02 5.18 8.97

p = 0.7 0.34 1.28 4.18 7.53

p = 0.8 0.10 0.72 3.18 6.28

p = 0.9 0.01 0.29 2.39 5.17

330

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

N = 10

Normalized output approximation

1 0.8 0.6 0.4 0.2 0 0

10 n

15

20

Normalized output approximation

5

1 0.8 0.6 0.4 0.2 0 0

10

20

30 n

40

50

60

Normalized output approximation

Normalized output approximation

p = 0.9

1 0.8 0.6 0.4 0.2

^ PR/PRss ^ WIP/WIPss

0 0

50

n

100

150

Normalized output approximation

N =5

Normalized output approximation

N =3

Normalized output approximation

N =1

Normalized output approximation

p = 0.7

1 0.8 0.6 0.4 0.2 0 0

100

200

n

300

400

500

1 0.8 0.6 0.4 0.2 0 0

5

10 n

15

20

1 0.8 0.6 0.4 0.2 0 0

10

20

30 n

40

50

60

1 0.8 0.6 0.4 0.2

^ PR/PRss ^ WIP/WIPss

0 0

50

n

100

150

1 0.8 0.6 0.4 0.2 0 0

100

200

n

300

400

500

d \ Figure 10.10: Comparison of transients of P R(n)/P Rss and W IP (n)/W IPss for M = 2

10.4. SETTLING TIME

331

M > 2-machine lines: For M > 2, the behavior of P R(n) and W IP (n) = PM −1 i=1 W IPi (n), where W IPi (n) is the occupancy of buffer i, is analyzed using numerical simulations of systems defined by conventions (a)-(e) of Subsection 4.2.1. A C++ code is constructed to perform the simulations. For each line considered, 5000 runs are carried out. Within each iteration, every buffer is initialized to be empty, and the status of the machines, up or down, is selected with probability p and 1 − p, respectively. Then the average is calculated over 5000 runs for the output at each time slot, resulting in a 90% confidence intervals of less than 0.01 for P R(n) and 0.05 for W IP (n). The results of these simulations are shown in Figure 10.11. This figure also supports the conclusions of Sections 10.2 and 10.3: the transients of PR and WIP become slower as N and M increase. M =3

0.6 0.4 0.2 10 n

15

0.4 0.2 5

10 n

15

0.4 0.2 10

20

n

30

40

0.6 0.4 PR/PRss WIP/WIPss

0.2 10

20

30 n

40

50

0.4 0.2 50

n

100

150

0.8 0.6 0.4 0.2 0 0

0.2 10

20

20

40

n

30

40

50

60

80

100

0.8 0.6 0.4 0.2

n

1 Normalized Outputs

Normalized Outputs

0.6

0.4

0 0

60

1

0.8

0.6

1

0.8

0 0

50

0.8

0 0

20

Normalized Outputs

0.6

1 Normalized Outputs

0.6

1 Normalized Outputs

Normalized Outputs

1

0 0

0.8

0 0

20

0.8

0 0

N =5

5

1 Normalized Outputs

0.8

0 0

N =3

M = 10

1 Normalized Outputs

1 Normalized Outputs

N =1

M =5

50

100 n

150

200

0.8 0.6 0.4 0.2 0 0

50

100

150 n

200

250

300

Figure 10.11: Transients of P R(n)/P Rss and W IP (n)/W IPss in lines with M > 2 for p = 0.9 Figure 10.11 exhibits one more interesting phenomenon of transients in serial production lines: the transportation delay. Unlike the dynamic delay, the transportation delay is not related to eigenvalues but to the time necessary for the input to reach the output. When M = 2 and the buffer is initially empty, the transportation delay for P R and W IP is 1 time slot. This is reflected in (10.33) and (10.39). When M > 2, the transportation delay for PM −1 W IP (n) = i=1 W IPi (n) remains 1, while for P R(n) it becomes M − 1. Thus, for large M the transportation delay for P R may be significant. This is shown in Figure 10.11, where the transients of P R and W IP for N = 1 are no

332

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

longer identical. However, as N increases, the effect of the dynamics becomes dominant, and the transportation delay does not alter the nature of the response in any significant manner (see the case of M = 10, N = 5 in Figure 10.11).

10.4.2

Analysis of tsP R and tsW IP

For M = 2, approximations (10.35)-(10.37) and (10.41)-(10.43) lead to: Theorem 10.2 In Bernoulli lines defined by conventions (a)-(e) of Subsection 4.2.1 with M = 2, the estimates of tsP R and tsW IP can be given as follows tˆsP R tˆsW IP

ln(20β) , ln λ1 ln(20γ) = 1− , ln λ1 = 1−

(10.45) (10.46)

where β and γ are defined in (10.37) and (10.43), respectively. In addition, tˆsP R = tˆsW IP ,

for N = 1,

(10.47)

tˆsP R < tˆsW IP ,

for N ≥ 2.

(10.48)

and

Proof: See Section 20.1. The behavior of estimates (10.45) and (10.46), along with the exact values, tsP R and tsW IP , is illustrated in Figure 10.12. Clearly, tsP R is monotonically decreasing as a function of p while tsW IP is convex, which is consistent with the behavior of the SLE and PEF analyzed in Sections 10.2 and 10.3. The accuracy in terms of ∆tsP R

=

∆tsW IP

=

tsP R − tˆsP R · 100%, tsP R tsW IP − tˆsW IP · 100%, tsW IP

(10.49) (10.50)

is quantified in Tables 10.3 and 10.4. Obviously, tˆsW IP is more accurate than tˆsP R . Also, both estimates become more accurate for larger p and less accurate for larger N .

10.5

Analysis of the Production Losses

10.5.1

Approach

In this section, we analyze production losses during a shift of duration T cycles. It turns out that these losses are relatively insensitive to T as long as T À 1.

10.5. PRODUCTION LOSSES

333

tsP R and tˆsP R

tsW IP and tˆsW IP

200

200

&t

sWIP

150 <

sWIP

100

100

t

t

sPR

& t

<

sPR

150

N =1

50

50

0 0

0.2

0.4

p

0.6

0.8

0 0

1

600

0.2

0.4

p

0.6

tsWIP 500

400

&t

<

sWIP

tˆsPR

sWIP

300 200

t

sPR

t

sPR

& t

<

100 0.2

0.4

p

0.6

0.8

300 200

0 0

1

1200

1000

1000

800

&t

<

sWIP

1200

sWIP

600 400

t

sPR

t

sPR

& t

<

tˆsWIP

400

100

0 0

N =5

1

600 tsPR

500

N =3

0.8

200 0 0

0.2

0.4

p

0.6

0.8

1

0.6

0.8

1

800 600 400 200

0.2

0.4

p

0.6

0.8

1

0 0

0.2

0.4

p

Figure 10.12: Behavior of settling time of P R and W IP as a function of p and N for M = 2

Table 10.3: Accuracy of estimate tˆsP R N =1 N =3 N =5

p = 0.6 0 -25.00 -66.66

p = 0.7 0 -28.57 -84.61

p = 0.8 0 -14.28 -100.00

p = 0.9 0 -25.00 -166.66

p = 0.95 0 0 0

334

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES Table 10.4: Accuracy of estimate tˆsW IP N =1 N =3 N =5

p = 0.6 0 -7.6923 -2.9412

p = 0.7 0 0 -2.7027

p = 0.8 0 0 -2.1739

p = 0.9 0 0 -2.5641

p = 0.95 0 0 -2.0833

Therefore, we assume that T = 500 minutes, which is typical for automotive assembly plants where the cycle time is around 1 minute and the shift is 8 hours. The production losses have been defined in Subsection 10.1.5 as LT (x1 (0), . . . , xM −1 (0); p, N, M ) =

T X £

¤ P Rss − P R(n; x1 (0), . . . , xM −1 (0); p, N, M ) , (10.51)

n=0

or, as the percent of loss, ΛT (x1 (0), . . . , xM −1 (0); p, N, M ) =

LT · 100%. T · P Rss

(10.52)

We evaluate LT and ΛT as follows: For M = 2 and M = 3 (with N = 1 and 2), P R(n) is calculated numerically by solving (10.5)-(10.7), respectively. For all other M and N , P R(n) is evaluated by simulations using the C++ program and the procedures described in Subsection 10.4.1. To evaluate P Rss , another 20 runs of simulations are performed with the first 10,000 time slots as the warm-up time and the following 100,000 time slots used for the evaluation, which results in a 95% confidence interval of less than 0.003. Using this approach, Subsection 10.5.2 presents the results on Λ0T (p, N, M ), h(0) while Subsection 10.5.3 explores ΛT (p, N, M ).

10.5.2

Percent of loss when the buffers are empty at the beginning of the shift

Figure 10.13 shows the behavior of Λ0T as a function of p, N , and M . In addition, the losses are quantified in Table 10.5 for M = 10. ¿From these data, we conclude: • Λ0T (p, N, M ) is a monotonically decreasing function of p with an almost constant rate (except for p’s close to 1). • Λ0T (p, N, M ) is a monotonically increasing function of N , again with an almost constant rate. • For large M and N and small p, Λ0T (p, N, M ) can be quite large. For instance, if M and N are 10 and p = 0.6, Λ0T (p, N, M ) is almost 12%.

10.5. PRODUCTION LOSSES

335

Λ0T vs p

Λ0T vs N

3

M =2

2 T

1.5

1

0.5

0.5

0.7

0.8 p

0.9

0

1

3

3

2.5

2.5

0 T

0.5

0.5 0.7

0.8 p

0.9

0

1

12

10

10

8

8 0 T

12

Λ

6

4

2

2 0.7

0.8 p

0.9

0

1

12

12

10

10

8

8

6

6

8

10

4

2

2 0.7

0.8 p

0.9

1

2

4

2

4

2

4

N

6

8

10

N

6

8

10

6

8

10

6

4

0 0.6

N

6

4

0 T

0 T

Λ Λ

0 T

M = 10

Λ

1

0 0.6

4

1.5

1

0 0.6

2

2

1.5

Λ

Λ

0 T

2

M =5

1.5

1

0 0.6

p=0.6 p=0.7 p=0.8 p=0.9 p=0.95 p=0.99

2.5

Λ0

Λ

0 T

2

M =3

3

N=1 N=3 N=5 N=10

2.5

0

N

Figure 10.13: Percent of production loss, Λ0T , as functions of p, N , and M

336

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES Table 10.5: Percent of loss Λ0T for ten-machine lines N =1 N =2 N =3 N =4 N =5 N =6 N =7 N =8 N =9 N = 10

10.5.3

p = 0.6 2.88 3.95 5.07 6.14 7.14 8.28 9.25 10.18 10.99 11.72

p = 0.7 2.49 3.39 4.24 5.15 6.08 7.02 7.90 8.69 9.35 9.84

p = 0.8 2.18 2.85 3.70 4.45 5.31 6.10 6.78 7.39 7.89 8.33

p = 0.9 1.92 2.52 3.23 3.95 4.62 5.21 5.70 6.10 6.40 6.67

p = 0.95 1.79 2.38 3.06 3.69 4.22 4.63 4.88 5.11 5.26 5.41

p = 0.99 1.71 2.25 2.76 3.01 3.17 3.25 3.30 3.31 3.32 3.32

Percent of loss when the buffers are not empty at the beginning of the shift h(0)

Figure 10.14 shows the behavior of ΛT From this figure we conclude:

as a function of h(0), p, N , and M .

h(0)

• ΛT (p, N, M ) is a monotonically decreasing function of h(0) with a decreasing rate. h(0)

• Roughly, ΛT (p, N, M ) = 0 if h(0) = dN/2e, where dxe is the smallest integer no less than x. Thus, Numerical Fact 10.3 Half full buffers provide the smallest initial buffer occupancy, which leads to practically zero losses due to transients.

10.6

Summary

• The transients of production systems can be characterized in terms of the following metrics: the second largest eigenvalue (SLE) of the transition matrix; pre-exponential factors (PEF); settling time; and production losses due to transients. • The SLE is a monotonically increasing function of the buffer capacity and the number of machines in the system. Thus, the transients of buffer occupancy in systems with large buffers and many machines are inherently long. • In production lines with buffer capacity larger than 1, the SLE is a nonmonotonic convex function of machine efficiency. This implies that the longest transients of buffer occupancy are in systems with machines having either low or high efficiencies.

10.6. SUMMARY

337

N =2

N =5

0.4 h(0) T

Λ

Λ

h(0) T

0.5 0.2

N = 10 1.5

p=0.6 p=0.7 p=0.8 p=0.9 p=0.95 p=0.99

1 h(0) T

M =2

1

0.5

Λ

0.6

0

0 0

1 h(0)

1.5

−0.5 0

2

1

M =3

h(0) T

3

h(0)

4

−0.5 0

0.5

1 h(0)

1.5

3

1

2

0.5

1

0

0

−0.5

−1

−1 0

2

2 1.5

−1 0

5

1.5

0

M =5

2

Λ

Λ

h(0) T

0.5

1

h(0) T

0.5

1

2

3

h(0)

4

−2 0

5

3

6

2

4

1

2

−0.5 −1 0

1 h(0)

1.5

0

2

6

M = 10

6

8

10

h(0) T

2

4

2

4

2

4

h(0)

6

8

10

6

8

10

6

8

10

−2

−1 0.5

h(0)

0

0

0

4

Λ

T

Λh(0)

Λ

h(0) T

1 0.5

2

Λ

−0.2 0

−0.5

1

2

h(0)

3

4

−4 0

5

8

h(0)

15

6 4

10 h(0) T

2

Λ

h(0) T

Λ

Λ

h(0) T

4 2

5

0 0

0 −2

−2 0

0.5

1 h(0)

1.5

2

−4 0

1

2

h(0)

3

h(0)

Figure 10.14: Percent of production loss, ΛT M

4

5

−5 0

h(0)

, as functions of h(0), p, N , and

338

CHAPTER 10. TRANSIENT BEHAVIOR OF BERNOULLI LINES

• Fortunately and, to a certain extent, unexpectedly, the transients of the production rate (PR) are considerably faster than those of the buffer occupancy. This is due to the fact that the transients of PR are defined not only by SLE but also by its pre-exponential factor, which happens to be relatively small. • The situation with the transients of the work-in-process (WIP ) is less fortunate: here, the pre-exponential factor is substantially larger, and the transients of WIP are roughly as long as those of the states. • Production losses due to transients can be quite substantial (up to 12% in systems with low efficiency machines). • In order to avoid production losses due to transients, the buffers at the beginning of the shift must be at least half full.

10.7

Problems

Problem 10.1 Consider the Bernoulli production line with two machines and buffer of capacity N . The transients of its states are described by the equation: x(n + 1) = Ax(n),

x ∈ RN +1 , A ∈ R(N +1)×(N +1) , n = 0, 1, . . . ,

where A is a stochastic matrix, i.e., the of the elements of every column is Psum PN N 1. Prove that if i=0 xi (0) = 1, then i=0 xi (n) = 1, ∀n. Problem 10.2 Consider the Bernoulli line with two non-identical machines. (a) Calculate the second largest eigenvalue of its transition matrix for p1 = 0.7, p2 = 0.95, N = 3. (b) Calculate the second largest eigenvalue of its transition matrix for p1 = 0.95, p2 = 0.7, N = 3. (c) Does the property of transients reversibility for the states take place or not? Problem 10.3 Repeat parts (a) and (b) of Problem 10.2 for (a) p1 = 0.9, p2 = 0.95, N = 3; (b) p1 = p2 = 0.95, N = 3. (c) Does the speed of state transients increase or decrease when the efficiencies of the machines become closer to each other? Problem 10.4 Plot the transients of P R(n) and W IP (n) by solving the equations: x(n + 1) P R(n)

= =

Ax(n), p

N X i=1

x(n) = [1, 0, 0, 0],

xi (n),

10.8. ANNOTATED BIBLIOGRAPHY

W IP (n) =

N X

339

ixi (n)

i=1

for the following systems: (a) p1 = 0.7, p2 = 0.95, N = 3; (b) p1 = 0.95, p2 = 0.7, N = 3. (c) Does the property of transients reversibility for the outputs take place or not? Problem 10.5 Repeat parts (a) and (b) of Problem 10.4 for (a) p1 = 0.9, p2 = 0.95, N = 3; (b) p1 = p2 = 0.95, N = 3. (c) Does the speed of the output transients increase or decrease when the efficiencies of the machines become closer to each other? Problem 10.6 Consider a two-machine Bernoulli line with the symmetric blocking convention. Assume both machines are identical with efficiency p and the buffer capacity is N . (a) Analyze the behavior of the SLE as a function of p and N . (b) Analyze the behavior of the PEFs as a function of p and N . (c) Formulate qualitative results of these analyzes with regard to the relative speeds of the transients of buffer occupancies, P R, and W IP . (d) Are these conclusions different from those obtained under the blocked before service convention.

10.8

Annotated Bibliography

The literature on transient behavior of production lines is scarce; only a few results are available. The issue of transient analysis in production-like systems has been raised in [10.1] S. Mocanu, “Numerical Algorithms for Transient Analysis of Fluid Queues,” Proceedings of Fifth International Conference on Analysis of Manufacturing Systems - Production Management, pp. 193-200, Zakynthos Island, Greece, May, 2005. The material of this chapter follows [10.2] S.M. Meerkov and L. Zhang, “Transient Behavior of Serial Production Lines with Bernoulli Machines,” IIE Transactions, vol. 40, pp. 297-312, 2008.

Chapter 11

Analysis of Exponential Lines Motivation: The motivation for developing analytical methods for analysis of serial lines with exponential machines is the same as in the Bernoulli case: the need for quick and easy methods for calculating P R, T P , W IPi , BLi , and STi . In this chapter, such methods are developed for both synchronous and asynchronous lines. Recall that in the synchronous case all machines have an identical cycle time, while in the asynchronous case cycle times are different (see Chapter 3). The exponential case is of importance because it can be used to model machining operations where the downtime of the machines is much longer than the cycle time and, therefore, the Bernoulli model is not applicable (unless exp-B and B-exp transformations are used). The geometric reliability model can also be applied to some of these systems. In fact, similar results have been derived for geometric machines as well. We concentrate, however, on the exponential model because, in our experience, it has a wider applicability. Overview: The approach of this chapter is similar to that of Chapter 4: first closed-form expressions for the performance measures of two-machine lines are derived and then an aggregation procedure for longer lines is proposed. The development, however, is more complicated since, unlike the Bernoulli case, the machines are dynamical systems with two states – up and down. In addition, the flow model is considered (see Chapter 3), which implies that the states of the buffers are continuous random variables. Thus, the mathematical description of the systems at hand leads to mixed state Markov processes. Also, since the machine state transitions can occur at any time moment (rather than at the beginning of a cycle time – as it is in the case of Bernoulli machines), the mathematical description is in terms of continuous time and differential equations, which require additional technical details. Thus, the development described in this chapter is more involved than in Chapter 4. Ideologically, 343

344

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

however, the two chapters are quite similar, and the reader not interested in details of the derivations may omit the technicalities. Using the above framework, Sections 11.1 and 11.2 below present performance analysis techniques for synchronous and asynchronous lines, i.e., lines {[exp, exp]1 , . . ., [exp, exp]M } and {[τ, exp, exp]1 , . . ., [τ, exp, exp]M }, respectively. Note that the aggregation techniques in these two cases are different: in the former they are based on the breakdown and repair rates λi and µi , while in the latter on the cycle time or, more precisely, on the machine capacity ci = 1/τi .

11.1

Synchronous Exponential Lines

11.1.1

Two-machine case

Mathematical description: The production system considered here is shown in Figure 11.1. Without loss of generality, it is assumed that c1 = c2 = 1 and, therefore, ci ’s are not indicated in this figure. Time is continuous and measured in unit of cycle time. Machines m1 and m2 obey the exponential reliability model, i.e., have breakdown and repair rates λi and µi , i = 1, 2, respectively. The buffer capacity is N .

λ1, µ

N

λ2 , µ

m1

b

m2

1

2

Figure 11.1: Synchronous exponential two-machine line Conventions: (a) Blocked before service. (b) The first machine is never starved; the last machine is never blocked. (c) Flow model, i.e., infinitesimal quantity of parts, produced during δt, are transferred to and from the buffers. (d) The state of each machine (up or down) is determined independently from all other machines. (e) Time-dependent failures. The flow model is assumed here to simplify the analysis. It turns out that it provides a sufficiently precise description of real, discrete event, systems. States of the system: Since the machines are exponential, each has two states: up = 1 and down = 0. Since the flow model is used, the state of the buffer is a real number between 0 and N . Thus, the system under consideration is described by a continuous time, mixed space Markov process. The state of

11.1. SYNCHRONOUS EXPONENTIAL LINES

345

the system is denoted by a triple (h, s1 , s2 ), where h is the state of the buffer and si , i = 1, 2, are the states of the first and the second machine, respectively. Statistical description of the states: It is convenient to separate the states into two groups: boundary (0, s1 , s2 ), (N, s1 , s2 ) and internal (h, s1 , s2 ), 0 < h < N . It turns out that the boundary states are described by “concentrated” or “mass” probabilities, i.e., by P [h(t) = 0, s1 (t) = i, s2 (t) = j] P [h(t) = N, s1 (t) = i, s2 (t) = j]

= =

P0,ij (t), PN,ij (t),

(11.1) (11.2)

while the internal states are described by “distributed” probabilities, i.e, h i ∆h ∆h P h− ≤ h(t) ≤ h + , s1 (t) = i, s2 (t) = j = fH,IJ (h, i, j, t)∆h 2 2 +o(∆h), 0 < h < N, (11.3) where ∆h ¿ 1. Obviously, 1 X 1 Z X i=0 j=0

N−

0+

fH,IJ (h, i, j, t)dh +

1 X 1 X

P0,ij (t) +

i=0 j=0

1 X 1 X

PN,ij (t) = 1.

i=0 j=0

Steady state probabilities: Boundary states: Using the method for analysis of continuous time, mixed state Markov processes, described in Section 2.3.3, the steady states of both “concentrated” and “distributed” probabilities can be evaluated (see Section 20.2). In this analysis, there emerges a function Q, which plays the same role as function Q in the Bernoulli case. For the exponential machines, this function takes the form:  (1−e )(1−φ) 1  , if µλ11 6= µλ22 ,  1−φe−βN   (11.4) Q(λ1 , µ1 , λ2 , µ2 , N ) = λ1 (λ1 +λ2 )(µ1 +µ2 )   (λ1 +µ1 )[(λ1 +λ2 )(µ1 +µ2 )+λ2 µ1 (λ1 +λ2 +µ1 +µ2 )N ] ,   if µλ11 = µλ22 , where ei

=

φ

=

β

=

µi , i = 1, 2, λi + µi e1 (1 − e2 ) , e2 (1 − e1 ) (λ1 + λ2 + µ1 + µ2 )(λ1 µ2 − λ2 µ1 ) . (λ1 + λ2 )(µ1 + µ2 )

(11.5) (11.6) (11.7)

In terms of this function, the stationary probabilities of the states can be expressed. For instance, P [{buffer is empty, m1 is down}|{m2 is up}]

= P [h = 0, s1 = 0|s2 = 1]

346

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

P [{buffer is full, m2 is down}|{m1 is up}]

=

Q(λ1 , µ1 , λ2 , µ2 , N ),(11.8)

= =

P [h = N, s2 = 0|s1 = 1] Q(λ2 , µ2 , λ1 , µ1 , N ).(11.9)

Similar to Bernoulli lines, function Q for synchronous exponential lines has the following monotonicity properties (which ensure the convergence of the recursive procedure for M > 2-machine lines described in Subsection 11.1.2): Lemma 11.1 Function Q(λ1 , µ1 , λ2 , µ2 , N ) defined by (11.4), with λi ∈ (0, ∞), µi ∈ (0, ∞), and N ∈ (0, ∞), takes values on (0, 1) and is • strictly decreasing in λ2 and µ1 , • strictly increasing in λ1 and µ2 , • strictly decreasing in N . Proof: See Section 20.2. Marginal pdf of buffer occupancy: The marginal steady state pdf of buffer occupancy can be expressed as follows (see Section 20.2 for derivations): fH (h) = A1 eKh + A2 δ(h) + A3 δ(h − N ),

h ∈ [0, N ],

(11.10)

where δ(·) is the δ-function, which accounts for the “concentrated” probabilities at h = 0 and h = N , and  (µ1 +µ2 +λ1 +λ2 )(λ2 µ1 −λ1 µ2 ) , for e1 6= e2 ,  (µ1 +µ2 )(λ1 +λ2 ) K =  0, for e1 = e2 , A1 A2 A3 D1 D2

=

2 + D1 +

1 D1

, D2 + D3 + D4 D3 = , D2 + D3 + D4 D4 = , D2 + D3 + D4 µ1 + µ2 = , λ 1 + λ2  KN  (2 + D1 + D11 ) e K−1 , =  (2 + D1 + D11 )N,

D3

=

D4

=

(11.11) for e1 6= e2 , for e1 = e2 ,

(λ1 + λ2 + µ1 + µ2 )(λ2 + µ1 ) + λ1 µ2 − λ2 µ1 , λ2 µ1 (λ1 + λ2 + µ1 + µ2 ) (λ1 + λ2 + µ1 + µ2 )(λ1 + µ2 ) + λ2 µ1 − λ1 µ2 KN e . λ1 µ2 (λ1 + λ2 + µ1 + µ2 )

This pdf is illustrated in Figure 11.2 for six serial lines defined below: Lines with identical ei ’s :

11.1. SYNCHRONOUS EXPONENTIAL LINES

347

µ1 = µ2 = 0.5, N = 20, L1 : λ1 = λ2 = 0.1, L2 : λ1 = λ2 = 0.01, µ1 = µ2 = 0.05, N = 20, Reversed lines with identical ei ’s : λ2 = 0.01, µ2 = 0.05, N = 20, L3 : λ1 = 0.1, µ1 = 0.5; (11.12) N = 20, L4 : λ1 = 0.01, µ1 = 0.05; λ2 = 0.1, µ2 = 0.5, Reversed lines with non-identical ei ’s : L5 : λ1 = 0.1, λ2 = 0.05, µ1 = µ2 = 0.5, N = 20, L6 : λ1 = 0.05, λ2 = 0.1, µ1 = µ2 = 0.5, N = 20. These figures indicate that: • the continuous part of the mixed random variable H is uniform for e1 = e2 and non-uniform for e1 6= e2 ; • the discrete part of the mixed random variable H is characterized by P [h = 0] = P [h = N ] if e1 = e2 and by P [h = 0] 6= P [h = N ] if e1 6= e2 ; • the lines with longer up- and downtime have larger probabilities of h = 0 and h = N than lines with the same e1 , e2 and N but with shorter upand downtime; • in reversed lines, the pdf’s are symmetrically reversed. Formulas for the performance measures: It is shown in Section 20.2 that P R, W IP , BL1 , and ST2 of a synchronous exponential two-machine line can be expressed as follows: Production rate: PR

= e2 [1 − Q(λ1 , µ1 , λ2 , µ2 , N )] = e1 [1 − Q(λ2 , µ2 , λ1 , µ1 , N )].

(11.13)

Work-in-process:    W IP =

 

D5 D2 +D3 +D4 , D

( 22 +D4 )N D2 +D3 +D4 ,

for e1 6= e2 , (11.14) for e1 = e2 ,

where D2 , . . . , D4 are given in (11.11) and D5

=

D2 [1 + (KN − 1)eKN ] + D4 N. K(eKN − 1)

(11.15)

Blockages and starvations: BL1 ST2

=

e1 Q(λ2 , µ2 , λ1 , µ1 , N ),

(11.16)

=

e2 Q(λ1 , µ1 , λ2 , µ2 , N ).

(11.17)

Clearly, these expressions are in agreement with (11.13), which now can be expressed as P R = e1 − BL1 = e2 − ST2 ,

348

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

f (h)

f (h)

H

H

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 0

5

10

15

20

h

0 0

5

(a) L1 0.4

0.4

H

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

20

15

20

15

20

h

f (h) H

0.05

0 0

5

10

15

20

h

0 0

5

(c) L3 0.5

H

0.4

0.3

0.3

0.2

0.2

0.1

0.1

5

10

(e) L5

10

h

(d) L4

f (h)

0.4

0 0

15

(b) L2

f (h)

0.35

0.5

10

15

20

h

0 0

fH(h)

5

10

(f) L6

Figure 11.2: Stationary distributions of buffer occupancy for Lines L1 -L6

h

11.1. SYNCHRONOUS EXPONENTIAL LINES

349

and with similar expressions derived in Chapter 4 for Bernoulli lines: P R = p1 − BL1 = p2 − ST2 . Effects of up- and downtime: Consider two lines, l1 and l2 , with el11 = el12 ,

el21 = el22 ,

N l1 = N l2 ,

(11.18)

i.e., with machines of identical efficiency and identical buffer capacity. Assume also that l1 l2 < Tdown,i , Tdown,i

i = 1, 2,

(11.19)

i.e., the machines in l1 have shorter up- and downtime than the machines in l2 . Theorem 11.1 In synchronous exponential two-machine lines defined by conventions (a)-(e), under assumptions (11.18), (11.19), P R l1 > P R l2 . Proof: See Section 20.2. Thus, shorter up- and downtime lead to a higher throughput than longer ones, even if the machines’ efficiency remains the same. This phenomenon takes place because finite buffers protect against shorter downtime better than against longer ones. Mathematically, this phenomenon is due to the fact that P [h = 0] and P [h = N ] are larger for machines with longer up- and downtime (see Figure 11.2). Another characterization of the effects of up- and downtime can be given as follows: Clearly, P R can be improved by either increasing the uptime of a machine or decreasing its downtime. Is it more beneficial to increase the uptime, say by a factor 1 + α, α > 0, or decrease its downtime by the same factor? When the production line consists of a single machine (in fact, with an arbitrary continuous time reliability model), the answer is obvious: the efficiency of the machine with either increased uptime or decreased downtime remains the same since in either case e=

1 1+

Tdown (1+α)Tup

.

Thus, for an isolated machine, increasing uptime or decreasing downtime by the same factor has the same effect. The situation is different in the case of more than one-machine systems. Specifically, Theorem 11.2 In synchronous exponential two-machine lines defined by assumptions (a)-(e), the P R has a larger increase when the downtime of a machine is decreased by a factor (1 + α), α > 0, than when the uptime is increased by the same factor.

350

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

Proof: See Section 20.2. This phenomenon also takes place because the buffer accommodates shorter downtimes more efficiently than longer ones. Although the effects of Tup and Tdown are formulated above for synchronous exponential two-machine lines, as it is indicated later in this chapter and in Chapter 12, they hold for synchronous and asynchronous exponential lines with M > 2 machines and for non-exponential lines as well. Asymptotic properties: Theorem 11.3 In synchronous exponential two-machine lines defined by assumptions (a)-(e), lim P R

N →∞

lim W IP

N →∞

lim BLi

N →∞

lim ST2

N →∞

= min(e1 , e2 ),  ∞,      (λ1 +λ2 )(µ1 +µ2 ) = (λ2 µ1 −λ1 µ2 )2 ,      limN →∞ N2 = ∞, = 0,   0, =  e2 − e1 ,

(11.20) if e1 > e2 , if e1 < e2 ,

(11.21)

if e1 = e2 , (11.22)

if e1 ≥ e2 , (11.23) if e1 < e2 .

Proof: See Section 20.2. This theorem is illustrated in Figure 11.3 for the six serial lines defined in (11.12) with N ∈ {1, 2, . . ., 100}. From this figure, the following conclusions can be made: • While P R is a saturating function of buffer capacity, W IP is increasing, practically linearly as N → ∞ (for e2 ≤ e1 ). Thus, there is no reason for having large N ’s. A specific value of reasonable N depends on the downtime of the machines. For instance, for lines 5 and 6, where Tdown = 2, N = 10 may be viewed as reasonable. For Line 1, a reasonable N is about 40, while for Line 2 it is over 100. The issue of reasonable (e.g., lean) buffer capacity is explored in details in Chapter 14. • Although for infinite buffers, P Rs in lines with identical machine efficiencies are the same, Figure 11.3 quantifies how much P R for shorter up- and downtime is larger than P R for longer ones. Indeed, Figure 11.3(a) shows that for N = 20, P RL1 = 0.8135, while P RL2 = 0.7465, which is almost 10% different. For smaller N ’s, the difference may be even larger. Results for two-machine lines are used below for analysis of M > 2-machine production systems.

11.1. SYNCHRONOUS EXPONENTIAL LINES

0.86 0.84

100 90

L 5 and L 6

L5

80

0.82

L 3 and L 4

L1

0.8

70 60

WIP

0.78

PR

351

0.76

L2

50

L4

40

0.74

L2

30 0.72 0.7

10

0.68 0

20

40

N

60

80

0

100

L6 10

20

30

(a) P R

50

N

60

70

80

90

100

0.16

0.14

L6

0.14

L5

0.12

0.12 0.1

ST2

0.1

BL1

40

(b) W IP

0.16

0.08

L2

0.06

L2 0.08 0.06

0.04

0.04

L1

0.02

L1

L 3 and L 4

20

40

60

N

(c) BL1

L 3 and L 4

0.02

L6 0 0

L3

L1

20

80

100

0 0

L5 20

40

60

80

100

N

(d) ST2

Figure 11.3: Performance measures of Lines L1 -L6 as a function of buffer capacity

352

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

11.1.2

M > 2-machine case

Mathematical description and aggregation preliminaries: The production system under consideration is shown in Figure 11.4. Time is continuous, and up- and downtime of machine mi , i = 1, . . . , M , are distributed exponentially with parameters λi and µi , respectively. The capacity of buffer i is Ni . λ1, µ1

N1

λ2, µ 2

N2

m1

b1

m2

b2

λM−1, µ

M−1

mM−1

N M−1 λ M, µ M

bM−1

mM

Figure 11.4: Synchronous exponential M -machine line Conventions: Remain the same as in the two-machine case, i.e., (a)-(e) of Subsection 11.1.1. States of the system: Each machine mi , i = 1, . . . , M , has two states and the state of each buffer bi , i = 1, . . . , M − 1, is defined on the continuum [0, Ni ]. Thus, the state of the system is (h1 , . . . , hM −1 , s1 , . . . , sM ), hi ∈ [0, Ni ], si ∈ {0, 1}. As in the Bernoulli case, the system is too complex for direct analysis, and a simplification is in order. Again, an aggregation approach is used. Aggregation concepts: Conceptually, the aggregation used here is identical to that of the Bernoulli case, i.e., the backward aggregation with machines mbM −1 (s = 1), . . ., mb1 (s = 1) is followed by the forward aggregation with machines mf2 (s = 1), . . ., mfM (s = 1), and so on for s = 2, 3, . . .. However, the parameters of the aggregated machines, which are now transition rates rather than transition probabilities, are selected differently. Backward aggregation: For the aggregated machine mbM −1 (1), the repair rate, µbM −1 (1), is assigned as follows: µbM −1 (1) = µM −1 (1 − Q(λM , µM , λM −1 , µM −1 , NM −1 )). This assignment accounts for the fact that mM −1 may be blocked and, therefore, its “average downtime” (including its real downtime and blockage time) is increased, and its repair rate is decreased. Clearly, the reduction in µ, quantified by 1 − Q, is related to the probability of blockage in two-machine lines. The breakdown rate of mbM −1 (1), i.e., λbM −1 (1), is assigned based on the following considerations: The production rate of the two-machine line being aggregated can be expressed using (11.13), as µM −1 [1 − Q(λM , µM , λM −1 , µM −1 , NM −1 ))]. λM −1 + µM −1 Selecting ebM −1 (1) equal to this production rate leads to the following equation

11.1. SYNCHRONOUS EXPONENTIAL LINES

353

with a single unknown, λbM −1 (1): µbM −1 (1) µM −1 = [1 − Q(λM , µM , λM −1 , µM −1 , NM −1 )]. b b λM −1 + µM −1 λM −1 (1) + µM −1 (1) (11.24) Solving it, we obtain the expression for the breakdown rate: λbM −1 (1) = λM −1 + µM −1 Q(λM , µM , λM −1 , µM −1 , NM −1 ). Thus, mbM −1 (1) is completely defined. All other machines in the backward aggregation are defined similarly. Forward aggregation: The repair rate of mf2 (1) is defined as µf2 (1) = µ2 (1 − Q(λ1 , µ1 , λb2 (1), µb2 (1), N1 )). The breakdown rate of mf2 (1) is calculated from an equation similar to (11.24), i.e., from µf2 (1) λf2 (1) resulting in

+

µf2 (1)

=

µ2 [1 − Q(λ1 , µ1 , λb2 (1), µb2 (1), N1 )], λ2 + µ2

λf2 (1) = λ2 + µ2 Q(λ1 , µ1 , λb2 (1), µb2 (1), N1 ).

All other machines in the forward aggregation are defined similarly. Aggregation equations and their properties: to the following Aggregation equations: µbi (s + 1)

=

λbi (s + 1)

=

µfi (s + 1)

=

λfi (s + 1)

=

The above arguments lead

µi (1 − Q(λbi+1 (s + 1), µbi+1 (s + 1), λfi (s), µfi (s), Ni )), i = 1, . . . , M − 1, λi + µi Q(λbi+1 (s + 1), µbi+1 (s + 1), λfi (s), µfi (s), Ni ), (11.25) i = 1, . . . , M − 1, f f b µi (1 − Q(λi−1 (s + 1), µi−1 (s + 1), λi (s + 1), µbi (s + 1), Ni−1 )), i = 2, . . . , M, λi + µi Q(λfi−1 (s + 1), µfi−1 (s + 1), λbi (s + 1), µbi (s + 1), Ni−1 ), i = 2, . . . , M, s = 1, 2, . . . ,

with initial conditions λfi (0) = λi ,

µfi (0) = µi ,

i = 2, . . . , M − 1,

and boundary conditions λf1 (s) = λ1 ,

µf1 (s) = µ1 ,

s = 1, 2, . . . ,

354

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES λbM (s) = λM ,

µbM (s) = µM ,

s = 1, 2, . . . ,

where function Q is defined by (11.4). Clearly, this procedure iterates λi ’s, µi ’s and Ni ’s, resulting in 4(M − 1) sequences of numbers λf2 (s), . . . , λfM (s), λb1 (s), . . . , λbM −1 (s), s

µf2 (s), . . . , µfM (s), µb1 (s), . . . , µbM −1 (s), = 1, 2, . . . .

The properties of these sequences and their interpretation are described next. Convergence: Theorem 11.4 Aggregation procedure (11.25) has the following properties: (i) The sequences, λf2 (s), . . ., λfM (s), µf2 (s), . . ., µfM (s), and λb1 (s), . . ., λbM −1 (s), µb1 (s), . . ., µbM −1 (s), s = 1, 2, . . ., are convergent, i.e., the following limits exist: lim λfi (s) =: λfi ,

s→∞

lim λbi (s) =: λbi ,

s→∞

lim µfi (s) =: µfi ,

s→∞

lim µbi (s) =: µbi .

s→∞

(11.26)

(ii) These limits are unique solutions of the steady state equations corresponding to (11.25), i.e., of µbi λbi µfi λfi

=

µi [1 − Q(λbi+1 , µbi+1 , λfi , µfi , Ni )],

=

λi + µi Q(λbi+1 , µbi+1 , λfi , µfi , Ni ), i µi [1 − Q(λfi−1 , µfi−1 , λbi , µfi , Ni−1 )], λi + µi Q(λfi−1 , µfi−1 , λbi , µfi , Ni−1 ),

= =

i = 1, · · · , M − 1, = 1, · · · , M − 1, i = 2, · · · , M, (11.27) i = 2, · · · , M.

(iii) In addition, these limits satisfy the relationships: efM

=

eb1

=

ebi+1 [1 − Q(λfi , µfi , λbi+1 , µbi+1 , Ni )]

=

efi [1

−

Q(λbi+1 , µbi+1 , λfi , µfi , Ni )],

(11.28) i = 1, . . . , M − 1,

where efi =

µfi λfi

+

µfi

,

ebi =

λbi

µbi , + µbi

i = 1, . . . , M.

Proof: See Section 20.2. Interpretation: As it follows from statement (iii) of Theorem 11.4, λfi , µfi and λbi , µbi can be given the following interpretation: From the point of view of each buffer bi , i = 1, . . . , M − 1, the upstream and downstream of the line are represented by exponential machines with parameters (λfi , µfi ) and (λbi+1 , µbi+1 ), respectively. Thus, taking into account (11.28), the M > 2-machine line can be represented as shown in Figure 11.5..

11.1. SYNCHRONOUS EXPONENTIAL LINES f

f

λi ,µi

f

mi

b

b

λ i +1, µ i +1

bi

b mi+1

=

f

f

b

λ M,µ M

λ 1, µ 1

b

Ni

355

= f

b m1

mM

i = 1,..., M − 1

Figure 11.5: Equivalent representation of synchronous exponential M > 2machine line through the aggregated machines Performance measure estimates: Based on the limits of the recursive aggregation procedure and the above equivalent representation, the performance measures of synchronous exponential M > 2-machine lines are defined, using the expressions for two-machine lines, as follows: Production rate: d P R

=

efM = eb1

=

ebi+1 [1 − Q(λfi , µfi , λbi+1 , µbi+1 , Ni )]

=

efi [1 − Q(λbi+1 , µbi+1 , λfi , µfi , Ni )], i = 1, . . . , M − 1.

(11.29)

Work-in-process:

\ W IP =

    

D5 D2 +D3 +D4 , D

( 22 +D4 )Ni D2 +D3 +D4 ,

for efi 6= ebi+1 , (11.30) for efi = ebi+1 ,

where K, Di , i = 2, ..., 5, are defined in (11.11) , (11.15) with λ1 , µ1 , λ2 , µ2 , N substituted by λfi , µfi , λbi+1 , µbi+1 , Ni , respectively. Blockages and starvations: di BL ci ST

=

ei Q(λbi+1 , µbi+1 , λfi , µfi , Ni ),

=

ei Q(λfi−1 , µfi−1 , λbi , µbi , Ni−1 ),

i = 1, . . . , M − 1,

(11.31)

i = 2, . . . , M.

(11.32)

Residence time: As usually, it can be evaluated using Little’s Law, i.e., \ d = W IP . RT d P R PSE Toolbox: Recursive procedure (11.25) and performance measures (11.29)(11.32) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Subsection 19.3.2.

356

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

Effects of up- and downtime: Using the aggregation procedure (11.25), it is possible to show that Theorems 11.1 and 11.2 hold for synchronous exponential lines with M > 2 machines as well. In other words, d • Shorter up- and downtime lead to larger P R than longer ones, even if the machine efficiencies remain the same. d R than increasing Tup • Decreasing Tdown by any factor leads to a larger P by the same factor. Asymptotic properties: The asymptotic properties of synchronous M > 2machine exponential lines are analyzed using systems similar to those studied in the Bernoulli case (4.43), i.e., L1 L2 L3 L4 L5 L6 L7 L8

: : : : : : : :

ei = 0.9, i = 1, . . . , 5, e = [0.9, 0.85, 0.8, 0.75, 0.7], e = [0.7, 0.75, 0.8, 0.85, 0.9], e = [0.9, 0.85, 0.7, 0.85, 0.9], e = [0.7, 0.85, 0.9, 0.85, 0.7], e = [0.7, 0.9, 0.7, 0.9, 0.7], e = [0.9, 0.7, 0.9, 0.7, 0.9], e = [0.75, 0.75, 0.95, 0.75, 0.75],

Ni Ni Ni Ni Ni Ni Ni Ni

= N, = N, = N, = N, = N, = N, = N, = N,

i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4, i = 1, . . . , 4.

(11.33)

To account for the effect of up- and downtime, each of these systems is considered with Tdown = 2 and Tdown = 10. The corresponding Tup ’s are calculated based on Tdown ’s and the efficiency of the machines. d di and ST c i as functions of N are \ The performance measures P R, W IP i , BL shown in Figures 11.6 - 11.13. Based on this information, the following can be concluded: As N → ∞, • • • •

d P R → mini ei . di → 0. BL c M → eM − P d ST R. The cases of Tdown = 10 consistently show performance levels clearly under those of Tdown = 2.

Thus, as one can see, these properties are similar to those obtained in Chapter 4 for the Bernoulli case but with a new feature: longer up- and downtime lead to a poorer performance. Accuracy of the performance measure estimates: The accuracy of formulas (11.29)-(11.32) has been investigated using a numerical approach similar to that of Chapter 4 in the case of Bernoulli machines. Specifically, we applied Simulation Procedure 4.1 to the production lines L1 - L8 defined in (11.33) c i and BL di for the case of both d \ and analyzed the accuracy of P R, W IP i , ST Tdown = 2 and Tdown = 10. The results are summarized in Figures 11.14 11.20 (where the errors are calculated using expression (4.45)-(4.48)). From this information, the following conclusions are derived:

11.1. SYNCHRONOUS EXPONENTIAL LINES

40

^ ,T =2 WIP 1 down ^ ,T WIP =2

30

^ ,T =2 WIP 3 down ^ ,T =2 WIP

PR

0.85 0.8

WIPi

<

<

0.75

0.65

T

2

down

4

down

2

down

4

down

20

^ ,T =10 WIP 1 down ^ ,T =10 WIP

10

^ ,T =10 WIP 3 down ^ ,T =10 WIP

0.7 Tdown=2

357

=10

down

0.6

10

20

N

30

40

0

50

10

20

d (a) P R 0.3

2

0.05 0

down

2

down

4

down

^ ,T BL =10 3 down ^ ,T =10 BL

0.1

10

20

N

50

30

40

<

2

down

4

down

2

down

^ ,T =10 ST 3 down ^ ,T =10 ST

0.1

0.05

50

down

^ ,T =10 ST 5 down ^ ,T =10 ST

0.15

0

4

^ ,T =2 ST 3 down ^ ,T =2 ST

0.2

^ ,T =10 BL 1 down ^ ,T =10 BL

0.15

40

^ ,T =2 ST 5 down ^ ,T ST =2

STi

BLi

4

<

0.3 0.25

down

^ ,T =2 BL 3 down ^ ,T =2 BL

0.2

30

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T =2 BL

0.25

N

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.6: Performance of Line L1

50 0.7

PR

40 0.6

30

WIPi

<

<

0.5

20

Tdown=2

0.4

T

10

^ ,T =2 WIP 1 down ^ ,T WIP =2 2

down

4

down

2

down

4

down

^ ,T =2 WIP 3 down ^ ,T =2 WIP ^ ,T =10 WIP 1 down ^ ,T =10 WIP ^ ,T =10 WIP 3 down ^ ,T =10 WIP

=10

down

0.3

10

20

N

30

40

0

50

10

d (a) P R

BL i

0.6

^ ,T =2 BL 3 down ^ ,T =2 BL

0.3

^ ,T =10 BL 1 down ^ ,T =10 BL

0

down

2

down

4

down

10

20

N

30

di (c) BL

40

50

down

2

down

4

down

2

down

^ ,T =10 ST 5 down ^ ,T ST =10

< 0.3

^ ,T ST =10 3 down ^ ,T =10 ST

0.2 0.1 0

4

^ ,T =2 ST 3 down ^ ,T =2 ST

0.4

^ ,T =10 BL 3 down ^ ,T =10 BL

0.1

^ ,T =2 ST 5 down ^ ,T =2 ST

0.5

down

0.4

0.2

30

STi

2

4

<

N

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T =2 BL

0.5

20

10

20

N

30

ci (d) ST

Figure 11.7: Performance of Line L2

40

50

358

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

40

^ ,T =2 WIP 1 down ^ ,T WIP =2

30

^ ,T =2 WIP 3 down ^ ,T =2 WIP

PR

0.7 0.6

T

down

2

down

4

down

10

^ ,T WIP =10 3 down ^ ,T =10 WIP

Tdown=2

0.4

4

^ ,T =10 WIP 1 down ^ ,T =10 WIP

WIP

0.5

down

< 20

i

<

2

=10

down

0.3

10

20

N

30

40

0

50

10

20

d (a) P R 2

0

10

20

N

^ ,T =2 ST 3 down ^ ,T ST =2

down

50

2

down

0.2

4

down

0.1

30

40

0

50

4

down

2

down

4

down

2

down

^ ,T ST =10 5 down ^ ,T =10 ST

< 0.3

^ ,T BL =10 3 down ^ ,T =10 BL

0.1

0.4

STi

BLi

0.2

^ ,T =2 ST 5 down ^ ,T ST =2

4

^ ,T =10 BL 1 down ^ ,T BL =10

0.3

<

40

0.5

down

^ ,T =2 BL 3 down ^ ,T =2 BL

0.4

30

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T =2 BL

0.5

N

^ ,T =10 ST 3 down ^ ,T =10 ST

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.8: Performance of Line L3

0.75

50

0.7

40

PR

0.65

WIP

<

0.55 0.5

Tdown=2

0.45 0.4

30

i

< 0.6

T

20 10

^ ,T =2 WIP 1 down ^ ,T WIP =2 down

4

down

2

down

4

down

^ ,T =10 WIP 1 down ^ ,T =10 WIP ^ ,T WIP =10 3 down ^ ,T =10 WIP

=10

down

10

20

N

30

40

0

50

10

d (a) P R 0.5

2

down

4

down

0.1

^ ,T BL =10 3 down ^ ,T =10 BL

BLi

0.2

20

N

30

di (c) BL

2

down

4

down

40

^ ,T =2 ST 5 down ^ ,T =2 ST

0.4

^ ,T =10 BL 1 down ^ ,T BL =10

10

30

50

0.3

down

2

down

4

down

2

down

^ ,T =10 ST 5 down ^ ,T ST =10

<

0.2

^ ,T ST =10 3 down ^ ,T =10 ST

0.1 0

4

^ ,T =2 ST 3 down ^ ,T =2 ST

ST i

0.3

N

0.5

^ ,T BL =2 3 down ^ ,T BL =2

<

20

\ (b) W IP i ^ ,T BL =2 1 down ^ ,T BL =2

0.4

0

2

^ ,T =2 WIP 3 down ^ ,T =2 WIP

10

20

N

30

ci (d) ST

Figure 11.9: Performance of Line L4

40

50

11.1. SYNCHRONOUS EXPONENTIAL LINES

50 0.7

PR

40 0.6

30

WIP

i

<

<

0.5 Tdown=2

0.4

T

20 10

359

^ ,T =2 WIP 1 down ^ ,T =2 WIP 2

down

4

down

2

down

4

down

^ ,T =2 WIP 3 down ^ ,T =2 WIP ^ ,T =10 WIP 1 down ^ ,T WIP =10 ^ ,T WIP =10 3 down ^ ,T =10 WIP

=10

down

0.3

10

20

N

30

40

0

50

10

20

d (a) P R

0.3

BLi

0.2

<

^ ,T =2 BL 1 down ^ ,T BL =2

0.4

^ ,T =2 ST 5 down ^ ,T =2 ST

^ ,T =2 BL 3 down ^ ,T =2 BL

0.3

^ ,T =2 ST 3 down ^ ,T =2 ST

2

down

10

20

N

50

4

down

4

down

2

down

4

down

2

down

^ ,T =10 BL 1 down ^ ,T =10 BL

< 0.2

^ ,T =10 ST 5 down ^ ,T ST =10

^ ,T BL =10 3 down ^ ,T =10 BL

0.1

^ ,T ST =10 3 down ^ ,T =10 ST

down

4

0

40

\ (b) W IP i

2

0.1

30

30

STi

0.4

N

down

40

0

50

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.10: Performance of Line L5

50 0.7

PR

40 0.6

30

WIP

i

<

<

0.5 Tdown=2

0.4

T

20 10

^ ,T =2 WIP 1 down ^ ,T WIP =2 2

down

4

down

2

down

4

down

^ ,T =2 WIP 3 down ^ ,T WIP =2 ^ ,T WIP =10 1 down ^ ,T =10 WIP ^ ,T =10 WIP 3 down ^ ,T =10 WIP

=10

down

0.3

10

20

N

30

40

0

50

10

d (a) P R 0.5

down

4

down

2

down

4

down

BLi

^ ,T BL =10 3 down ^ ,T =10 BL

0.1

10

20

N

30

di (c) BL

40

^ ,T =2 ST 5 down ^ ,T ST =2

0.4

^ ,T =10 BL 1 down ^ ,T BL =10

0.2

0

2

50

0.3

down

2

down

4

down

2

down

^ ,T ST =10 5 down ^ ,T =10 ST

<

0.2

^ ,T =10 ST 3 down ^ ,T =10 ST

0.1 0

4

^ ,T =2 ST 3 down ^ ,T ST =2

STi

0.3

30

0.5

^ ,T =2 BL 3 down ^ ,T =2 BL

<

N

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T =2 BL

0.4

20

10

20

N

30

ci (d) ST

Figure 11.11: Performance of Line L6

40

50

360

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

50 0.7 40 30

i

<

0.6

WIP

PR

0.65

0.55

<

0.5 0.45

Tdown=2

0.4 0.35

T

20 10

^ ,T =2 WIP 1 down ^ ,T =2 WIP 2

4

down

2

down

4

down

^ ,T =10 WIP 1 down ^ ,T WIP =10 ^ ,T WIP =10 3 down ^ ,T =10 WIP

=10

down

10

20

N

30

40

0

50

10

20

d (a) P R 0.6

2

down

2

down

4

down

0.2

BLi

10

20

N

30

40

50

down

2

down

4

down

2

down

^ ,T =10 ST 3 down ^ ,T =10 ST

0.1 0

4

^ ,T ST =10 5 down ^ ,T =10 ST

< 0.3

^ ,T =10 BL 3 down ^ ,T =10 BL

0.1

50

^ ,T =2 ST 3 down ^ ,T ST =2

0.4

STi

4

0.2

40

^ ,T =2 ST 5 down ^ ,T ST =2

0.5

^ ,T =10 BL 1 down ^ ,T =10 BL

< 0.3

30

0.6

down

^ ,T =2 BL 3 down ^ ,T BL =2

0.4

N

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T BL =2

0.5

0

down

WIP3, Tdown=2 ^ ^ ,T =2 WIP

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.12: Performance of Line L7

0.8

50

0.7

40 30

WIP

i

PR

< 0.6

<

0.5 Tdown=2

0.4

T

20 10

^ ,T =2 WIP 1 down ^ ,T =2 WIP 2

down

4

down

2

down

4

down

^ ,T =2 WIP 3 down ^ ,T =2 WIP ^ ,T =10 WIP 1 down ^ ,T WIP =10 ^ ,T WIP =10 3 down ^ ,T =10 WIP

=10

down

0.3

10

20

N

30

40

0

50

10

d (a) P R ^ ,T BL =2 1 down ^ ,T BL =2

0.4

^ ,T =2 ST 5 down ^ ,T ST =2

^ ,T BL =2 3 down ^ ,T BL =2

0.3

^ ,T =2 ST 3 down ^ ,T ST =2

4

0.2

down

2

4

10

20

N

30

di (c) BL

<

down

40

50

down

2

down

4

down

2

down

^ ,T =10 ST 3 down ^ ,T =10 ST

0.1 0

4

^ ,T ST =10 5 down ^ ,T =10 ST

0.2

down

^ ,T BL =10 3 down ^ ,T =10 BL

0.1 0

down

^ ,T =10 BL 1 down ^ ,T BL =10

BL i

<

30

\ (b) W IP i 2

0.3

N

ST i

0.4

20

10

20

N

30

ci (d) ST

Figure 11.13: Performance of Line L8

40

50

11.2. ASYNCHRONOUS EXPONENTIAL LINES

361

d • The accuracy of P R is high, typically within 1%. Oscillatory patterns of ei ’s allocation lead to somewhat larger errors. di and ST c i is lower and is comparable to that \ • The accuracy of W IP i , BL observed in the Bernoulli case. System-theoretic properties: Based on Lemma 11.1, aggregation equations (11.25) and expressions (11.29)-(11.32), it is possible to prove the following: Theorem 11.5 The performance measures of a synchronous exponential line, L, and its reverse, Lr , are related as follows: d P R

L

Lr \ W IP ir L di BL

Lr

=

d P R

=

\ NM −i − W IP M −i ,

=

Lr c (M ST −i+1)r ,

, L

i = 1, . . . , M − 1,

i = 1, . . . , M − 1.

Theorem 11.6 In synchronous exponential lines defined by assumptions d d (a)-(e) of Subsection 11.1.1, P R=P R(λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM −1 )) is • strictly monotonically increasing in µi , i = 1, . . . , M , and Ni , i = 1, . . . , M − 1; • strictly monotonically decreasing in λi , i = 1, . . . , M . d Note that while P R is monotonic in λi and µi , it is non-monotonic in ei . Indeed, if ei is increased and Tup,i and Tdown,i are also increased, the resulting d P R may decrease (see Problem 11.5).

11.2

Asynchronous Exponential Lines

11.2.1

Two-machine case

Mathematical description: The production system considered here is shown in Figure 11.21. As in Subsection 11.1.1, each machine is defined by its breakdown and repair rates, λi and µi ; in addition, each machine is characterized by its cycle time, τi or, equivalently, capacity, ci = 1/τi . Since the cycle times of the machines are not identical, the λi ’s and µi ’s are not in units of 1/cycle time, as they are in the synchronous case, but rather in parts/unit of time, for instance, parts/sec or parts/min. Also, although the efficiency of each machine in isolation remains the same, i.e., ei = µi /(λi + µi ), its performance in isolation is denoted as the throughput, T Pi [parts/unit of time]: T Pi = ci

µi . λi + µi

Similarly, the performance of the system as a whole is measured by T P parts/unit of time.

362

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES 0.9

Line 1 PR

0.8

0.7

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10

0.6 0.5

1

2

3

4

Machine i

5

0.4 0

5

10

N

15

²P R (%)

ei

1

0.6

1.5

0.8

0 0

PR

0.5 0.4 0.3

2

3

4

Machine i

5

0.2 0

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5 10 15 20

²P R (%)

ei

1

1

1.5

0.6

0.8 0.6

0.5

20

0.7

Line 2

²P R with Tdown = 2 ²P R with Tdown = 10

1

5

10

N

15

20

²P R with Tdown = 2 ²P R with Tdown = 10

1 0.5 0 0

5

N

10

N

15

20

0.7

Line 3

0.6

PR

0.8 0.6

0.5 0.4 0.3

1

2

3

4

Machine i

5

0.2 0

1 d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5 10 15 20

²P R (%)

ei

1

0.5

0 0

PR

0.5 0.4 0.3

1

2

3

4

Machine i

5

0.2 0

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5

10

N

15

²P R (%)

ei

1

0.6

1

0.6

0.8

PR

0.5 0.4 0.3

2

3

4

Machine i

5

0.2 0

Line 6

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5 10 15 20

N

Machine i

5

ei

0.4

0.2 0

PR

0.5 0.4 0.3

2

3

4

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5 10 15 20

5

0.2 0

N

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5

10

N

15

20

0.6

ei

PR

1 0.8

0.4

2

3

4

Machine i

5

0.2 0

d P R with Tdown = 2 P R with Tdown = 2 d P R with Tdown = 10 P R with Tdown = 10 5

10

N

15

20

²P R (%)

Line 8

1

²P R with Tdown = 2 ²P R with Tdown = 10 5 10 15 20

N

1 0 0

0.8

0.6

20

2

0.6

0.8

Machine i

15

1

3

Line 7

1

N

3

²P R (%)

4

1

0.6

0.5

0.3

3

10

2

0 0

²P R (%)

PR

ei

0.8 2

5

0.6

1

1

20

3

0.7

0.6

15

²P R with Tdown = 2 ²P R with Tdown = 10

0 0

²P R (%)

ei

1

1

N

0.6

0.8 0.6

10

0.5

20

0.7

Line 5

5

N

0.7

Line 4

²P R with Tdown = 2 ²P R with Tdown = 10

²P R with Tdown = 2 ²P R with Tdown = 10 5 10 15 20

N

²P R with Tdown = 2 ²P R with Tdown = 10

2 1 0 0

5

3

²P R with Tdown = 2 ²P R with Tdown = 10

10

N

15

20

2 1 0 0

5

10

N

15

20

d Figure 11.14: Accuracy of P R for Tdown = 2 and Tdown = 10

11.2. ASYNCHRONOUS EXPONENTIAL LINES

15

Wd IP 2

3

4

Machine i

5

0 0

Wd IP

4

2

3

4

Machine i

5

0 0

Wd IP

ei

1

1

2

3

4

Machine i

5

10

Wd IP

ei

0.8 1

2

3

4

Machine i

5

6

Wd IP

0.8

5

10

N

15

0 0

20

20

d W IP 1 d W IP 2

15

d W IP 3 d W IP 4

10

5

10

N

15

20

2

3

4

Machine i

5

15

d W IP 3 d W IP 4

5

0 0

0 0

10

20 15

Wd IP

1 0.8

5

10

N

15

20

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

15

2

3

4

Machine i

5

10

0 0 20

Line 8

15

Wd IP

1 0.8

5

10

N

15

10

2

3

4

Machine i

5

0 0

10

0 0

20

20

d W IP 1 d W IP 2

15

d W IP 3 d W IP 4

²W IP (%) 10

N

15

10

N

15

10

5

10

N

15

0 0

20

10

N

15

20

d W IP 1 d W IP 2

15

d W IP 3 d W IP 4

10

N

15

10

N

15

20

0 0

10

N

15

10

N

N

15

15 10

5

10

N

10

N

15

20

20

²W IP1 ²W IP2 ²W IP3 ²W IP4

5

10

15

20

10

15

20

15

20

10

15

20

10

15

20

N

²W IP1 ²W IP2 ²W IP3 ²W IP4

5

N

²W IP1 ²W IP2 ²W IP3 ²W IP4

5 0 0

5

0 0 15

10

15

5 0 0

20

20

1 0 0

20

15

²W IP1 ²W IP2 ²W IP3 ²W IP4

2

10

10

20

1

15

W IP1 W IP2 W IP3 W IP4

5

5

2

0 0

20

5 5

0 0

20

10

15

²W IP1 ²W IP2 ²W IP3 ²W IP4

1

20

W IP1 W IP2 W IP3 W IP4

5

N

3

W IP1 W IP2 W IP3 W IP4

5

10

0.5

20

W IP1 W IP2 W IP3 W IP4

5

5

3

W IP1 W IP2 W IP3 W IP4

5

1

1.5

20

W IP1 W IP2 W IP3 W IP4

5

2

2

W IP1 W IP2 W IP3 W IP4

5

²W IP1 ²W IP2 ²W IP3 ²W IP4

3

0 0

20

5

5

1

N

15

5

5

1

10

5

0 0

Line 7

10

20

d W IP 1 d W IP 2

5

1

4

0 0

15

1

ei

8

5

20

Line 6

ei

0 0

20

0 0

15

1

ei

15

5

20

Line 5

10

2

15

0.8

0.6

N

W IP

1

Line 4

0.6

10

2

20

0.6

5

W IP

ei

1

10

5

5

d W IP 1 d W IP 2 d W IP 3 d W IP 4

6

0.8

0.6

15

W IP

1

Line 3

0.6

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

5

8

0.6

0 0

20

W IP

0.6

10

N

15

W IP

ei

1 0.8

10

²W IP (%)

20

Line 2

5

²W IP (%)

0 0

²W IP (%)

Machine i

5

²W IP (%)

4

5

²W IP (%)

3

4

W IP1 W IP2 W IP3 W IP4

²W IP (%)

2

10

²W IP (%)

1

5

W IP

0.6

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

W IP

Wd IP

ei

1 0.8

10

W IP

15

Line 1

363

10

5

10

N

²W IP1 ²W IP2 ²W IP3 ²W IP4

5

N

²W IP1 ²W IP2 ²W IP3 ²W IP4

5 0 0

\ Figure 11.15: Accuracy of W IP i for Tdown = 2

5

N

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

20

Line 2

15

Wd IP 4

5

0 0

ei

Wd IP 1

2

3

4

Machine i

5

5

0 0 20

Line 4

15

Wd IP

ei

1 0.8 1

2

3

4

Machine i

5

10

Wd IP

ei

1 0.8 1

2

3

4

Machine i

5

Wd IP

ei

1

1

2

3

4

Machine i

10

15

Wd IP

ei

1 0.8

10

1

2

3

4

Machine i

5

0 0

Line 8 Wd IP

1

ei

10

d W IP 1 d W IP 2 d W IP 3 d W IP 4

5

0 0

20

10

N

15

0.8 1

2

3

4

Machine i

5

5

0 0

20

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

15

5

10

N

15

20

10

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

5

10

N

15

20

10

10

N

15

10

²W IP (%) N

15

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

10

N

15

10

N

15

10

N

15

15 10

10

N

15

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

5

0 0

20

N

15

10

N

15

20

10

N

15

1

5

10

N

10

N

15

N

15

15

20

5

10

N

15

20

15

20

²W IP1 ²W IP2 ²W IP3 ²W IP4

3 2 1 0 0

5

10

N

²W IP1 ²W IP2 ²W IP3 ²W IP4

2 1 0 0

5

10

N

15

20

²W IP1 ²W IP2 ²W IP3 ²W IP4

2 1.5 1 0.5 0 0

20

20

20

0.5

5

10

N

15

20

²W IP1 ²W IP2 ²W IP3 ²W IP4

3

10

15

²W IP1 ²W IP2 ²W IP3 ²W IP4

1

2.5

W IP1 W IP2 W IP3 W IP4

5

N

0.5

0 0

20

5 0 0

10

3

10

20

²W IP1 ²W IP2 ²W IP3 ²W IP4

0 0

20

W IP1 W IP2 W IP3 W IP4

5

5

1.5

20

5 5

0 0

4

10

15

²W IP1 ²W IP2 ²W IP3 ²W IP4

1.5

W IP1 W IP2 W IP3 W IP4

5

N

0.5

20

W IP1 W IP2 W IP3 W IP4

5

10

2

W IP1 W IP2 W IP3 W IP4

5

5

1

20

W IP1 W IP2 W IP3 W IP4

5

1

1.5

W IP1 W IP2 W IP3 W IP4

5

²W IP1 ²W IP2 ²W IP3 ²W IP4

2

0 0

20

5 0 0

20

5 0 0

10

5

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

5

10

0 0

5

15

0.6

15

5

20

Line 7

N

5

0 0

5

10

0 0

15

0.8

5

0 0

10

5

5

5

0 0

Line 6

10

5

15

Line 5

0.6

15

W IP

3

0.8

0.6

20

d W IP 1 d W IP 2 d W IP 3 d W IP 4

W IP

2

Machine i

1

0.6

0 0

20

W IP

1

10

0.6

15

5

Line 3

0.6

N

W IP

0.6

10

10

W IP

ei

1 0.8

5

²W IP (%)

0 0

²W IP (%)

5

²W IP (%)

4

5

²W IP (%)

3

3

W IP1 W IP2 W IP3 W IP4

²W IP (%)

2

Machine i

10

²W IP (%)

1

5

W IP

0.6

15

d W IP 1 d W IP 2 d W IP 3 d W IP 4

²W IP (%)

Wd IP

ei

1 0.8

10

W IP

15

Line 1

W IP

364

2 1 0 0

\ Figure 11.16: Accuracy of W IP i for Tdown = 10

5

10

N

15

20

11.2. ASYNCHRONOUS EXPONENTIAL LINES

d BL 0.1

15

20 d1 BL d2 BL d3 BL d4 BL

0.3 0.2

3

4

5

0 0

5

10

N

15

d BL 0.1

2

3

4

Machine i

5

0 0

5

10

N

15

0.4

Line 4 ei

d BL

0.8 0.6

1

2

3

4

Machine i

5

0 0

d BL

ei

0.8 2

3

4

Machine i

N

15

0 0

d BL

0.8 2

3

4

Machine i

5

5

10

N

15

ei

d BL

1

0.6

5

10

N

15

2

3

4

Machine i

5

0.2

5

10

N

15

d BL

ei

0.8 0.6

0.1

1

2

3

4

Machine i

5

0 0

15

5

10

15

N

0 0

8

x 10

20

²BL1 ²BL2 ²BL3 ²BL4

0.01

20

5

10

N

15

5

10

15

N

20

−3

²BL1 ²BL2 ²BL3 ²BL4

6 4

0 0

20 BL1 BL2 BL3 BL4

0.15 0.1

5

10

N

15

0.06 0.04

5

10

N

15

20

²BL1 ²BL2 ²BL3 ²BL4

0.02 0 0

20

BL1 BL2 BL3 BL4

0.2

0 0

5

10

N

15

5

10

N

15

0.04

20

²BL1 ²BL2 ²BL3 ²BL4

0.02

0.3

0 0

20 BL1 BL2 BL3 BL4

0.4

0.2

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.2

1

0.005 0 0

5

10

N

15

10

N

15

20

N

15

20

²BL1 ²BL2 ²BL3 ²BL4

0.02

BL1 BL2 BL3 BL4

0.2

0 0

20

0.1

5

10

0.01

0.3

0 0

5

0.03

0.1

0.3

Line 8

N

0.3

0.1

1

²BL1 ²BL2 ²BL3 ²BL4

2

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.3

0 0

20

0.1

0.4

0.8

10

0.2

20 d1 BL d2 BL d3 BL d4 BL

0.5

Line 7

15

0.05

0.2

0 0

5

0.2

0.1

1

N

0.02

BL1 BL2 BL3 BL4

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.3

1

ei

10

0.1

Line 6

0.6

5

0.15

5

10

0.01

20 BL1 BL2 BL3 BL4

0.3

0.05

1

15

0.1

0.2

1

N

0.4

0.1

Line 5

0.6

0.2

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.3

1

10

0.1

BL

1

5

0.2

BL

0.6

5

0.015

0.2

0.3

BL

ei

0.8

0.3

0 0

20 BL1 BL2 BL3 BL4

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.2

1

15

0.1

0.3

Line 3

N

BL

2

Machine i

10

0.4

0.1

1

5

²BL

d BL

1

ei

N

0.4

0.8 0.6

10

4 2

²BL

Line 2

5

0 0

²BL

0 0

²BL

Machine i

5

²BL

4

²BL

3

²BL1 ²BL2 ²BL3 ²BL4

6

5

10

N

15

0.04

20

²BL1 ²BL2 ²BL3 ²BL4

0.03

²BL

2

BL

1

x 10

8

0.1

BL

0.6

0.2

BL

ei

0.8

−3

BL1 BL2 BL3 BL4

BL

0.2

1

0.3

d1 BL d2 BL d3 BL d4 BL

²BL

0.3

Line 1

365

0.02 0.01

5

10

N

15

20

0 0

di for Tdown = 2 Figure 11.17: Accuracy of BL

5

10

N

15

20

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

0.1

5

10

N

15

0.6

Line 2 d BL

0.2

2

3

4

Machine i

5

0 0

5

10

N

15

0.4

Line 3 d BL 1

2

3

4

Machine i

5

0 0

5

10

N

15

ei

d BL

0.8 0.6

2

3

4

Machine i

5

0 0

5

10

N

15

d BL

ei

0.8 0.6

0.2

2

3

4

Machine i

5

0 0

5

10

N

15

ei

d BL

0.8 2

3

4

0.3 0.2

Machine i

5

0 0

5

10

N

15

d BL

ei

1

0.6

2

3

4

Machine i

5

0 0

Line 8 d BL

ei

0.8 0.6

5

10

N

15

2

3

4

Machine i

5

d1 BL d2 BL d3 BL d4 BL

0.3 0.2

0 0

BL1 BL2 BL3 BL4

5

10

N

15

5

10

N

15

20

²BL1 ²BL2 ²BL3 ²BL4

0.02

0 0

5

10

N

15

5

10

N

15

5

10

N

15

10

N

15

20

5

10

N

15

20

²BL1 ²BL2 ²BL3 ²BL4

0.02 5

10

N

15

0.08

20

²BL1 ²BL2 ²BL3 ²BL4

0.06 0.04

BL1 BL2 BL3 BL4

0.4 0.3

0 0

0.2

5

10

N

15

0.06

20

²BL1 ²BL2 ²BL3 ²BL4

0.04 0.02 0 0

20

5

10

N

15

0.08

20

²BL1 ²BL2 ²BL3 ²BL4

0.06 0.04 0.02

0.1 5

20

²BL1 ²BL2 ²BL3 ²BL4

0.04

0 0

20

BL1 BL2 BL3 BL4

0.4

0 0

15

0.02

0.6

0 0

N

0.01

0 0

20

0.2

0 0

10

0.06

BL1 BL2 BL3 BL4

0.3

5

0.02

20 BL1 BL2 BL3 BL4

0.4

20

0.1

1

20

²BL1 ²BL2 ²BL3 ²BL4

0.04

20

0.2

0.4

1

15

0.5

0.2

1

N

0.2

0 0

20

d1 BL d2 BL d3 BL d4 BL

0.4

0.8

15

0.02

0 0

0.1

0.6

Line 7

10

0.3

0.1

1

5

0.4

20 d1 BL d2 BL d3 BL d4 BL

0.4

1

N

0.03

0.1

0.5

Line 6

0.6

0 0

0.1

1

10

0.04

20 BL1 BL2 BL3 BL4

0.4

20 d1 BL d2 BL d3 BL d4 BL

0.3

1

15

0.2

0.4

Line 5

N

0.6

0.2

1

10

0.2

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.4

1

5

0.1

0.6

Line 4

5

0.3

0.1

0 0

0.01

BL

0.6

0.2

0.005

20 BL1 BL2 BL3 BL4

0.4

BL

ei

0.8

0 0

20 d1 BL d2 BL d3 BL d4 BL

0.3

1

15

0.2

BL

1

N

0.4

BL

0.6

10

BL

ei

1 0.8

5

0.6

d1 BL d2 BL d3 BL d4 BL

0.4

0 0

20

²BL1 ²BL2 ²BL3 ²BL4

0.01

²BL

0 0

²BL

5

²BL

4

²BL

3

²BL

2

Machine i

BL

1

0.1

BL

0.6

0.015

²BL

BL

ei

0.8

BL1 BL2 BL3 BL4

0.2

²BL

0.2

1

0.3

d1 BL d2 BL d3 BL d4 BL

²BL

0.3

Line 1

BL

366

5

10

N

15

20

0 0

di for Tdown = 10 Figure 11.18: Accuracy of BL

5

10

N

15

20

11.2. ASYNCHRONOUS EXPONENTIAL LINES

5

10

15

N

0.25

c ST

0.15 0.1 0.05

2

3

4

Machine i

5

0 0

5

10

15

N

0.5

Line 3 c ST

0.3 0.2

2

3

4

Machine i

0 0

5

10

15

ei

c ST

1

0.6

2

3

4

Machine i

5

N

Line 5 ei

c ST

0.8 0.6

2

3

4

Machine i

0 0

ei

N

15

c ST

0.8 2

3

4

Machine i

5

5

10

N

15

ei

c ST

1

0.6

5

10

N

15

2

3

4

Machine i

5

5

10

N

15

ei

c ST

0.8 0.6

0.1

1

2

3

4

Machine i

5

0 0

N

15

²ST2 ²ST3 ²ST4 ²ST5

5

10

N

15

20

²ST2 ²ST3 ²ST4 ²ST5

0.01

0 0

5

10

N

15

20

0.01

0 0

20 ST2 ST3 ST4 ST5

0.15 0.1

5

10

N

15

²ST2 ²ST3 ²ST4 ²ST5

5

10

N

15

0.1

20

²ST2 ²ST3 ²ST4 ²ST5

0.05

0 0

20

ST2 ST3 ST4 ST5

0.2

0 0

5

10

N

15

0.4

5

10

N

15

20

²ST2 ²ST3 ²ST4 ²ST5

0.04

0.02

0.2

0 0

20 ST2 ST3 ST4 ST5

0.3

0 0

20 d2 ST d3 ST d4 ST d5 ST

0.2

1

20

0.01

0 0

5

10

N

15

0.3

0.1

5

10

N

15

20

0 0

10

N

15

20

²ST2 ²ST3 ²ST4 ²ST5

0.02 0.01 0 0

20 ST2 ST3 ST4 ST5

0.2

5

0.03

0.1

0.3

Line 8

10

0.3

0.1

1

5

0 0

20 d2 ST d3 ST d4 ST d5 ST

0.2

0 0

15

0.1

0.3

0.8

N

0.02

20

0.2

20 d2 ST d3 ST d4 ST d5 ST

0.4

Line 7

10

0.05

0.2

0 0

15

ST2 ST3 ST4 ST5

0 0

0.1

1

N

0.2

20 d2 ST d3 ST d4 ST d5 ST

0.3

1

10

0.3

0.1

Line 6

0.6

10

0.15

5

5

0.4

0.05

1

5

0.02

0.1 5

0.2

1

ST2 ST3 ST4 ST5

0 0

d2 ST d3 ST d4 ST d5 ST

0.2

0 0

2

20

0.2

0.1

1

15

0.3

20

0.3

0.8

N

4

0.1

5

0.4

Line 4

10

0.4

0.1

1

5

²ST2 ²ST3 ²ST4 ²ST5

6

0 0

20

0.1

0.5

ST

0.6

15

ST2 ST3 ST4 ST5

0 0

ST

ei

0.8

N

0.15

20

d2 ST d3 ST d4 ST d5 ST

0.4

1

10

0.05

ST

1

5

0.2

ST

ei

0.8 0.6

d2 ST d3 ST d4 ST d5 ST

0.2

1

20

²P R (%)

Line 2

0 0

x 10

²P R (%)

0 0

²ST

Machine i

5

²ST

4

²ST

3

5

10

N

15

0.04

20

²ST2 ²ST3 ²ST4 ²ST5

0.03

²ST

2

0.1

ST

1

8

²ST (%)

c ST 0.1

ST

0.6

0.2

ST

ei

0.8

−3

ST2 ST3 ST4 ST5

ST

0.2

1

0.3

d2 ST d3 ST d4 ST d5 ST

²P R (%)

0.3

Line 1

367

0.02 0.01

5

10

N

15

20

0 0

c i for Tdown = 2 Figure 11.19: Accuracy of ST

5

10

N

15

20

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

0 0

5

10

N

15

0.4

Line 2 c ST

0.1

2

3

4

Machine i

5

0 0

5

10

N

15

c ST 2

3

4

Machine i

5

0 0

5

10

N

15

ei

c ST

0.8 0.6

1

2

3

4

Machine i

5

0

0

5

10

N

15

ei

c ST 2

3

4

Machine i

5

0 0

ei

5

10

N

15

c ST

0.8 2

3

4

Machine i

5

0.2

0 0

5

10

N

15

ei

c ST

1

0.6

2

3

4

Machine i

5

0 0

5

10

N

15

ei

c ST

0.8 0.6

0.2 0.1

1

2

3

4

Machine i

5

0 0

5

10

N

15

0.2

0 0

5

10

N

15

5

10

N

15

10

N

15

20

20

²ST2 ²ST3 ²ST4 ²ST5

5

10

N

15

20

²ST2 ²ST3 ²ST4 ²ST5

5

10

N

15

20

²ST2 ²ST3 ²ST4 ²ST5

²P R (%)

0.02 5

10

N

15

0.08

20

²ST2 ²ST3 ²ST4 ²ST5

0.06 0.04

0.4

ST2 ST3 ST4 ST5

0.3 0.2

0 0

5

10

N

15

0.06

20

²ST2 ²ST3 ²ST4 ²ST5

0.04 0.02 0 0

20

5

10

N

15

0.08

20

²ST2 ²ST3 ²ST4 ²ST5

0.06 0.04 0.02

0.1 5

15

0.04

0 0

20 ST2 ST3 ST4 ST5

0.4

0 0

N

0.02

0.6

0 0

10

0.06

20 ST2 ST3 ST4 ST5

0.3

20 d2 ST d3 ST d4 ST d5 ST

0.3

1

ST2 ST3 ST4 ST5

5

0.01

0 0

20

²ST2 ²ST3 ²ST4 ²ST5

0.02

20

0.2

0.4

Line 8

15

0.4

0.2

1

N

0.2

0 0

20 d2 ST d3 ST d4 ST d5 ST

0.4

0.8

10

15

0.02

0 0

0.1

0.6

Line 7

5

0.3

0.1

1

0

N

0.04

20 ST2 ST3 ST4 ST5

0.4

20 d2 ST d3 ST d4 ST d5 ST

0.3

1

15

10

0.02

0 0

0.1

0.4

Line 6

0.6

0.2

0

0.1

1

N

0.4

20 d2 ST d3 ST d4 ST d5 ST

0.3

0.8

10

0.2

0.4

1

5

0.6

0.2

Line 5

0.6

0 0

20 d2 ST d3 ST d4 ST d5 ST

0.4

1

ST2 ST3 ST4 ST5

5

0.04

20

0.2

0.6

Line 4

N

15

0.4

0.2

1

10

ST

0.6

5

0.6

ST

ei

0.8

0.2

0 0

20 d2 ST d3 ST d4 ST d5 ST

0.4

1

ST2 ST3 ST4 ST5

²ST2 ²ST3 ²ST4 ²ST5

0.01

0 0

20

0.1

0.6

Line 3

15

ST

1

N

0.3

ST

0.6

0.2

10

ST

ei

1 0.8

5

0.4

d2 ST d3 ST d4 ST d5 ST

0.3

0 0

20

²P R (%)

5

²ST

4

²ST

3

²ST

2

Machine i

ST

1

0.1

0.02

²P R (%)

c ST 0.1

ST2 ST3 ST4 ST5

0.2

ST

ei

0.8

0.3

²P R (%)

0.2

1

0.6

d2 ST d3 ST d4 ST d5 ST

²ST

0.3

Line 1

ST

368

5

10

N

15

20

0 0

c i for Tdown = 10 Figure 11.20: Accuracy of ST

5

10

N

15

20

11.2. ASYNCHRONOUS EXPONENTIAL LINES

λ 1, µ1 , c1

m1

N

λ2, µ 2 , c 2

b

m2

369

Figure 11.21: Asynchronous exponential two-machine line Otherwise, the mathematical description of the asynchronous system of Figure 11.21 remains the same as the synchronous one of Figure 11.1. Specifically, the conventions of system operation remain the same (see (a)-(e) of Subsection 11.1.1), the states are also the same, and their description is carried out in terms of the probabilities (11.1), (11.2) and density (11.3). The calculations to determine these probabilities are more involved; they are included in Section 20.2. Marginal pdf of buffer occupancy: Based on these analyzes, the following steady state pdf of buffer occupancy are calculated: fH (h) = B1 eK1 h + B2 eK2 h + B3 δ(h) + B4 δ(h − N ),

h ∈ [0, N ],

(11.34)

where K1 K2 B1

B2 B3 B4 E1 E2 E3 E4

p (E1 − E4 )2 + 4E2 E3 (E1 + E4 ) + , = 2 2 p (E1 − E4 )2 + 4E2 E3 (E1 + E4 ) − , = 2 2 ³ ´ F2 F5 , = C0 F4 − 2M2  ³ ´ F1 F3 F5  C e(K1 −K2 )N , for c1 < c2 , − F F  0 1 4 2E2  = ³ ´    C0 F1 F3 F5 − F1 F4 , for c1 > c2 , 2E2 = F6 C0 , = F7 C0 , λ2 (c1 µ1 + c2 µ2 ) − µ2 (c2 − c1 )(µ1 + µ2 + λ1 + λ2 ) , = c1 (c2 − c1 )(µ1 + µ2 ) −λ2 (c1 µ1 + c2 µ2 ) , = c1 (c2 − c1 )(µ1 + µ2 ) −λ1 (c1 µ1 + c2 µ2 ) , = c2 (c2 − c1 )(µ1 + µ2 ) λ1 (c1 µ1 + c2 µ2 ) + µ1 (c2 − c1 )(µ1 + µ2 + λ1 + λ2 ) , = c2 (c2 − c1 )(µ1 + µ2 )

(11.35)

(11.36)

370

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES C0 F1

=

=

F2 F3

= =

F4

=

F5

=

F6

=

F7

=

F8

=

F9

=

1 , F4 F8 + F5 F9 + F6 + F7  c (µ +µ +λ )(E −E −K +K )+2λ c E 2 1 2 1 1 4 1 2 1 1 2   c (µ +µ +λ )(E −E +K −K )+2λ c E ,

for c1 < c2 ,

 

for c1 > c2 ,

2

1

2

1

1

4

1

2

1 1

2

c2 λ2 (E1 −E4 −K1 +K2 )+2(µ1 +µ2 +λ2 )c1 E2 c2 λ2 (E1 −E4 +K1 −K2 )+2(µ1 +µ2 +λ2 )c1 E2 ,

(11.37)

E1 − E4 − (K1 − K2 ), E1 − E4 + (K1 − K2 ), λ1 c2 + , µ1 + µ2 c2 − c1 λ2 c1 − , µ1 + µ2 c2 − c1  h   2E2 c1 (λ1 + µ1 )(λ2 + µ1 + µ2 )(1 − F1 e(K1 −K2 )N )   i     +c2 λ2 (λ2 + µ2 )(−F2 + F1 F3 e(K1 −K2 )N ) .h i (11.38) 2E2 λ2 µ1 (λ1 + λ2 + µ1 + µ2 ) , for c1 < c2 ,         c2 (F1 F3 − F2 )/(2µ1 E2 ), for c1 > c2 ,  c1 (1 − F1 )eK1 N /µ2 , for c1 < c2 ,       h   K1 N c2 (λ2 + µ2 )(λ1 + µ1 + µ2 )(−F2 + F1 F3 e(K2 −K1 )N ) e i   +2E2 c1 λ1 (λ1 + µ1 )(1 − F1 e(K2 −K1 )N )    .h i    2E2 λ1 µ2 (λ1 + λ2 + µ1 + µ2 ) , for c1 > c2 , ( K1 N K2 N e −1 − e K2−1 · F1 e(K1 −K2 )N , for c1 > c2 K1 K2 N eK1 N −1 − e K2−1 · F1 , for c1 < c2 , K1 ( (K2 −K1 )N K1 N K2 N F2 1−e · 2E + e K2−1 · F1 F3 e 2E2 , for c1 > c2 K1 2 F2 1−eK1 N eK2 N −1 F1 F3 · + · , for c1 < c2 . K1 2E2 K2 2E2

These pdf’s are illustrated in Figure 11.22 for the following production lines: = 1, c2 = 0.9, N = 20, = 0.9, c2 = 1, N = 20, = 1, c2 = 0.9, N = 20, = 0.9, c2 = 1, N = 20 = 1, c2 = 0.8, N = 20, = 0.8, c2 = 1, , N = 20. (11.39) Lines L1 and L2 are selected in order to illustrate the reversibility with respect to ci ’s. Line L3 is similar to Line 1 but with longer up- and downtime. Line L4 consists of machines with non-equal capacities and non-equal throughputs in isolation. In Line L5 , the machine efficiencies and capacities are selected L1 L2 L3 L4 L5 L6

: : : : : :

λ1 λ1 λ1 λ1 λ1 λ1

= λ2 = 0.1, µ 1 = µ2 = λ2 = 0.1, µ 1 = µ2 = λ2 = 0.01, µ 1 = µ2 = 0.01, λ2 = 0.1, µ1 = µ2 = 0.25, λ2 = 0.1, µ1 = µ2 = 0.1, λ2 = 0.25, µ1 = µ2

= 0.5, = 0.5, = 0.05, = 0.5, = 0.5, = 0.5,

c1 c1 c1 c1 c1 c1

11.2. ASYNCHRONOUS EXPONENTIAL LINES

371

so that the throughput of each machine in isolation is the same. Line L6 is a reverse of Line L5 . From Figure 11.22, we conclude the following: • The reversibility property takes place for asynchronous lines as well. • Longer up- and downtime may qualitatively change the pdf of buffer occupancy as compared with shorter ones. Formulas for performance measures: It is shown in Section 20.2 that the performance measures, T P , W IP , BL1 and ST2 of an asynchronous twomachine line with exponential machines can be expressed as follows: Throughput: G4 + G5 eK1 N + G6 eK2 N , (11.40) TP = G1 + G2 eK1 N + G3 eK2 N where G1 G2

G3

G4

G5

G6 G7

= µ1 G27 + µ1 G7 [c1 (µ1 + µ2 + λ2 ) − c2 (µ1 + µ2 + λ1 )],   µ2 λ1 c2 [(c1 − c2 )(µ1 − µ2 ) − (c2 λ1 + c1 λ2 ) − G7 ], for c1 < c2 , =  µ1 λ2 c1 [(c1 − c2 )(µ1 − µ2 ) − (c2 λ1 + c1 λ2 ) + G7 ], for c1 > c2 ,  e (c −c e )G +c e (1−e )G 2 2 1 1 1 1 1 2 2  , for c1 < c2 ,  c1 e1 (e2 −1) =   e1 (c1 −c2 e2 )G1 +c2 e2 (1−e1 )G2 , for c > c , 1 2 c2 e2 (e1 −1)   c2 e2 G1 , for c1 < c2 , =  c1 e1 G1 , for c1 > c2 ,   c1 e1 G2 , for c1 < c2 , =  c2 e2 G2 , for c1 > c2 ,   c1 e1 G3 , for c1 < c2 , =  c2 e2 G3 , for c1 > c2 , p = [c1 (µ1 + µ2 + λ2 ) − c2 (µ1 + µ2 + λ1 )]2 + 4c1 c2 λ1 λ2 .

Work-in-process: W IP =

F4 F10 + F5 F11 + F7 N , F4 F8 + F5 F9 + F6 + F7

(11.41)

where F4 , . . . , F9 are given in (11.38) and  1+(K N −1)eK1 N K2 N 1  + 1+(K2 NK−1)e · F1 e(K1 −K2 )N , for c1 < c2 , 2  K12 2 F10 =  K2 N  1+(K1 N −1)eK1 N − 1+(K2 NK−1)e · F1 , for c1 > c2 , 2 K2 1

2

372

0.5

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

0.5

fH(h)

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0

5

10

15

20

h

f (h) H

0 0

5

(a) L1 0.5

10

15

20

15

20

15

20

h

(b) L2

f (h)

0.4

H

f (h) H

0.35

0.4

0.3 0.25

0.3

0.2

0.2

0.15 0.1

0.1

0.05

0 0

5

10

15

20

h

0 0

5

(c) L3 0.12

fH(h)

0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02 5

10

(e) L5

h

(d) L4 0.12

0.1

0 0

10

15

20

h

0 0

fH(h)

5

10

(f) L6

Figure 11.22: Stationary distribution of buffer occupancy for Lines L1 -L6

h

11.2. ASYNCHRONOUS EXPONENTIAL LINES     =

F11

F2 [1+(K1 N −1)eK1 N ] 2E2 K12

+

[1+(K2 N −1)eK2 N ]F1 F3 e(K1 −K2 )N 2E2 K22

 K N   − [1+(K1 N −1)e2 1 ]F2 + 2E2 K 1

373 ,

[1+(K2 N −1)eK2 N ]F1 F3 , 2E2 K22

for c1 < c2 , for c1 > c2 .

Blockages and starvations: The general expressions for these performance measures remain the same as in the synchronous case, i.e., BL1 ST2

= P [{m1 up} ∩ {buffer full} ∩ {m2 down}], = P [{m2 up} ∩ {buffer empty}].

However, since function Q does not explicitly emerge in the formulas for T P , the specific expressions for BL1 and ST2 are derived in a different manner. Namely, represent T P as c2 µ2 P [{buffer not empty}|{m2 up}] TP = λ2 + µ2 and denote P [{buffer empty}|{m2 up}] =: st2 . Then TP =

(11.42)

c2 µ2 (1 − st2 ) λ2 + µ 2

and st2 = 1 −

TP c2 µ2 λ2 +µ2

.

Therefore, using the conditional probability formula, ST2

= P [{m2 up} ∩ {buffer empty}] = P [{buffer empty}|{m2 up}]P [{m2 up}] µ2 TP = (1 − c2 µ2 ). λ2 + µ2 λ2 +µ2

Thus, finally,

TP e2 c2 − T P = . c2 c2 Similarly, introducing the notation ST2 = e2 −

bl1 := P [{buffer full}|{m1 up}]

(11.43)

(11.44)

and taking into account that TP =

c1 µ1 (1 − bl1 ), λ1 + µ1

it is possible to show that BL1 = e1 −

TP e1 c1 − T P = . c1 c1

(11.45)

Note that (11.43) and (11.45) with c1 = c2 coincide with the corresponding expressions for the synchronous case.

374

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

Effects of up- and downtime: Using expression (11.40), it is possible to show that the effects of Tup and Tdown on T P in two-machine asynchronous lines remain the same as in synchronous ones. Asymptotic properties: The behavior of the performance measures (11.40)(11.45) as functions of N is illustrated in Figure 11.23 for lines L1 -L4 defined in (11.39). From these data, the following can be concluded: 50 0.8

L4

40 L1 and L2 WIP

TP

0.75 0.7

30 L1

20

L3

L3 0.65

10

0.6

0

10

20

N

30

40

50

L2

L4 10

(a) P R

40

50

40

50

0.15

0.15

2

L3

ST

1

BL

30

0.2

0.2

L1

0.1

0

N

(b) W IP

0.25

0.05

20

0.1

L4

0.05

L2 10

L2 L3

20

N

30

40

50

(c) BL1

0

L4

L1 10

20

N

30

(d) ST2

Figure 11.23: Performance measures of Lines L1 -L4 as functions of buffer capacity

• W IP may grow almost linearly in N , while T P is always saturating; thus, large N ’s are not advisable (see Chapter 14 below). • Reverse lines have identical T P . • Longer up- and downtime result in lower T P than shorter up- and downtime; in some cases, the difference is as large as 25%.

11.2.2

M > 2-machine case

11.2. ASYNCHRONOUS EXPONENTIAL LINES

375

Aggregation procedure and its properties: The production system considered here is shown in Figure 11.24. Each machine is characterized by λi , µi [parts/unit of time] and τi [units of time/part] or ci = 1/τi [parts/unit of time]. To analyze the performance of this system, we develop another recursive aggregation procedure. It turns out that, in this case, it is more convenient to iterate ci ’s, bli ’s and sti ’s rather than λi ’s and µi ’s. λ1, µ1 , τ1

m1

N1 λ2, µ 2 , τ2 m2

b1

λM−1, µ

N2

b2

M−1,

τ M−1 N M−1λ M, µ M, τM

mM−1

bM−1

mM

Figure 11.24: Asynchronous exponential M > 2-machine line Aggregation equations: Using (11.40), (11.43), and (11.45), introduce the following aggregation procedure: bli (s + 1)

=

cbi (s + 1)

=

sti (s + 1)

=

cfi (s + 1)

=

ei cfi (s) − T P (λi , µi , cfi (s), λi+1 , µi+1 , cbi+1 (s + 1), Ni )

ci [1 − bli (s + 1)],

ei cfi (s) 1 ≤ i ≤ M − 1, 1 ≤ i ≤ M − 1,

,

ei cbi (s + 1) − T P (λi−1 , µi−1 , cfi−1 (s + 1), λi , µi , cbi (s + 1), Ni−1 ) , ei cbi (s + 1) 2 ≤ i ≤ M, ci [1 − sti (s + 1)],

2 ≤ i ≤ M,

(11.46)

with initial conditions cfi (0) = ci ,

i = 2, . . . , M − 1,

and boundary conditions cf1 (s) = c1 ,

cbM (s) = cM ,

s = 0, 1, 2, . . . ,

where T P (λ1 , µ1 , c1 , λ2 , µ2 , c2 , N ) is calculated according to (11.13) or (11.40), whichever is applicable. Theorem 11.7 Aggregation procedure (11.46) has the following properties: (i) The sequences, cf2 (s), . . ., cfM (s), cb1 (s), . . ., cbM −1 (s), and bl1 (s), . . ., blM −1 (s), st2 (s), . . ., stM (s), s = 1, 2, . . ., are convergent, i.e., the following limits exist: lim bli (s) =: bli ,

s→∞

i = 1, . . . , M − 1,

lim cbi (s) =: cbi ,

s→∞

i = 1, . . . , M,

lim sti (s) =: sti ,

s→∞

lim cf (s) s→∞ i

=: cfi ,

i = 2, . . . , M, i = 1, . . . , M.

376

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

(ii) These limits are unique solutions of the steady state equations corresponding to (11.46), i.e., of bli

=

cbi

=

sti

=

cfi

=

ei cfi − T P (λi , µi , cfi , λi+1 , µi+1 , cbi+1 , Ni ) ei cfi ci [1 − bli ], ei cbi

,

1 ≤ i ≤ M − 1,

1 ≤ i ≤ M − 1,

− T P (λi−1 , µi−1 , cfi−1 , λi , µi , cbi , Ni−1 ) , ei cbi

ci [1 − sti ],

2 ≤ i ≤ M,

2 ≤ i ≤ M.

(iii) In addition, these limits satisfy the relationships: cfM eM

=

cb1 e1

=

T P (λi , µi , cfi , λi+1 , µi+1 , cbi+1 , Ni ), i = 1, . . . , M − 1,

(11.47)

where T P is defined in (11.40) with G1 − G7 having c1 replaced by cfi , c2 by cbi+1 , and N by Ni . Proof: See Section 20.2. Interpretation of cfi and cbi : Remains similar to that of the synchronous lines: From the point of view of each buffer bi , i = 1, . . . , M − 1, the upstream and downstream of the line are represented by exponential machines with parameters (λi , µi , cfi ) and (λi+1 , µi+1 , cbi+1 ), respectively. Thus, the M > 2-machine line can be represented as shown in Figure 11.25. f

bi

mi

b

m ib+1

= f λi , µ i, c i

Ni

f

m1

λi+1 , µ i+1, cib+1

mM

= λ1, µ1, c b 1

f

λM , µ M , c M

i =1,..., M −1

Figure 11.25: Equivalent representation of asynchronous exponential M > 2machine line through the aggregated machines

Performance measure estimates: Based on the above interpretation, the performance measures of the asynchronous M > 2-machine lines are defined as follows: Throughput: From the last statement of Theorem 11.7, d T P = eM cfM = e1 cb1 .

(11.48)

11.2. ASYNCHRONOUS EXPONENTIAL LINES

377

d Alternatively, based on the interpretation of Figure 11.25, T P can be calf b culated using expressions (11.40) with ci and ci+1 substituted for c1 and c2 , respectively, and Ni for N . \ Work-in-process: Using the interpretation of Figure 11.25, W IP i can be defined by expressions (11.41) with c1 and c2 substituted by cfi and cbi+1 , respectively. Then, the total W IP can be evaluated as \ W IP =

M −1 X

\ W IP i .

(11.49)

i=1

Blockages and starvations: Based on (11.43) and (11.45), di BL ci ST

= ei bli ,

i = 1, . . . , M − 1,

(11.50)

= ei sti ,

i = 2, . . . , M.

(11.51)

Residence time: \ d = W IP . RT d T P PSE Toolbox: Recursive procedure (11.46) and performance measures (11.48)(11.51) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Subsection 19.3.3. Effects of up- and downtime: Using the aggregation procedure (11.46), it d is possible to show that the effects of Tup and Tdown on T P remain the same as in two-machine lines. Asymptotic properties: The asymptotic properties of asynchronous M > 2machine exponential lines are illustrated using the following systems (Ni = N , i = 1, . . . , 4): L1 L2 L3 L4 L5 L6

: : : : : :

ei = 0.9, i = 1, . . . , 5, ce = [2, 1.75, 1.5, 1.25, 1], e = [0.75, 0.8, 0.85, 0.9, 0.95], ce = [2, 1.75, 1.5, 1.25, 1], e = [0.9, 0.775, 0.65, 0.525, 0.4], ce = [2, 1.75, 1.5, 1.25, 1], ei = 0.9, i = 1, . . . , 5, ce = [1, 1.25, 1.5, 1.75, 2], ei = 0.9, i = 1, . . . , 5, ce = [2, 1.75, 1.5, 1.25, 1], ei = 0.9, i = 1, . . . , 5, ce = [2, 1.5, 1, 1.5, 2].

(11.52)

The downtime of each machine is selected as either Tdown = 2 or Tdown = 10. d di and ST c i as functions of N are shown in \ The behavior of T P, W IP i , BL Figures 11.26 - 11.31. Based on this information, we conclude that the behavior of asynchronous lines is similar to that of the synchronous ones.

378

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

50

1 WIPi

40

TP

<

Tdown=2 T 0.9

=10

2

30 20

WIP ^ 1,Tdown=10 ^ ,T =10 WIP

10

^ ,T =10 WIP 3 down ^ ,T =10 WIP

4

down

10

20

N

30

40

0

50

down

4

down

10

20

2

down

4

down

2

down

0.1

^ ,T =10 BL 3 down ^ ,T BL =10

BLi 0.2

^ ,T BL =10 1 down ^ ,T BL =10

0

4

10

20

N

40

50

30

^ ,T ST =2 5 down ^ ,T ST =2

0.4 0.3

40

2

down

4

down

2

down

0.2 0.1

^ ,T ST =10 3 down ^ ,T ST =10

0

50

down

^ ,T =10 ST 5 down ^ ,T =10 ST

<

down

4

^ ,T ST =2 3 down ^ ,T =2 ST

STi

0.3

30

0.5

^ ,T BL =2 3 down ^ ,T =2 BL

<

N

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T BL =2

0.4

down

2

d (a) T P 0.5

down

WIP3,Tdown=2 ^ ^ ,T =2 WIP

<

0.95

^ ,T =2 WIP 1 down ^ ,T =2 WIP

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.26: Performance of Line L1

50

1 40

TP

<

WIPi

0.95

T

0.85 0.8

30

=2

down

Tdown=10 10

20

N

30

40

2

down

4

down

0.1

^ ,T BL =10 3 down ^ ,T =10 BL

BLi 0.2

2

4

10

20

N

30

di (c) BL

4

down

10

20

N

30

down

down

40

^ ,T =2 ST 5 down ^ ,T ST =2

0.4

^ ,T =10 BL 1 down ^ ,T =10 BL

0

down

50

4

down

2

down

4

down

^ ,T ST =2 3 down ^ ,T =2 ST

0.3 0.2

^ ,T ST =10 5 down ^ ,T ST =10

0.1

^ ,T =10 ST 3 down ^ ,T ST =10

STi

0.3

2

^ ,T WIP =10 3 down ^ ,T WIP =10

0.5

^ ,T =2 BL 3 down ^ ,T =2 BL

<

down

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T BL =2

0.4

4

10

d (a) T P 0.5

down

20

0

50

2

^ ,T WIP =2 3 down ^ ,T =2 WIP ^ ,T =10 WIP 1 down ^ ,T =10 WIP

<

0.9

^ ,T WIP =2 1 down ^ ,T WIP =2

<

0

2

10

20

N

30

ci (d) ST

Figure 11.27: Performance of Line L2

down

40

50

11.2. ASYNCHRONOUS EXPONENTIAL LINES

50

1

40

WIPi

0.8 TP

<

30

Tdown=2 Tdown=10

0.4 10

20

N

30

40

BLi

<

^ ,T =2 BL 3 down ^ ,T =2 BL

0.5

4

0.3

2

0.1

4

10

20

N

2

down

4

down

10

20

N

30

40

50

30

down

<

50

2

down

4

down

2

down

^ ,T ST =10 3 down ^ ,T ST =10

0.1 0

down

^ ,T =10 ST 5 down ^ ,T =10 ST

0.4

0.2

down

4

^ ,T ST =2 3 down ^ ,T =2 ST

0.3

down

40

^ ,T ST =2 5 down ^ ,T ST =2

0.6

^ ,T =10 BL 3 down ^ ,T =10 BL

0.2

0

0.7

down

^ ,T =10 BL 1 down ^ ,T =10 BL

0.4

down

^ ,T =10 WIP 3 down ^ ,T =10 WIP

^ ,T =2 BL 1 down ^ ,T =2 BL 2

0.5

4

\ (b) W IP i

STi

0.6

down

10

d (a) T P 0.7

2

^ ,T =2 WIP 3 down ^ ,T =2 WIP ^ ,T =10 WIP 1 down ^ ,T =10 WIP

0

50

^ ,T =2 WIP 1 down ^ ,T =2 WIP

20

<

0.6

379

10

di (c) BL

20

N

30

40

50

ci (d) ST

Figure 11.28: Performance of Line L3

6

1.02

WIPi

0.98

< 0.96 TP

<

0.94

10

20

N

30

40

10

2

down

4

down

0.1

^ ,T BL =10 3 down ^ ,T BL =10

BLi 0.2

2

4

10

20

N

30

di (c) BL

30

down

4

down

40

50

down

down

40

^ ,T ST =2 5 down ^ ,T ST =2

0.4

^ ,T =10 BL 1 down ^ ,T =10 BL

0

N

50

4

down

2

down

4

down

2

down

^ ,T ST =2 3 down ^ ,T =2 ST

0.3 0.2

^ ,T =10 ST 5 down ^ ,T =10 ST

0.1

^ ,T ST =10 3 down ^ ,T ST =10

STi

0.3

20

0.5

^ ,T BL =2 3 down ^ ,T =2 BL

<

2

^ ,T =10 WIP 3 down ^ ,T =10 WIP

\ (b) W IP i ^ ,T BL =2 1 down ^ ,T BL =2

0.4

down

^ ,T WIP =10 1 down ^ ,T =10 WIP

d (a) T P 0.5

4

3

0

50

down

^ ,T =2 WIP 3 down ^ ,T =2 WIP

1

Tdown=10

2

4

2

Tdown=2

0.92 0.9

^ ,T =2 WIP 1 down ^ ,T WIP =2

5

1

<

0

10

20

N

30

ci (d) ST

Figure 11.29: Performance of Line L4

40

50

380

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

50 1 40

TP

<

WIPi

0.95

T

0.85

=2

down

Tdown=10 0.8

30

10

20

N

30

40

down

4

down

2

down

4

down

^ ,T =10 WIP 1 down ^ ,T WIP =10

10

^ ,T WIP =10 3 down ^ ,T =10 WIP

0 0

50

2

^ ,T =2 WIP 3 down ^ ,T =2 WIP

20

<

0.9

^ ,T =2 WIP 1 down ^ ,T =2 WIP

10

d (a) T P 2

0.4

^ ,T =2 BL 3 down ^ ,T =2 BL

0.3

^ ,T =10 BL 1 down ^ ,T =10 BL

4

0.2

0

down

2

down

4

down

<

10

20

N

40

50

30

40

^ ,T =2 ST 5 down ^ ,T ST =2 down

2

down

4

down

2

down

^ ,T =2 ST 3 down ^ ,T =2 ST

0.3

^ ,T =10 ST 5 down ^ ,T =10 ST ^ ,T ST =10 3 down ^ ,T =10 ST

0.1 0

50

4

0.4

0.2

^ ,T BL =10 3 down ^ ,T =10 BL

0.1

30

0.5

down

STi

BLi

<

N

\ (b) W IP i ^ ,T =2 BL 1 down ^ ,T BL =2

0.5

20

10

20

di (c) BL

N

30

40

50

40

50

ci (d) ST

Figure 11.30: Performance of Line L5

50 1 40

TP

<

WIPi

0.95

Tdown=2

0.85

Tdown=10 0.8

30

10

20

N

30

40

^ ,T BL =2 1 down ^ ,T BL =2

BLi

down

0.3

^ ,T =10 BL 1 down ^ ,T =10 BL

0

down

2

down

4

down

^ ,T BL =10 3 down ^ ,T BL =10 10

20

N

30

di (c) BL

40

50

STi

2

^ ,T BL =2 3 down ^ ,T =2 BL

0.1

2

down

4

down

^ ,T =10 WIP 3 down ^ ,T =10 WIP 10

20

N

30

^ ,T =2 ST 5 down ^ ,T =2 ST

0.5

0.4

0.2

down

\ (b) W IP i

4

<

4

10

d (a) T P 0.5

down

20

0

50

2

^ ,T =2 WIP 3 down ^ ,T WIP =2 ^ ,T WIP =10 1 down ^ ,T =10 WIP

<

0.9

^ ,T =2 WIP 1 down ^ ,T WIP =2

<

4

down

2

down

4

down

2

down

0.4

^ ,T =2 ST 3 down ^ ,T =2 ST

0.3

^ ,T =10 ST 5 down ^ ,T =10 ST

0.2

^ ,T =10 ST 3 down ^ ,T =10 ST

0.1 0

10

20

N

30

ci (d) ST

Figure 11.31: Performance of Line L6

40

50

11.3. CASE STUDIES

381

Accuracy of the estimates: The accuracy of formulas (11.48)-(11.51) has been investigated using a numerical approach similar to that of Chapter 4. It turned out that all measures of accuracy, defined by (4.45)-(4.48), are worse than in the Bernoulli and synchronous exponential lines. Specifically, ²T P is on the average within 5%, ²W IPi within 8% and ²STi and ²BLi are within 0.03. System-theoretic properties: As in the synchronous case, it is possible to show that asynchronous exponential lines possess the properties of reversibility and monotonicity, including strict monotonicity with respect to ci , i = 1, . . . , M . Note, however, that if one assumes that Tup,i is a monotonically decreasing d P becomes non-monotonic with respect to ci (see Problem function of ci , T 11.14). Such a situation takes place when machine reliability deteriorates if its speed is increased.

11.3

Case Studies

11.3.1

Automotive ignition coil processing system

The Bernoulli model of this system has been validated and investigated in “what if” scenarios in Subsection 4.4.1. Here, similar analyses are carried out using the exponential model constructed in Subsection 3.10.1. To take into account starvations by pallets, the efficiency and the downtime of Op. 1 have been modified, respectively, to 0.9226 and 19.12 for Period 1 and 0.916 and 12.92 for Period 2. d Model validation: Using expression (11.29), we calculate P R and convert it d to T P based on 3600 d d · P R. T P = τ The results are shown in Table 11.1, where the error is evaluated using (4.53). Clearly, the accuracy of the exponential model is similar to that of the Bernoulli case (see Table 4.2). Table 11.1: Exponential model validation Period 1 Period 2

d P R 0.8553 0.8512

d T P (parts/hr) 481.1 504.8

T Pmeas (parts/hr) 472 472.6

Error (%) 1.93 6.81

Effect of starvations by pallets: Assuming that Op. 1 is not starved for d pallets, we recalculate T P and evaluate the expected improvements. The results, shown in Table 11.2, are similar to those obtained in Subsection 4.4.1

382

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

for the Bernoulli case: eliminating starvations for pallets leads to insignificant improvement of the throughput. Table 11.2: System performance without starvation of Op.1 for pallets Period 1 Period 2

d T P (parts/hr) 482.7 507.3

Improvement (%) 0.33 0.50

Effect of increasing buffer capacity and m9−10 efficiency: The system throughput with increased buffer capacity and efficiency of the worst machine, m9−10 , is shown in Tables 11.3 and 11.4, respectively. (The efficiency of m9−10 has been increased by decreasing its downtime in accordance with Table 4.7.) As in the Bernoulli case, each of these “what if” scenarios leads to a significant increase in system productivity. Table 11.3: Performance with increased buffer capacity Increment (%) 100 200 300

d T P (parts/hr) 490.8 495.6 498.8

Improvement (%) 2.02 3.01 3.68

(a). Period 1 Increment (%) 100 200 300

d T P (parts/hr) 514.7 518.1 520.3

Improvement (%) 1.96 2.63 3.07

(b). Period 2

Bernoulli vs. exponential analyses: This case study illustrates that the analysis of exponential models can be carried out either directly (using the aggregation procedures developed in this chapter) or indirectly (using the expB transformation of Chapter 3). Our experience indicates that in most cases the results are quite similar. Since the Bernoulli analysis is simpler, the indirect approach may be preferable, especially for asynchronous systems in which the aggregations procedure accuracy is relatively low.

11.3. CASE STUDIES

383

Table 11.4: Performance with increased m9−10 efficiency Decrement of Tdown (%) 25.11 42.24 82.17

Tdown 1.18 0.91 0.28

d T P (parts/hr) 492.3 499.7 512.8

Improvement (%) 2.33 3.87 6.59

(a). Period 1 Decrement (%) 25.59 42.08 80.87

Tdown 1.53 1.19 0.39

d T P (parts/hr) 517.6 525.4 537.4

Improvement (%) 2.54 4.08 6.46

(b). Period 2

11.3.2

Crankshaft production line

This case study was carried out at the design stage of the production system. The goal was to evaluate the efficacy of the initial design and predict the expected performance. System description and layout: The layout of the initial design is shown in Figure 11.32. It includes parallel machines (Stations 2 and 3) and consecutive dependent machines (Stations 6, 7, and 12). The expected reliability and capacity of each station are given in Table 11.5 and the buffer capacity is provided in Table 11.6. With these parameters, the system was expected to produce 62 parts/hour. Sta. 1

Ends prep transfer Sta. 8

Brunish thurst wall

Sta. 2

Sta. 3

Sta. 4

Sta. 5

Sta. 6

Clearance cut mill

Broach mains

Mill pins

Fillet roll

Drill oil holes

Sta. 9

Grind mains

Sta. 7

Ends work transfer

Sta. 10

Sta. 11

Sta. 12

Sta. 13

Sta.14

Grind pins

Grind pins

DIS assembly

Balancer

Polisher

Sta. 15

Sta. 16

Washer

Final inspection

Figure 11.32: Layout of crankshaft production line

384

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES Table 11.5: Expected station performance

Sta. Ave. uptime (m) Ave. downtime (m) Capacity (pts/hr) Sta. Ave. uptime (m) Ave. downtime (m) Capacity (pts/hr)

1 104 6.89 75.9 9 9 1.85 73.9

2 3 4 5 6 7 8 700 491 857 718 131 81 76 35.56 26.29 49.74 29.91 7.66 10.93 7.01 78.6 90.6 78.4 120 75.9 75.9 110.1 10 11 12 13 14 15 16 107 107 102 168 132 46500 3000 0.42 0.42 3 13.44 5.99 120 120 76.8 76.8 75 119 120 120 120

Table 11.6: Buffer capacity bi Ni

1 15

2 20

3 20

4 30

Structural modeling: shown in Figure 11.33. Sta. 1

Sta. 2

Sta. 3

Sta. 9

Sta. 10

Sta. 11

5 13

6 41

7 11

8 20

9 30

10 6

11 6

12 6

13 20

14 11

15 10

Clearly, the structural model can be represented as

Sta. 4

Sta. 5

Sta. 6

Sta. 7

Sta. 12

Sta. 13

Sta.14

Sta. 15

Sta. 8

Sta. 16

Figure 11.33: Structural model of crankshaft production line

Machine parameters: Based on Table 11.5, machine parameters are given in Table 11.7. This implies that the system is modeled as an asynchronous exponential line. Performance analysis: Using the data of Tables 11.6 and 11.7 and the recursive aggregation procedure (11.46), we calculate the expected throughput to be 63 parts/hour, which exceeds the design specification by 1 part/hour. Thus, from this point of view the design can be viewed as satisfactory.

11.3. CASE STUDIES

385 Table 11.7: Machine parameters

Sta. λi (1/min) µi (1/min) ei ci (pts/min) Sta. λi (1/min) µi (1/min) ei ci (pts/hr)

1 2 3 4 0.0096 0.0014 0.0020 0.0012 0.1451 0.0281 0.0380 0.0201 0.9379 0.9517 0.9492 0.9451 1.265 1.31 1.5083 1.3067 9 10 11 12 0.1111 0.0093 0.0093 0.0098 0.5405 2.3810 2.3810 0.3333 0.8295 0.9961 0.9961 0.9714 2.4053 1.28 1.28 1.25

5 6 7 8 0.0014 0.0076 0.0123 0.0132 0.0334 0.1305 0.0915 0.1427 0.9600 0.9448 0.8811 0.9156 2 1.265 1.265 1.835 13 14 15 16 0.0060 0.0076 0.0000 0.0003 0.0744 0.1669 0.0083 0.0083 0.9259 0.9566 0.9974 0.9615 1.9833 2 2 2

Sensitivity to buffer capacity: The throughput with increased buffer capacities is shown in Table 11.8. Based on these data, buffer capacity cannot be viewed as a resource, if the throughput would need to be increased. Table 11.8: Sensitivity to buffer capacity Increment d T P (pts/hr)

25% 64

50% 64

100% 65

Sensitivity to machine efficiency: Among all stations, Sta. 7 has the smallest throughput in isolation (67 parts/hour). If its efficiency is decreased (by decreasing its uptime), the resulting throughput is shown in Table 11.9. Decreasing in a similar manner the efficiency of all machines in the system results in the throughput shown in Table 11.10. Table 11.9: Sensitivity to efficiency of Sta. 7 Decrease d T P (pts/hr)

2% 62

5% 60

7% 59

10% 57

Conclusion on the proposed design: The above analyses indicate that although the proposed design does meet the productivity specification, it is quite sensitive to machine efficiency. Particular attention during system operation should be given to Sta. 7.

386

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES Table 11.10: Sensitivity to efficiency of all machines Decrease d T P (pts/hr)

11.4

2% 61

5% 58

7% 56

10% 54

Summary

• The performance measures of serial lines with exponential machines can be evaluated using the same approach as in the Bernoulli case. Specifically, two-machine lines can be evaluated by closed-form expressions, and aggregation procedures can be used to analyze longer lines. However, all analytical expressions are more involved, especially in the asynchronous case. • Marginal pdf’s of buffer occupancy contain δ-functions, which account for buffers being empty and full. • The accuracy of the resulting performance measure estimates in the synchronous case is similar to that of the Bernoulli case. In the asynchronous case, the accuracy is lower. • Shorter up- and downtime lead to a higher production rate (or throughput) than longer ones, even if machine efficiency remains constant. • A decrease in downtime leads to higher throughput of a serial line than an equivalent increase in uptime. • Exponential lines observe the usual monotonicity and reversibility properties.

11.5

Problems

Problem 11.1 Consider a two-machine synchronous exponential line defined by assumptions (a)-(e) of Subsection 11.1. (a) Assume e1 = 0.95, e2 = 0.6, Tdown,1 = 2 and Tdown,2 = 6. Calculate and plot P R, W IP , BL1 and ST2 as functions of N for N from 2 to 60. Based on these plots, select the buffer capacity, which would be reasonable for this system. (b) Assume now that ei , i = 1, 2, remain the same as above but Tdown,1 = 1 and Tdown,2 = 3. Again, calculate and plot the performance measures and select the buffer capacity. (c) Compare and interpret the results of (a) and (b). (d) Compare the results of (a) and (b) with those obtained in part (a) of Problem 4.2. Problem 11.2 Consider a two-machine synchronous exponential line with e1 = 0.9, e2 = 0.95, Tdown,1 = 10, Tdown,2 = 3 and N = 15.

11.5. PROBLEMS (a) (b) (c) (d)

387

Calculate its P R. Assume that Tup,1 is increased twice and calculate P R again. Assume that Tdown,1 is decreased twice and calculate P R again. Compare and interpret the results of (a)-(c).

Problem 11.3 Consider a 5-machine synchronous exponential line defined by assumptions (a)-(e) of Subsection 11.1. (a) Assume λi = 0.3 and µi = 0.5, i = 1, . . . , 5, and all the buffers are of equal capacity Ni = 1, i = 1, . . . , 4. Calculate the machine efficiency in isolation d and the P R of the system as a whole. (b) Assume now that λi = 0.03 and µi = 0.05, i = 1, . . . , 5, and the buffers are d as above. Again calculate the machine efficiency in isolation and the P R of the system. (c) Determine the smallest buffer capacity for the system in case (b) so that d the resulting P R is close to that of case (a). Problem 11.4 Consider a 5-machine synchronous exponential line defined by assumptions (a)-(e) of Subsection 11.1. d \ R, W IP i , (a) Assume ei = 0.9, Tdown,i = 1, i = 1, ..., 5. Calculate and plot P d c ST i and BLi as functions of N for N from 1 to 10. Based on these results, select the buffer capacity for this system. (b) Assume again that ei = 0.9, Tdown,i = 2, i = 1, ...5. Select the buffer capacity for this case as well. (c) Repeat (a) and (b) for ei = 0.7, i = 1, ..., 5. (d) Compare and interpret the results of (a)-(c). (e) Compare and interpret the results of (a)-(d) with those obtained in Problem 4.9. Problem 11.5 Consider a 5-machine synchronous line with ei = 0.8, Tdown,i = 1, i = 1, ..., 5, Ni = 1, i = 1, ..., 4. Assume that one machine can be replaced by another one with e = 0.85. d (a) Which machine should be replaced so that P R is maximized? d (b) With the new machine, will P R always be larger than that of the original line? (c) If the answer is in the negative, find a sufficient condition under which the answer is in the positive. Problem 11.6 Consider again the 5-machine line introduced in Problem 11.5. Assume that the capacity of two buffers can be increased by a factor of 2. d (a) Which buffer capacity should be increased so that P R is maximized? d (b) With the new buffers, will P R always be larger than in the original line?

388

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES

Problem 11.7 Consider the following approach to performance analysis of synchronous exponential lines: • Assume that the buffer capacity is infinite. Determine the production rate, i.e., P R∞ , and W IPi , i = 1, . . . , M − 1. • Select buffer capacities equal to W IPi . Expect that the systems with these finite buffers will have roughly the same P R as the original system with infinite buffers, i.e., P RW IPi ≈ P R∞ . (a) Verify if this approach is viable. For this purpose, use the system defined d by λi = 0.3 and µi = 0.5, i = 1, . . . , 5, and calculate its P R with the buffers defined by the second bullet above. (b) If this approach does not work, calculate the error between the expected d R. P RW IPi and P Note: To evaluate W IPi in the system with infinite buffers, assume Ni = 100. Problem 11.8 Consider a two-machine asynchronous exponential line defined by assumptions (a)-(e) of Subsection 11.1. (a) Assume T P1 = 0.95, T P2 = 0.6, Tdown,1 = 2, Tdown,2 = 6, c1 = 2, c2 = 1. Calculate and plot T P , W IP , BL1 , ST2 as functions of N for N from 2 to 60. Based on these plots, select a reasonable buffer capacity for the system. (b) Assume that all the parameters of the system remain the same, except for Tdown,1 = 1 and Tdown,2 = 3. Again, calculate and plot the performance measures and select the buffer capacity. (c) Compare the results to those obtained in Problems 11.1 and 4.2. Problem 11.9 Consider a two-machine asynchronous line defined by assumptions (a)-(e) of Subsection 11.1. Assume e1 = 0.9, e2 = 0.7, Tdown,1 = 1 min, Tdown,2 = 1.5 min, τ1 + τ2 = 2 min and N = 3. (a) Determine τ1 and τ2 , which maximize T P . (You may use either the trial and error approach or an educated guess.) (b) With this τ1 and τ2 , calculate T P , W IP , BL1 and ST2 . What can you say about the nature of W IP ? Problem 11.10 For a five-machine serial line with λi = 0.3 and µi = 0.5, d R, i = 1, . . . , 5, determine the smallest Ni , i = 1, . . . , 4, that results in the P which is 95% of the machine efficiency. Problem 11.11 Consider a production line with two exponential machines defined by m1 : m2 : b:

Tup,1 = 9 min,

Tdown,1 = 1 min,

c1 = 1 part/min,

Tup,2 = 8 min,

Tdown,2 = 3 min,

c2 = 1 part/min,

N = 12.

11.6. ANNOTATED BIBLIOGRAPHY

389

Assume that c2 can be increased as desired. Determine the smallest c2 so that the T P of this system is the same as that of the two-machine production line where each machine is identical to m1 . Problem 11.12 Consider the exponential model of the production system of Figure 3.38 defined in Problem 3.3. (a) (b) (c) (d)

Calculate the production rate of this system. Calculate the average occupancy of each buffer. Calculate the probabilities of blockages and starvations of all machines. Compare the results to those obtained in Problem 4.11.

Problem 11.13 Repeat the steps of Problem 11.12 for the exponential model given in Problem 3.4 (compare the results to those of Problem 4.12). Problem 11.14 Assume that Tup of an exponential machine is a function of c, i.e., Tup = f (c), where f (·) is a non-strictly decreasing function (e.g., there exists c∗ such that Tup is constant for c < c∗ and strictly decreasing for c > c∗ ). Such a machine is referred to as having reliability-capacity coupling (RCC). Consider a synchronous line with two exponential RCC machines and analyze the monotonicity properties of T P as a function of machine capacity.

11.6

Annotated Bibliography

Serial production lines with exponential machines are analyzed in numerous publications using both queueing theory and system-theoretic (decomposition) approaches. The queueing theory approach can be found, for example, in [11.1] H.T. Papadopoulos, C. Heavey and J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design, Chapman & Hall, London, 1993. [11.2] J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993. [11.3] H.G. Perros, Queueing Networks with Blocking, Oxford University Press, New York, 1994. The decomposition approach is presented in [11.4] S.B. Gershwin, Manufacturing Systems Engineering, Prentice Hall, Englewood Cliffs, NJ, 1994. [11.5] Y. Dallery, R. David and X.L. Xie, “An Efficient Algorithm for Analysis of Transfer Lines with Unreliable Machines and Finite Buffers,” IIE Transactions, vol. 20, pp. 280-283, 1988.

390

CHAPTER 11. ANALYSIS OF EXPONENTIAL LINES The material of this chapter is based on

[11.6] D.A. Jacobs, Improvability in Production Systems: Theory and Case Studies, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1993. [11.7] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “c-Bottleneck in Serial Production Lines,” Mathematical Problems in Engineering, vol. 7, pp. 543578, 2001. [11.8] J. Li, “Performance Analysis of Production Systems with Rework Loops,” IIE Transactions, vol. 36, pp. 755-765, 2004. Similar results for analysis of production lines with the geometric machine reliability model have been obtained in [11.9] J. Li, Production Variability in Manufacturing Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 2000.

Chapter 12

Analysis of Non-Exponential Lines Motivation: The analysis of serial lines with non-exponential machines is of importance because, as empirical evidence indicates, the coefficients of variation of up- and downtime of manufacturing equipment on the factory floor are often less than 1 and, thus, the corresponding pdf’s cannot be exponential. Therefore, a method for performance analysis of non-exponential lines is of importance. Since these pdf’s are typically unknown, to be useful in practice, this method should be “insensitive” to particular up- and downtime distributions and depend mostly on their first two moments. Such a method is provided in this chapter. Overview: In the case of non-exponential machines, the approach to performance analysis is quite different from that in the exponential case. This is because, as it was pointed out on a number of occasions above, there are no analytical methods for performance analysis of serial lines when the machines are non-exponential and, therefore, the resulting systems are not Markovian. (One exception, involving machines with the so-called co-axial, e.g., Erlang, reliability model, leads to an extremely demanding numerical procedure and could hardly be used for realistic systems.) Thus, no exact analytical formulas for P R, W IPi , BLi , and STi can be derived, even in the two-machine case. Therefore, we resort to an empirical approach. Within this approach, we derive an approximate analytical formula for one performance measure - throughput (T P ) - using numerical simulations of systems at hand. To accomplish this, we first study production lines {[ftup , ftdown ]1 , . . ., [ftup , ftdown ]M } and {[τ, ftup , ftdown ]1 , . . ., [τ, ftup , ftdown ]M }, where ftup,i and ftdown,i are either Weibull (W) or gamma (ga) or log-normal (LN) distributions. It turns out, fortunately, that T P is rather insensitive to particular distributions involved and depends mainly on their coefficients of variation, i.e., the ratio of the standard deviation and the expected value. Based on this observation, we introduce an analytical formula that provides an estimate of T P as a function of the coefficients of variation 391

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CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

of up- and downtime. Although this formula is derived for lines with machines obeying W, ga, and LN reliability models, it is conjectured that it holds for any unimodal distribution of up- and downtime. In this sense, this chapter provides a method for calculating T P in serial production lines with a general model of machine reliability.

12.1

Systems Considered

12.1.1

Mathematical description

System parameters and conventions: The production system considered here is shown in Figure 12.1. f t up,1 f t down,1 τ1

m1

N1

f t up,2 f t down,2 τ2

N2

b1

m2

b2

f t up,M−1

f t down,M−1 N M−1 τM−1

mM−1

bM−1

f t up,M f t down,M τM

mM

Figure 12.1: M -machine production line with non-exponential machines Each machine is defined by the pdf’s of its up- and down time, ftup,i (t) and ftdown,i (t). In addition, each machine is characterized by its cycle time, τi or, equivalently, its capacity, ci = 1/τi . In a special case, all τi ’s may be the same, say, τi = 1, implying that a synchronous system is considered. As in all previous chapters, the efficiency of each machine is ei =

Tup,i , Tup,i + Tdown,i

i = 1, . . . , M,

and its performance in isolation is characterized by the production rate, P Ri = ei [parts/cycle time] in the synchronous case or by the throughput, T Pi = ci ei [parts/unit of time] in the asynchronous case. Otherwise, the mathematical description of the production line of Figure 12.1 remains the same as in Sections 11.1 and 11.2, i.e., conventions (a)-(e) of Subsection 11.1.1 take place. Unfortunately, however, due to the non-Markovian nature of this system, it cannot be analyzed using the techniques of the previous chapters. Reliability models: in this section:

The following machine reliability models are considered

• Weibull (3.4), • Gamma (3.5), and

12.1. SYSTEMS CONSIDERED

393

• Log-normal (3.6). These distributions are selected because, being defined by two parameters, they allow us to place their coefficients of variation at will (unlike, say, Rayleigh and Erlang distributions, which admit only a unique or discrete value of CV ). For a fixed coefficient of variation, the shape of these distributions may vary significantly, from almost uniform to almost δ-function, as illustrated in Figure 12.2 (for CV = 0.5). Thus, they provide a considerable variety of options for system modeling and analysis. 0.06

ga, T = 20 down ga, Tdown = 100 W, Tdown = 20 W, Tdown = 100 LN, Tdown = 20 LN, Tdown = 100

0.05

f(t)

0.04

0.03

0.02

0.01

0

0

50

100

150

t

200

250

300

350

Figure 12.2: Different distributions with identical coefficients of variation (CVdown = 0.5)

12.1.2

Second-order-similar production lines

Consider synchronous {[ftup , ftdown ]1 , . . ., [ftup , ftdown ]M } or asynchronous {[τ, ftup , ftdown ]1 , . . ., [τ, ftup , ftdown ]M } lines with up- and downtime distributions belonging to the set {W, ga, LN }. To present the results, introduce the notion of similarity of production lines. Definition 12.1 Two serial production lines, L1 and L2 , defined by {[τ, ftup , ftdown ]1 , . . . , [τ, ftup , ftdown ]M }L1 and {[τ, ftup , ftdown ]1 , . . . , [τ, ftup , ftdown ]M }L2 , are referred to as second-order-similar (SOS) if they have the same number of machines, (12.1) M (L1 ) = M (L2 ),

394

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

the corresponding buffers in both lines are of equal capacity, Ni (L1 ) = Ni (L2 ),

i = 1, . . . , M − 1,

(12.2)

the corresponding machines have identical cycle time, τi (L1 ) = τi (L2 ),

i = 1, . . . , M,

(12.3)

and, finally, the first two moments of the pdf ’s of up- and downtime of the corresponding machines are identical, which implies that Tup,i (L1 ) = Tup,i (L2 ),

Tdown,i (L1 ) = Tdown,i (L2 ),

i = 1, . . . , M,

(12.4)

and CVup,i (L1 ) = CVup,i (L2 ),

CVdown,i (L1 ) = CVdown,i (L2 ),

i = 1, . . . , M. (12.5)

Thus, according to this definition, two lines are SOS if the only difference between them is the reliability model of the machines, constrained by the fact that the first two moments are the same. Clearly, when the lines are synchronous, condition (12.3) is superfluous. For a given M -machine line, the set of all SOS production lines is denoted as AM . Production lines from AM will be denoted as A, B, etc.

12.2

Sensitivity of P R and T P to Machine Reliability Model

Consider the class AM of SOS lines. Assume that each buffer in these lines has capacity satisfying the condition Ni ≥ max ci Tdown,i .

(12.6)

i

In the case of synchronous machines, this condition becomes Ni ≥ max Tdown,i .

(12.7)

i

Conditions (12.6) and (12.7) are introduced in order to allow an analytical evaluation of P R and T P in systems with CVup,i = CVdown,i = 0 (see Section 12.3 below). These conditions can be removed at the cost of forfeiting this opportunity. In addition to (12.6) and (12.7), assume also that all CV ’s belong to the interval [0, 1], i.e., CVup,i ∈ [0, 1],

CVdown,i ∈ [0, 1],

i = 1, . . . , M.

(12.8)

The reason for this assumption is that, as it has been pointed out on a number of occasions above, manufacturing equipment on the factory floor often has

12.2. SENSITIVITY TO MACHINE RELIABILITY

395

coefficients of variation less than 1 (see also Subsection 2.2.4 for a theoretical justification of this fact). Condition (12.8) can also be removed at the expense of somewhat complicating the empirical formulas for T P and P R derived in Section 12.3. Introduce a measure of sensitivity of T P to the machine reliability model as follows: |T P A − T P B | . (12.9) ²M = max A,B∈AM TPA Clearly, small ²M would imply that T P (or, in the synchronous case, P R) depends mainly on the first two moments of the pdf’s of up- and downtime, rather than on the complete probability distributions. Numerical Fact 12.1 In serial production lines defined by conventions (a)(e) of Subsection 11.1.1 with machine reliability models from the set {W, ga, LN }, under assumptions (12.6) and (12.8) the following takes place: • For SOS lines, the sensitivity measure, ²M , is small, i.e., ²M ¿ 1.

(12.10)

• The monotonicity properties of ²M are as follows: M↑

⇒

²M ↑, CV ↑ ⇒ ²M ↑, N ↑ ⇒ ²M ↓, e ↑ ⇒ ²M ↓, where CV ↑, N ↑, or e ↑ denote that either CVup,i , CVdown,i , or Ni , or ei increases. • T P (or P R) is a monotonically decreasing function of CVup,i and CVdown,i . Justification: This statement is justified using simulations of both synchronous and asynchronous lines. In each case, three groups of systems are considered: Group 1: Lines with identical machines and with CVup = CVdown =: CV Weibull line: gamma line: log-normal line:

{[W (λ, Λ), W (µ, M)]i , i = 1, . . . , M }, {[ga(λ, Λ), ga(µ, M)]i , i = 1, . . . , M }, {[LN (λ, Λ), LN (µ, M)]i , i = 1, . . . , M }.

(12.11)

This group mimics the exponential case, {[exp(λ), exp(µ)]i , i = 1, . . . , M }, with the only difference that the coefficient of variation, CV , is not necessarily 1, but can take arbitrary values in [0, 1]. Group 2: Lines with machines having identical reliability models and with CVup,i =: CVup , CVdown,i =: CVdown , i = 1, . . . , M : Weibull line: {[W (λ1 , Λ1 ), W (µ1 , M1 )], . . . , [W (λM , ΛM ), W (µM , MM )]},

396

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES gamma line: {[ga(λ1 , Λ1 ), ga(µ1 , M1 )], . . . , [ga(λM , ΛM ), ga(µM , MM )]},

log-normal line: {[LN (λ1 , Λ1 ), LN (µ1 , M1 )], . . . , [LN (λM , ΛM ), LN (µM , MM )]}. (12.12) Obviously, the machines here may have different Tup and Tdown but their reliability models are identical and, in addition, all CVup ’s are the same as are CVdown ’s. This group also mimics the exponential case, {[exp(λ1 ), exp(µ1 )], . . ., [exp(λM ), exp(µM , )]}, but allows for different coefficients of variation of uptime and downtime. Group 3: Mixed reliability models: This group is formed by selecting reliability models of the machines randomly and equiprobably from the set {W, ga, LN }. In this manner, the following ten-machine lines have been formed: Mixed line 1:

{[ga, W ], [LN, LN ], [W, ga], [ga, LN ], [ga, W ],

Mixed line 2:

[LN, ga], [W, W ], [ga, ga], [LN, W ], [ga, LN ]}, {[W, LN ], [ga, W ], [LN, W ], [W, ga], [ga, LN ], [ga, W ], [W, W ], [LN, ga], [ga, W ], [LN, LN ]}. (12.13)

Obviously, Tup,i , Tdown,i , CVup,i , and CVdown,i in this case may differ from one machine to another. The parameters of lines (12.11)-(12.13) have been selected randomly and equiprobably from the following sets: M e CVup

Tdown and CVdown Ni Ni ci

∈ ∈ ∈ ∈ ∈ = ∈

{3, 5, 10}, {0.55, 0.65, 0.75, 0.85, 0.9, 0.95}, {10, 20}, {0.1, 0.25, 0.5, 0.75, 1}, (12.14) {Tdown,i , 3Tdown,i }(in the synchronous case) and (max ci )(max Tdown,i ) (in the asynchronous case), {1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2}.

The parameters of the reliability models have been selected as shown in Table 12.1. More precisely, Table 12.1 defines the pdf’s of the downtime for each reliability model. The pdf’s of the uptime were selected according to the group of production lines considered. Namely, in the case of Group 1, for a given machine efficiency, e, the parameters of the uptime pdf were selected as Tup CVup

e Tdown , 1−e = CVdown , =

(12.15)

and, thus, the pdf is completely defined. In Group 2, Tup was selected as above but CVup and CVdown were selected randomly and equiprobably from the set {0.1, 0.25, 0.5, 0.75, 1}. In Group 3, ei , Tdown,i , CVup,i and CVdown,i , i = 1, . . . , M , were all selected randomly and equiprobably from the sets {0.55, 0.65,

12.2. SENSITIVITY TO MACHINE RELIABILITY

397

Table 12.1: Downtime distributions considered CVdown 0.1 0.25 0.5 0.75 1.00

Tdown = 10 W (0.096, 12.15), ga(10, 100), LN (2.30, 0.1) W (0.092, 4.54), ga(1.6, 16), LN (2.27, 0.25) W (0.088, 2.1), ga(0.4, 4), LN (2.19, 0.47) W (0.092, 1.35), ga(0.18, 1.78), LN (2.08, 0.67) LN (1.96, 0.83)

CVdown 0.1 0.25 0.5 0.75 1.00

Tdown = 20 W (0.048, 12.15), ga(5, 100), LN (2.99, 0.1) W (0.046, 4.54), g(0.8, 16), LN (2.97, 0.25) W (0.044, 2.1), ga(0.2, 4), LN (2.88, 0.47) W (0.046, 1.35), ga(0.09, 1.78), LN (2.77, 0.67) LN (2.65, 0.83)

0.75, 0.85, 0.9, 0.95}, {10, 20}, and {0.1, 0.25, 0.5, 0.75, 1}, respectively. Finally, in the asynchronous case, machine capacity, ci , was selected either randomly and equiprobably from the set {1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2} or according to the allocations defined in Table 12.2, i.e, according to the bowl, inverted bowl, increasing, or oscillating patterns. CV s were selected randomly and equiprobably from the set {0.2, 0.4, 0.6, 0.8, 1}. Table 12.2: Machine capacity allocation in asynchronous case bowl inv. bowl incr. oscil.

c1 1.32 1 1 1.06

c2 1.22 1.06 1.03 1.32

c3 1.14 1.14 1.05 1

c4 1.06 1.22 1.08 1.14

c5 1 1.32 1.11 1.06

c6 1 1.32 1.16 1.32

c7 1.06 1.22 1.2 1.14

c8 1.14 1.14 1.23 1.22

c9 1.22 1.06 1.27 1

c10 1.32 1 1.32 1.22

In each of the production lines thus constructed, P R or T P was evaluated numerically using Simulation Procedure 4.1. The results of simulations for Group 1 are shown in Figures 12.3 and 12.4 (the role of the broken lines in these and the subsequent figures will become clear later on). In the interest of space, we included here only a sample of typical results; in particular, only the case of synchronous lines is illustrated. The graphs for k = 1 and 3 correspond to N = Tdown and 3Tdown , respectively. The sensitivity, ²M , of P R and T P to the type of reliability model, calculated according to (12.9), is given in Tables 12.3 and 12.4. Clearly, ²M ≤ 0.02 for CV ≤ 0.5 and ²M ≤ 0.06 for 0.5 < CV ≤ 1, supporting Numerical Fact 12.1. Also, examining these data, one can see that properties (12.10) take place. To simulate the lines of Group 2, the following three parameter sets have been formed,

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

M=3

M=5

0.8 0.7 0.6 0.5

k=1

0.3

0.3

0.2

0.2

0

0.2

0.4

0.6

CV

0.8

1

0.6

0.8

CV

1

0.3

0.3 0

0.2

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0

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1

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1

0.5

Weibull Gamma Log−normal Line 1 Line 2 Empirical law 0

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k=1

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CV

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1

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1

0.8

1

0.5

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0

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1

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0

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1

k=1

PR

PR

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1

0.6

CV

k=3

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1

k=1

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1

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0.7

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0

1

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1

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1

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1

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1 0.9

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0

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PR 0

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1

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e=0.95

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e=0.9

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e=0.55

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PR

398

0

0.2

0.4

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CV

0.8

1

0.5

0

0.2

0.4

0.6

CV

Figure 12.3: Production rate as a function of CV for lines of Group 1: Tdown = 10

12.2. SENSITIVITY TO MACHINE RELIABILITY

M=3

M=5 0.8

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Weibull Gamma Log−normal Line 1 Line 2 Empirical law 0

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0.6

0.6

0.4

PR

k=1

0.7

CV

0.2

0.8

0.6

0.4

0

0.9

k=3

0.6

0.2

0.8

0.3

0.7

0

1

k=1

0.4

0.7

0.5

0.8

k=3

1

PR

0.8

PR

0.9

0.6

CV

0.5

1

k=3

0.4

PR

1 0.9

0.2

0.7

0.6

0

0

0.6

0.7

0.6

1

k=1

0.8

PR

PR

k=1

0.8

PR 0

0.8

0.7

1

0.3

0.9

k=3

0.8

0.8

k=3

0.4

1

0.9

0.6

CV

0.5

k=1

0.5

0.4

0.4

0.6

PR

PR

k=1

0.5

0.2

0.7

k=3

0.8

0.7

0

PR

0

0.2

0.8

PR

PR

0.4

1

e=0.95

0.2

0.5

k=1

0.4

0.8

e=0.9

0

k=1

0.3

0.6

0.5

0.5

k=1

0.7

0.6

e=0.85

0.4

0.8

0.7

e=0.75

0.5

0.4

0.2

k=3

PR

0.5

0.3

0.2

0.6

k=3

PR

k=3

0.5 0.4

e=0.65

M=10

0.8

PR

e=0.55

399

0

0.2

0.4

0.6

CV

0.8

1

0.5

0

0.2

0.4

0.6

CV

Figure 12.4: Production rate as a function of CV for lines of Group 1: Tdown = 20

400

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

Table 12.3: Sensitivity ²M of production rate to the type of up- and downtime distributions (in units of 100²M %): Tdown = 10

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.1 5 10 0.06 0.02

CV = 0.25 3 5 10 0.22 0.30 0.33

3 1.10

0.02 0.03

0.04 0.35

0.02 0.36

0.04 0.33

0.17 1.59

0.22 1.86

0 1.65

0.03 0.01

0.02 0.06

0.05 0.33

0.05 0.28

0 0.33

0.31 1.13

0.38 1.47

0.15 1.68

0.04 0.02

0.03 0.05

0 0.09

0.03 0.17

0.01 0.34

0.01 0.39

0.20 0.78

0.18 0.77

0.24 1.28

N = 3Tdown N = Tdown

0.01 0.03

0.01 0.02

0.01 0.01

0.06 0.15

0.05 0.28

0.05 0.33

0.15 0.56

0.05 0.81

0.24 0.87

N = 3Tdown N = Tdown

0.02 0.02

0.02 0.01

0 0.01

0.03 0.06

0.09 0.18

0.03 0.13

0.16 0.21

0.08 0.41

0.08 0.47

N = 3Tdown

0.01

0

0.01

0.02

0.01

0.03

0.04

0.07

0.06

M N = Tdown

CV = 0.75 3 5 10 2.93 3.54 3.68

3 4.50

N = 3Tdown N = Tdown

0.47 3.11

0.74 3.87

0.68 3.75

1.21 4.44

1.77 5.54

1.64 6.37

N = 3Tdown N = Tdown

0.84 2.20

1.05 2.91

1.01 3.02

1.66 3.19

1.56 5.63

2.09 5.86

N = 3Tdown N = Tdown

0.63 1.29

0.57 2.00

0.06 2.69

1.44 2.00

2.25 3.65

1.97 4.29

N = 3Tdown N = Tdown

0.35 0.97

0.47 1.30

0.69 1.87

1.03 1.38

1.62 1.84

1.43 3.62

N = 3Tdown N = Tdown

0.36 0.40

0.27 0.87

0.50 1.17

0.75 0.54

0.64 1.23

1.26 1.79

N = 3Tdown

0.14

0.26

0.26

0.22

0.60

0.59

M N = Tdown

3 0.06

N = 3Tdown N = Tdown

0.05 0.05

0.02 0.03

N = 3Tdown N = Tdown

0.02 0.07

N = 3Tdown N = Tdown

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.5 5 10 1.73 1.40

CV = 1 5 10 6.25 6.08

12.2. SENSITIVITY TO MACHINE RELIABILITY

401

Table 12.4: Sensitivity ²M of production rate to the type of up- and downtime distributions (in units of 100²M %): Tdown = 20

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.1 5 10 0.11 0.06

CV = 0.25 3 5 10 0.12 0.34 0.37

3 1.46

0.05 0.03

0.24 0.33

0.06 0.48

0.09 0.47

0.31 1.27

0.12 1.39

0.16 1.92

0.03 0

0.03 0.06

0 0.34

0.13 0.31

0.02 0.30

0.23 1.16

0.08 1.13

0.37 1.88

0.01 0.01

0.04 0.05

0.01 0.04

0.11 0.17

0.07 0.29

0.05 0.32

0.18 0.61

0.07 1.00

0.36 1.45

N = 3Tdown N = Tdown

0.01 0.03

0.02 0.03

0.01 0.02

0.05 0.20

0.01 0.34

0.06 0.39

0.17 0.46

0.14 0.94

0.33 1.06

N = 3Tdown N = Tdown

0.02 0.02

0.01 0.06

0.01 0.05

0.03 0.14

0.06 0.16

0.09 0.14

0.16 0.31

0.10 0.47

0.13 0.52

N = 3Tdown

0.01

0.01

0.01

0.06

0.03

0.02

0.12

0.01

0.06

M N = Tdown

CV = 0.75 3 5 10 3.12 3.68 3.74

3 5.34

N = 3Tdown N = Tdown

0.86 2.69

0.62 3.65

0.94 3.38

1.91 4.22

4.02 6.27

1.66 5.94

N = 3Tdown N = Tdown

0.55 1.47

0.80 2.77

0.75 3.45

1.64 3.44

2.18 4.39

1.62 5.43

N = 3Tdown N = Tdown

0.32 1.42

0.81 2.27

0.90 2.67

1.67 2.10

1.61 3.04

1.49 4.00

N = 3Tdown N = Tdown

0.54 0.93

0.58 1.52

0.80 1.89

0.09 1.57

1.04 2.23

1.43 3.06

N = 3Tdown N = Tdown

0.36 0.64

0.45 0.88

0.30 0.98

0.75 1.01

1.00 1.00

1.04 2.07

N = 3Tdown

0.30

0.39

0.24

0.65

0.47

0.81

M N = Tdown

3 0.06

N = 3Tdown N = Tdown

0.09 0.08

0.04 0.05

N = 3Tdown N = Tdown

0.05 0.04

N = 3Tdown N = Tdown

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.5 5 10 1.56 1.25

CV = 1 5 10 6.70 6.05

402

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

PS1: Tdown = [20, 10, 10, 20, 20, 10, 20, 20, 10, 10], e = [0.95, 0.85, 0.9, 0.95, 0.85, 0.9, 0.95, 0.9, 0.9, 0.85], PS2: Tdown = [10, 20, 20, 20, 10, 10, 20, 20, 10, 10],

(12.16)

e = [0.9, 0.85, 0.85, 0.95, 0.9, 0.95, 0.9, 0.85, 0.9, 0.95], PS3: Tdown = [10, 20, 10, 20, 10, 20, 20, 10, 10, 20], e = [0.55, 0.75, 0.9, 0.85, 0.65, 0.9, 0.55, 0.95, 0.65, 0.75],

(12.17) (12.18)

and the CV ’s of up- and down time have been selected as {0.1, 0.25, 0.5, 0.75, 1}. The results are shown in Figure 12.5. In this figure, the argument CVave is M =3

0.8

0.7

0.7

TP

0.8

0.7

TP

0.8

TP

0.9

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.2

0.4

0.6

CVave

0.8

1

0.9

0.9

0.8

0.8

0.7

0.7

0

0.2

0.6

0.6

0.5

0.5

0.4

0.4

0

0.2

0.4

0.6

CVave

0.8

1

0.4

0.6

CVave

0.8

1

0

0.2

0.4

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

CVave

0.9 0.8 0.7

Weibull Gamma Lognormal Line 1 Line 2 Empirical law 0

0.2

0.4

0.6 0.5

0.6

CVave

0.8

1

0.9

0.4

CVave

0.9 0.9

0.8

0.8

0.8

0.7

TP

TP

0.7

0.7

TP

PS3

0.4

TP

0

TP

PS2

M = 10

0.9

TP

PS1

M =5

0.9

0.6

0.6

0.6

0.5 0.4

0.5

0.5

0

0.2

0.4

0.6

CVave

0.8

1

0.4

0.4 0

0.2

0.4

0.6

CVave

0.8

1

CVave

Figure 12.5: Throughput as a function of CVave for lines of Group 2 defined as CVave =

M 1 X CVup,i . M i=1

Again, the results support Numerical Fact 12.1. Finally, the lines of Group 3 have been simulated with the following sequences of CVup and CVdown : S1:

CVup = [1, 0.25, 0.75, 0.5, 1, 0.75, 0.5, 0.1, 1, 0.5],

12.2. SENSITIVITY TO MACHINE RELIABILITY

403

CVdown = [0.25, 0.5, 0.75, 0.75, 0.25, 0.5, 1, 0.1, 0.75, 0.1],

(12.19)

CVup = [1, 0.5, 1, 0.75, 0.25, 1, 0.75, 1, 1, 1], CVdown = [1, 1, 0.25, 1, 0.75, 1, 1, 0.75, 1, 0.5],

(12.20)

S3:

CVup = [0.1, 0.25, 0.1, 0.5, 0.1, 0.1, 0.1, 0.75, 0.1, 0.1], CVdown = [0.75, 0.1, 0.25, 0.1, 0.5, 0.1, 0.1, 0.25, 0.1, 0.1],

(12.21)

S4:

CVup = [0.25, 0.1, 0.1, 0.1, 0.25, 0.1, 0.1, 0.5, 0.1, 0.1], CVdown = [0.1, 0.25, 0.1, 0.1, 0.1, 0.1, 0.5, 0.1, 0.1, 0.1], CVup = [0.75, 1, 1, 0.75, 1, 1, 1, 0.5, 1, 1],

S2:

S5: S6: S7:

(12.22)

CVdown = [1, 1, 0.75, 1, 1, 1, 0.5, 1, 1, 1], CVup = [1, 0.75, 0.5, 1, 1, 0.25, 1, 0.75, 1, 0.1], CVdown = [0.1, 0.25, 0.5, 0.1, 0.1, 0.75, 0.1, 1, 0.1, 0.1], CVup = [0.1, 0.25, 0.1, 0.1, 0.5, 0.1, 1, 0.1, 0.1, 0.75], CVdown = [1, 0.5, 0.75, 1, 0.5, 1, 0.75, 0.25, 0.1, 1],

(12.23) (12.24) (12.25)

The results are shown in Figure 12.6. The argument CVave in this figure is M =3

0.7

TP

0.8

0.7

TP

0.8

0.7

TP

0.8

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0

0.2

0.4

0.6

CVave

0.8

1

0

0.2

0.4

0.6

CVave

0.8

1

0.4

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7 0.6

0.5

0.5

0.4

0.4

0

0.2

0.4

0.6

CVave

0.8

1

0

0.2

0.4

0.8

1

0.4

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.6

0.8

1

0.2

0.4

0

0.2

0.4

0.8

1

0.6

0.8

1

0.6

0.8

1

CVave

TP

0.8

CVave

0

0.6

CVave

TP 0.6

CVave

0.9

0.4

0.4

0.5

0.9

0.2

0.2

0.6

0.9

0

0

0.7

Weibull Gamma lognormal Line 1 Line 2 Empirical law

TP

0.6

TP

PS3

0.9

TP

PS2

M = 10

0.9

TP

PS1

M =5

0.9

0

0.2

0.4

0.6

CVave

0.8

1

0.4

CVave

Figure 12.6: Throughput as a function of CVave for lines of Group 3 defined as CVave =

M 1 X (CVup,i + CVdown,i ). 2M i=1

(12.26)

404

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

Clearly, these results are also in agreement with Numerical Fact 12.1. Thus, Numerical Fact 12.1 takes place in serial lines with reliability models from {W, ga, LN }. General reliability model: Clearly, simulation of production lines with every machine reliability model is impossible. Therefore, and taking into account that the analysis described above involves a wide range of reliability models, we conjecture that: Numerical Fact 12.1 takes place in serial production lines with any unimodal machine reliability model.

12.3

Empirical Formulas for P R and T P

The data of Figures 12.3 - 12.6 indicate that P R and T P are close to linear functions of CV or CVave . These functions are indicated in Figures 12.3 - 12.6 by broken lines. Two points on these lines can be calculated analytically – for CV = 0 and CV = 1. Indeed, for CV = 0, the following holds: Theorem 12.1 Under assumption (12.6), P R of a synchronous production line with CVup,i = CVdown,i = 0 is given by P R = min i

Tup,i . Tup,i + Tdown,i

Proof: See Section 20.2. For asynchronous lines, a similar fact is proven analytically only for M = 2. For the general M > 2-machine case this fact is verified by simulations: Numerical Fact 12.2 Under assumption (12.6), TP of an asynchronous production line with CVup,i = CVdown,i = 0 is given by T P ≈ min i

ci Tup,i . Tup,i + Tdown,i

For the case of CVup,i = CVdown,i = 1, P R and T P can be estimated using aggregation procedures (11.25) and (11.46). However, since the resulting estimates have limited precision, to evaluate the accuracy of the empirical formulas introduced below, we use simulations to determine P R and T P at CVup,i = CVdown,i = 1. Based on the above arguments, we achieve the goal of this chapter by introducing the following empirical formulas for the production rate and throughput in synchronous and asynchronous serial production lines, respectively: d P R d T P

=

emin − (emin − P Rexp )CVave ,

= (ce)min − ((ce)min − T P

exp

)CVave ,

(12.27) (12.28)

12.3. EMPIRICAL FORMULAS

405

where emin (ce)min

Tup,i , i Tup,i + Tdown,i ci Tup,i = min , i Tup,i + Tdown,i = min

P Rexp and T P exp are the production rate and throughput of the system if all machines were exponential, respectively, and CVave is defined by (12.26). To evaluate the accuracy of these formulas, introduce the following measures: ∆syn

=

∆asyn

=

d |P RA − P R| , A A∈AM PR d |T P A − T P| max . A A∈AM TP max

(12.29) (12.30)

Numerical Fact 12.3 In serial production lines with machine reliability models belonging to the set {W, ga, LN }, under assumptions (12.6) and (12.7) the following takes place: ∆syn ¿ 1,

∆asyn ¿ 1.

Justification: This justification is carried out in the same manner as that for Numerical Fact 12.1 except that quantities (12.29) and (12.30) are calculated (instead of (12.9)). The results are shown in Tables 12.5 - 12.8. Clearly, in most cases, the accuracy of the empirical formulas is within 0.02, with only a few cases within 0.06. Thus, expressions (12.27) and (12.28) allow us to evaluate P R and T P in serial lines with up- and downtime obeying any distribution from the set {W, ga, LN}. For other distributions, we conjecture that: Numerical Fact 12.3 takes place in serial production lines with any unimodal machine reliability model. Effects of up- and downtime: As it follows from (12.27) and (12.28), the d d up- and downtime affect P R and T P through emin , P Rexp , T P exp , and CV ’s. To increase emin , as it has been shown in Chapter 11, it is better to decrease Tdown than to increase Tup . To increase P Rexp and T P exp , as it has been described in Chapter 11, it is better to have shorter up- and downtime than longer ones. Finally, decreasing CVup and CVdown always leads to increased d d P R and T P. PSE Toolbox: Empirical formulas (12.27) and (12.28) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Subsection 19.3.6.

406

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

Table 12.5: Accuracy of empirical law (12.27) for lines in Group 1 (in units of 100∆syn %): Tdown = 10

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.1 5 10 1.33 1.34

CV = 0.25 3 5 10 1.36 1.76 1.72

3 0.61

2.09 0.08

2.64 0.40

3.51 0.38

4.09 0.42

3.04 1.79

4.22 2.23

4.67 2.28

1.58 0.35

1.90 0.43

2.27 1.18

3.06 1.24

4.03 1.27

2.70 2.18

3.56 2.36

4.40 2.61

0.96 0.53

1.31 0.63

1.62 0.79

1.78 1.30

2.48 1.43

3.15 1.40

2.01 1.91

2.91 1.78

3.74 1.64

k=3 k=1

0.62 0.48

0.90 0.65

1.20 0.81

1.14 0.98

1.72 1.17

2.33 1.06

1.18 1.25

1.91 1.46

2.81 0.99

k=3 k=1

0.45 0.36

0.66 0.49

0.87 0.62

0.86 0.63

1.29 0.75

1.76 0.55

0.97 0.76

1.41 0.71

2.10 0.35

k=3

0.22

0.34

0.49

0.40

0.64

1.01

0.40

0.74

1.27

M k=1

3 1.14

k=3 k=1

1.40 0.17

1.82 0.16

k=3 k=1

1.22 0.33

k=3 k=1

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = .75 5 10 1.86 2.99

3 4.23

CV = 0.5 5 10 1.10 1.00

CV = 1 5 10 5.85 5.59

M k=1

3 1.52

k=3 k=1

2.20 2.36

3.21 2.79

3.49 3.19

0.91 4.04

2.01 5.28

1.76 5.88

k=3 k=1

2.05 1.92

2.85 2.36

3.45 2.17

1.46 3.06

1.65 5.03

1.95 5.16

k=3 k=1

1.64 1.40

2.14 1.81

2.72 1.71

1.35 1.68

2.04 3.13

1.72 3.76

k=3 k=1

0.95 0.91

1.54 1.23

2.24 0.99

0.83 1.52

1.29 1.79

1.30 3.51

k=3 k=1

0.80 0.55

1.14 0.52

1.74 0.80

0.82 0.46

0.65 1.27

1.33 1.89

k=3

0.30

0.61

1.09

0.25

0.49

0.75

12.3. EMPIRICAL FORMULAS

407

Table 12.6: Accuracy of empirical law (12.27) for lines in Group 1 (in units of 100∆syn %): Tdown = 20

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = 0.1 5 10 1.44 1.49

CV = 0.25 3 5 10 1.33 1.85 1.86

3 0.74

2.09 0.27

2.79 0.17

3.47 0.31

4.11 0.14

3.13 1.45

3.91 1.83

5.06 2.14

1.64 0.14

1.90 0.24

2.42 1.10

3.12 1.06

3.78 1.07

2.76 2.07

3.58 1.99

4.59 2.46

1.00 0.45

1.35 0.44

1.62 0.61

1.84 1.21

2.56 1.26

3.28 1.15

2.09 1.62

2.93 1.66

3.98 1.57

k=3 k=1

0.65 0.42

0.94 0.58

1.19 0.69

1.16 0.93

1.82 1.14

2.42 0.98

1.25 1.12

2.07 1.43

3.03 0.88

k=3 k=1

0.49 0.31

0.66 0.48

0.89 0.58

0.93 0.58

1.29 0.72

1.87 0.48

1.02 0.59

1.48 0.76

2.28 0.34

k=3

0.25

0.35

0.45

0.46

0.66

1.00

0.58

0.82

1.25

M k=1

3 1.23

k=3 k=1

1.47 0.35

1.78 0.32

k=3 k=1

1.34 0.23

k=3 k=1

e = 0.55 e = 0.65 e = 0.75 e = 0.85 e = 0.9 e = 0.95

CV = .75 5 10 1.99 2.64

3 4.65

CV = 0.5 5 10 0.99 0.84

CV = 1 5 10 6.25 5.94

M k=1

3 1.25

k=3 k=1

2.76 1.58

2.77 2.46

3.94 2.55

1.64 4.47

1.91 5.52

2.09 5.88

k=3 k=1

2.30 1.82

2.89 1.97

3.50 2.30

1.90 3.12

2.07 5.04

1.81 5.72

k=3 k=1

1.38 1.36

2.37 1.51

3.13 1.36

1.52 1.67

2.12 3.15

1.98 4.28

k=3 k=1

1.07 0.83

1.73 1.06

2.66 1.01

0.82 1.64

1.27 1.93

1.73 3.00

k=3 k=1

0.90 0.67

1.29 0.58

1.85 0.84

1.06 1.00

0.82 1.19

1.30 1.84

k=3

0.66

0.67

1.10

0.67

0.56

0.69

408

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

Table 12.7: Accuracy of empirical law (12.28) for lines in Group 2 (in units of 100∆asyn %)

PS1

PS2

PS3

PS1

PS2

PS3

CVave M =3

0.1 0.34

0.175 0.73

0.25 0.87

0.3 1.21

0.375 1.25

0.425 1.12

M =5 M = 10 M =3

0.64 1.12 0.13

1.16 1.91 0.44

1.44 2.37 0.45

1.61 2.57 1.06

1.90 3.02 0.93

1.73 2.91 1.76

M =5 M = 10 M =3

0.06 0.47 0.82

0.32 0.81 1.43

0.17 1.00 1.97

0.99 1.22 2.43

0.53 1.59 2.80

1.91 1.70 2.76

M =5 M = 10

1.07 1.82

1.88 3.05

2.65 4.17

3.13 4.68

3.67 5.66

3.90 5.43

CVave M =3

0.5 1.37

0.55 1.67

0.625 1.23

0.75 1.02

0.875 0.65

1 0.81

M =5 M = 10 M =3

2.06 3.50 1.28

1.62 3.17 2.35

2.21 3.86 1.69

2.20 3.81 1.20

2.07 3.27 1.04

1.64 2.55 1.27

M =5 M = 10 M =3

1.21 2.15 3.10

2.73 3.51 2.79

1.93 2.56 2.99

1.13 3.05 2.56

1.55 3.06 1.86

2.07 2.15 1.35

M =5 M = 10

4.21 6.16

3.96 5.52

4.25 6.58

4.30 6.35

3.28 5.11

2.04 2.72

Table 12.8: Accuracy of empirical law (12.28) for lines in Group 3 (in units of 100∆asyn %) M =3

M =5

M = 10

CVave PS1 PS2 PS3 CVave PS1 PS2 PS3 CVave PS1 PS2 PS3

0.15 0.60 0.18 1.22 0.145 0.82 0.05 1.55 0.1625 2.45 0.06 2.28

0.2583 0.26 0.16 2.00 0.275 0.67 0.47 2.73 0.2275 1.77 0.93 4.09

0.45 0.22 1.62 1.96 0.48 0.45 1.40 3.35 0.4975 0.70 0.09 1.12

0.5167 1.51 1.12 2.61 0.53 1.98 0.98 3.93 0.5225 2.20 0.97 3.89

0.5833 1.22 1.03 3.00 0.6 1.59 0.60 3.81 0.565 2.10 1.24 3.17

0.7917 1.18 0.61 0.85 0.75 2.65 1.28 2.12 0.825 2.19 1.53 1.98

0.9167 0.47 0.93 1.22 0.925 1.23 1.95 1.94 0.9225 1.81 2.64 3.34

12.4. SUMMARY

12.4

409

Summary

• The performance measures of serial lines with non-exponential machines cannot be evaluated in closed form, even in the two-machine case. Therefore, the approach of Chapters 4 and 11 is not applicable here. • The throughput of serial lines with Weibull, gamma, and log-normal pdf’s of up- and downtime is practically independent of the particular distribution involved and depends, mainly, on its coefficient of variation (CV ). • For CV < 1 (which is the practical case - see Subsection 2.2.4), the throughput of a serial line with Weibull, gamma, and log-normal distributions of up- and downtime is a monotonically decreasing function of CV . Moreover, this function is practically linear. • Based on this observation, empirical formulas are introduced, which can be used for T P evaluation in serial lines with any unimodal pdf of up- and downtime.

12.5

Problems

Problem 12.1 Consider a two-machine synchronous non-exponential line defined by assumptions (a)-(e) of Subsection 11.1.1. (a) Assume e1 = 0.95, e2 = 0.6, Tdown,1 = 2, Tdown,2 = 6, and CVave = 0.5. d Calculate and plot P R as a function of N for N from 2 to 60. (b) Based on this plot, determine a reasonable buffer capacity for this system. (c) Compare the result with that obtained in part (a) of Problem 11.1 and formulate a hypothesis on how lean buffering depends on CVave . Problem 12.2 Consider a production line with two identical exponential machines defined by Tup = 9min,

Tdown = 1min,

c = 1part/min,

and buffer capacity N = 1. Assume that pdf’s of up- and downtime can be modified by preventative maintenance, so that CV s are less than 1. How much should the CV s (of either up- or downtime or both) be decreased so that P R is increased by 10%? Problem 12.3 Consider the exponential model of the production system of Figure 3.38 defined in Problem 3.3. Assume that Tup and Tdown of each machine remain the same but the reliability model of each machine is no longer exponential, having CVup,i = CVdown,i = 0.5, i = 1, . . . , 5. (a) Calculate the production rate of this system. (b) Compare the production rate with that obtained under the exponential assumption (see Problem 11.12) and suggest managerial actions that could lead to decreased CVup and CVdown .

410

CHAPTER 12. ANALYSIS OF NON-EXPONENTIAL LINES

Problem 12.4 Repeat Problem 12.3 for the exponential line defined in Problem 3.4 (compare the results with those of Problem 11.13). Problem 12.5 Derive an empirical formula for P R of a serial line with identical machines and buffers, assuming that CVup = CVdown ∈ [0, 3].

12.6

Annotated Bibliography

Using Queueing Theory, performance of non-exponential serial lines has been analyzed, for instance, in [12.1] J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993. [12.2] H. Tempelmeier and M. Burger, “Performance Evaluation of Unbalanced Flow Lines with General Distributed Processing Times, Failures and Imperfect Production,” IIE Transactions, vol. 33, pp. 293-302, 2001. The material of this chapter is based on [12.3] J. Li and S.M. Meerkov, “Evaluation of Throughput in Serial Production Lines with Non-exponential Machines,” in Analysis, Control and Optimization of Complex Dynamic Systems, ed. E.K. Boukas and R. Malhame, pp. 55-82, Kluwer Academic Publishers, 2005.

Chapter 13

Improvement of Continuous Lines Motivation The motivation here is the same as that of Chapter 5: the need for rigorous analytical methods for design of continuous improvement projects with predictable results. In this chapter, such methods are developed for synchronous and asynchronous serial lines with machines having continuous time models of machine reliability.

Overview Both constrained and unconstrained improvability problems are addressed. In the case of the former, improvability with respect to buffer capacity and cycle time re-allocations are considered; in the latter, bottlenecks with respect to uptime, downtime, and cycle time are analyzed. For each of these cases, improvability indicators are formulated and continuous improvement procedures are introduced.

13.1

Constrained Improvability

Production lines considered in this chapter are shown in Figures 13.1 - 13.3. In each of these systems, the machines are characterized by their average up- and downtime, Tup,i and Tdown,i , and their cycle time, τi ; in addition, in the case of the general reliability model, the distributions of up- and downtime, ftup,i (t) and ftdown,i (t), are involved. Each buffer is characterized by its capacity, Ni . The conventions of systems operation remain the same as in Chapters 11 and 12, i.e., assumptions (a)-(e) of Subsection 11.1.1 hold. 411

412

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

λ1, µ1

N1

λ2, µ 2

N2

m1

b1

m2

b2

λM−1, µ

M−1

mM−1

N M−1 λ M, µ M

bM−1

mM

Figure 13.1: M -machine synchronous exponential production line

λ1, µ1 , τ1

m1

N1 λ2, µ 2 , τ2

b1

m2

N2

b2

λM−1, µ

M−1,

τ M−1 N M−1λ M, µ M, τM

mM−1

bM−1

mM

Figure 13.2: M -machine asynchronous exponential production line

f t up,1

f t up,2

f t up,M

f t up,M−1

f t down,1 τ1

N1

f t down,2 τ2

N2

m1

b1

m2

b2

f t down,M f t down,M−1 N M−1 τM τM−1

mM−1

bM−1

mM

Figure 13.3: M -machine asynchronous general production line

13.1. CONSTRAINED IMPROVABILITY

13.1.1

413

Resource constraints and definitions

For the systems under consideration, direct analogues of constraints (5.1) and (5.2) are: M −1 X

Ni = N ∗ ,

(13.1)

i=1 M Y

ei = e ∗ ,

(13.2)

i=1

where, as before, ei =

Tup,i . Tup,i + Tdown,i

(13.3)

While (13.1) remains meaningful, constraint (13.2) is less justified because, as it follows from (13.3), ei is not an independent variable. Therefore, instead of (13.2), the following constraints may be considered: M X

∗ Tup,i = Tup ,

i=1 M X

∗ Tdown,i = Tdown .

(13.4)

i=1

However, in practice Tup,i and Tdown,i are difficult or even impossible to “reallocate” from one machine to another. Therefore, constraints (13.4) are not pursued here. Instead, we consider the cycle time constraint defined as M X

τi = τ ∗ ,

(13.5)

i=1

where τ ∗ is a given constant. This implies that the total processing time of a part by all machines in the system is τ ∗ . Improvability with respect to the cycle time would mean, then, that some of the machines could be sped-up and others slowed-down so that the performance of the system as a whole is improved, while the total processing time remains constant. In practice, such a re-allocation is often possible since the speed of most machining and assembly operations can be adjusted relatively easily, typically within ±10% of their nominal values. In the case of synchronous lines, this would mean that they may be “de-synchronized”, if this leads to improved performance. Based on the above arguments, this section considers constrained improvability with respect to buffer capacity (BC) and cycle time (CT) defined as follows: Definition 13.1 A serial production line with a continuous time model of machine reliability is:

414

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES 0 • improvable with respect to BC if there exists a sequence N10 , . . ., NM −1 such that M −1 X

Ni0 = N ∗

i=1

and 0 T P (N10 , . . . , NM −1 ) > T P (N1 , . . . , NM −1 );

(13.6)

0 such • improvable with respect to CT if there exists a sequence τ10 , . . ., τM that M X

τi0 = τ ∗

i=1

and 0 T P (τ10 , . . . , τM ) > T P (τ1 , . . . , τM ).

(13.7)

(For the sake of brevity, expressions (13.6) and (13.7) include only the variables that are being adjusted for improvability.) Analytical proofs of the criteria for improvability of serial lines with continuous time models of machine reliability are all but impossible to derive (due to the complexity of the expressions for their T P – see Chapter 11). Therefore, motivated by similar results for Bernoulli machines described in Chapter 5, we introduce heuristic formulations of these criteria and then justify them by calculations and simulations.

13.1.2

Improvability with respect to CT

CT-Improvability Indicator 13.1: A serial production line with machines having their reliability models in the set {exp, W, ga, LN} is unimprovable with respect to CT if each buffer is, on average, close to being half full. Based on this indicator, introduce

13.1. CONSTRAINED IMPROVABILITY

415

CT-Continuous Improvement Procedure 13.1: (1) By calculations or by measurements, evaluate the buffer occupancy, W IPi , i = 1, . . ., M − 1. (2) Determine the buffer for which |W IPi − Ni /2| is the largest. Assume this is buffer k. (3) If W IPk − Nk /2 is positive, re-allocate a small fraction of cycle time, ∆τ , from mk+1 to mk ; if negative, re-allocate from mk to mk+1 . (4) Return to step (1). (5) Continue this process until maxi=1,...,M −1 |W IPi −Ni /2| is sufficiently small. Using this procedure, we carry out Justification of CT-Improvability Indicator 13.1: For exponential lines, this indicator is justified by calculations (using recursive aggregation procedure (11.46)) and for non-exponential lines by simulations (using Simulation Procedure 4.1). In either case, the parameters of the machines and buffers have been selected randomly and equiprobably from the following sets Tdown,i ∈ [5, 15], ei ∈ [0.60, 0.95], Ni ∈ {8, 9, 10, 11, 12}, M ∈ {2, ..., 5}, τ ∗ = M. For non-exponential lines, additional parameters have been selected from fup,i , fdown,i CVup,i , CVdown,i

∈ {W, ga, LN, exp}, ∈ [0.2, 1].

Examples of production lines, thus formed, are shown in Tables 13.1-13.3. Table 13.1: Examples of exponential two-machine lines line 1 2 3 4 5 6 7 8 9 10

Tdown,1 9.0571 6.2105 7.5477 5.1527 9.1865 10.9153 14.3424 11.4583 9.5735 11.2731

Tdown,2 14.3547 9.5075 13.6560 12.4679 13.4622 6.1975 7.6445 14.6689 9.5069 11.9908

e1 0.9209 0.8506 0.6813 0.7558 0.7838 0.6133 0.6561 0.8327 0.7443 0.7390

e2 0.7436 0.9125 0.8817 0.9261 0.6709 0.7605 0.9055 0.9046 0.9156 0.7448

N 10 9 12 8 9 12 9 8 8 11

416

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

Table 13.2: Examples of non-exponential five-machine lines Line 1

2

3

4

5

i 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

fup,i exp LN ga LN exp LN W W LN ga W W W ga exp LN exp exp W exp LN exp LN LN exp

CVup,i 1 0.3093 0.6549 0.7256 1 0.5815 0.9049 0.5657 0.6877 0.4799 0.9008 0.9912 0.6037 0.4175 1 0.5304 1 1 0.8826 1 0.8901 1 0.7500 0.7803 1

fdown,i W W exp LN LN LN W LN W W ga exp exp exp exp LN ga ga W LN exp LN ga exp LN

CVdown,i 0.4825 0.2819 1 0.6059 0.3797 0.6455 0.8062 0.8767 0.3263 0.6014 0.6693 1 1 1 1 0.7327 0.6717 0.8811 0.6796 0.2921 1 0.9212 0.6525 1 0.4667

Tdown,i 7.1518 13.5381 9.8212 14.9183 14.4976 7.3337 11.6122 9.9696 12.0850 6.7101 5.8872 7.0249 11.2444 8.2783 7.4815 5.9019 10.0097 6.2931 11.6755 14.8140 9.7771 11.6542 13.1047 11.8892 10.2316

ei 0.7558 0.6606 0.8251 0.7732 0.6018 0.7088 0.7192 0.6021 0.6295 0.7475 0.6963 0.8819 0.7414 0.7662 0.7695 0.9040 0.7870 0.6391 0.6385 0.9062 0.9013 0.7576 0.8582 0.9168 0.9146

Ni 9 10 9 9 10 9 12 12 9 11 9 11 12 10 8 11 11 9 8 8 -

13.1. CONSTRAINED IMPROVABILITY

417

Table 13.3: Examples of non-exponential five-machine lines (cont.) Line 6

7

8

9

10

i 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

fup,i ga W W LN LN LN LN LN exp ga LN LN ga LN W W ga ga LN exp W W LN LN exp

CVup,i 0.9627 0.8345 0.2034 0.7935 0.7688 0.3430 0.5469 0.5438 1 0.6858 0.2739 0.2686 0.7471 0.8232 0.8045 0.6782 0.2417 0.7362 0.6096 1 0.2522 0.3015 0.8226 0.3720 1

fdown,i ga LN W exp LN exp exp ga LN ga exp ga exp exp ga LN W W exp exp LN LN exp ga exp

CVdown,i 0.5137 0.7744 0.4894 1 0.3289 1 1 0.3047 0.9525 0.8286 1 0.3830 1 1 0.6328 0.4681 0.2227 0.8836 1 1 0.2597 0.7171 1 0.9374 1

Tdown,i 8.0893 11.7826 8.4346 12.0255 11.9677 12.1406 8.1310 5.5005 7.9897 11.8815 13.7011 14.7001 8.4040 13.6312 10.7755 7.4056 5.0504 5.0268 13.2138 6.8641 13.3459 8.2771 13.2628 8.2221 12.8257

ei 0.7525 0.8013 0.7184 0.7683 0.7499 0.7576 0.7540 0.7481 0.6998 0.7705 0.6370 0.6986 0.8633 0.6883 0.9015 0.8688 0.8649 0.7387 0.8867 0.7909 0.6406 0.6389 0.6520 0.9192 0.7916

Ni 10 10 9 10 10 9 10 9 12 10 10 9 9 9 12 11 10 10 10 11 -

418

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

For each of the lines considered, CT-Continuous Improvement Procedure 13.1 has been carried out and τiunimp along with T P unimp have been obtained. To evaluate the accuracy of these allocations, a full search in τ ∗ (with ∆τ = 0.001) has been carried out to determine τiopt and T P opt . As measures of accuracy, the following have been used: ¯ ¯ ¯ τ unimp − τ opt ¯ ¯ i ¯ i (13.8) ∆τ = max ¯ ¯ · 100%, opt i=1,...,M ¯ ¯ τi ¯ ¯ ¯ T P unimp − T P opt ¯ ¯ ¯ (13.9) ∆T P = ¯ ¯ · 100%. ¯ ¯ T P opt The results are illustrated in Tables 13.4 and 13.5. In all cases analyzed, the unimprovable allocations were within 5% of the optimal ones. Thus, we consider CT-Improvability Indicator 13.1 justified. Table 13.4: Accuracy of CT-Improvability Indicator 13.1 for two-machine exponential lines Line 1 2 3 4 5 6 7 8 9 10

T P opt 0.7260 0.8170 0.6811 0.7568 0.5932 0.5711 0.6448 0.7814 0.7257 0.6249

τ1opt 1.0390 0.9742 0.8865 0.9177 1.0080 0.9466 0.9385 0.9838 0.9499 1

τ2opt 0.9610 1.0258 1.1135 1.0823 0.9920 1.0534 1.0615 1.0162 1.0501 1

T P unimp 0.7241 0.8169 0.6810 0.7567 0.5902 0.5694 0.6402 0.7809 0.7245 0.6245

τ1unimp 1.0593 0.9694 0.8815 0.9128 1.0497 0.9092 0.8958 0.9757 0.9336 0.9951

τ2unimp 0.9407 1.0306 1.1185 1.0872 0.9503 1.0908 1.1042 1.0243 1.0664 1.0049

∆T P 0.2586 0.0196 0.0091 0.0136 0.5065 0.3117 0.7099 0.0620 0.1598 0.0663

∆τ 2.1124 0.4927 0.5640 0.5339 4.2036 3.9510 4.5498 0.8233 1.7160 0.4900

An illustration of this indicator and CT-Continuous Improvement Procedure 13.1 is given in Tables 13.6 and 13.7 for a five-machine exponential line with N = [10, 10, 10, 11] and τ ∗ = 5. Initially, the cycle time of the machines was allocated according to a decreasing pattern. Using CT-Continuous Improvement Procedure 13.1, the unimprovable allocation was obtained as a bowl, resulting in T P = 0.5149, whereas the initial T P was 0.4381, i.e., a 17% improvement. As far as the general, G, machine reliability model is concerned, the following conjecture is formulated: CT-Improvability Indicator 13.1 and CT-Continuous Improvement Procedure 13.1 can be used for serial production lines with any unimodal distribution of up- and downtime.

13.1.3

Improvability with respect to BC

BC-Improvability Indicator 13.1: A serial production line with machines having their reliability models in the set {exp, W, ga, LN} is unimprovable with

13.1. CONSTRAINED IMPROVABILITY

419

Table 13.5: Accuracy of CT-Improvability Indicator 13.1 for five-machine nonexponential lines Line 1 2 3 4 5 6 7 8 9 10

T P opt 0.5075 0.5003 0.5782 0.5409 0.6702 0.5750 0.5702 0.5376 0.6942 0.5199

T P unimp 0.4989 0.4962 0.5630 0.5507 0.6570 0.5610 0.5438 0.5178 0.6910 0.5149

∆T P 1.6946 0.8195 2.6288 1.8118 1.9696 2.4348 4.6300 3.6830 0.4610 0.9617

respect to BC if the buffer capacity allocation is such that max |W IPi−1 − (Ni − W IPi )| i

is minimized over all sequences N1 , . . ., NM −1 such that

PM i=1

Ni = N ∗ .

Based on this indicator, introduce BC-Continuous Improvement Procedure 13.1: (1) By calculations or measurements, evaluate the average occupancy of each buffer, W IPi , i = 1, . . . , M − 1. (2) Determine the buffer for which |W IPi−1 − (Ni − W IPi )| is the largest. Assume this is buffer k. (3) If W IPi−1 − (Ni − W IPi ) is positive, transfer a unit of buffer capacity from bk to bk+1 ; if W IPi−1 − (Ni − W IPi ) is negative, re-allocate a unit of buffer capacity from bk+1 to bk . (4) Return to step (1). (5) Continue this process until arriving at a limit cycle (i.e., when Ni ’s repeat themselves) and choose the buffer capacity allocation on the limit cycle, which maximizes T P . Using this procedure, we carry out Justification of BC-Improvability Indicator 13.1: The approach of Subsection 13.1.2 is used again except that the full search is carried out with respect to Ni ’s, rather than τi ’s. The buffer capacity allocations, obtained using BCImprovability Indicator 13.1 and the full search, are denoted as Niunimp and

420

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

Table 13.6: Illustration of CT-Continuous Improvement Procedure 13.1 W IP1 5.40 5.41 5.28 5.30 5.21 5.17 5.15 5.14 5.13 5.11 5.13 5.13 5.16 5.15 5.15 5.17 4.52 3.96 3.97 3.54 3.22 4.21 3.82 3.81 3.92 5.11 5.06 5.02 4.96 4.93 4.88 4.84 4.79 4.77 4.75 4.70 4.70

W IP2 4.04 4.04 3.57 3.58 3.21 2.98 2.97 2.79 2.76 2.63 2.63 2.52 2.56 2.46 2.40 2.48 2.68 2.89 2.82 3.02 3.28 3.54 3.94 3.80 3.95 4.19 4.23 4.30 4.35 4.41 4.44 4.50 4.54 4.62 4.58 4.63 4.61

W IP3 0.94 0.87 1.01 0.93 1.06 1.55 1.27 1.74 1.48 1.92 1.72 2.09 2.00 2.32 2.69 2.63 2.74 2.84 3.30 3.35 3.34 3.69 3.67 4.31 4.23 4.57 4.59 4.62 4.64 4.63 4.63 4.64 4.62 4.64 4.63 4.60 4.69

W IP4 1.35 1.52 1.49 1.71 1.68 1.66 1.91 1.87 2.21 2.16 2.72 2.50 3.30 3.09 2.82 3.80 4.00 4.14 3.82 3.87 3.90 4.33 4.38 3.93 5.43 5.86 5.88 5.92 5.96 5.94 5.96 5.96 5.96 6.00 5.84 5.82 5.80

τ1 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.249 1.149 1.149 1.149 1.149 1.049 1.049 1.049 1.049 1.049 1.049 1.049 1.049 1.049 1.049 1.049 1.049

τ2 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.160 1.060 0.960 0.960 0.860 0.760 0.860 0.760 0.760 0.760 0.860 0.850 0.840 0.830 0.820 0.810 0.800 0.790 0.780 0.780 0.770 0.770

τ3 1.122 1.122 1.022 1.022 0.922 0.822 0.822 0.722 0.722 0.622 0.622 0.522 0.522 0.422 0.322 0.322 0.422 0.522 0.422 0.522 0.622 0.622 0.722 0.622 0.622 0.622 0.632 0.642 0.652 0.662 0.672 0.682 0.692 0.702 0.702 0.712 0.702

τ4 0.926 0.826 0.926 0.826 0.926 1.026 0.926 1.026 0.926 1.026 0.926 1.026 0.926 1.026 1.126 1.026 1.026 1.026 1.126 1.126 1.126 1.126 1.126 1.226 1.126 1.126 1.126 1.126 1.126 1.126 1.126 1.126 1.126 1.126 1.136 1.136 1.146

τ5 0.543 0.643 0.643 0.743 0.743 0.743 0.843 0.843 0.943 0.943 1.043 1.043 1.143 1.143 1.143 1.243 1.243 1.243 1.243 1.243 1.243 1.243 1.243 1.243 1.343 1.343 1.343 1.343 1.343 1.343 1.343 1.343 1.343 1.343 1.333 1.333 1.333

TP 0.4381 0.4377 0.4425 0.4424 0.4460 0.4476 0.4481 0.4481 0.4491 0.4497 0.4495 0.4494 0.4484 0.4484 0.4487 0.4471 0.4555 0.4615 0.4613 0.4644 0.4665 0.4862 0.4886 0.4895 0.4846 0.4998 0.4998 0.4999 0.5008 0.5009 0.5018 0.5015 0.5021 0.5016 0.5028 0.5035 0.5033

13.1. CONSTRAINED IMPROVABILITY

421

Table 13.7: Illustration of CT-Continuous Improvement Procedure 13.1 (cont.) W IP1 4.65 4.62 4.75 4.73 4.74 4.69 4.82 4.78 4.78 4.91 4.89 4.85 4.97 4.94 4.96

W IP2 4.65 4.70 4.74 4.73 4.73 4.77 4.80 4.86 4.84 4.86 4.83 4.89 4.93 5.00 4.99

W IP3 4.66 4.66 4.71 4.69 4.77 4.75 4.80 4.78 4.86 4.91 4.89 4.88 4.93 4.91 4.98

W IP4 5.76 5.78 5.85 5.66 5.62 5.62 5.69 5.65 5.62 5.67 5.52 5.51 5.56 5.55 5.49

τ1 1.049 1.049 1.039 1.039 1.039 1.039 1.029 1.029 1.029 1.019 1.019 1.019 1.009 1.009 1.009

τ2 0.760 0.750 0.760 0.760 0.760 0.750 0.760 0.750 0.750 0.760 0.760 0.750 0.760 0.750 0.750

τ3 0.712 0.722 0.722 0.722 0.712 0.722 0.722 0.732 0.722 0.722 0.722 0.732 0.732 0.742 0.732

τ4 1.146 1.146 1.146 1.156 1.166 1.166 1.166 1.166 1.176 1.176 1.186 1.186 1.186 1.186 1.196

τ5 1.333 1.333 1.333 1.323 1.323 1.323 1.323 1.323 1.323 1.323 1.313 1.313 1.313 1.313 1.313

TP 0.5039 0.5039 0.5047 0.5058 0.5055 0.5062 0.5074 0.5069 0.5074 0.5089 0.5095 0.5096 0.5107 0.5107 0.5149

Niopt , respectively, while the corresponding production rates are T P unimp and T P opt . The accuracy of the indicator is quantified in terms of ¯ ¯ ¯ T P unimp − T P opt ¯ ¯ ¯ (13.10) ∆T P = ¯ ¯ · 100%. ¯ ¯ T P opt The machine characteristics are chosen from the sets fup,i , fdown,i CVup,i , CVdown,i Tdown,i ei

∈ {exp, W, ga, LN }, i = 1, . . . , 5, ∈ [0.2, 1], i = 1, . . . , 5, ∈ [5, 15], i = 1, . . . , 5, ∈

[0.55, 0.9],

i = 1, . . . , 5.

Examples of five-machine lines, thus formed, are shown in Tables 13.8 and 13.9. For each of these lines, the total buffer capacity was selected as N ∗ = 40.

The results obtained are shown in Table 13.10. Clearly, BC-Improvability Indicator 13.1 leads to near-optimal buffer capacity allocations. Thus, we consider BC-Improvability Indicator 13.1 justified. To extend the above conclusion to the general machine reliability model, the following conjecture is formulated:

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CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

Table 13.8: Examples of non-exponential five-machine lines Line 1

2

3

4

5

i 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

fup,i W LN LN LN ga W W ga exp W LN W exp W LN LN exp ga exp exp ga ga exp exp LN

CVup,i 0.6039 0.4710 0.5337 0.7313 0.9756 0.2757 0.7803 0.2000 1 0.4197 0.5160 0.4301 1 0.3231 0.6939 0.4501 1 0.2325 1 1 0.3038 0.2776 1 1 0.7272

fdown,i LN ga W LN W ga exp ga W ga ga exp LN W LN exp W LN W exp exp ga ga exp ga

CVdown,i 0.8776 0.8442 0.8306 0.5625 0.2070 0.8722 1 0.4855 0.3779 0.5051 0.6645 1 0.3658 0.9833 0.7332 1 0.2281 0.3670 0.3368 1 1 0.3081 0.8797 1 0.5039

Tdown,i 5.5421 7.2775 10.9987 8.1183 5.2817 6.3129 6.5742 13.7374 9.9440 5.7966 9.0213 13.3912 14.9227 13.2212 14.0609 7.9909 5.1004 13.1779 11.2724 8.9908 8.2323 12.9370 10.5801 9.5964 11.4333

ei 0.7187 0.8406 0.7913 0.7419 0.5747 0.6700 0.9294 0.7316 0.5501 0.9293 0.8148 0.6511 0.8979 0.6001 0.7638 0.8214 0.7080 0.7916 0.6076 0.6075 0.5980 0.6775 0.8697 0.7087 0.7193

Ni 1 1 1 37 1 1 19 19 1 1 19 19 1 1 1 37 37 1 1 1 -

13.1. CONSTRAINED IMPROVABILITY

423

Table 13.9: Examples of non-exponential five-machine lines (cont.) Line 6

i 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

7

8

9

10

fup,i LN exp W W LN ga W exp LN LN ga ga W ga ga LN exp exp LN W LN exp W exp W

CVup,i 0.9261 1 0.8824 0.6162 0.9431 0.8113 0.8571 1 0.7246 0.3233 0.2811 0.5989 0.2322 0.9558 0.8231 0.6844 1 1 0.2234 0.5345 0.2634 1 0.6076 1 0.8402

fdown,i W ga exp W LN W Ga LN Ga exp exp LN W exp LN exp Ga exp exp W exp exp LN LN exp

CVdown,i 0.7585 0.3701 1 0.3185 0.9746 0.2739 0.2524 0.2965 0.5987 1 1 0.6521 0.9889 1 0.3146 1 0.5187 1 1 0.7578 1 1 0.5718 0.4914 1

Tdown,i 11.9306 12.1931 10.7019 8.6013 5.5913 12.0217 12.8855 12.8081 5.5566 10.4924 8.8016 13.1424 7.5795 11.9192 12.8043 11.4923 13.2304 14.7083 11.7865 12.2501 13.6322 13.4672 8.5633 10.6609 7.8958

ei 0.8868 0.8921 0.8922 0.7186 0.8698 0.6670 0.6443 0.5745 0.9269 0.7000 0.5500 0.7613 0.5942 0.6543 0.8907 0.8900 0.5529 0.8941 0.8665 0.8188 0.7782 0.5604 0.7672 0.5709 0.9324

Table 13.10: Accuracy of BC-Improvability Indicator 13.1 Line 1 2 3 4 5 6 7 8 9 10

N unimp [5, 7, 11, 17] [5, 7, 17, 11] [7, 9, 13, 11] [4, 8, 13, 15] [12, 11, 9, 8] [4, 7, 16, 13] [10, 15, 11, 4] [11, 12, 12, 5] [14, 16, 7, 3] [9, 14, 11, 6]

T P unimp 0.5511 0.5218 0.4946 0.4930 0.5141 0.6706 0.4780 0.4548 0.5104 0.4403

N opt [5, 7, 11, 17] [9, 6, 18, 7] [6, 8, 16, 8] [4, 8, 13, 15] [12, 11, 8, 9] [4, 7, 16, 13] [10, 16, 9, 5] [11, 11, 17, 1] [12 20 3 5] [10, 14, 14, 2]

T P opt 0.5511 0.5300 0.4961 0.4930 0.5142 0.6706 0.4786 0.4569 0.5122 0.4463

∆T P 0 1.55 0.30 1.34 0.02 0 0.13 0.46 0.35 1.34

Ni 1 1 19 19 1 19 19 1 37 1 1 1 19 19 1 1 19 19 1 1 -

424

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

BC-Improvability Indicator 13.1 and BC-Continuous Improvement Procedure 13.1 can be used for serial production lines with any unimodal distribution of up- and downtime. Concluding this subsection, we compare the buffer capacity allocation obtained by BC-Improvability Indicator 13.1 to Goldratt’s approach. According to this approach, all available buffer capacity is allocated in front of the machine with the smallest efficiency. The corresponding production rate is denoted as T PG . We compare the unimprovable and Goldratt’s buffer capacity allocation by T P unimp − T PG · 100%, ∆G = T PG using the ten lines defined in Tables 13.8 and 13.9. The results, given in Table 13.11, show that BC-Improvability Indicator 13.1 leads to production rates from 3% to over 30% higher than those of Goldratt’s approach. Table 13.11: Comparison with Goldratt’s approach Line 1 2 3 4 5 6 7 8 9 10

T P unimp 0.5511 0.5218 0.4946 0.4930 0.5141 0.6706 0.4780 0.4548 0.5104 0.4403

T PG 0.4204 0.4616 0.4334 0.3338 0.3508 0.6234 0.3873 0.2969 0.3834 0.2684

∆G 23.78 11.79 8.85 32.28 31.80 2.90 12.18 34.71 3.05 11.75

The operation of BC-Continuous Improvement Procedure 13.1 is illustrated in Table 13.12 using Line 1 of Table 13.8 starting from Goldratt’s buffer capacity allocation. Clearly, it results in the production rate 23% higher than the initial one.

13.2

Unconstrained Improvability

In this section, we again consider the serial production lines shown in Figures 13.1 - 13.3 and address the issues of bottleneck machine and bottleneck buffer identification.

13.2.1

Definitions

Bottleneck machines: Since, unlike the Bernoulli case, each machine is defined by three independent variables, i.e., Tup,i , Tdown,i , ci , different types of

13.2. UNCONSTRAINED IMPROVABILITY

425

Table 13.12: Illustration of BC-Continuous Improvement Procedure 13.1 Step 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

N1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 3 4 4 4 5 5 5 5

N2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 4 5 5 6 7 7 8 7 6 7 7 6 7 7 6 7 7 7

N3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 14 13 14 13 12 13 12 11 12 11 11 11 10 11 11 10 11 11 10 11 10

N4 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 23 23 22 22 22 21 21 21 20 20 20 20 20 19 19 19 18 18 18 17 18

T P unimp 0.4204 0.4378 0.4511 0.4636 0.4719 0.4799 0.4854 0.4883 0.4916 0.4954 0.4968 0.4980 0.4994 0.5020 0.5004 0.5123 0.5190 0.5186 0.5261 0.5296 0.5307 0.5352 0.5378 0.5376 0.5390 0.5441 0.5443 0.5458 0.5461 0.5466 0.5492 0.5493 0.5507 0.5502 0.5511 0.5498

426

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

bottlenecks are possible. Definition 13.2 Machine mi , i ∈ {1, . . . , M }, is: • uptime bottleneck (UT-BN) if ∂T P ∂T P > , ∂Tup,i ∂Tup,j

∀j 6= i;

(13.11)

• downtime bottleneck (DT-BN) if ¯ ∂T P ¯ ¯ ∂T P ¯ ¯ ¯ ¯ ¯ ¯ ¯>¯ ¯, ∂Tdown,i ∂Tdown,j

∀j 6= i;

(13.12)

• bottleneck (BN-m) if it is simultaneously UT-BN and DT-BN; • c-bottleneck (c-BN) if ∂T P ∂T P > , ∂ci ∂cj

∀j 6= i.

(13.13)

Thus, mi is UT-BN if increasing its uptime results in the largest increase of the T P ; DT-BN if decreasing its downtime leads to the largest increase of T P ; BN-m if these properties take place simultaneously; and c-BN if decreasing its cycle time leads to the largest increase of T P , compared to decreasing the cycle time of any other machine in the system. In addition, the following notion is useful: Definition 13.3 Let mi be a BN-m. Then it is referred to as the uptime preventative maintenance BN (UTPM-BN) if ¯ ∂T P ¯ ∂T P ¯ ¯ >¯ ¯; ∂Tup,i ∂Tdown,i

(13.14)

if the inequality is reversed, it is referred to as the downtime preventative maintenance bottleneck (DTPM-BN). In other words, if the machine is UTPM-BN, it is more effective to increase its uptime than to decrease its downtime; the opposite is true for DTPM-BN. Machines with the smallest Tup or largest Tdown are not necessarily the bottlenecks (in the sense of Definition 13.2). Examples are given in Figures 13.4 and 13.5, where all machines have ci = 1, the numbers in the circles are the machine efficiency, ei , and the numbers in the rectangles are the buffer capacity, Ni . (In these¯ and the¯ subsequent figures the estimate of partial derivatives, i.e., ¯ ∆T P ¯ ∆T P ∆Tup,i and ¯ ∆Tdown,i ¯, are obtained by simulation.) Also, machines with the smallest efficiency, Tup,i ei = , Tup,i + Tdown,i

13.2. UNCONSTRAINED IMPROVABILITY

427

or the smallest capacity, ci , or the smallest throughput in isolation, T P i = ci

Tup,i , Tup,i + Tdown,i

are not necessarily the bottlenecks (see Figures 13.6 - 13.8, respectively). Therefore, the identification of bottlenecks is a non-trivial problem, since none of the partial derivatives of Definition 13.2 can be easily evaluated on the factory floor. Thus, indirect methods, based on “measurable” or “calculable” quantities, are necessary. Such methods are described in this section. UTBN m1

b1

m2

b2

m3

b3

m4

0.85

13

0.8

33

0.9

33

0.85

T up,i

66.667

50

20

66.667

Tdown,i

11.765

12.5

2.222

11.765

0.002

0.0036

0.0017

0.0016

∆ TP ∆ Tup,i

Figure 13.4: Machine with the shortest Tup,i is not necessarily UT-BN

DTBN m1

b1

m2

b2

m3

b3

m4

0.8

3

0.8

3

0.8

3

0.8

T up,i

10

5

12.5

20

Tdown,i

2.5

1.25

3.125

5

0.0307

0.061

0.0297

0.0192

∆ TP ∆ Tdown,i

Figure 13.5: Machine with the longest Tdown,i is not necessarily DT-BN

Bottleneck buffers: Bottleneck buffers in production lines with continuous time models of machine reliability are defined similarly to those in the Bernoulli case:

428

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES UTBN DTBN m1

b1

m2

b2

m3

b3

m4

0.85

13

0.85

13

0.9

13

0.85

T up,i

66.667

51

20

66.667

Tdown,i

11.765

9

2.222

11.765

0.0008

0.0013

0.0017

0.0005

0.006

0.0105

0.0197

0.0041

∆ PR ∆ Tup,i ∆ PR ∆ Tdown,i

Figure 13.6: Machine with the smallest efficiency is not necessarily BN-m c−BN m1

b1

m2

b2

m3

b3

m4

b4

2.53

100

2.592

80

2.732

150

2.886

100

m5

b5

m6

2.572

100

2.581

Tup,i

30

35

14

40

36

14

Tdown,i

4

10

6.5

8.5

12

3.5

ci

2.8761

3.3333

4.0

3.5

3.43

3.2258

0.079

0.1142

0.067

0.0292

0.085

0.0613

∆ PR ∆ ci

Figure 13.7: Machine with the smallest capacity is not necessarily c-BN c−BN m1

b1

m2

b2

m3

b3

m4

0.7

6

0.69

3

0.9

9

0.681

12

0.8

6

0.682

T up,i

3.333

5.714

1.786

6.667

5

2.857

Tdown,i

1.429

2.564

0.694

3.125

1.25

1.333

ci

1

1

1.25

1

1

1

0.7

0.69

0.9

0.681

0.8

0.682

0.1177

0.0181

0.1068

0.0139

0.0503

c ie

i

∆ TP ∆c

i

0.0824

Figure 13.8: Machine with the smallest throughput in isolation is not necessarily c-BN

13.2. UNCONSTRAINED IMPROVABILITY

429

Definition 13.4 Buffer bi , i ∈ {1, . . . , M − 1}, is the bottleneck buffer (BNb) of a serial line if T P (N1 , . . . , Ni + 1, . . . , NM −1 ) > T P (N1 , . . . , Nj + 1, . . . , NM −1 ), ∀j 6= i. (13.15) As in the Bernoulli lines, the buffer with the smallest capacity is not necessarily BN-b (see Figure 13.9). A method for BN-b identification is also described in this section. BN−b

T up,i T

down,i

ci PR(N i+1)

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

0.7

6

0.74

3

0.9

6

0.681

4

0.8

4

m6 0.682

0.3

0.175

0.56

0.15

0.2

0.35

0.7

0.49

1.44

0.32

0.8

0.75

1

1

1.25

1

1

1

0.5745

0.5758

0.5756

0.5780

0.5768

Figure 13.9: Buffer with the smallest capacity is not necessarily BN-b

13.2.2

One-machine lines

In the case of a single-machine production system, the only applicable notion of bottleneck is that of UTPM- vs. DTPM-BN. When either of them takes place is characterized below. Theorem 13.1 A single machine with any reliability model is UTPM-BN if Tup < Tdown and DTPM-BN if Tup > Tdown . Proof: See Section 20.2. Since for a single machine with any reliability model TP = c

Tup , Tup + Tdown

the quantities involved in Definition 13.3 can be easily calculated: ¯ ∂T P ¯ Tup ¯ ¯ , ¯ = c ¯ ∂Tdown (Tup + Tdown )2 Tdown ∂T P = c , ∂Tup (Tup + Tdown )2 leading to the conclusion of Theorem 13.1.

430

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

Since in most practical situations, Tup > Tdown , the above theorem implies that it is more beneficial to focus preventative maintenance efforts on decreasing machine’s downtime than increasing its uptime, irrespective of the model of machine reliability.

13.2.3

Two-machine synchronous exponential lines

Machines with identical Tup and Tdown : Consider a synchronous line with two exponential machines satisfying the conditions: Tup,i Tdown,i

=: Tup , i = 1, 2, =: Tdown , i = 1, 2.

(13.16)

Theorem 13.2 In a two-machine exponential production line satisfying (13.16), both machines are DTPM-BNs if e > 0.5

(i.e., Tdown < Tup )

and both machines are UTPM-BN if e < 0.4315

(i.e., Tdown > 1.3175Tup ).

Proof: See Section 20.2. As it has been shown above, in the case of a single-machine production system, the threshold for switching between UTPM-BN and DTPM-BN is 0.5. In the case of a two-machine exponential line satisfying (13.16), this threshold depends on the buffer capacity N , i.e., has the form f (N ). Theorem 13.2 states, therefore, that 0.4315 < f (N ) < 0.5. (13.17) When N tends to infinity, f (N ) tends to 0.5. The minimum threshold occurs for N = 0.8Tdown , which results in the lower bound of (13.17). Thus, in two-machine lines with machines having identical average up-time and downtime, the primary attention of the preventative maintenance should be given to decreasing the downtime if Tdown < Tup ; if Tup < 0.759Tdown , attention should be concentrated on increasing the uptime. Machines with identical efficiencies: Consider now the case when the exponential machines may have non-identical up- and downtime but identical efficiencies, i.e., i = 1, 2. (13.18) ei =: e, Theorem 13.3 In a two-machine exponential production line satisfying (13.18), the machine with the shortest average downtime (and, thus, the shortest average uptime) is the BN-m.

13.2. UNCONSTRAINED IMPROVABILITY

431

Proof: See Section 20.2. This conclusion is somewhat unexpected. Indeed, it has been shown in Chapter 11 that, given a fixed ratio Tup,i /Tdown,i , machines with longer upand downtime are more detrimental to the system’s production rate than those with shorter up- and downtime. In view of this fact, one might expect that the machine with the longer downtime is the BN-m. Theorem 13.3, however, states that the opposite is the case. The reason for this is that an improvement of the machine with a shorter downtime leads to a better utilization of the disturbance rejection capabilities of the buffer and, thus, to the largest increase of the T P . When is the BN-m, identified in Theorem 13.3, UTPM-BN and when is it DTPM-BN? The answer is as follows: Theorem 13.4 Assume that the BN-m, identified by Theorem 13.3 is machine mi , and for this machine Tdown,i < Tup,i . Then it is DTPM-BN. If its Tdown,i is sufficiently larger than Tup,i , specifically, k+1 Tdown,i , > Tup,i k where k=

Tup,j , Tup,i

j 6= i,

(13.19)

(13.20)

then it is UTPM-BN. Proof: See Section 20.2. This result is in agreement with Theorem 13.1: the smaller of up- and downtime of the bottleneck machine defines its nature as far as preventative maintenance is concerned. In most practical situations, machines’ efficiency is greater than 0.5. Therefore, taking into account Theorems 13.2 and 13.4, we concentrate below on DT-BN. Machines with non-identical efficiencies: Assume that a synchronous exponential line consists of two machines with non-equal efficiencies: e1 6= e2 . For this case, we formulate the following DT-BN Indicator 13.1: In synchronous exponential lines consisting of two machines with non-equal efficiencies, m1 is the DT-BN if BL1 Tup,1 Tdown,1 < ST2 Tup,2 Tdown,2 ;

(13.21)

if the inequality is reversed, the DT-BN is m2 . This indicator has been justified by calculations using expression (11.13). Two typical results are shown in Figure 13.10. Very few counterexamples have been found.

432

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

m1 0.9

DT−BN

DT−BN

b1

m2

m1

b1

m2

1.2

0.85

0.9

1.2

0.85

T up,i

5

6.667

T up,i

5

6.667

Tdown,i

1.11

1.176

Tdown,i

0.556

1.176

ST i T up,iTdown,i

0

0.364

ST i T up,iTdown,i

0

0.296

0

BLi T up,iTdown,i

BLi T up,iTdown,i ∆ TP ∆ Tdown,i

1.071 0.062

0.097

∆ TP ∆ Tdown,i

(a)

0.2438 0.116

0 0.097

(b)

Figure 13.10: DT-BN identification in synchronous exponential two-machine lines

13.2.4

Asynchronous exponential lines

Our practical experience indicates that c-BN’s are more important in applications than all other types of bottlenecks. Therefore, we concentrate below on this type of bottlenecks for M ≥ 2-machine lines. Consider a serial production line with exponential machines and assume that BLi and STi of all machines are measured during normal system operation. Alternatively, if they are not measured but the parameters λi , µi , ci , and Ni are c i , can be calculated usdi and ST known, the estimates of these probabilities, BL ing aggregation procedure (11.46). It turns out that these probabilities (or their estimates) can be used for c-BN and BN-b identification in the same manner as they are used in the case of Bernoulli machines. Specifically, (a) Assign arrows pointing from one machine to another using Arrow Assignment Rule 5.1, i.e., an arrow is pointing from mi to mi+1 if BLi > STi+1 and from mi+1 to mi if BLi < STi+1 (see Figures 13.11 - 13.13). (b) For every machine mj with no emanating arrows, calculate its severity, Sj S1 SM

= |STj+1 − BLj | + |STj − BLj−1 |, = |ST2 − BL1 |, = |STM − BLM −1 |.

j = 2, . . . , M − 1, (13.22)

di , the severity is denoted as Sbj . c i , BL In the case of calculated quantities ST c-BN Indicator 13.1: In an asynchronous exponential line with M ≥ 2machines,

13.2. UNCONSTRAINED IMPROVABILITY

433

• if there is a single machine with no emanating arrows, it is the c-BN of the system; • if there are multiple machines with no emanating arrows, the primary c-BN (i.e., Pc-BN) is the one with the largest severity; • the BN-b is one of the buffers surrounding Pc-BN; specifically, if Pc-BN is machine mi , then BN-b is bi if BLi > STi or bi−1 if BLi < STi . Justification: Carried out using the same approach as in the case of the Bernoulli lines. Both simulations and calculations have been used. The calc i and BL di , identifying c-BN and BN-b using culations consist of evaluating ST d c-BN Indicator 13.1, and then verifying the conclusions using ∆T P /∆ci and d T P (Ni + 1). The simulation approach is carried out in a similar manner but using STi , BLi , ∆T P/∆ci and T P (Ni + 1) estimated numerically based on Simulation Procedure 4.1. In both calculations and simulations, ∆ci is selected as 0.03. In the majority of systems analyzed, c-BN Indicator 13.1 leads to correct c-BN’s. Typical examples are shown in Figures 13.11-13.13. According to c-BN Indicator 13.1, the c-BN of System 1 (Figure 13.11) is m3 . When this machine is improved by increasing c3 , the c-BN shifts to m4 (Figure 13.12). When m4 is protected by larger buffers (b3 and b4 , Figure 13.13), the Pc-BN shifts again, now to m2 . All the above conclusions are supported by ∆T P/∆ci . Along with numerous supporting examples, a few counterexamples have been discovered. One of them is shown in Figure 13.14, where m2 is identified as the c-BN, while the partial derivatives reveal m3 as the true c-BN. To quantify the “frequency” of counterexamples, a statistical experiment, similar to that for Bernoulli lines in Subsection 5.2.3, has been carried out. The results showed that the accuracy of c-BN Indicator 13.1 is roughly the same as that of BN Indicator 5.1 (see Figures 5.11 and 5.13). Thus, we conclude that c-BN Indicator 13.1 provides a sufficiently accurate tool for bottleneck identification.

13.2.5

M ≥ 2-machine non-exponential lines

The c-Bottleneck Indicator 13.1 has been applied to serial lines with M ≥ 2-machines having mixed reliability models and the distributions of up- and downtime from the set {exp, W, ga, LN}. The results of this simulation study showed that it identified the c-BN and BN-b with roughly the same accuracy as in the case of M > 2-machine exponential lines. Based on the above, we conjecture that: c-BN Indicator 13.1 can be used for serial lines with machines having arbitrary unimodal distributions of up- and downtime. PSE Toolbox: The c-BN Indicator 13.1 for exponential and non-exponential machines is implemented in the Bottleneck Identification function of the toolbox (see Subsections 19.5.2 and 19.5.3).

434

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES c−BN BN−b

λ µ

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

0.7

6

0.69

3

0.72

3

0.681

6

0.8

3

0.682

0.3

i

0.7

i

0.175

0.56

0.15

0.2

0.35

0.39

1.44

0.32

0.8

0.75

ci

1

1

1

1

1

1

STi

0

0.017

0.0777

0.1194

0.2113

0.1241

0.1423

0.1186

0.0949

0.0046

0.0421

0

0.0119

0.07

0.2727

0.0687

0.0035

0.0008

BLi ∆ TP ∆ ci

STi

0.0175

0.0876

0.1333

0.1996

0.1537

0.1734

0.1451

0.1055

0.0208

0.0741

0

0.028

0.0708

0.1067

0.0729

0.0098

0

BLi ∆ TP ∆ ci

0.0321

Figure 13.11: c-BN identification in exponential lines: System 1

13.2.6

Buffering potency and measurement-based management

The notion of buffering potency, introduced in Definition 5.4 for Bernoulli lines, is equally applicable to serial lines with continuous time models of machine reliability. In other words, buffering is potent if and only if the machine with the smallest T Pi is the c-BN, and T P is sufficiently close to this T Pi . Clearly, continuous improvement projects in serial lines with any reliability model can be designed using the procedure of Figure 5.15. Similarly, the method of measurement-based management, described in Section 5.3 for Bernoulli lines, is applicable to serial lines with continuous time models of machine reliability. This implies that monitoring blockages and starvations of the machines and taking managerial decisions using the arrow-based method is a productive way for system management in any large volume manufacturing environment.

13.3

Case Study

In this section, we consider the automotive ignition coil processing system, the mathematical model of which was discussed and validated in Subsection 11.3.1. Below, we carry out bottleneck analysis using the method developed in this

13.3. CASE STUDY

435

c−BN

λi µ

i

BN−b Pc−BN

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

0.7

6

0.69

3

0.9

3

0.681

6

0.8

3

0.3 0.7

m6 0.682

0.175

0.56

0.15

0.2

0.35

0.39

1.44

0.32

0.8

0.75

ci

1

1

1.25

1

1

1

STi

0

0.0396

0.1263

0.0587

0.1355

0.0773

0.049

0.1355

0.0193

0.0722

0

BLi

0.0955

Si ∆ TP ∆ ci

STi BLi

0.0585

0 0.1677

Si ∆ TP ∆ ci

0.191

0.1332

0.1134

0.0806

0.0881

0.0277

0.0157

0.0238

0.1325

0.1074

0.1747

0.138

0.1231

0.1534

0.031

0.0834

0

0.017

0.0546

0.1897

0.1432

0.0433

0.1032

0.0236

0.1063

Figure 13.12: c-BN identification in exponential lines: System 2

436

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES Pc−BN BN−b m1 0.7

λ

c−BN

b1

m2

b2

m3

b3

m4

b4

m5

b5

6

0.69

3

0.9

9

0.681

12

0.8

3

m6 0.682

0.3

0.175

0.56

0.15

0.2

0.35

0.7

0.39

1.44

0.32

0.8

0.75

ci

1

1

1.25

1

1

1

STi

0

0.0526

0.1912

0.0544

0.1036

0.0621

0.0194

0.045

0.0073

0.0881

0

µ

i

i

BLi

0.0803

Si

0.026

0.1994

∆ TP

STi

0

BLi

0.0350

0.2137

0.1690

∆ ci

0.1146

Si

0.0189

0.0035

0.0027

0.0385

0.1816

0.0713

0.1069

0.0949

0.0653

0.0699

0.021

0.1081

0

0.0132

0.1923

∆ TP ∆ ci

0.0181

0.1177

0.0824

0.1068

0.0139

0.0503

Figure 13.13: c-BN identification in exponential lines: System 3 m1

b1

m2

b2

m3

b3

m4

b4

m5

0.72

3

0.65

3

0.665

3

0.84

3

0.88

λi

0.1

0.35

0.05

0.3

0.2

µi

0.4

0.6502

0.9497

0.6998

0.8

ci

0.9

1

0.7

1.2

1.1

STi

0

0.0469

0.1285

0.2231

0.2984

BLi

0.1868

0.0514

0.0317

0.0149

0

0.073

0.142

0.24

0.018

0.0073

∆ TP ∆ ci

Figure 13.14: Counterexample of c-BN identification in exponential lines

13.4. SUMMARY

437

chapter. Based on the model for Period 1, we calculate the probabilities of blockages and starvations of all machines in the system and identify the bottleneck using c-BN Indicator 13.1 (see Figure 13.15). Clearly, it is machine m9−10 . BN−m m1

^

ST

i

0

^ BLi 0.0671

b1

m2

b2

m3

b3

m5

b5

m

6

b

6

m7

b7

m 9−10b 9−10m 11

b 11 m 14

b 14 m 15 b 15 m 16

0.0664

0.0531

0.0082

0.0061

0.0313

0.0218

0.1052

0.1200

0.1348

0.1348

0.0754

0.0936

0.1365

0.1120

0.1106

0.0224

0.0022

0.0052

0.0006

0

Figure 13.15: Bottleneck identification in ignition coil processing system To protect the bottleneck, we double and triple the capacities of the buffers d d P = 488.6 parts/hour and T P = 493.4 surrounding m9−10 . This leads to T parts/hour, i.e., a 1.56% and 2.56% improvement, respectively. Still, machine d m9−10 is the c-BN. Increasing its capacity by 10%, we obtain T P = 516.8 parts/hour, a 7.42% improvement. The bottleneck is now m1 , which is largely due to its starvations by pallets. When these starvations are removed, the throughput becomes 522.2 parts/hour, and the bottleneck shifts to m11 . All these actions lead to a 8.54% throughput improvement. Similar improvements can be demonstrated for Period 2 as well.

13.4

Summary

• In serial lines with continuous time models of machine reliability, reallocating cycle time (CT) and buffer capacity (BC) are efficient methods of continuous improvement under constraints. • The CT- and BC-Improvability Indicators, developed in this chapter, can be applied based on measurements of buffer occupancies during normal system operation. • Although several types of bottlenecks in serial lines with continuous time models of machine reliability can be introduced, the most important one for practical applications is the c-BN. • c-BN Indicator 13.1 can be applied to a wide variety of serial lines: synchronous and asynchronous, exponential and non-exponential.

13.5

Problems

Problem 13.1 Consider a 5-machine exponential line with ei = 0.9, Tdown,i = P5 2, i = 1, . . . , 5, and Ni = 3, i = 1, . . . , 4. Assume i=1 τi = 5. (a) Determine the unimprovable allocation of cycle time and the resulting throughput.

438

CHAPTER 13. IMPROVEMENT OF CONTINUOUS LINES

(b) Comment on the qualitative features of the unimprovable allocation obtained in (a). Problem 13.2 Consider the 5-machine line with machine and buffer parameters given in Problem 13.1 but with a mixed reliability model of the machines. Specifically, assume that the uptime of all machines obeys the Weibull distribution with coefficient of variation 0.3, while the downtime is gamma-distributed with coefficient of variation 0.5. (a) Determine the unimprovable allocation of cycle time and the resulting throughput. (b) Did the qualitative features of this allocation change compared to those obtained in Problem 13.1? Problem 13.3 Consider a 5-machine exponential line with ei = 0.9, Tdown,i = P4 2 and τi = 1, i = 1, . . . , 5. Assume i=1 Ni = 20. (a) Determine the unimprovable allocation of buffer capacity and the resulting throughput. (b) Comment on the qualitative features of this buffer capacity allocation. Problem 13.4 Consider the 5-machine line with machine and buffer parameters given in Problem 13.3 but with a mixed reliability model of the machines. Specifically, assume that the uptime of all machines obeys the Weibull distribution with coefficient of variation 0.3, while the downtime is gamma-distributed with coefficient of variation 0.5. (a) Determine the unimprovable allocation of buffer capacity and the resulting throughput. (b) Did the qualitative features of this allocation change compared to those obtained in Problem 13.3? Problem 13.5 Identify c-BN’s and BN-b’s in unimprovable lines obtained in Problems 13.1-13.4. Problem 13.6 Consider the exponential serial line of Problem 3.3. (a) (b) (c) (d)

Identify its c-BN. Is the buffering potent or not? If not, design a potent buffering. Determine the smallest increase in c-BN capacity of the resulting system so that the bottleneck shifts to another machine.

Problem 13.7 Repeat Problem 13.6 for the exponential line defined in Problem 3.4.

13.6. ANNOTATED BIBLIOGRAPHY

13.6

439

Annotated Bibliography

The notion of DT-BN is introduced and investigated in [13.1] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “Bottlenecks in Markovian Production Lines: A Systems Approach,” IEEE Transactions on Robotics and Automation, vol. 14, pp. 352-359, 1998. [13.2] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “DT-Bottleneck in Serial Production Lines: Theory and Application,” IEEE Transactions on Robotics and Automation, vol. 16, pp. 567-580, 2000. The c-BN is introduced in [13.3] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “c-Bottleneck in Serial Production Lines,” Mathematical Problems in Engineering, vol. 7, pp. 543578, 2001. More details can be found in [13.4] S.-Y Chiang, Bottlenecks in Production Systems with Markovian Machines: Theory and Applications, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1999. A description of Goldaratt’s approach, mentioned in Subsection 13.1.3, can be found in [13.5] E.M. Goldratt and J. Cox, The Goal, North River Press Inc, Croton-onHudson, NY, 1984.

Chapter 14

Design of Lean Continuous Lines Motivation: The motivation here is the same as that of Chapter 6: the need for rigorous engineering methods for selecting lean buffering in serial lines. In this chapter, such methods are developed for serial lines with continuous time models of machine reliability. Overview: The development is carried out in terms of a parametrization, which is a generalization of that used in Chapter 6 for Bernoulli lines. Closedform expressions for lean buffering in serial lines with exponential machines are derived using an analytical approach. For systems with non-exponential machines, empirical formulas are obtained using simulations. The cases of both identical and non-identical machines are considered and synchronous and asynchronous lines are discussed. In addition, a simple rule-of-thumb is formulated, which may be used by factory floor personnel managing production systems and their inventories.

14.1

Parametrization and Problem Formulation

The production lines addressed in this chapter are shown in Figures 13.1-13.3. The conventions of their operation remain the same as in Subsection 11.1.1. As in the case of Bernoulli lines, it is convenient to introduce parameter normalizations in order to formulate and solve the problem at hand. To define these normalizations, assume for a moment that the system is synchronous and the up- and downtime of all machines are the same, i.e., Tup,i =: Tup ,

Tdown,i =: Tdown ,

i = 1, . . . , M,

(14.1)

where Tup and Tdown are in units of cycle time. In addition, assume that the 441

442

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

capacity of all buffers is the same: Ni =: N,

i = 1, . . . , M − 1.

(14.2)

These assumptions are introduced to limit combinatorial explosion of cases that would arise otherwise. However, the results obtained under these assumptions are generalized, later in this chapter, to systems with non-identical machines and buffers. Clearly, assumption (14.1) implies that the efficiency, e, of all machines is the same, Tup , e= Tup + Tdown and assumption (14.2) excludes the possibility of the “inverted bowl” buffer capacity allocation. However, since the inverted bowl leads to only a marginal increase in the throughput (within 1% in the majority of realistic scenarios), for the purposes of simplicity we forgo this possibility. Using assumptions (14.1) and (14.2), introduce the following two dimensionless parameters, which are at the center of the methods developed in this chapter: Level of buffering (k) – capacity of the buffer in units of average downtime, i.e., N . (14.3) k := Tdown Obviously, the level of buffering is a non-negative real number. Since Tdown is in units of cycle time, k indicates how many downtimes can be accommodated by N . For instance, k = 1.75 implies that the buffer is capable of accommodating 1.75 average downtimes. Line efficiency (E) – production rate of the line in units of the largest possible production rate of the system. As it is clear from Chapter 11, the largest production rate is obtained when all buffers are of infinite capacity. Denote this production rate as P R∞ . Let P R denotes the production rate of the same system for any other buffer capacity allocation. Then the line efficiency for this allocation is defined as E :=

PR . P R∞

(14.4)

Clearly, 0 < E < 1. Lean level of buffering, LLB (kE ) – the smallest level of buffering that ensures line efficiency E. In other words, kE is the level of buffering that is necessary and sufficient to obtain the desired P R quantified by E. In terms of the above dimensionless parametrization, the problems addressed in this chapter are: • Develop a method for calculating kE for exponential lines. • Investigate the sensitivity of kE to the type of up- and downtime distributions in non-exponential lines.

14.2. LEAN BUFFERING FOR IDENTICAL MACHINES

443

• Based on the above, provide empirical formulas for calculating kE for non-exponential lines. • Generalize the results to systems with non-identical machines and buffers, i.e., when (14.1) and (14.2) do not hold. • Generalize the results to asynchronous lines.

14.2

Lean Buffering in Synchronous Lines with Identical Exponential Machines

14.2.1

Two-machine lines

For the case of a two-machine line satisfying (14.1), λ1 = λ2 =: λ,

µ1 = µ2 =: µ,

and, therefore, function Q(λ1 , µ1 , λ2 , µ2 , N ), defined by (11.4) becomes: Q(λ1 , µ1 , λ2 , µ2 , N ) = Q(λ, µ, N ) =

2λ . (λ + µ)[2 + (λ + µ)N ]

(14.5)

Hence, as it follows from (11.13) and from the monotonicity of P R with respect to N , the buffer capacity, NE , which is necessary and sufficient to ensure line efficiency E, is defined by the following equation: P R = Ee = e[1 − Q(λ, µ, NE )].

(14.6)

Solving for NE , we obtain:   NE =



2(1−e)(E−e) , λ(1−E)

if e < E,

0,

otherwise.

(14.7)

Thus, we have Theorem 14.1 The lean level of buffering for a line with two identical exponential machines is given by the following expression:  2e(E−e)  1−E , if e < E, exp (M = 2) = (14.8) kE  0, otherwise. Figures 14.1(a) and 14.2(a) illustrate the behavior of kE as a function of e and E, respectively. From these figures and expression (14.8) we observe: • LLB does not depend on λ or µ explicitly. • LLB is a quadratic function of e, reaching its maximum at e = 0.5E.

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

20

15

15

10 5 0 0

20

E=0.85 E=0.90 E=0.95

15

kexp E

20

kexp E

kexp E

444

10 5

0.2

0.4

e

0.6

0.8

0 0

1

(a) M = 2

10 5

0.2

0.4

e

0.6

0.8

0 0

1

(b) M = 3

0.2

0.4

e

0.6

0.8

1

(c) M = 10

20

15

15

10 5 0 0.8

10 5

0.85

E

0.9

(a) M = 2

0.95

0 0.8

20

e=0.75 e=0.85 e=0.95

15

kexp E

20

kexp E

kexp E

Figure 14.1: Lean level of buffering for exponential lines with identical machines as a function of machine efficiency

10 5

0.85

E

0.9

(b) M = 3

0.95

0 0.8

0.85

E

0.9

0.95

(c) M = 10

Figure 14.2: Lean level of buffering for exponential lines with identical machines as a function of line efficiency

14.2. LEAN BUFFERING FOR IDENTICAL MACHINES

445

• If E > e, the lean system must have a buffer. For instance, if e = 0.85 and E = 0.95, a buffer capacity of approximately three downtimes is required. • If E ≤ e, the system does not need a buffer. For instance, if e = 0.85 and E = 0.85, just-in-time operation is acceptable. • LLB is a monotonically increasing function of E with a hyperbolically increasing rate, approaching infinity as E → 1.

14.2.2

Three-machine lines

exp Since no closed formula for P R in three-machine lines is available, kE (M = 3) cannot be evaluated using the simple arguments of Subsection 14.2.1. However, based on the aggregation procedure (11.25), the following can be proved:

Theorem 14.2 In three-machine lines under consideration, the lean level of buffering is given by

exp (M = 3) = kE

        

√ √ e(1+ E)(e+e E−2) √ 2(1− E)

³ ln

´ √ 1−e E √ , (1−e)(1+ E) if e <

0,

√

E,

otherwise. (14.9)

Proof: See Section 20.2. exp Figures 14.1(b) and 14.2(b) show the behavior of kE (M = 3) as a function of e and E, respectively. From these Figures and (14.9) we observe: • LLB is again independent of λ or µ explicitly. exp is not quadratic, its qualitative behavior is similar to a • Although kE √ parabola with the maximum at e = 0.5 E. √ • If E > e, a lean system must have buffers. For example, if e = 0.85 and E = 0.95, LLB is roughly 6. If e = 0.85 and E = 0.85, LLB is close to 1; note that for M = 2 this case requires no buffering. √ • If √ E ≤ e, lean systems need no buffering. Since 0 < E < 1 and, therefore, E > E, the region of e’s where no buffer is needed is smaller than that for M = 2.

• LLB is monotonically increasing as a function of E, with increasing rate.

14.2.3

M > 3-machine lines

Exact expression: be proved:

Using aggregation procedure (11.25), the following can

446

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Theorem 14.3 In M > 3-machine lines under consideration, the lean level of buffering is given by ´ ³  e(2−Q)(2e−eQ−2) E−eE+eEQ−1+e−2eQ+eQ2 +Q  , ln  2Q (1−e−Q+eQ)(E−1)   1 exp if e < E M −1 , (M > 3) = kE     0, otherwise, (14.10) where (14.11) Q = Q(λfM −2 , µfM −2 , λbM −1 , µbM −1 , NE ) is defined in (11.4) and λfM −2 ,, µfM −2 , λbM −1 , µbM −1 are the steady states of aggregation procedure (11.25). Proof: See Section 20.2. exp expressed Unfortunately, (14.10) is not a closed formula since it gives kE not only in terms of e and E but also Q. Thus, to calculate LLB, the value of Q must be evaluated. A method to accomplish this is described below. Approximation of Q: As it follows from Subsection 11.1.2, function (14.11) is the probability that buffer bM −2 is empty and machine mfM −2 is down, given that machine mbM −1 is up. This probability can be evaluated using aggregation procedure (11.25). The resulting behavior of Q as a function of e for several values of E and M = 10 is illustrated in Figure 14.3 by solid lines. In order to “convert” (14.10) into a closed-form expression, we approximate Q as follows: ½ ¾ b = a + b exp − α − e , Q (14.12) β where a, b, and α, β are to be selected as functions of E and M so that the

0.1

0.05

0 0

0.15

Q ^ Q

<

Q or Q

<

Q or Q

<

0.15

Q or Q

0.15

0.1

0.05

0.25

0.5

e

0.75

(a) M = 10,E = 0.85

1

0 0

0.1

0.05

0.25

0.5

e

0.75

(b) M = 10,E = 0.9

1

0 0

0.25

0.5

e

0.75

1

(c) M = 10,E = 0.95

b versus e. Figure 14.3: Behavior of functions Q and Q approximation is sufficiently accurate. We select these functions based on the following arguments:

14.2. LEAN BUFFERING FOR IDENTICAL MACHINES

447

For M = 3 and any e, function Q can be calculated in closed form as (see the proof of Theorem 14.2 in Section 20.2) Q=1−

√

E.

(14.13)

1

For any M and e = E M −1 (i.e., when, according to (14.10), no buffering is required), function Q again can be easily calculated to be M −2

Q = 1 − E M −1 .

(14.14) 1

exp (e) is parabolic with the maximum at e = 12 E M −1 , Finally, if we assume that kE function Q can be evaluated as 1

M −3 M/4

Q ≈ 1 − E 2 [1+( M −1 )

]

.

(14.15)

Using these three expressions, we select the three unknowns, a, b, and α as follows: a = b = α

=

1

M −3 M/4

1 − E 2 [1+( M −1 ) E E

M −3 M/4 1 ] 2 [1+( M −1 ) 1 M −1

]

,

−E

.

(14.16) M −2 M −1

,

(14.17) (14.18)

To select β, we observe that smaller values of E require larger “time constants” of the exponential behavior of Q as a function of e. By trial and error, we find that this can be captured by selecting β =1−

√

E.

(14.19)

Combining (14.12) with (14.16)-(14.19), the following approximation of Q is obtained: ´ n ³ E M1−1 − e ´o ³ 1 £ ¡ M −3 ¢M/4 ¤ M −2 √ . + E 2 1+ M −1 −E M −1 exp − 1− E (14.20) 1 Note that for M = 3 this expression coincides with (14.13) and for e = E M −1 with (14.14). b is illustrated in Figure 14.3 by broken lines. QuantitaThe behavior of Q tively, the accuracy of this approximation, in units of

b = 1 − E 12 Q

£

1+

¡

M −3 M −1

¢M/4 ¤

∆Q =

b Q−Q · 100%, Q

(14.21)

is characterized in Table 14.1. As one can see, in most cases, it is within 1-2%.

448

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

b Table 14.1: Accuracy of approximating Q by Q e 0.1 0.2 0.3 0.4 (a) M = 5 0.5 0.6 0.7 0.8 0.9

E = 0.85 -2.65 -2.55 -2.75 -2.36 -2.28 -2.12 -1.88 -1.53 -1.70

E = 0.9 -3.15 -2.85 -2.56 -2.56 -2.56 -2.27 -2.00 -1.59 -1.69

E = 0.95 -1.92 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.22 -1.52

e 0.1 0.2 0.3 0.4 (b) M = 10 0.5 0.6 0.7 0.8 0.9

E = 0.85 0.54 0.12 0.62 0.45 0.77 1.02 1.32 -1.21 0.49

E = 0.9 0.37 0.25 0.37 0.50 0.63 0.88 1.21 1.51 0.85

E = 0.95 0.86 0.35 0.35 0.35 0.35 0.60 0.60 1.11 1.75

e 0.1 0.2 0.3 0.4 (c) M = 20 0.5 0.6 0.7 0.8 0.9

E = 0.85 0.32 0.32 0.64 0.72 1.10 1.34 1.66 1.48 -0.27

E = 0.9 0.69 0.44 0.57 0.69 0.94 1.18 1.49 1.85 1.09

E = 0.95 1.13 0.63 0.63 0.63 0.63 0.88 1.13 1.36 2.17

12

10

10

8

8

6

6

4

4

2

2

0

5

10

15

M

20

25

30

0

449

12

E=0.85 E=0.90 E=0.95

10 8

kexp E

12

kexp E

kexp E

14.2. LEAN BUFFERING FOR IDENTICAL MACHINES

6 4 2

5

(a) e = 0.85

10

15

M

20

25

30

(b) e = 0.9

0

5

10

15

M

20

25

30

(c) e = 0.95

Figure 14.4: Lean level of buffering as a function of M (solid lines - exact; dashed lines - approximation) Closed formula for LLB: Since the accuracy of the approximation of Q is commensurable with the accuracy of the data available in most practical b substituted for Q, i.e., applications, expression (14.20) with Q ´ ³  b b b b Q b 2 +Q b e(2−Q)(2e−e Q−2) E−eE+eE Q−1+e−2e Q+e  , ln  b b b 2Q (1−e−Q+eQ)(E−1) exp 1 b kE (M > 3) = if e < E M −1 ,   0, otherwise, (14.22) provides a closed, however approximate, formula for LLB in the M > 3-machine case. Note that for M = 3, expressions (14.20), (14.22) coincide with (14.9). Expressions (14.20), (14.22) quantify LLB in terms of three variables: e, E, and M . For every fixed M , this function is qualitatively similar to those for M = 2 and 3. The broken lines in Figures 14.1(c) and 14.2(c) illustrate this function for M = 10; the solid lines, which almost coincide with the broken ones, show the exact values of kE (calculated according to aggregation procedure (11.25) and (14.10), (14.11)). Obviously, LLB is increased significantly and the area of no buffering is decreased in comparison with M = 3. For instance, if E = 0.9, the smallest e that ensures this E without buffers is 0.9884. exp as a Expressions (14.20), (14.22) can be used to analyze the behavior of b kE function of M . This is illustrated in Figure 14.4 (broken lines) where the values of kE , calculated according to (11.25) and (14.10), (14.11)), are also shown (solid lines). The errors, quantified in terms of ∆k E =

b kE − kE · 100%, kE

(14.23)

are given in Table 14.2. From this table and Figure 14.4 the following can be concluded: • The closed formulas (14.20), (14.22) provide an estimate of LLB typically within a 5% error; in a few cases, the error may be more than 10%. However, when this error is large, (14.20) and (14.22) typically provide a more conservative estimate.

450

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Table 14.2: Accuracy of approximating kE by b kE M 5 10 (a) e = 0.85 15 20 25 30

E = 0.85 3.41 -2.32 -2.97 -2.88 -2.80 -2.54

E = 0.9 2.49 -3.33 -4.14 -4.29 -4.32 -4.08

E = 0.95 2.43 -2.13 -3.05 -3.15 -3.09 -2.99

M 5 10 (b) e = 0.9 15 20 25 30

E = 0.85 4.86 0.63 0.68 1.04 1.31 1.78

E = 0.9 3.16 -2.33 -3.37 -3.36 -3.15 -2.83

E = 0.95 2.36 -3.05 -4.10 -4.30 -4.28 -4.13

M 5 10 (c) e = 0.95 15 20 25 30

E = 0.85 10.04 13.28 16.10 18.30 19.70 21.32

E = 0.9 6.91 4.75 5.47 6.59 7.29 8.15

E = 0.95 2.93 -2.57 -3.60 -3.37 -3.32 -2.94

14.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

451

• LLB is an increasing function of M . • However, the rate of the increase is decreasing, leveling off at about M = 10. Thus, the LLB necessary for M = 10 is practically acceptable for any M > 10. Based on this observation, the following can be formulated: Rule-of-thumb for selecting lean buffering: LLB in synchronous exponential lines with M ≥ 10 can be selected as shown in Table 14.3. Table 14.3: Rule-of-thumb for selecting Lean Level of Buffering in synchronous exponential lines with 10 or more machines e 0.85 0.90 0.95

E = 0.85 3.4 2.7 1.6

E = 0.90 5 3.9 2.4

E = 0.95 9.8 7.2 4.3

Discussion: Some of the entries of Table 14.3 are quite large. For instance, if e = 0.85 and E = 0.95, LLB is close to 10. Does this imply that the buffer is exp = 10 may represent physically large? Such a question is ill-posed. Indeed, kE either a large or a small buffer, depending on the value of Tdown : If Tdown is, say, 100 cycle times, then the buffer capacity, NE , is 1000 and, therefore, it may be viewed as large; if Tdown = 1, then NE = 10, and the buffer is small. Equally ill-posed is the question: Is buffer capacity NE = 100 large or small? exp = 1, and the buffer is very small and, moreover, If Tdown = 100, then kE insufficient if e ≤ 0.95 and E ≥ 0.85 (see Table 14.3). If, however, Tdown = 2, exp = 50, and the buffer is an order of magnitude larger than required by then kE the rule-of-thumb. PSE Toolbox: The method for LLB calculation is implemented in the Lean Buffer Design function of the toolbox (see Section 19.6).

14.3

Lean Buffering in Synchronous Lines with Non-identical Exponential Machines

For non-identical machines, condition (14.1) clearly does not take place. Therefore, instead of (14.3), we normalize the capacity of the i-th buffer by the largest downtime of the surrounding machines, i.e., ki :=

Ni , max(Tdown,i , Tdown,i+1 )

(14.24)

and denote the smallest ki necessary and sufficient to achieve line efficiency E as ki,E , i = 1, . . . , M − 1. This section provides a characterization of ki,E in synchronous exponential lines.

452

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CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Two-machine lines

In the case of e1 6= e2 , the equation that defines the lean buffer capacity NE , as it follows from (11.4)(11.7) and (11.13), becomes h (1 − e1 )(1 − φ) i , P R = E · P R∞ = e2 [1 − Q(λ1 , µ1 , λ2 , µ2 , NE )] = e2 1 − 1 − φe−βNE (14.25) where P R∞ = min(e1 , e2 ) and φ = β

=

e1 (1 − e2 ) , e2 (1 − e1 ) (µ1 + µ2 + λ1 + λ2 )(λ1 µ2 − λ2 µ1 ) . (µ1 + µ2 )(λ1 + λ2 )

Solving (14.25) for NE and normalizing by max(Tdown,1 , Tdown,2 ), we obtain Theorem 14.4 Lean level of buffering in a line with two non-identical exponential machines is given by ( o n min(µ1 ,µ2 ) 2 −E·P R∞ ) , if min(e1 , e2 ) < E, ln φ (e exp β (e −E·P R ) 1 ∞ kE (M = 2) = 0 otherwise. (14.26) Unlike the case of identical machines, this kE depends explicitly not only on E and e’s but also on the up- and downtimes of the machines. However, similar to the Bernoulli case, this expression is reversible: Theorem 14.5 In two-machine exponential lines, LLB is reversible, i.e., exp exp kE (λ1 , µ1 ; λ2 , µ2 ) = kE (λ2 , µ2 ; λ1 , µ1 ).

Proof: See Section 20.2. exp The behavior of kE is illustrated in Figures 14.5 and 14.6 as a function of e1 and E, respectively.

14.3.2

M > 2-machine lines

As in the case of Bernoulli machines, exact analytical expressions for LLB in M > 2-machine exponential lines are all but impossible to derive. Therefore, we use the approaches of Chapter 6 to provide estimates for ki,E . Specifically, we consider the following approaches:

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14.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

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Figure 14.5: Lean level of buffering for two non-identical exponential machine lines as a function of e1 .

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Figure 14.6: Lean level of buffering for two non-identical exponential machine lines as a function of E.

454

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Closed formula approaches: They are similar to those of Chapter 6. Namely, I. Local pair-wise approach. For each pair of consecutive machines, mi and I , i = 1, . . . , M − mi+1 , i = 1, . . . , M −1, use formula (14.26) to calculate ki,E 1. II. Global pair-wise approach. It is based on applying formula (14.26) to all possible pairs of machines (not necessarily consecutive) and then selecting the LLB of each buffer equal to the largest of all kj,E obtained by this proce2 dure. Since there are (M −1) 2+(M −1) possible combinations of two-machine 2

lines, index j ranges from 1 to (M −1) 2+(M −1) . Thus, this approach leads to II . The actual capacity, Ni,E , of each buffers with equal LLB, denoted as kE II by max(Tdown,i−1 , Tdown,i ). buffer i is selected then by multiplying kE III. Local upper bound approach. Instead of each pair of consecutive machines, mi and mi+1 , i = 1, . . . , M − 1, consider a two-machine line with identical machines specified by ebi = min(ei , ei+1 ),

µ bi = min(µi , µi+1 ),

bi = µ λ bi (1 − ebi )/b ei (14.27)

III and select LLB, denoted as ki,E , using formula (14.26). This results in a sequence of LLBs III III , . . . , kM k1,E −1,E .

IV. Global upper bound approach. Instead of the original line, consider a line with M identical machines specified by eb = min(e1 , . . . , eM ),

µ b = min(µ1 , . . . , µM ),

b=µ λ b(1 − eb)/b e (14.28)

IV and select LLB, denoted as kE , using formulas (14.20) and (14.22). This clearly provides an upper bound of ki,E , i = 1, . . . , M .

The efficacy of these approaches has been investigated using the same method as in Chapter 6. Specifically, 100,000 lines, Ls , s ∈ {1, . . . , 100, 000}, have been simulated with the parameters selected randomly and equiprobably from the following sets: Ms

∈

{5, 10, 15, 20, 25, 30},

(14.29)

ei,s

∈ ∈ ∈

[0.7, 0.97], {5, 10, . . . , 50}, [0.8, 0.98].

(14.30) (14.31) (14.32)

Tdown,i,s Es

The results obtained are quite similar to those of Chapter 6. Namely, • The local pair-wise approach in almost all cases leads to a lower line efficiency than desired. • The global pair-wise approach outperforms approach III for M ≤ 15. For M > 15, the local upper bound approach is recommended. • The global upper bound approach substantially overestimates LLB.

14.3. LEAN BUFFERING FOR NON-IDENTICAL MACHINES

455

Recursive approaches: The following two recursive methods have been investigated: V. Full search approach. Start from all buffers of capacity 1. Increase the capacity of the first buffer by 1 and, using the aggregation procedure (11.25), calculate the production rate of the system. Return the first buffer capacity to its initial value, increase the second buffer capacity by 1 and calculate the resulting production rate. Repeat the same procedure for all buffers, determine the buffer that leads to the largest production rate, and permanently increase its capacity by 1. Then, repeat the process again until the desired line efficiency is reached. This will result in the sequence of buffer capacities V V , . . . , NM N1,E −1,E . V Normalizing Ni,E by max{Tdown,1 , . . . , Tdown,M }, results in V V k1,E , . . . , kM −1,E .

VI. Bottleneck-based approach. Consider a production line with the buffer capacity calculated according to approach I. This buffering often leads to line efficiency less than desired. Therefore, to improve the line efficiency, increase the buffering according to the following procedure: Using the technique described in Chapter 13, identify the c-bottleneck (or, when applicable, primary c-bottleneck) and increase the capacity of both buffers surrounding this machine by 1. Repeat this procedure until the desired line efficiency is reached. This will result in the sequence of buffer capacities denoted as VI VI , . . . , NM N1,E −1,E . VI by max{Tdown,1 , . . . , Tdown,M }, results in Normalizing Ni,E VI VI k1,E , . . . , kM −1,E .

These two approaches also have been analyzed using the same procedure as in Chapter 6. As a result, the following conclusions have been obtained: • The full search approach, as expected, results in the smallest level of buffering and the longest computation time. • Approach VI provides a good tradeoff between the computation time and the level of buffering. It is up to two orders of magnitude faster than approach V and results in average level of buffering slightly larger than that of approach V (about 20% difference). It is, of course, slower than approaches I-IV but gives buffering 2-6 times smaller than approaches II-IV. PSE Toolbox: The six approaches of lean buffering calculation for exponential lines are implemented in the Lean Buffer Design function of the toolbox (see Chapter 19).

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CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Generalization: The above results can be generalized to asynchronous exponential lines by reducing them to synchronous ones. This can be accomplished by modifying µi so that all ci ’s are the same and T Pi ’s remain unchanged (see Problem 14.6).

14.4

Lean Buffering in Synchronous Lines with Non-exponential Machines

14.4.1

Approach

General considerations: Since LLB depends on line efficiency E, one has to know P R in order to evaluate kE . Although an empirical formula for P R in production lines with non-exponential machines has been derived in Chapter 12, due to its approximate nature we use below numerical simulations rather than an analytical approach. As a final result, however, we obtain an empirical formula, exp and the coefficients of variation, CV , which approximates kE in terms of kE of the machines in question. To accomplish this, similar to Chapter 12, we • • • •

define a representative set of up- and downtime distributions, evaluate P R and kE using Simulation Procedure 4.1, investigate the sensitivity of kE to machine reliability models, and derive an empirical formula for kE .

Each of these steps is described below. Systems analyzed: Two groups of production lines are considered. The first one is given by {[W (λ, Λ), W (µ, M)]i , {[ga(λ, Λ), ga(µ, M)]i , {[LN (λ, Λ), LN (µ, M)]i ,

i = 1, . . . , 10}, i = 1, . . . , 10}, i = 1, . . . , 10},

(14.33)

and is referred to as the case of identical reliability models. The second, Line 1:

{(ga, W ), (LN, LN ), (W, ga), (ga, LN ), (ga, W ),

Line 2:

(LN, ga), (W, W ), (ga, ga), (LN, W ), (ga, LN )}, {(W, LN ), (ga, W ), (LN, W ), (W, ga), (ga, LN ), (ga, W ), (W, W ), (LN, ga), (ga, W ), (LN, LN )},

(14.34)

is referred to as the case of mixed reliability models. We use the notations A ∈ {(14.33)}, A ∈ {(14.34)}, or A ∈ {(14.33), (14.34)} to indicate that line A is one of (14.33), or one of (14.34), or one of (14.33) or (14.34), respectively.

14.4. LEAN BUFFERING FOR NON-EXPONENTIAL MACHINES

457

The CVs of up- and downtime in each of these groups are selected either as CVup,i = CVdown,i =: CV,

i = 1, . . . , M,

(14.35)

or as CVup,i =: CVup ,

CVdown,i =: CVdown ,

i = 1, . . . , M,

(14.36)

while the up- and downtime of all machines are identical, i.e., Tup,i =: Tup , Parameters selected: follows:

Tdown,i =: Tdown ,

i = 1, . . . , M.

(14.37)

The values of M , E, and e have been selected as

exp is not very sensitive to M • Since, as it is shown in Subsection 14.2.3, kE for M ≥ 10, the number of machines in the system is selected as 10.

• In practice, production lines often operate close to their maximum capacity. Therefore, E was selected from the set {0.85, 0.9, 0.95}. • Although in practice e may have widely different values, to obtain a manageable set of systems for simulation, e was selected from the set {0.85, 0.9, 0.95}. • Tup was selected as 20 or 100. As it is shown below, kE is not sensitive to Tup and, therefore, Tup = 20 was used in most of the simulations. Reliability models considered: The set of downtime distributions analyzed is given in Table 12.1. The uptime distributions have been selected in a manner similar to that of Chapter 12. Specifically, for a given e and Tdown , the average uptime was chosen as e Tdown . Tup = 1−e Then CVup was selected as CVup = CVdown , when the case of identical coefficients of variation was addressed. In the case of non-identical coefficients of variation, CVup was selected from the set {0.1, 0.25, 0.5, 0.75, 1}. Finally, using these Tup and CVup , the distribution of uptime was selected according to either (14.33) or (14.34). For instance, if ftdown (t) = ga(0.018, 1.8) and e = 0.9, the uptime distribution was selected as ( ga(0.002, 1.8) for CVup = CVdown , ftup (t) = ga(0.0044, 4) for CVup = 0.5, or

( LN (6.69, 0.47) ftup (t) = LN (2.88, 0.49)

for for

CVup = CVdown , CVup = 0.5.

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CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Evaluation of P R and LLB: The P R of every system under consideration was evaluated using Simulation Procedure 4.1. The kE was evaluated using the following procedure: For each model of serial line (14.33), (14.34), the production rate was evaluated first for N = 0, then for N = 1, and so on, until the production rate P R = E · P R∞ was achieved. Then, kE was determined by dividing the resulting NE by the machine average downtime (in units of cycle time).

14.4.2

Sensitivity of kE to machine reliability models

The case of CVup = CVdown =: CV : Using the approach described above, kE for systems (14.33) and (14.34) have been evaluated for CV ∈ {0.1, 0.25, 0.5, 0.75, 1}. The results are shown in Figures 14.7 and 14.8 for the case of systems (14.33) and in Figures 14.9 and 14.10 for the case of systems (14.34). (The role of broken lines in these figures will become clear in Section 14.4.3.) These results lead to the following conclusions: E=0.85

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Figure 14.7: LLB versus CV for systems (14.33) with Tdown = 20

• kE is a monotonically increasing function of CV . In addition, kE (CV ) is convex, which implies that reducing larger CV s leads to a larger reduction of kE than reducing smaller CV s. • For every pair (E, e), the corresponding graphs of Figures 14.7 and 14.8 and of Figures 14.9 and 14.10 are practically identical. Thus, similar to

14.4. LEAN BUFFERING FOR NON-EXPONENTIAL MACHINES E=0.85 8

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Figure 14.8: LLB versus CV for systems (14.33) with Tdown = 100 E=0.85

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Figure 14.9: LLB versus CV for systems (14.34) with Tdown = 20

460

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES E=0.85

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Figure 14.10: LLB versus CV for systems (14.34) with Tdown = 100 exp kE , LLB in the non-exponential case is independent of Tup and Tdown , as long as their ratio (i.e., e) remains constant.

• The most important conclusion, however, is that kE is practically insenA (CV ) sitive to the type of up- and downtime distributions. Indeed, let kE denote LLB for line A ∈ {(14.33)} with CV ∈ {0.1, 0.25, 0.5, 0.75, 1.0}. Then the sensitivity of kE to up- and downtime distributions can be characterized by ²1 (CV ) =

¯ k A (CV ) − k B (CV ) ¯ ¯ E ¯ E ¯ ¯ · 100%. A (CV ) A,B∈{(14.33)} kE max

(14.38)

Function ²1 (CV ) is illustrated in Figure 14.11. As one can see, in most cases it takes values within 10%, which support the above assertion.

The case of CVup 6= CVdown : This case allows us to analyze whether CVup or CVdown has a larger effect of kE . To investigate this issue, introduce two functions: kE (CVup |CVdown = α) (14.39) and kE (CVdown |CVup = α),

(14.40)

14.4. LEAN BUFFERING FOR NON-EXPONENTIAL MACHINES E=0.9

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Figure 14.11: Sensitivity of LLB to the nature of up- and downtime distributions for systems (14.33) where α ∈ {0.1, 0.25, 0.5, 0.75, 1.0}.

(14.41)

Expression (14.39) is intended to describe kE as a function of CVup given that CVdown = α, while (14.40) describes kE as a function of CVdown given CVup = α. If for all α and CV ∈ (0, 1] kE (CVdown = CV |CVup = α) < kE (CVup = CV |CVdown = α),

(14.42)

it must be concluded that CVdown has a larger effect on kE than CVup . If the inequality is reversed, CVup has a stronger effect. If (14.42) holds for some α’s from (14.41) and does not hold for others, the conclusion would be that, in general, neither has a dominant effect. In order to determine which of these situations takes place, we evaluated functions (14.39) and (14.40) for all values of α from (14.41) using the approach described above. Some of the results for Weibull distributions are shown in Figure 14.12. Similar results are obtained for other distributions as well. These results lead to the following conclusions: • The sensitivity to reliability models is low. • Since for all α’s from (14.41), the curve kE (CVdown |CVup = α) is below kE (CVup |CVdown = α), CVdown has a larger effect on kE than CVup .

462

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES E=0.85 8

8

6

6

6

0.9

k

k

4

4

4

2

2

2

0

0

0.4 or CV

0.6 or CV

down

0.8

1

0

CV

up

10

8

8

6

6 k

k

4

2

2

0

0

CV

up

0.4 or CV

0.6 or CV

down

0.8

0.6 or CV

down

0.8

1

0

1

0

0.2 CV

eff

up

0.4 or CV

0.6 or CV

down

0.8

0

1

0.4 or CV

0.6 or CV

0.8

1

CV

eff

0.95

4

4

2

2

2

0

0

1

down

k

k

k

0.9

0.85

6

4

0.8

1

0.2 up

8

eff

0

eff

6

0.6 or CV

0.6 or CV

2

6

down

0.4 or CV

0.8

CV

4

10

0.4 or CV

1

eff

6

empirical law

8

up

0.6 or CV

down

8

10

CV

0.4 or CV

0.8

CV 10

8

0.2

0.2 up

10

0

0

eff

down

4

0.2

0.4 or CV

CVup = 0.1 CVdown = 0.1 CVup = 0.5 CVdown = 0.5 CVup = 1 =1 CV

0.9

0.85

10

0

0.2

eff

0.95

0.2

k

0

up

e=0.95

0.95

8

0.85

10

CV

e=0.9

E=0.95

10

k

e=0.85

E=0.9

10

0

0.2 CV

up

0.4 or CV

0.6 or CV

down

eff

0.8

1

0

0

0.2 up

down

eff

Figure 14.12: LLB versus CV for M = 10 Weibull machines • However, since the two curves corresponding to the same α are close to each other, the difference in the effects is not too dramatic. To quantify this difference, introduce the function ²A 2 (CV |CVup = CVdown = α) =

A A kE (CVup = CV |CVdown = α) − kE (CVdown = CV |CVup = α) · 100%, A kE (CVup = CV |CVdown = α) (14.43)

where A ∈ {W, g, LN }. The behavior of this function is illustrated in Figure 14.13 for A = W . Clearly, it takes values within 30% and, thus, the effects of CVdown and CVup on kE are not too different. Similar results were obtained for gamma and log-normal distributions as well.

14.4.3

Empirical formulas for kE

The need for analytical expressions: The simulations reported above provide a characterization of kE for several values of E and e and M = 10. How can kE be determined for other values of these parameters? Obviously, simulations for all values are impossible. Therefore, an analytical method for calculating kE for all values of E, e, and M is desirable. Since, as it shown above, kE depends

14.4. LEAN BUFFERING FOR NON-EXPONENTIAL MACHINES

=α)

80

ε (CV|CV =CV

60

ε (CV|CV =CV

60

20

0.4

0.6

0.8

0

1

0

0.2

0.4

60

ε (CV|CV =CV

60

0.4

α = 0.1 α = 0.5 α=1

0.6

0.8

40 20 0

1

0

0.2

0.4

80 60

0.6

0.8

=α)

=α)

ε (CV|CV =CV

60

ε (CV|CV =CV

60

0.6 CV

0.8

1

40

W 3

W 3

0.4

1

0.2

0.4

0.6

0.8

1

0.6

0.8

1

up

up

up

W 3

down

=α)

80

60

down

100

80

down

100

20

0.8

CV

80

0.2

0

CV

40

0.6

20 0

1

100

0

0.4

40

ε (CV|CV =CV

CV

0

0.2

CV

W 3

W 3

20

0.2

0

100

up

up

40

0

0

1

down

=α)

80

W 3

0.8

up

=α)

80

down

100

down

100

0

e=0.95

0.6

20

CV

ε (CV|CV =CV

CV

40

=α)

0.2

20

ε (CV|CV =CV

0

40

W 3

W 3

40

up

up

up

down

=α)

80

60

down

=α)

80

down

100

0

e=0.9

E=0.95

100

W 3

e=0.85

E=0.9

100

ε (CV|CV =CV

E=0.85

463

20 0

0

0.2

0.4

0.6

0.8

1

40 20 0

0

0.2

CV

0.4 CV

Figure 14.13: Function ²W 2 (CV |CVup = CVdown = α) on CV s, this method might be based on the function kE = kE (M, E, e, CV )

(14.44)

for the case of CVup = CVdown = CV , or on the function kE = kE (M, E, e, CVup , CVdown )

(14.45)

for the case of CVup 6= CVdown . Below, we introduce upper bounds of these expressions and referred to them as the empirical formulas. General considerations: Only a single point on all curves of Figures 14.7 exp . 14.10 can be calculated analytically – that for CV = 1, i.e., kE (CV = 1) = kE exp Clearly, kE , calculated using the method developed in Section 14.2, depends on M , E, and e. Thus, the dependence of (14.44) and (14.45) on M , e, and exp and the only additional dependence that E can be accounted for through kE must be taken into account is the dependence of kE on CV or CVup and CVdown . Figures 14.7 - 14.10 indicate that this dependence might be approximated by polynomials of appropriate orders. However, the coefficients of these polynomials will be “parameter-dependent”, i.e., again dependent on M , E, and e in some unknown, complex manner. Therefore, these approximations are not pursued. Instead, piece-wise linear approximations are used. They are derived next.

464

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Empirical formula for the case of CVup = CVdown = CV : This empirical formula is based on the fact that all curves of Figures 14.7 - 14.10 lie below the exp for CV ∈ (0.25, 1] and below a constant linear function of CV with slope kE exp 0.25kE for CV ∈ (0, 0.25]. This is illustrated in Figures 14.7 - 14.10 by broken lines. Thus, kE for all CV ≤ 1 can be quantified by  exp  kE (M, E, e) · CV, if 0.25 < CV ≤ 1, (14.46) kE (M, E, e, CV ) ≤  exp (M, E, e), if 0 ≤ CV ≤ 0.25, 0.25kE or, equivalently, exp kE (M, E, e, CV ) ≤ max{0.25, CV }kE (M, E, e).

The tightness of this bound can be characterized by the function ∆1kE (CV ) =

upper kE

max

bound

A − kE

A kE

A∈{(14.33),(14.34)}

· 100%,

CV ≤ 1,

(14.47)

upper bound is the right-hand-side of (14.46). Function ∆1kE is illustrated where kE in Figure 14.14. Although, as one can see, the empirical formula is quite conservative, its utilization still yields to up to 400% reduction of buffering, as compared to that based on the exponential assumption (see Figures 14.7 - 14.10).

E=0.9

150

150

150

0

100

2

2

100

ε (CV)

200

50

50

0

0.2

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1

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0

200

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150

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50

50

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0

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ε (CV)

200

0

0

0.2

0.4

0.6

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0

1

100

2

2

ε (CV)

150

ε (CV)

200

150

50

50

0.4

0.6 CV

0.8

1

0

0.8

1

0.2

0.4

0.6

0.8

1

0.6

0.8

1

CV

200

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0

CV

100

0.6

50

150

0

0.4

100

200

0

0.2

CV

200

ε (CV)

2

ε (CV)

2

ε (CV)

0.6 CV

CV

e=0.95

100

50

CV

e=0.9

E=0.95

200

ε (CV)

2

e=0.85

ε (CV)

E=0.85 200

100

50

0

0.2

0.4

0.6

0.8

1

0

0

0.2

CV

Figure 14.14: Tightness of empirical formula (14.46)

0.4 CV

14.4. LEAN BUFFERING FOR NON-EXPONENTIAL MACHINES

465

Empirical formula for the case of CVup 6= CVdown : Since the upper bound (14.46) is not too tight (and, hence, may accommodate additional uncertainties) and the effects of CVup and CVdown on kE are not dramatically different, the following empirical formula for the case of CVup 6= CVdown is suggested: kE (M, E, e, CVup , CVdown ) max{0.25, CVup } + max{0.25, CVdown } exp ≤ kE (M, E, e), 2 CVup ≤ 1, CVdown ≤ 1, (14.48) exp where, as before, kE is defined by (14.22). If CVup = CVdown , (14.48) reduces to (14.46); otherwise, it takes into account different values of CVup and CVdown . If the first factor in the right-hand-side of (14.48) is denoted as

max{0.25, CVup } + max{0.25, CVdown } , 2

CVef f :=

(14.49)

expression (14.48) can be rewritten as exp kE ≤ k E (M, E, e) · CVef f .

(14.50)

This expression is illustrated in Figure 14.12 by broken lines. The utilization of this formula can be illustrated as follows: Suppose CVup = 0.1 and CVdown = 1. Then CVef f = 0.635 and, according to (14.50), exp kE < 0.625kE (M, E, e).

To investigate the validity of empirical formula (14.48), consider the following function: ∆2kE (M, E, e) = min min A∈{(14.33)} CVup ,CVdown ∈{(14.41)} h i upper bound A kE (M, E, e, CVef f ) − kE (M, E, e, CVup , CVdown ) , (14.51) upper where kE

bound upper kE

is the right-hand-side of (14.48), i.e., bound

exp (M, E, e, CVef f ) = kE (M, E, e) · CVef f .

If for all values of its arguments, function ∆2kE (M, E, e) is positive, the righthand-side of inequality (14.48) is an upper bound. The values of ∆2kE (M, E, e) for E ∈ {0.85, 0.9, 0.95} and e ∈ {0.85, 0.9, 0.95} are shown in Table 14.4. As one can see, ∆2kE (M, E, e) indeed takes positive values. Thus, empirical formula (14.48) holds for all distributions and parameters analyzed. The tightness of this bound can be evaluated by the function ∆3kE (CVef f ) = upper kE

bound

max

max

A∈{(14.33)} CVup ,CVdown ∈{(14.41)}

A (M, E, e, CVef f ) − kE (M, E, e, CVup , CVdown ) · 100%. (14.52) A kE (M, E, e, CVup , CVdown )

466

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES

Table 14.4: ∆2kE (10, E, e) for all CVup 6= CVdown cases considered E=0.85 0.1016 0.0425 0.0402

e = 0.85 e = 0.9 e = 0.95

E=0.9 0.0386 0.1647 0.0488

E=0.9

150

150

150

0

50

0

0.2

0.4

0.6

0.8

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1

50

0

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CV

0

1

150

50

50

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0

1

50

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0

1

1

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1

100

4

4

4

100

50

0.8

0.8

eff

ε (CV )

eff

eff

ε (CV )

150

0.6

0.6 eff

200

eff

0.4 CV

150

CV

0.2

eff

200

0.4

0

CV

150

0.2

1

100

200

0

0.8

50

eff

100

0.6

4

100

4

4

100

0.4

0.4

eff

eff

ε (CV )

200

150 ε (CV )

200

0.2

0.2

eff

150

0

0

CV

200

eff

ε (CV )

0.8

eff

CV

ε (CV )

0.6 CV

eff

e=0.95

100

4

100

4

100

eff

eff

eff

ε (CV )

200

ε (CV )

200

50

e=0.9

E=0.95

200

4

e=0.85

ε (CV )

E=0.85

E=0.95 0.0687 0.1625 0.1200

50

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

CV

eff

Figure 14.15: Tightness of the empirical formula (14.48)

CV

eff

14.5. SUMMARY

467

Figure 14.15 illustrates the behavior of this function. Comparing this with Figure 14.14, we conclude that the tightness of bound (14.48) appears to be similar to that of (14.46). Although the above results have been obtained only for Weibull, gamma, and log-normal reliability models, we conjecture that: Empirical formulas (14.46) and (14.48) hold for any unimodal reliability model of machine up- and downtime. Generalizations: The results described above have been derived under assumptions (14.35)-(14.37), i.e., all machines have identical average up- and downtime and identical coefficients of variation. How should ki,E be selected if these assumptions are not satisfied? To account for different up- and downtime, LLB can be selected as discussed j , i = 1, . . . , M − 1, j = I, . . . , V I. To account for in Section 14.3, i.e., as ki,E different coefficients of variation, CVef f , can be defined as CVef f,i :=

max{0.25, CVup,i } + max{0.25, CVdown,i } , 2

i = 1, . . . , M.

Thus, an empirical formula for ki,E in serial lines with non-identical, nonexponential machines can be given as follows: j,non−exp ≤ ki,E

CVef f,i + CVef f,i+1 j · ki,E , 2

i = 1, . . . , M − 1,

(14.53)

where j ∈ {I, . . . , V I} is the same for all i.

14.5

Summary

• Lean buffering in serial lines with continuous time models of machine reliability can be characterized in terms of two dimensionless parameters: level of buffering (which quantifies buffer capacities in units of downtime) and line efficiency (which quantifies the desired production rate in units of the largest production rate available in the system). • Closed formulas for the lean level of buffering (LLB) have been derived for exponential lines with identical machines. • For exponential lines with non-identical machines, a closed formula is presented only for the two-machine case. For longer lines, LLB approximations, based on either closed formulas or recursions, have been developed. • Among the closed formula approaches, the so-called global pair-wise approach is recommended in lines with fewer than 15 machines; otherwise, the local upper bound approach is deemed the best. Among the recursive approaches, the bottleneck-based one is preferred. • For lines with non-exponential machines, it is shown that LLB is not sensitive to the shapes of up- and downtime distributions and depends mainly on their coefficients of variation (CV ).

468

CHAPTER 14. DESIGN OF LEAN CONTINUOUS LINES • Based on this observation, empirical formulas are proposed, which provide an upper bound on LLB as a function of up- and downtime CV ’s, given that both of them are less than 1.

14.6

Problems

Problem 14.1 Consider a production line with five identical exponential machines having e = 0.95. (a) Determine the largest line efficiency that can be achieved in the JIT fashion (i.e., with kE = 0). (b) Determine kE with E increased by 3% as compared to the one obtained in (a). (c) Which E would you recommend for practical implementation and why? Problem 14.2 Investigate the accuracy of the rule-of-thumb given in Table 14.3. Specifically, consider an exponential line with 15 machines having e = 0.81. Assume that the desired line efficiency is 0.93. (a) Using expressions (14.20) and (14.22), calculate b kE . (b) Using the rule-of-thumb, determine the lean level of buffering for this system. (c) Compare the two results and make a recommendation as to the applicability of the rule-of-thumb for values of e and E, which are not included in Table 14.3. Problem 14.3 Consider the exponential model of the ignition production line of Problem 3.3. (a) Using any approach of Subsection 14.3.3, design the lean level of buffering for this system with E = 0.9. (b) Using the same approach as in (a), design the LLB for this system with E = 0.97. (c) Compare the two results and recommend which line efficiency should be pursued in practice. Problem 14.4 Consider the exponential model of the serial line given in Problem 3.4. Assume that the desired line efficiency is 0.97. (a) Determine the estimate of LLB and the lean buffer capacity using the global upper bound approach. (b) Determine the estimate of LLB and the lean buffer capacity using the bottleneck-based approach. (c) Compare the two results and state which of the approaches is preferable (from all points of view).

14.7. ANNOTATED BIBLIOGRAPHY

469

Problem 14.5 Consider a production line with exponential machines having e = [0.88, 0.95, 0.9, 0.92], Tdown = [2, 1, 2, 1]. (a) Select ki,E and Ni,E for E = 0.95 using the following methods: global pairwise approach, local upper bound approach, bottleneck-based approach, and full search approach. (b) Compare the results and comment on the advantages and disadvantages of each of these approaches. Problem 14.6 Consider a two-machine asynchronous exponential line with N = 50 and m1 : m2 :

λ1 = 0.01, µ1 = 0.1, c1 = 1, λ2 = 0.011, µ2 = 0.09, c2 = 1.03.

(a) By modifying µ2 , reduce m2 to a machine with c2 = 1 and T P2 unchanged. (b) Calculate T P of the original and modified system. (c) Calculate kE and NE , E = 0.97, for the modified system using expression (14.26). (d) Calculate kE and NE , E = 0.97, for the original system using the trial and error approach. (e) Compare the two results and make a conclusion as to the efficacy of this approach to LLB calculation in serial asynchronous exponential lines. Problem 14.7 Consider a five-machine non-exponential serial line with e = 0.92, Tdown = 3, CVup = 0.8, and CVdown = 0.2. Determine the upper bound for kE and NE in this system.

14.7

Annotated Bibliography

Buffer capacity allocation has been studied in numerous publications. Several examples can be given as follows: [14.1] T. Altiok and S.S. Stidham, “The Allocation of Interstage Buffer Capacities in Production Lines,” IIE Transactions, vol. 15, pp. 292–299, 1983. [14.2] M. Caramanis, “Production Line Design: A Discrete Event Dynamic System and Generalized Benders Decomposition Approach,” International Journal of Production Research, vol. 25, pp. 1223–1234, 1987. [14.3] R. Conway, W. Maxwell, J.O. McClain and L.J. Thomas, “The Role of Work-in-process Inventory in Serial Production Lines,” Operations Research, vol. 25, pp. 229–241, 1988.

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[14.4] S.B. Gershwin and Y. Goldis, “Efficient Algorithms for Transfer Line Design,” Lab for Manufacturing and Productivity, MIT, Cambridge, MA, Rep. LMP-95-005, 1995. [14.5] H. Yamashita and T. Altiok, “Buffer Capacity Allocation for a Desired Throughput of Production Lines,” IIE Transactions, vol. 30, pp. 883–891, 1998. [14.6] J.H. Harris and S.G. Powell, “An Algorithm for Optimal Buffer Placement in Reliable Serial Lines,” IIE Transactions, vol. 31, pp. 287–302, 1999. [14.7] S.B. Gershwin and J.E. Schor, “Efficient Algorithms for Buffer Space Allocation,” Annals Operations Research, vol. 93, pp. 117–144, 2000. [14.8] H. Tempelmeier, “Practical Considerations in the Optimization of Flow Production Systems,” International Journal of Production Research, vol. 41, pp. 149–170, 2003. [14.9] J.M. Smith and F.R.B. Cruz, “The Buffer Allocation Problem for General Finite Buffer Queueing Networks,” IIE Transactions, vol. 37, pp. 343– 365, 2005. The material of Section 14.2 is based on [14.10] E. Enginarlar, J. Li and S.M. Meerkov, “How Lean Can Lean Buffers Be?” IIE Transactions, vol. 37, pp. 333-342, 2005. The results included in Section 14.3 have been derived in [14.11] S.-Y. Chiang, A.B. Hu and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Non-identical Exponential Machines,” IEEE Transactions on Automation Science and Engineering, vol. 5, pp. 298-306, 2008. The presentation of Section 14.4 follows [14.12] E. Enginarlar, J. Li and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Non-exponential Machines,” OR Spectrum, vol. 27, pp. 195-219, 2005.

Chapter 15

Customer Demand Satisfaction in Continuous Lines Motivation: The motivation here is the same as in Chapter 9: the need for rigorous engineering methods for analysis and design of serial production lines from the point of view of customer demand satisfaction. In this chapter, such methods are developed for serial lines with continuous time models of machine reliability. Overview: The development is carried out in terms of a parametrization, which is a generalization of that used in Chapter 9. Analytical methods for calculating the due time performance (DT P ) in exponential lines are derived. A lower bound on DT P in non-exponential lines is given. In addition, effects of demand variability on DT P are explored. Finally, system-theoretic properties of DT P are discussed, and it is shown that there exists a so-called law of conservation of filtering, according to which the amount of filtering in space multiplied by the amount of filtering in time is practically constant. This law can be used for designing lean production lines from the point of view of customer demand satisfaction.

15.1

Modeling and Parametrization

15.1.1

Production-inventory-customer (PIC) system

The manufacturing system considered in this chapter is shown in Figure 15.1. It consists of three subsystems: the production subsystem (PS), the inventory subsystem (IS), and the customer subsystem (CS). Each of them and their interactions are defined as follows: 471

472

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES Production Subsystem

T α , T β , CV α, CV

Inventory Subsystem

Customer Subsystem

N FGB

D,T

β

Figure 15.1: Production-inventory-customer system Production subsystem: (i) The production subsystem is intended to produce one part during a fixed cycle time. Due to machine breakdowns, this may or may not happen, depending on the status of the last machine in the system (up or down) and the buffer occupancy in front of it (empty or not). Therefore, the PS may be in two states: active or passive. When active, a part is produced during each cycle time; when passive, no parts are produced. (ii) The time intervals, during which the production system is active or passive, are random variables, tα and tβ , with expected values and coefficients of variation Tα , CVα and Tβ , CVβ , respectively. Inventory subsystem: (iii) The inventory subsystem consists of a finished goods buffer (FGB) with capacity 0 < NF GB < ∞. Parts produced by PS are immediately transferred to FGB. Interaction between PS and IS: (iv) The PS is blocked at time t if the FGB is full at time t and the customer subsystem does not take out parts from the FGB at time t. Customer subsystem: (v) The customer requires D parts to be shipped during each shipping period. The duration of the shipping period is T cycles, where, as in all previous chapters, the cycle is the time necessary to process a part. To avoid triviality, it is assumed that D ≤ T · P R,

(15.1)

where P R is the production rate of the PS in isolation, i.e., when neither IS nor CS is present Interactions among the PS, IS, and CS: (vi) At the beginning of each shipping period i, parts are removed from the FGB in the amount of min{H(i − 1), D}, where H(i − 1) is the number of

15.1. MODELING AND PARAMETRIZATION

473

parts in the FGB at the end of the previous, (i − 1)-th, shipping period. If H(i − 1) ≥ D, the shipment is complete; if H(i − 1) < D, the balance of the shipment, i.e., D − H(i − 1) parts, is to be produced by the PS during the i-th shipping period. The parts produced are immediately removed from the FGB and prepared for shipment, until the shipment is complete, i.e., D parts are available. If the shipment is complete before the end of the shipping period, the PS continues operating, but with the parts being accumulated in the FGB, either until the end of the shipping period or until the PS is blocked, whichever occurs first. If the shipment is not complete by the end of the shipping period, an incomplete shipment is sent to the customer. No backlog is allowed. Similar to Chapter 9, the assumption that the demand D is constant is introduced to simplify the presentation; in Section 15.5, the results are generalized to the case of a random demand.

15.1.2

DT P definition

The definition of DT P remains the same as in Chapter 9, i.e., DT P = P [tˆ(i) + H(i − 1) ≥ D],

(15.2)

where tˆ(i) is the random variable representing the number of parts produced by the PS during the i-th shipping period and H(i − 1) is the number of parts in the FGB at the end of the (i − 1)-th shipping period. Given the production-inventory-customer system (i)-(vi), the goal of this chapter is to develop a method for DT P calculation and analyze its systemtheoretic properties.

15.1.3

Parametrization

The above PIC system is specified by (Tα , CVα ), (Tβ , CVβ ), NDT P , D, and T . In order to characterize the regimes, in which PIC is operating, similar to the Bernoulli case, introduce normalized values of these parameters. Relative FGB capacity (ν) – the value of NF GB in units of D, i.e., ν=

NF GB . D

(15.3)

Clearly, ν characterizes regimes of operation from the point of view of filtering in space. Load factor (L) – the value of D in units of average production during T , i.e., D L= . (15.4) T · PR In addition, for the continuous time model of PS, we introduce

474

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES Relative shipping period (τ ) – the quantity of T in units of Tα + Tβ , i.e., τ=

T . Tα + Tβ

(15.5)

The value of Tα + Tβ is referred to as the reliability cycle. Clearly, τ quantifies filtering in time: when τ is large, the shipping period offers a significant level of filtering in time; small τ implies a regime with insignificant time filtering. The parameters ν, τ , and L are used throughout this chapter to characterize the regime of operation of various PIC systems considered, to establish the law of conservation of filtering, and to describe a method for selecting lean FGB.

15.2

DT P Evaluation in PIC Systems with Exponential Machines

The PIC system considered here is shown in Figure 15.2. Each machine mi is defined by parameter λi and µi , i = 1, . . . , M , and each in-process buffer bi by its capacity Ni , i = 1, . . . , M − 1. The conventions of PS operation are the same as in Chapter 11. As in the Bernoulli case, we first address the issue of DT P for M = 1 and then extend it to an arbitrary M . Inventory Subsystem

Production Subsystem

λ 1 , µ1

λ2, µ 2

N1

λ M−1, µ M−1 N M−1

Customer Subsystem

D,T

λ M, µ M

N FGB

Figure 15.2: Production-inventory-customer system with exponential machines

15.2.1

DT P in PIC systems with one-machine PS

Let t(i) be the number of parts produced by PS during a shipping period if IS were of infinite capacity. Introduce the following quantities: P(x) = P [t(i) ≥ x], x ∈ {0, 1, . . . , T }, rk,l = P [t(i) = D + k − l], k = 1, . . . , NF GB − 1, l = 0, 1, . . . , NF GB , rbNF GB ,l

=

P [t(i) ≥ D + NF GB − l], l = 0, 1, . . . , NF GB .

These quantities can be calculated as follows: P(x) =

· ¸ j−2 ∞ X X (λx)j−1 ³ [µ(T − x)]k ´ µe−λx −µ(T −x) 1−e 1+ λ+µ (j − 1)! k! j=2 k=0

15.2. PIC SYSTEM WITH EXPONENTIAL MACHINES

475

· ¸ j−1 ∞ X [µ(T − x)]k λe−λx X (λx)j−1 −µ(T −x) + 1−e , λ + µ j=1 (j − 1)! k!

(15.6)

k=0

rk,l rbNF GB ,l

= P(D + k − l) − P(D + k − l + 1),

(15.7)

= P(D + NF GB − l).

(15.8)

Introduce a nonsingular matrix R and vector Z0 defined by  r1,2 − r1,0 ... r1,N − r1,0 r1,1 − r1,0 − 1  r2,1 − r2,0 r − r − 1 . . . r2,N − r2,0 2,2 2,0 R=  ... ... ... ... rbN,2 − rbN,0 . . . rbN,N − rbN,0 − 1 rbN,1 − rbN,0

  , 

 r1,0  r2,0   Z0 =   ··· . rbN,0

(15.9)



(15.10)

Theorem 15.1 Under assumptions (i)-(vi) with M = 1 and an exponential machine, NX F GB zk P(D − k), (15.11) DT P = k=0

where zk = P [H(i − 1) = k], k = 0, 1, . . . , N , and vector Z = [z1 , z2 , . . . , zN ]T is calculated according to Z = −R−1 Z0 . (15.12) Proof: See Section 20.2. The behavior of DT P as a function of L and ν is illustrated in Figures 15.3 and 15.4. Clearly, DT P is monotonically decreasing in L and monotonically increasing in ν and τ . As it follows from Theorem 15.1, the calculation of DT P involves evaluations of infinite sums in (15.6) and matrix inversion in (15.12). In our calculations, the infinite sums have been evaluated by truncating higher order terms so that the first neglected term is less than 10−10 . (The terms are decreasing due to the factorials in the denominators of (15.6).) The inverse of R exists due to the ergodicity of the Markov process at hand.

15.2.2

DT P in PIC systems with M -machine PS

Theorem 15.1 can be used to obtain a lower bound on DT P in PIC systems with more than one exponential machine. Specifically, similar to the Bernoulli case, let DT PM be the DT P of an M -machine PIC system and DT P1 be the duetime performance in the one-machine case, where the machine is the aggregation of the M -machine system with the aggregation carried out using the method in Chapter 11. Then,

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

λ=0.1, µ=0.9, N

=3

FGB

1

DTP

0.8 0.6 0.4

T=40 T=20 T=12 T=4

0.2 0 0

0.2

0.4

0.6 L

0.8

1

1.2

Figure 15.3: DT P as a function of load factor, L

λ=0.1, µ=0.9, T=20, D=15, L=0.8333 1.01 1 0.99 DTP

476

0.98 0.97 0.96 0.95 0

0.2

0.4

ν

0.6

0.8

1

Figure 15.4: DT P as a function of relative finished goods buffering, ν

15.3. PIC SYSTEM WITH NON-EXPONENTIAL PS

477

Numerical Fact 15.1 Under assumptions (i)-(vi) with exponential machines, the following inequality holds: DT PM ≥ DT P1 .

(15.13)

This fact has been established by extensive numerical simulations. No counterexamples have been discovered. Figure 15.5 illustrates this statement for a five-machine line, where DT PM is evaluated numerically and DT P1 is calculated using (15.11).

λ =0.1, µ =0.6, N =2, T=20, D=12 i

1

i

i

DTP

5

DTP

0.9 DTP

1

0.8 0.7 0.6 0

0.5

νDTP

1

1.5

Figure 15.5: Illustration of DT PM and its lower bound DT P1

15.3

DT P in PIC Systems with Non-exponential PS

For the case of non-exponential PS, DT P has been analyzed using the approach similar to that of Chapter 12. Specifically, under the assumption that the pdf’s of tα and tβ are in the set {W, ga, LN }, numerous PIC systems defined by assumptions (i)-(vi) have been simulated, leading to the observation that DT P depends mainly on the coefficients of variation of tα and tβ and is practically independent of the higher order moments. This is illustrated in Figures 15.6 and 15.7 for systems with Tα + Tβ e CVα = CVβ

=

25,

(15.14)

=

0.825,

(15.15)

=: CV

(15.16)

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

τ = 1, ν = 0.4 1

DTP

0.96 0.92

0.88 Lognormal Gamma Weibull

0.84 0.8 0

0.2

0.4 CV

0.6

0.8

1

Figure 15.6: DT P for non-exponential PS: System 1

τ = 3, ν = 0.4 1 0.98 DTP

478

0.96 0.94 Lognormal Gamma Weibull

0.92 0.9 0

0.2

0.4 CV

0.6

0.8

1

Figure 15.7: DT P for non-exponential PS: System 2

15.4. CONSERVATION OF FILTERING

479

and τ and ν as indicated in the figures. Although, unlike Chapter 12, the resulting curves are far from being linear, they can be used to define a lower bound on DT P . Indeed, for CV = 0, DT P is clearly 1; for CV = 1, DT P (denoted as DT P exp ) can be calculated using Theorem 15.1; for all intermediate values of CV , DT P can be characterized by a lower bound as follows: DT P ≥ 1 − (1 − DT P exp )CV.

(15.17)

We conjecture that: If ν > 0, bound (15.17) holds for any unimodal pdf of tα and tβ and for CVα 6= CVβ ; in the latter case, CV in (15.17) should be substituted by CVave =

CVα + CVβ . 2

15.4

Lean FGB and Conservation of Filtering in PIC Systems with Exponential Machines

15.4.1

Lean FGB

Consider a PIC system with one exponential machine. The DT P of this system can be calculated using Theorem 15.1. Assume that the desired value of DT P is denoted as DT Pdes . The smallest ν, which ensures DT Pdes , is referred to as lean and is denoted as νDT P . In this section, the relationship between the filtering in space and time is described and a method for selecting νDT P is presented. A “brute-force” method for calculating νDT P is as follows: For given Tα , Tβ , D and T , evaluate L and τ . Then, assuming NF GB = 0, use Theorem 15.1 and calculate DT P , i.e., DT P (NF GB = 0). If it is larger than DT Pdes , the lean FGB capacity is 0. Otherwise, assume NF GB = 1, calculate DT P (NF GB = 1) and again compare it to the desired DT P . Continue this process until arriving at the smallest NF GB , denoted as NDT P , for which DT P is larger than DT Pdes . Then, evaluate νDT P as follows: νDT P =

NDT P . D

(15.18)

Thus, for given L and τ , the value of νDT P is determined.

15.4.2

Conservation of filtering in lean systems

The typical behavior of νDT P as a function of τ and L, i.e., νDT P = FDT P (τ, L), is illustrated in Figure 15.8. Three regimes of system operation are presented: heavy load (L = 0.97), medium load (L = 0.92), and light load (L = 0.71).

480

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

These graphs are calculated, using the approach described above, for the production system characterized by e = 0.825 and Tα + Tβ = 25, i.e., Tα = 20.625, and Tβ = 4.375. Also, the desired DT P is assumed to be 0.99, which is a typical desired level of customer demand satisfaction in the automotive industry. Since in most practical situations, the shipping period is longer than the reliability cycle, we assume τ ≥ 1.

5

L=0.97 L=0.92 L=0.71

ν

DTP

4 3 2 1 0 1

2

4

τ

6

8

Figure 15.8: Typical behavior of νDT P vs. τ Clearly, the graphs of Figure 15.8 exhibit a tradeoff between filtering in time and filtering in space. For instance, in the heavy load regime, τ = 5 requires νDT P ∼ = 1, while for τ = 1, the νDT P ∼ = 4.5 is necessary. Similar tradeoffs take place in the medium and light load regimes. However, the amount of filtering in space, necessary to achieve the desired DT P , drops down significantly when the load is decreased. For example, in the light load regime and τ = 5, no finished goods buffer is necessary, and the deliveries can be just-in-time, while this is impossible for medium and heavy loads. This dramatic improvement in the acceptable leanness, ensured by low loads, may be a justification for operating in light load regimes: it leads to a high level of customer demand satisfaction with a small (if any) finished goods inventory. Effect of e and (Tα , Tβ ): From the graphs of Figure 15.8, it is clear that νDT P strongly depends on the load factor L. The question arises: How does νDT P depend on e and (Tα , Tβ )? An answer is suggested by Figure 15.9, where νDT P = FDT P (τ, L) is plotted for various values of e and Tα + Tβ . Clearly, the effect of e is significant, while the effect of Tα +Tβ is not. Indeed, except for the case of τ close to 1, the values of νDT P are almost independent of the

15.4. CONSERVATION OF FILTERING

481

specific values of Tα and Tβ as long as e (i.e., Tα /Tβ ) is the same: the maximum difference, ¯ ¯ ¯ ¯ ∆1 = max ¯νDT P (Tα + Tβ = 50, τ, L, e) − νDT P (Tα + Tβ = 25, τ, L, e)¯, L,e,DT P

is less than 0.09 for τ ≥ 2 (except for the case of L = 0.97, where ∆1 is 0.30 and 0.89 for e = 0.825 and e = 0.74, respectively). Thus, we conclude that, for any DT P , νDT P depends mainly on τ , L and e, i.e., νDT P = FDT P (τ, L, e).

Heavy load L = 0.97

Medium load L = 0.92 1.5

Tα+Tβ=25, DTP=0.99

0

8

4

Tα+Tβ=25, DTP=0.95

3

Tα+Tβ=50, DTP=0.95

νDTP

2

8

T +T =25, DTP=0.99 α

Tα+Tβ=25, DTP=0.95

Tα+Tβ=50, DTP=0.95

8 6

Tα+Tβ=50, DTP=0.95

2

τ

6

2

8

4

τ

6

0

Tα+Tβ=50, DTP=0.99

2

0

8

T +T =25, DTP=0.99 α

2

Tα+Tβ=25, DTP=0.95

α

Tα+Tβ=50, DTP=0.95

τ

6

8

β

Tα+Tβ=25, DTP=0.95

10

Tα+Tβ=50, DTP=0.99

4

T +T =25, DTP=0.99

12

β

6

2

8

1

8

4

4

6

3

2 2

τ

3

4

0

4

Tα+Tβ=25, DTP=0.99

Tα+Tβ=25, DTP=0.95

Tα+Tβ=50, DTP=0.99

10

Tα+Tβ=50, DTP=0.99

2

5

Tα+Tβ=25, DTP=0.99

12

β

0

8

4

0

6

6

Tα+Tβ=50, DTP=0.95

0

τ

τ

Tα+Tβ=25, DTP=0.95

1 4

4

4

1 2

2

5

Tα+Tβ=50, DTP=0.99

Tα+Tβ=50, DTP=0.95

0.5

νDTP

6

νDTP

νDTP

τ

Tα+Tβ=25, DTP=0.99

10

νDTP

4

Tα+Tβ=50, DTP=0.99

1

νDTP

2

12

e = 0.74

Tα+Tβ=50, DTP=0.95

0.5

5

e = 0.825

Tα+Tβ=25, DTP=0.95

Tα+Tβ=50, DTP=0.99

1

νDTP

νDTP

Tα+Tβ=50, DTP=0.95

0.5

0

Tα+Tβ=25, DTP=0.99

Tα+Tβ=25, DTP=0.95

Tα+Tβ=50, DTP=0.99

1

1.5

Tα+Tβ=25, DTP=0.99

Tα+Tβ=25, DTP=0.95

e = 0.91

Light load L = 0.71

νDTP

1.5

(15.19)

Tα+Tβ=50, DTP=0.99

Tα+Tβ=50, DTP=0.95

8 6 4 2

2

4

τ

6

8

0

2

4

τ

6

8

Figure 15.9: Effect of e and (Tα , Tβ ) on νDT P

Conservation law: The nature of the curves in Figures 15.8 and 15.9 suggests that νDT P and τ could be related in a hyperbolic manner, i.e., τ · νDT P = const.

(15.20)

If this were the case, the constant in the right hand side of (15.20) would depend on L and e only. To evaluate this constant, one can use the approach of Section 15.2 to calculate F (1, L, e) and then, combining (15.19) and (15.20), obtain: τ · νDT P = FDT P (1, L, e).

(15.21)

482

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

The accuracy of this expression is illustrated in Figure 15.10 where νDT P = F (τ, L, e) is plotted along with νDT P = 1/F (1, L, e). Clearly, the two curves are quite close; with the exception of the case L = 0.97 and e = 0.74, the maximum error, ¯ FDT P (1, L, e) ¯¯ ¯ ∆2 = max ¯FDT P (τ, L, e) − ¯, τ,L,e τ is less than 0.18 for DT P = 0.99 and 0.14 for DT P = 0.95. Although for the case of L = 0.97 and e = 0.74 the maximum error is relatively large (1.05 for DT P = 0.99 and 0.22 for DT P = 0.95), we conclude that for all practical purposes (15.21) approximates (15.19) sufficiently well. Expression (15.21) can be interpreted as a conservation law: Numerical Fact 15.2 For lean PIC systems defined by assumptions (i)(vi) with exponential machines, the amount of filtering in time (in units of τ ) multiplied by the amount of filtering in space (in units of νDT P ) is practically constant. Expression (15.21) implies, in particular, that • changing the shipping period by a factor of k requires a change in the lean FGB capacity by a factor of 1/k; • the lean FGB capacity can be evaluated as NDT P = νDT P D =

FDT P (1, L, e) FDT P (1, L, e) D= (Tα + Tβ )D. τ T (15.22)

15.5

DT P in the Case of Random Demand

15.5.1

Random demand modeling

In the previous sections, it has been assumed that the demand D per shipping period T is constant. Although strictly speaking this is the case in many manufacturing systems with long-term contractual relationships between the manufacturer and the customer, in reality D may have some variability. In this section, we model this variability by assuming that D is a random variable. More precisely, instead of assumption (v) of Subsection 15.1.1, suppose the following: (v0 ) At the end of each shipping period i, D(i) parts must be shipped to the customer, where D(i), i = 1, 2 . . ., is a sequence of independent, identically distributed random variables defined by the probability mass function P [D = Dj ] = Pj , j = 1, . . . , J. Thus, the customer is defined by the triplet (D, P, T ), where D = [D1 , . . . , DJ ], P = [P1 , . . . , PJ ], and T is the shipping period. It is assumed that D ≤ T · P R, where D is the expected value of D(i) and P R is the production rate of the PS in isolation. Below, a method for calculating DT P for this demand is provided.

15.5. DT P FOR RANDOM DEMAND Heavy load L = 0.97 1.5

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

0.5

0.5

0

8

0

0

6

8

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

10

νDTP

2

2

0

2

4

τ

6

0

8

4

τ

6

8

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

4

2

4

τ

6

3 2

0

8

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

2

4

τ

6

8

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

12 10

6 4

2

5

8

4

0

8

1

10

6

6

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

12

8

τ

2 1

τ

4

3

1 4

2

4

2

2

0.5

5

νDTP

νDTP

6

3

12

νDTP

τ

ν =F (τ) DTP DTP νDTP=FDTP(1)/τ

4

e = 0.74

4

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

1

νDTP

2

5

e = 0.825

1.5

νDTP

0

Light load L = 0.71

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

1

νDTP

1

νDTP

Medium load L = 0.92

νDTP

1.5

e = 0.91

483

8 6 4 2

2

4

τ

6

0

8

2

4

τ

6

8

(a). DTP= 0.99 Heavy load L = 0.97 νDTP=FDTP(τ) νDTP=FDTP(1)/τ

0.1

0.1

τ

6

0

8

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

1

2

4

τ

6

τ

6

0

8

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

2

νDTP

4

τ

6

2 1.5

1

8

0

2

4

τ

6

8

τ

6

8

2

0.5

6

4

1.5

1

τ

8

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

3

0.5 4

2

2.5

1

2

6

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

3.5

0.5 0

τ

1

0

8

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

3 2.5

2

4

0.5

3.5

1.5

2

1.5

1

0

8

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

3 2.5

νDTP

4

νDTP

0

0.3

0.1 2

0.5

3.5

0.4

0.2

1.5

νDTP

νDTP

4

0.5

e = 0.74

0.3 0.2

2

0.5

0.4

0.2

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

0.6

νDTP

0.3

1.5

e = 0.825

0.5

0.4

0

νDTP=FDTP(τ) νDTP=FDTP(1)/τ

0.6

νDTP

νDTP

0.5

Light load L = 0.71

νDTP

0.6

e = 0.91

Medium load L = 0.92

0

2

4

τ

6

8

(b). DTP= 0.95 Figure 15.10: Accuracy of the conservation law: Functions νDT P = FDT P (τ ) and νDT P = FDT P (1)/τ

484

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

15.5.2

DT P for random demand in PIC systems with exponential machines

Again, introduce the following quantities: rk,l,j

=

P [t(i) = D(i) + k − l], k = 1, . . . , NF GB − 1, l = 0, 1, . . . , NF GB , D(i) = Dj , j = 1, · · · , J,

rbNF GB ,l,j

=

P [t(i) ≥ D(i) + NF GB − l],

l = 0, 1, . . . , NF GB , D(i) = Dj , j = 1, . . . , J.

These quantities can be calculated as follows: rk,l,j = P(Dj + k − l) − P(Dj + k − l + 1),

(15.23)

and rbNF GB ,l,j = P(Dj + NF GB − l),

(15.24)

where P(x) is given in (15.6). Introduce matrix R and vector Z0 defined by R=  PJ j=1 (r1,1,j − r1,0,j )Pj − 1 P J  j=1 (r2,1,j − r2,0,j )Pj   ... PJ rN,1,j − r bN,0,j )Pj j=1 (b

PJ (r − r1,0,j )Pj PJ j=1 1,2,j j=1 (r2,2,j − r2,0,j )Pj − 1 ... PJ rN,2,j − r bN,0,j )Pj j=1 (b

... ... ... ...

PJ (r1,N,j − r1,0,j )Pj Pj=1 J j=1 (r2,N,j − r2,0,j )Pj ... PJ rN,N,j − r bN,0,j )Pj − 1 j=1 (b

  , 

(15.25)

 PJ j=1 r1,0,j Pj  PJ  j=1 r2,0,j Pj Z0 =   ··· PJ bN,0,j Pj j=1 r

   . 

(15.26)

Theorem 15.2 Under assumptions (i)-(iv), (v0 ), (vi) with M = 1 and an exponential machine, DT P =

NX F GB

J X

zk P(Dj − k)Pj ,

(15.27)

k=0 j=1

where zk = P [H(i−1) = k], k = 0, 1, . . . , NF GB , and vector Z = [z1 , z2 , . . . , zNF GB ]T is calculated according to (15.12). Proof: See Section 20.2. Again, Theorem 15.2 can be useful for DT P evaluation in productioninventory-customer systems with more than one exponential machine because, as it is to show, (15.28) DT PM ≥ DT P1 , where DT PM is the due-time performance of an M -machine PIC system and DT P1 is the due-time performance of the aggregated one-machine system.

15.5. DT P FOR RANDOM DEMAND

15.5.3

485

DT P degradation as a function of demand variability

To analyze the behavior of DT P as a function of demand variability, consider five uniform demand distributions given in Table 15.1. All of them have D = 15 and coefficients of variation (CV ) ranging from 0.1 to 0.8. Table 15.1: Uniform pmf’s considered CV 0.1 0.25 0.4 0.6 0.8

Di 12, 13, . . ., 17, 18 9, 10, . . ., 20, 21 5, 6, . . ., 24, 25 4, 15, 26 3, 27

Pj 1/7 1/13 1/21 1/3 1/2

To analyze DT P degradation for various FGB capacities, we consider an IS with N taking values 1, 5, 15, and 45. To investigate the properties of DT P for various levels of customer demand, we consider three load factors: low (L = 0.9159), medium (L = 0.95114), and high (L = 0.9892). The parameters of the three production subsystems and the shipping periods that result in the above mentioned load factors are shown in Table 15.2. To ensure fairness, the efficiency of each PS is the same, e = 0.6066, as is the relative shipping period, τ = 1.0918. Table 15.2: Systems analyzed System 1 System 2 System 3

Load Factor Low L = 0.9159 Medium L = 0.9511 High L = 0.9892

λ

µ

T

0.0667

0.1028

27

0.0692

0.1067

26

0.0720

0.1110

25

The behavior of DT P as a function of the demand CV is shown in Figures 15.11 - 15.13. These figures include also the case of deterministic demand (CV = 0). Examining these data, we conclude the following: • As expected, for all values of LF and N , larger CV s lead to lower DT P s. Similarly, larger LFs lead to lower DT P s. Finally, for any CV and LF, larger N results in larger DT P . • Unexpectedly, the curves representing DT P as function of CV for all LF with identical N are practically congruent. This implies that the percent

486

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

− p=0.0667, r=0.1028, e=0.6066, T=27, D=15, τ=1.0918, L=0.9159 1 0.95 0.9

N

FGB

=45

0.85

DTP

0.8 N

0.75

=15

FGB

0.7 0.65 N 0.6 N

0.55 0.5

0

0.1

0.2

0.3

FGB

FGB

0.4 CV

=5

=1

0.5

0.6

0.7

0.8

Figure 15.11: DT P degradation as a function of demand randomness: System 1, low LF case

− p=0.0692, r=0.1067, e=0.6066, T=26, D=15, τ=1.0918, L=0.9511 0.95 0.9 N

FGB

=45

0.85 0.8 0.75 DTP

N

FGB

=15

0.7 0.65 N

0.6

FGB

=5

0.55 N

FGB

=1

0.5 0.45

0

0.1

0.2

0.3

0.4 CV

0.5

0.6

0.7

0.8

Figure 15.12: DT P degradation as a function of demand randomness: System 2, medium LF case

15.5. DT P FOR RANDOM DEMAND

487

− p=0.072, r=0.111, e=0.6066, T=25, D=15, τ=1.0918, L=0.9892 0.9

0.85

N

FGB

=45

0.8

0.75 N

FGB

=15

DTP

0.7

0.65

0.6 N

FGB

=5

0.55 N

0.5

0.45

0

0.1

0.2

0.3

FGB

=1

0.4 CV

0.5

0.6

0.7

0.8

Figure 15.13: DT P degradation as a function of demand randomness: System 3, high LF case of DT P degradation as a function of CV is almost independent of LF, as long as N remains the same.

15.5.4

Effect of demand pdf on DT P

The method developed in Subsection 15.5.2 implies that DT P is a functional of pmf of the demand. Below, using examples, we show that DT P for various types of pmf’s remains practically the same as long as their CV s are identical. This implies that DT P can be viewed as a function of CV , no matter what the shape of pmf may be. The pmf’s that we analyze are uniform, triangular, and “ramp” (see Figure 15.14). We consider three groups of these distributions, with CV s equal to 0.1, 0.25, 0.4, and 0.8. The expected values of all distributions are 15. Using these pmf’s and Theorem 15.2, we calculate DT P as a function of N for the three systems defined in Table 15.2. The results are shown in Figures 15.15 - 15.17. From these data we conclude: • For every value of N , DT P for all pmf’s considered are almost the same, as long as CV s are identical. • The largest differences occur for the smallest N and the largest CV . Table 15.3 quantifies these differences, which take place due to the effects of higher moments of corresponding distribution. Based on these results, we conjecture that:

488

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES CV=0.1

CV=0.25

Uniform Triangular Ramp

0.3 0.25

0.1 0.08 j

P

j

0.2 P

Uniform Triangular Ramp

0.12

0.15

0.06

0.1

0.04

0.05

0.02

0 10

12

14

16

18

0

20

5

10

15 Dj

D

j

CV=0.4

0.05

Uniform Triangular Ramp

0.4

P

j

0.3

j

0.04 P

25

CV=0.8 0.5

Uniform Triangular Ramp

0.06

20

0.03

0.2

0.02 0.1

0.01 0

0

10

20

0

30

5

10

Dj

15 Dj

20

25

Figure 15.14: Probability mass functions considered _ p=0.0667, r=0.1028, e=0.6066, T=27, D=15, τ=1.0918, L=0.9159 1 0.95 0.9 0.85

DTP

0.8 Deterministic Uniform CV=0.1 Triangular CV=0.1 Ramp CV=0.1 Uniform CV=0.25 Triangular CV=0.25 Ramp CV=0.25 Uniform CV=0.4 Triangular CV=0.4 Ramp CV=0.4 Uniform CV=0.8 Triangular CV=0.8 Ramp CV=0.8

0.75 0.7 0.65 0.6 0.55 0.5

0

5

10

15

20

25 N FGB

30

35

40

45

Figure 15.15: DT P as function of demand CV : System 1, low LF case

15.5. DT P FOR RANDOM DEMAND

489

_ p=0.0692, r=0.1067, e=0.6066, T=26, D=15, τ=1.0918, L=0.9511 0.95 0.9 0.85 0.8 Deterministic Uniform CV=0.1 Triangular CV=0.1 Ramp CV=0.1 Uniform CV=0.25 Triangular CV=0.25 Ramp CV=0.25 Uniform CV=0.4 Triangular CV=0.4 Ramp CV=0.4 Uniform CV=0.8 Triangular CV=0.8 Ramp CV=0.8

DTP

0.75 0.7 0.65 0.6 0.55 0.5 0.45

0

5

10

15

20

25 N FGB

30

35

40

45

Figure 15.16: DT P as function of demand CV : System 2, medium LF case

_ p=0.072, r=0.111, e=0.6066, T=25, D=15, τ=1.0918, L=0.9892 0.9

0.85

0.8

0.75

Deterministic Uniform CV=0.1 Triangular CV=0.1 Ramp CV=0.1 Uniform CV=0.25 Triangular CV=0.25 Ramp CV=0.25 Uniform CV=0.4 Triangular CV=0.4 Ramp CV=0.4 Uniform CV=0.8 Triangular CV=0.8 Ramp CV=0.8

DTP

0.7

0.65

0.6

0.55

0.5

0.45

0

5

10

15

20

25 N FGB

30

35

40

45

Figure 15.17: DT P as function of demand CV : System 3, high LF case

490

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

Table 15.3: DT P for different distributions of the demand (CV = 0.8, N = 1) High LF Medium LF Low LF

Uniform 0.4709 0.4949 0.5308

Triangular 0.4782 0.4992 0.5256

“Ramp” 0.4708 0.4948 0.5307

In the production-inventory-customer systems defined by assumptions (i)(iv), (v0 ), (vi), DT P for all demand distributions with identical coefficients of variation remains practically the same.

15.6

Summary

• The DT P in serial lines with continuous models of machine reliability can be characterized in terms of three dimensionless parameters: the relative finished goods buffer capacity, the relative shipping period, and the load factor. • In terms of these parameters, DT P can be calculated relatively easily for the case of one-machine production systems; for longer lines, a lower bound for DT P can be calculated. • Similar to Bernoulli lines, the following conservation law takes place for systems with exponential machines: in lean serial lines, the product of filtering in space (quantified by the relative finished goods buffer capacity) and filtering in time (quantified by the relative shipping period) is practically constant. • Demand variability leads to deterioration of DT P , and the extent of this deterioration can be calculated using the method provided. • The DT P in systems with random demand depends mainly on the coefficient of variation of the demand and is practically independent of the higher order moments.

15.7

Problems

Problem 15.1 Consider the PIC system defined by assumptions (i)-(vi) of Subsection 15.1.1 with T = 40, NF GB = 6 and the PS consisting of one exponential machine with e = 0.9. (a) Assume Tβ = 3 and calculate and plot DT P as a function of the load factor L. (b) Assume Tβ = 15 and again calculate and plot DT P as a function of L.

15.7. PROBLEMS

491

(c) Compare the two results and formulate a conjecture about the behavior of DT P as a function of L for various Tβ . Problem 15.2 Consider the PIC system defined by assumptions (i)-(vi) of Subsection 15.1.1 with T = 40, D = 35 and one exponential machine having e = 0.9. (a) Assume Tβ = 3 and calculate and plot DT P as a function of the finished goods buffer capacity NF GB . (b) Assume Tβ = 15 and again calculate and plot DT P as a function of NF GB . (c) Compare the two results and formulate a conjecture about the behavior of DT P as a function of NF GB for various Tβ . Problem 15.3 Consider the PIC system defined by assumptions (i)-(vi) of Subsection 15.1.1 with T = 40, D = 35, NF GB = 6 and one exponential machine having e = 0.9. (a) Calculate and plot DT P as a function of Tβ , for Tβ ∈ [3, 30]. (b) Formulate a conjecture on DT P behavior as a function of Tβ . Problem 15.4 Consider the PIC system defined by assumptions (i)-(vi) of Subsection 15.1.1 and assume T = 20, D = 16 and one exponential machine with Tβ = 5. Suppose the required DT P is 0.99. (a) For e = 0.9 determine the smallest capacity of the finished goods buffer that ensures the desired DT P . (b) Repeat the same calculation for e = 0.7 and e = 0.8. (c) Compare the results and draw a conclusion about how to keep the finished goods buffer small. Problem 15.5 Consider the PIC system with T = 15 and one exponential machine having e = 0.7, Tβ = 5 and with (1) constant demand, D = 10, (2) random demand with a uniform pmf defined on {7, 8, 9, 10, 11, 12, 13}, (3) random demand with a triangle pmf, which takes values on {6, 7, 8, 9, 10, 11, 12, 13, 14} with the probabilities {0.04, 0.08, 0.12, 0.16, 0.2, 0.16, 0.12, 0.08, 0.04}, respectively. Calculate: (a) (b) (c) (d)

The load factor, L, of the system. The CV of the random demands. The DT P as a function of NF GB ; plot this function. Summarize and interpret the results.

492

CHAPTER 15. DEMAND SATISFACTION IN CONTINUOUS LINES

15.8

Annotated Bibliography

The material of this chapter is based on [15.1] J. Li, E. Enginarlar and S.M. Meerkov, “Random Demand Satisfaction by Unreliable Production-Inventory-Customer Systems,” Annals of Operations Research, vol. 126, pp. 159-175, 2004. [15.2] J. Li, E. Enginarlar and S.M. Meerkov, “Conservation of Filtering in Manufacturing Systems with Unreliable Machines and Finished Goods Buffers,” Mathematical Problems in Engineering, vol. 2006, pp. 1-12, 2006. More details can be found in [15.3] J. Li, Production Variability in Manufacturing Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 2000.

Chapter 16

Assembly Systems with Bernoulli Model of Machine Reliability Motivation: As in the case of serial lines, assembly systems with Bernoulli machines are easier for analytical investigation than assembly systems with other reliability models. Therefore, we begin the study of assembly systems with the Bernoulli case and then extend the results to continuous time models. Overview: The simplest assembly system consists of at least three machines – two producing components and one assembling them into a product. Similar to serial lines with three machines, no closed formulas for performance evaluation of such a system can be derived. Thus, unlike serial lines, even the simplest assembly system requires a simplification procedure for its evaluation. Such a procedure is developed in Section 16.1 and then, using the aggregation approach of Chapter 4, generalized to larger assembly systems. The subsequent Sections 16.2 - 16.4 present methods for continuous improvement, design, and customer demand satisfaction in assembly systems, respectively. A case study is described in Section 16.5.

16.1

Analysis of Bernoulli Assembly Systems

16.1.1

Three-machine assembly systems

Mathematical description: The production system considered here is shown in Figure 16.1. The time is slotted with the slot duration equal to the cycle time of the machines. Machines m1 , m2 and m0 are up during each time slot with probability p1 , p2 , and p0 , respectively. The buffer capacities are N1 and N2 , both being finite. 495

496

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

p1

N1

m1

b1

p2

N2

m2

b2

p0

m0

Figure 16.1: Three-machine Bernoulli assembly system

States: Since the machines are memoryless, the states of the system are defined by a pair (h1 , h2 ) where h1 ∈ {0, 1, . . . , N1 } is the occupancy of buffer b1 and h2 ∈ {0, 1, . . . , N2 } is the occupancy of b2 . Conventions: The conventions of system operation remain the same as in Chapter 4, i.e., (a)-(e) of Subsections 4.2.1, along with an additional assumption: (f) The mating machine m0 is starved when either b1 , or b2 , or both are empty. State transition diagram: Under the above assumptions, the system at hand is described by a Markov chain. Its state transition diagram is shown in Figure 16.2. 0N2

13

02

01

00

12

23

11

10

N−1N 1 2

22

21

20

N1N2

32

N1N−1 2

23

N10

Figure 16.2: Transition diagram of three-machine Bernoulli assembly system

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS

497

Transition probabilities: Denote the transition probabilities of the Markov chain of Figure 16.2 as Pij,lm = P [h1 (n + 1) = i, h2 (n + 1) = j|h1 (n) = l, h2 (n) = m]. Then, using the formulas for the union and intersection of independent or mutually exclusive events, these transition probabilities can be written as follows: P0j,0j P0N2 ,0N2 P1j,0j P1N2 ,0N2 P11,00 Pi0,i0 PN1 0,N1 0 Pi1,i0 PN1 1,N1 0 Pij,ij PN1 j,N1 j PiN2 ,iN2 Pij,(i−1,j−1) P1j,(0,j−1) Pi1,(i−1,0) Pij,(i−1,j) Pij,(i,j−1) Pij,(i+1,j+1) Pij,(i+1,j) Pij,(i,j+1)

= (1 − p1 )(1 − p2 ), = 1 − p1 , = p1 (1 − p2 ), = p1 , = p1 p2 , = = = = = = = = = = = = =

j = 0, 1, . . . , N2 − 1,

(1 − p1 )(1 − p2 ), i = 0, 1, . . . , N1 − 1, 1 − p2 , (1 − p1 )p2 , i = 0, 1, . . . , N1 − 1, p2 , p1 p2 p0 + (1 − p1 )(1 − p2 )(1 − p0 ), i = 1, . . . , N1 − 1, j = 1, . . . , N2 − 1, p1 p2 p0 + (1 − p2 )(1 − p0 ), j = 1, . . . , N2 − 1, p1 p2 p0 + (1 − p1 )(1 − p0 ), i = 1, . . . , N1 − 1, (16.1) p1 p2 (1 − p0 ), i = 2, . . . , N1 , j = 2, . . . , N2 , p1 p2 , j = 1, . . . , N2 , p1 p2 , i = 1, . . . , N1 , p1 (1 − p2 )(1 − p0 ), i = 2, . . . , N1 , j = 1, . . . , N2 − 1, (1 − p1 )p2 (1 − p0 ), i = 1, . . . , N1 − 1, j = 2, . . . , N2 , (1 − p1 )(1 − p2 )p0 , i = 0, 1, . . . , N1 − 1, j = 0, 1, . . . , N2 − 1, (1 − p1 )p2 p0 , i = 0, 1, . . . , N1 − 1, j = 1, . . . , N2 , p1 (1 − p2 )p0 , i = 1, . . . , N1 , j = 0, 1, . . . , N2 − 1, p1 (1 − p0 ), i = 2, . . . , N1 ,

PiN2 ,(i−1,N2 )

= = =

PN1 j,(N1 ,j−1)

= p2 (1 − p0 ),

PN1 N2 ,N1 N2

j = 0, 1, . . . , N2 − 1,

j = 2, . . . , N2 ,

= p1 p2 p0 + 1 − p0 .

Steady state probabilities: Let Pij denote the steady state probabilities of the ergodic Markov chain with transition probabilities (16.1), i.e., Pij = P [h1 = i, h2 = j],

i ∈ {0, . . . , N1 }, j ∈ {0, . . . , N2 }.

(16.2)

498

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

Then, the balance equations for the Markov chain under consideration (see Section 2.3) are: Pij =

N2 N1 X X

Pij,lm Plm ,

i, l ∈ {0, 1, . . . , N1 }, j, m ∈ {0, 1, . . . , N2 }. (16.3)

l=0 m=0

Unfortunately, equation (16.3) cannot be solved analytically in a closed form. Therefore, the performance analysis of the system at hand requires a simplification. This simplification is described next. Approximations: Let Pi1 and Pj2 be the steady state probabilities that there are i parts in buffer b1 and j parts in buffer b2 , respectively. Clearly, these probabilities are the marginals of the joint steady state probabilities Pij , i.e., Pi1

:=

N2 X

Pij ,

i = 0, 1, . . . , N1 ,

Pij ,

j = 0, 1, . . . , N2 .

j=0

Pj2

:=

N1 X

(16.4)

i=0

It turns out that, although the joint probabilities, Pij , cannot be evaluated analytically, their marginals, Pi1 and Pj2 , can. This is carried out using the following recursive procedure: Assume that an estimate of the probability that b2 is empty at the s-th step of this recursive procedure is known; denote this probability as Pb02 (s), s = 0, 1, . . .. Consider the two-machine serial line {m1 , b1 , m00 } (see Figure 16.1), where the efficiency of m1 is p1 , the efficiency of the “virtual” machine m00 is p0 [1 − Pb02 (s)], and the capacity of b1 is N1 . For this line, using the results of Chapter 4, it is easy to calculate the probability that b1 is empty. Specifically, this probability is Q(p1 , p0 [1 − Pb02 (s)], N1 ), where function Q(x, y, N ) is defined by (4.14). Denote this probability as Pb01 (s + 1) and consider another serial line, {m2 , b2 , m000 }, with the efficiencies of m2 and the “virtual” machine m000 given by p2 and p0 [1 − Pb01 (s + 1)], respectively. Again, calculate the probability that b2 is empty and, thus, obtain Pb02 (s + 1). As a result, we obtain the following recursive procedure: Pb01 (s + 1) Pb2 (s + 1)

= =

Q(p1 , p0 [1 − Pb02 (s)], N1 ), Q(p2 , p0 [1 − Pb1 (s + 1)], N2 ),

Pb02 (0)

=

0,

0

0

(16.5)

s = 0, 1, 2, . . . .

Theorem 16.1 Recursive procedure (16.5) is convergent, i.e., the following limits exist: lim Pb01 (s) =: Pb01 ,

s→∞

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS lim Pb02 (s)

s→∞

=: Pb02 .

499 (16.6)

Moreover, sequences Pb01 (s) and Pb02 (s) are monotonically decreasing and increasing, respectively. Proof: See Section 20.3. Using the limits Pb01 , Pb02 and formula (4.8), the steady state probabilities of buffer occupancies can be evaluated as: Pbi1

=

Pbi2

=

α1i

Pb01 , 1 − p0 (1 − Pb02 ) α2i Pb02 , 1 b 1 − p0 (1 − P )

i = 1, . . . , N1 , i = 1, . . . , N2 ,

(16.7)

0

where α1

=

p1 [1 − p0 (1 − Pb02 )] , p0 (1 − Pb2 )(1 − p1 )

=

p2 [1 − p0 (1 − Pb01 )] . p0 (1 − Pb1 )(1 − p2 )

0

α2

(16.8)

0

Thus, the pmf’s of buffer occupancies in a three-machine Bernoulli assembly system are evaluated. An illustration of these pmf’s is given in Figure 16.3. The accuracy of these estimates can be characterized as follows: First, we investigate the accuracy of approximating the joint probability distributions, Pi0 and P0j , by their marginals. For this purpose, introduce δi1 δj2 δ

:= |Pi0 − Pi1 P02 |, := |P0j − P01 Pj2 |, =:

i = 0, . . . , N1 , j = 0, . . . , N1 ,

(16.9)

max{δi1 , δj2 }. i,j

Numerical Fact 16.1 For the assembly system under consideration, δ ¿ 1. Justification: Justification is carried out using numerical analysis of the Markov chain describing the system at hand. Specifically, the transition matrix based on (16.1) has been iterated upon until the stationary probability distribution, Pij , has been practically reached. The Pi0 and P0j , thus calculated, have been compared to the products of their marginals Pi1 P02 and P01 Pj2 . Finally, the value of δ has been calculated. In every case analyzed, it was found that δ ¿ 1. Several typical examples are shown in Table 16.1. Based on this investigation, we conclude that Numerical Fact 16.1 holds. Next, using the δ investigated above, we formulate

500

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

1

2

Pi

Pi

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

0

1

2

3

4

5 i

0 −1

0

1

2

3

4

5 i

3

4

5 i

4

5 i

(a) p1 = p2 = p0 = 0.9, N1 = N2 = 5 1

2

Pi

Pi

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

0

1

2

3

4

5 i

0 −1

0

1

2

(b) p1 = p2 = 0.9, p0 = 0.95, N1 = N2 = 5 2

P1 0.8

i

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 −1

0

1

2

3

4

5 i

0 −1

Pi

0

1

2

3

(c) p1 = 0.9, p2 = 0.6, p0 = 0.7, N1 = 2, N2 = 5 Figure 16.3: Stationary pmf of buffer occupancies in three-machine Bernoulli assembly systems

Table 16.1: Illustration of δ in three-machine Bernoulli assembly systems p1 0.7 0.8 0.7 0.6 0.9

p2 0.6 0.7 0.6 0.6 0.9

p0 0.8 0.8 0.8 0.99 0.6

N1 1 1 2 3 2

N2 1 2 2 3 2

δ 0.0032 0.0024 0.0108 0.029 0.0013

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS

501

Theorem 16.2 For the three-machine Bernoulli assembly system defined by assumptions (a)-(f ), PiL = PbiL + O(δ),

i = 0, 1, . . . , NL , L ∈ {1, 2}.

(16.10)

Proof: See Section 20.3. Thus, limits (16.6) and expressions (16.7) indeed approximate the probabilities of buffer occupancy in Bernoulli assembly systems. Formulas for performance measures: Although Pbi1 and Pbi2 are the marginals rather than the joint probabilities, it is shown below that they are sufficient to evaluate all the performance measures of the assembly system under consideration. Production rate: In terms of the joint steady state probabilities Pij , the production rate of the three-machine Bernoulli assembly system can be expressed as N1 X N2 N1 N2 h i X X X (16.11) Pij = p0 1 − P00 − Pi0 − P0j . P R = p0 i=1 j=1

i=1

j=1

Taking into account expressions (16.10), we approximate the joint probabilities involved in the above expression by a product of marginals, i.e., PR

=

N1 N2 h i X X Pi1 P02 − P01 Pj2 + O(δ) p0 1 − P00 − i=1

=

p0 [1 −

Therefore,

P01 P02

− (1 −

j=1

P01 )P02

− P01 (1 − P02 )] + O(δ).

P R = p0 (1 − P01 )(1 − P02 ) + O(δ).

(16.12)

Based on the above and keeping in mind Numerical Fact 16.1, we introduce the estimate of P R as follows: d P R = p0 (1 − Pb01 )(1 − Pb02 ).

(16.13)

Formulas for all other performance measures follow directly from the corresponding expressions for two-machine serial lines derived in Sections 4.1. Namely, Work-in-process: \ W IP 1

=

N1 h X p1 [1 − p0 (1 − Pb02 )] ii i , 1 − p0 (1 − Pb02 ) i=1 p0 (1 − Pb02 )(1 − p1 )

\ W IP 2

=

N2 h X p2 [1 − p0 (1 − Pb01 )] ii i . 1 − p0 (1 − Pb01 ) i=1 p0 (1 − Pb01 )(1 − p2 )

Pb01 Pb02

Blockages and starvations: d1 BL

=

p1 Q(p0 (1 − Pb02 ), p1 , N1 ),

(16.14)

502

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS d2 BL 1 c ST 0 2 c ST 0 1

=

p2 Q(p0 (1 − Pb01 ), p2 , N2 ),

=

p0 Q(p1 , p0 (1 −

Pb02 ), N1 ),

=

p0 Q(p2 , p0 (1 − Pb01 ), N2 ),

(16.15)

(16.16)

2

c and ST c 0 represent the estimates of starvation probabilities of m0 where ST 0 due to component 1 and component 2, respectively. Accuracy of the estimates: Theorem 16.3 The accuracy of the performance measures estimates (16.13)(16.16) is O(δ). Proof: See Section 20.3. Concluding this subsection, we illustrate the performance measures as a function of buffer capacity N1 = N2 =: N for the following assembly systems (Figure 16.4): A1 : A2 :

p1 = p2 = p0 = 0.9, p1 = p2 = 0.9, p0 = 0.95,

A3 :

p1 = p2 = 0.95, p0 = 0.9.

Clearly, the asymptotic properties of three-machine assembly systems remain qualitatively the same as those for two-machine serial lines.

16.1.2

M > 3-machine assembly systems

Idea of the approach: The approach to analysis of assembly systems with more than three machines combines the approach for three-machine assembly systems described above with the aggregation procedure for serial lines developed in Section 4.2. Specifically, consider the serial production line consisting of the component machines m11 , . . . , m1M1 , the merge machine m01 , the additional processing machines m02 , . . . , m0M0 , and the buffers separating the machines (see Figure 16.5). A recursive procedure for analysis of such a line is given by (4.30). In order to use this procedure for the serial line at hand, assume that the efficiency of m01 is modified to account for the other component line. Specifically, similarly to the approach of Section 16.1.1, introduce a virtual assembly machine, denoted as m001 , with the efficiency defined by p01 Prob{buffer b2M2 is not empty}. If Prob{buffer b2M2 is not empty} were known and if this probability were independent of the occupancy of buffer b1M1 , the recursive procedure (4.30) would result in the production rate of the assembly system. Since this probability is unknown and the dependence does exist, we introduce iterations as follows: At the first step, assume that Prob{buffer b2M2 is not empty} = 1. Then the serial production line {m11 , b11 . . . , m1M1 , b1M1 , m001 , b01 , . . . , m0M0 }, which we refer to as the upper line, is completely defined and, using recursive procedure

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS

503

10 Wd IP 1 or Wd IP 2

A3

0.9 d P R

A2 A1

0.85 0.8 0.75 0.7

2

4

N

6

8

8 6 A3 4 2 0

10

0.15

0.1

0.1

0

c 1 or ST c2 ST

d1 or BL d2 BL

0.15

A3 A2 2

4

2

4

N

6

10

0.05 A2

A1 A1

A3 N

8

\ \ (b) W IP 1 or W IP 2

(a) P R

0.05

A1 A2

6

8

0

10

2

4

N 1

d1 or BL d2 (c) BL

6

8

10

2

c or ST c (d) ST 0 0

Figure 16.4: Performance measure estimates of three-machine Bernoulli assembly systems as functions of buffer capacity

p11

N11

p12

p1M

m11

b11

m12

m1M

N21

p22

p2M

p21

m21

b21

m22

N1M 1

1

1

2

m2M

2

b1M 1 N2M 2

b2M

p01

N01

p02

p0M

m01

b01

m02

m0M

2

Figure 16.5: M > 3-machine Bernoulli assembly system

0

0

504

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

(4.30), calculate Prob{buffer b1M1 is not empty}. Consider next what is referred to as the lower line {m21 , b21 , . . . , m2M2 , b2M2 , m0001 , b0l , . . . , m0M0 }, where m0001 is another virtual machine with the efficiency p01 Prob{buffer b1M1 is not empty} and, again using the recursive procedure (4.30), calculate Prob{buffer b2M2 is not empty}. Use now this probability for the second iteration in the analysis of the upper line and continue this process, alternating between the upper and lower lines. As it turns out, these iterations are convergent and result in the estimates of Prob{buffer biMi is not empty}, i = 1, 2, which lead to the possibility of calculating all performance measures for the assembly system at hand. Aggregation equations and their properties: In order to use the aggregation procedure (4.30), introduce the following notations: ½ pui

= ½

Niu

= ½

pli

= ½

Nil

=

p1i , p0(i−M1 ) ,

i = 1, . . . , M1 , i = M1 + 2, . . . , M1 + M0 ,

N1i , i = 1, . . . , M1 , N0(i−M1 ) , i = M1 + 1, . . . , M1 + M0 − 1, p1i , p0(i−M2 ) ,

i = 1, . . . , M2 , i = M2 + 2, . . . , M2 + M0 ,

(16.17)

N2i , i = 1, . . . , M2 , N0(i−M2 ) , i = M2 + 1, . . . , M2 + M0 − 1,

where u and l stand for “upper” and “lower” lines, respectively. Using these notations and combining aggregation procedure (4.30) with iterations (16.5), we arrive at the following aggregation procedure for the assembly system of Figure 16.5: Equation for the efficiency of virtual machine m001 : puM1 +1 (s + 1) = p01 [1 − Pb02M2 (s)],

(16.18)

where Pb02M2 (s) is the probability that buffer b2M2 is empty at step s; Equations for aggregation of the upper line: pub i (s + 1) puf i (s

+ 1)

=

uf u pui (s + 1)[1 − Q(pub i+1 (s + 1), pi (s), Ni )],

=

pui (s

+ 1)[1 −

i = 1, . . . , M1 + M0 − 1 u + 1), pub i (s + 1), Ni−1 )], (16.19) i = 2, . . . , M1 + M0 ;

Q(puf i−1 (s

Equation for the probability of buffer b1M1 being empty: ub u Pb01M1 (s + 1) = Q(puf M1 (s + 1), pM1 +1 (s + 1), NM1 );

(16.20)

Equation for the efficiency of virtual machine m0001 : plM2 +1 (s + 1) = p01 [1 − Pb01M1 (s + 1)];

(16.21)

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS

505

Equations for aggregation of the lower line: plb i (s + 1)

=

lf l pli (s + 1)[1 − Q(plb i+1 (s + 1), pi (s), Ni )],

plf i (s + 1)

=

i = 1, . . . , M2 + M0 − 1 lb l pli (s + 1)[1 − Q(plf i−1 (s + 1), pi (s + 1), Ni−1 )], (16.22) i = 2, . . . , M2 + M0 ;

Equation for the probability of buffer b1M1 being empty: lb l Pb02M2 (s + 1) = Q(plf M2 (s + 1), pM2 +1 (s + 1), NM2 ),

s = 0, 1, . . . .

(16.23)

The boundary conditions for this aggregation procedure are: u puf 1 (s) = p1 ,

plf 1 (s)

=

pl1 ,

u pub M1 +M0 (s) = pM1 +M0 ,

s = 0, 1, . . . ,

plb M2 +M0 (s)

s = 0, 1, . . . ,

=

plM2 +M0 ,

(16.24)

and the initial conditions are: u puf i (0) = pi ,

i = 1, . . . , M1 + M0 ,

l plf i (0) = pi , Pb02M2 (0) = 0.

i = 1, . . . , M2 + M0 ,

Function Q in (16.19)-(16.23), as usually, is given by ( (1−x)(1−y) , x= 6 y, N 1− x yα Q(x, y, N ) = 1−x x = y, N +1−x , α

=

x(1 − y) . y(1 − x)

(16.25)

(16.26) (16.27)

Theorem 16.4 Aggregation procedure (16.18)-(16.25) has the following properties: ub l (i) The sequences puM1 +1 (s), puf i (s), pi (s), i = 1, . . . , M1 +M0 , and pM2 +1 (s), lf lb pi (s), pi (s), i = 1, . . . , M2 + M0 , s = 1, 2, . . . , are convergent, i.e., the following limits exist:

lim puM1 +1 (s) =:

puM1 +1 ,

lim puf (s) s→∞ i lim pub i (s) s→∞ u lim pM2 +1 (s) s→∞ lim plf (s) s→∞ i

=:

puf i ,

i = 1, . . . , M1 + M0 ,

=:

pub i ,

i = 1, . . . , M1 + M0 ;

=:

puM2 +1 ,

=:

plf i ,

i = 1, . . . , M2 + M0 ,

=:

plb i ,

i = 1, . . . , M2 + M0 .

s→∞

lim

s→∞

plb i (s)

(16.28)

506

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

(ii) These limits are unique solutions of the steady state equations corresponding to (16.18)-(16.23). (iii) In addition, these limits satisfy the relationships: puf M1 +M0

=

lf lb pub 1 = pM2 +M0 = p1

=

uf ub u pub i+1 [1 − Q(pi , pi+1 , Ni )]

=

uf ub u puf i [1 − Q(pi+1 , pi , Ni )],

=

lf lb l plb i+1 [1 − Q(pi , pi+1 , Ni )] lf lb l plf i [1 − Q(pi+1 , pi , Ni )],

=

i = 1, ..., M1 + M0 − 1, (16.29) i = 1, ..., M2 + M0 − 1.

Proof: See Section 20.3. lf ub lb Illustrations of the behavior of puf i (s), pi (s) and pi (s), pi (s) are given below. To define systems considered in these illustrations, it is convenient to introduce the following vectors: p1 p2 p0 N1 N2 N0

= = = = = =

[p11 , p12 , . . . , p1M1 ], [p21 , p22 , . . . , p2M2 ], [p01 , p02 , . . . , p0M0 ], [N11 , N12 , . . . , N1M1 ], [N21 , N22 , . . . , N2M2 ], [N01 , N02 , . . . , N0M0 −1 ].

(16.30)

In terms of these vectors, we consider the following assembly systems: A1 :

A2 :

A3 :

A4 :

M1 = M2 = M0 = 2, p1 = p2 = p0 = [0.9, 0.9], N1 = N2 = [3, 3], N0 = 3; M1 = M2 = M0 = 2, p1 = p2 = [0.8, 0.8], p0 = [0.9, 0.9], N1 = N2 = [3, 3], N0 = 3; M1 = 2, M2 = 3, M0 = 2, p1 = [0.9, 0.9], p2 = [0.8, 0.8, 0.8], p0 = [0.85, 0.85], N1 = [1, 1], N2 = [3, 3, 3], N0 = 2; M1 = M2 = M0 = 2, p1 = [0.8, 0.85], p2 = [0.9, 0.85], p0 = [0.9, 0.95], N1 = [3, 2], N2 = [2, 3], N0 = 2.

Their trajectories are shown in Figure 16.6. Clearly, the convergence is quite fast, typically within 5-15 steps.

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS

507

0.9

0.86

ub 3

lb

0.82

0.85 0.84

uf

ub

lb

p3 , p2

p3

lb

lf 3

p , p2

ub

uf

uf

0.86

ub

lf

lf

ub

uf

ub

p3

0.84

p

0.87

pi ,pi ,pi ,pi

lb

0.88

0.88

lf

p2

uf 2

p

pi ,pi ,pi ,pi

0.89

lb

p3

0.8

p2

uf p2

0.76

p4 , p1

ub

0.78 plf 2

0.74 uf

0.83

lb

lf

0.72

p4, p1

0.82 0

1

2

3

4

5

uf

p3

ub

p4 , p1

0.7 0

6

lb p2

lf

lf

lb

p4, p1

p3

5

Iteration number

10

15

Iteration number

(a) Assembly system A1

(b) Assembly system A2 0.9

uf 2

p

0.82

0.88

lb

p2

0.76

uf

p3

lb p 4 lf p4

0.7 0.68 puf, pub 4

0.66 0

uf

ub p 2

0.74 0.72

ub

lf

lf

p3

1

1

2

uf

p3

0.8

lb

p3

2

0.76

lb 2

lb

ub

p2

0.82

0.78 puf

lb 3

p p

lf 5

p , p1

lf

p2

0.84

pi ,pi ,pi ,pi

lf

ub

uf

p3

lf

0.78

ub

p3

0.86

ub

pi ,pi ,pi ,pi

lb

0.8

0.74 3

4

Iteration number

(c) Assembly system A3

5

0

uf p4 ,

ub p1 lf p 3

1

lb

p2 lf

lb

p4, p1 2

3

4

5

6

7

Iteration number

(d) Assembly system A4

lf ub lb Figure 16.6: Illustration of the dynamics of puf i (s), pi (s), pi (s) and pi (s)

8

508

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

Interpretation: The interpretation of the limits defined in (16.28) is similar to that for serial lines (see Figure 4.10). Specifically, (16.29) implies that aggregation procedure (16.18)-(16.25) reduces the assembly system of Figure 16.5 to the two-machine lines shown in Figure 16.7. uf

ub

u

pi

pi

Ni

lf

pi

+1

i =1, ..., M 1+M 0−1

l

Ni

lb

p i +1

i =1, ..., M 2+M 0−1

Figure 16.7: Representation of the assembly system through the aggregated machines Formulas for performance measure estimates: Using the limits (16.28) and the interpretation of Figure 16.7, the estimates of the performance measures of the assembly system of Figure 16.5 are defined as follows: Production rate: d P R

=

uf ub u pub i+1 [1 − Q(pi , pi+1 , Ni )]

=

uf ub u puf i [1 − Q(pi+1 , pi , Ni )],

=

lf lb l plb i+1 [1 − Q(pi , pi+1 , Ni )] lf lb l plf i [1 − Q(pi+1 , pi , Ni )],

=

i = 1, ..., M1 + M0 − 1 (16.31) i = 1, ..., M2 + M0 − 1.

Work-in-process: ³ ´ u  ub 1−αNi (puf puf i ,pi+1 ) u Niu uf ub i  − N α (p , p ) ,  uf uf N u uf ub i i+1 i ub ub  1−α(pi ,pi+1 ) pi+1 −pi α i (pi ,pi+1 )   u ub if puf \ i 6= pi+1 , (16.32) W IP i =     Niu (Niu +1)  ub , if puf i = pi+1 . 2(N u +1−puf ) i

l

\ W IP i

i

i = 1, . . . , M1 + M0 − 1,  ³ ´ l lb 1−αNi (plf plf  i ,pi+1 ) l Nil lf lb i  − N α (p , p ) lf l  i i+1 , i lf N lf lb lb  1−α(pi ,plb i+1 )   pi+1 −pi α i (pi ,pi+1 ) lb if plf (16.33) = i 6= pi+1 ,     l u  lb  Ni (Ni +1) , if plf i = pi+1 . 2(N l +1−plf ) i

i

i = 1, . . . , M2 + M0 − 1, Blockages and starvations: u

d BL i

uf u = pui Q(pub i+1 , pi , Ni ),

i = 1, . . . , M1 + M0 − 1,

(16.34)

16.1. ANALYSIS OF BERNOULLI ASSEMBLY SYSTEMS u

c ST i

l di BL u ci ST

509

=

ub u pui Q(puf i−1 , pi , Ni−1 ),

i = 2, . . . , M1 + M0 ,

(16.35)

=

lf l pli Q(plb i+1 , pi , Ni ),

i = 1, . . . , M2 + M0 − 1,

(16.36)

=

lb l pli Q(plf i−1 , pi , Ni−1 ),

i = 2, . . . , M2 + M0 .

(16.37)

Accuracy: The accuracy of estimates (16.31)-(16.37) is roughly the same as that for the Bernoulli serial lines evaluated in Section 4.2. Monotonicity: Using the steady states of the recursive aggregation procedure described above, it is possible to show that the Bernoulli assembly systems possess the property of monotonicity with respect to machine efficiency and buffer capacity. In other words, increasing any pij or Nij always leads to an d increase P R, albeit with a possibility of diminishing returns. Reversibility: Consider an assembly system A and its reverse, i.e., the disassembly system Ar (see Figure 16.8), where the parameters of machine mrij and buffer brij , i = 0, 1, 2, j = 1, . . . , Mi , are the same as the parameters of machine mi,Mi −j+1 and buffer bi,Mi −j+1 of the original system A. Note that in the disassembly system, machine mr0M0 splits one part into two and is blocked when either buffer b11 or b21 is full. p11

N11

p12

p1M

m11

b11

m12

m1M

p21

N21

p22

p2M

m21

b21

1

b1M 1 N2M 2

2

m2M

m22

N1M 1

1

b2M

2

r

0

m 01

N01

p02 r

m 0M

0

p01

r

−1

b 0M

0 −1

r

b11

m0Mr

N01

p02

p0M

m01

b01

m02

m0M

0

0

2

N1M 1 p0M

p01

N2M 2

0

r

b21

p1M

1

r

m11 p2M

2

r

m21

p12 r

m 1M

1 −1

p22 r

m 2M

2

N11

p11

b1M 1

r

m 1M

N21

p21

r

−1

b2M

r

1

r

2

m 2M

2

Figure 16.8: Bernoulli assembly system and its reverse (i.e., disassembly system) The analysis of the disassembly system is analogous to that of assembly systems except that the probabilities that buffers bi Mi are empty, Pb0iMi , i = 1, 2, are replaced by the probabilities that buffers bi1 are full, PbNi1i , i = 1, 2. Similar

510

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

to the case of serial lines, it is possible to show that the performance measures of an assembly system and its reverse are related as follows: d P R

A

A d BL ij

Ar

=

d P R

=

Ar c [i(M ST r, i −j+1)]

, i = 0, 1, 2, j = 1, . . . , Mi ,

(16.38)

i.e., the reversibility holds. Assembly systems with complex structure: Assembly systems with more than two component lines and more than one merge machine can also be analyzed using the approach described above. The only difference is that more than two “constituent” serial lines must be considered and iterated upon. An example of more than two component lines is given in Figure 16.9. For this system, the following constituent serial lines must be considered: p1

N1

m1

b1

p2

N2

p0

m2

b2

m0

p3

N3

m3

b3

Figure 16.9: Bernoulli assembly system with more than two component lines Line 1: Line 2: Line 3:

{m1 , b1 , m00 }, {m2 , b2 , m000 }, {m3 , b3 , m000 0 },

where m00 , m000 and m000 0 are the virtual machines with their efficiencies defined by p0 [1 − Pb02 Pb03 ], p0 [1 − Pb01 Pb03 ] and p0 [1 − Pb01 Pb02 ], respectively, where Pb0i , i = 1, 2, 3, is the estimate of the probability that buffer bi is empty. An example of a system with more than one merge machine is given in Figure 16.10. For this system, three constituent serial lines must be considered: Line 1: Line 2:

{m1 , b1 , m001 , b01 , m002 }, {m2 , b2 , m0001 , b01 , m002 },

16.2. IMPROVEMENT OF BERNOULLI ASSEMBLY SYSTEMS p1

N1

m1

b1

p2

N2

p001

N01

p 02

m2

b2

m 001

b 01

m 02

p3

N3

m3

b3

511

Figure 16.10: Bernoulli assembly system with more than one merge machine {m3 , b3 , m0002 },

Line 3:

and m001 , m0001 , m002 , m0002 are the virtual machines with efficiencies: m0001 :

p01 [1 − Pb001,2 ], p01 [1 − Pb01,1 ],

m002 m0002

p02 [1 − Pb002,3 ], p02 [1 − Pb02,01 ],

m001 : : :

0

0

and Pb001,i , i = 1, 2, are the probabilities buffer bi is empty, and Pb002,3 and Pb002,01 are the probabilities buffer b3 or b01 is empty, respectively. Applying aggregation procedure (4.30) to each of these lines in a round-robin manner, one can evaluate all the performance measures of the assembly system at hand. PSE Toolbox: Recursive procedure (16.18)-(16.25) and performance measures (16.31)-(16.37) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Subsection 19.3.5.

16.2

Continuous Improvement of Bernoulli Assembly Systems

16.2.1

Constrained improvability

Resource constraints and definitions: The resource constraints for Bernoulli assembly systems are similar to those for Bernoulli serial lines (see Chapter 5).

512

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

Specifically, the buffer capacity (BC) constraint is M1 X

N1i +

i=1

M2 X

N2i +

M 0 −1 X

i=1

N0i = N ∗ ,

(16.39)

i=1

while the workforce (WF) constraint is modeled as M1 Y

p1i

i=1

M2 Y

p2i

i=1

M0 Y

p0i = p∗ .

(16.40)

i=1

Using the notations (16.30), introduce Definition 16.1 An assembly system with Bernoulli machines is: • improvable with respect to BC if there exist vectors N01 , N02 , N00 such that the BC constraint (16.39) is satisfied and P R(p1 , p2 , p0 , N01 , N02 , N00 ) > P R(p1 , p2 , p0 , N1 , N2 , N0 ); • improvable with respect to WF if there exist vectors p01 , p02 , p00 such that the WF constraint (16.40) is satisfied and P R(p01 , p02 , p00 , N1 , N2 , N0 ) > P R(p1 , p2 , p0 , N1 , N2 , N0 ); • improvable with respect to BC and WF simultaneously if there exist vectors N01 , N02 , N00 and p01 , p02 , p00 such that the constraints (16.39) and (16.40) are satisfied and P R(p01 , p02 , p00 , N01 , N02 , N00 ) > P R(p1 , p2 , p0 , N1 , N2 , N0 ). Improvability with respect to WF:

Necessary and sufficient conditions:

Theorem 16.5 A Bernoulli assembly system defined by assumptions (a)-(f ) of Section 16.1 is unimprovable with respect to WF if and only if

puf i ,

piuf

= pub i+1 ,

i = 1, . . . , M1 + M0 − 1,

plf i

= plb i+1 ,

i = 1, . . . , M2 + M0 − 1,

plf i

(16.41)

where pub and plb i , i are the steady states of the recursive aggregation procedure (16.18)-(16.25). Proof: Similar to the proof of Theorem 5.1. Corollary 16.1 Under conditions (16.41), \ W IP 1i

=

\ W IP 2i

=

\ W IP 0i

=

N1i (N1i + 1) , 2(N1i + 1 − puf i ) N2i (N2i + 1) , 2(N2i + 1 − plf i ) N0i (N0i + 1) 2(N0i + 1 − puf M1 +i )

i = 1, . . . , M1 , i = 1, . . . , M2 , ,

i = 1, . . . , M0 − 1.

16.2. IMPROVEMENT OF BERNOULLI ASSEMBLY SYSTEMS

513

From here, similar to the case of serial lines, we obtain: WF-Improvability Indicator 16.1: An assembly system with Bernoulli machines is practically unimprovable with respect to the workforce if each buffer is, on the average, close to being half full. Thus, WF-Continuous Improvement Procedure 5.1 of Section 5.1 can be used for assembly systems as well. Unimprovable allocation of pij : Introduce ∗

d P R = p1 ,p2 ,p0 ;

QM1

max Q

∗ i=1 p1i

M2 i=1

p∗ 2i

QM0

∗ ∗ i=1 p0i =p

d P R(p1 , p2 , p0 , N1 , N2 , N0 ) (16.42)

and consider the equation: x(s + 1)

=

! M +M2 +M 1 2 0 N1i + x(s) (p ) N1i + 1 i=1 2 ! ! M +M2 +M Ã Ã M1 +M2 +M0 M0 −1 M2 1 2 0 Y Y N2i + x(s) N0i + x(s) × (16.43) , N2i + 1 N0i + 1 i=1 i=1 ∗

1 M1 +M2 +M0

M1 Y

Ã

x(0) ∈ (0, 1). PM1 −1 PM2 −1 PM0 −1 −1 Theorem 16.6 Assume i=1 N1i + i=1 N2i + i=0 N0i < (M1 + M2 + M0 )/2. Then (16.43) is a contraction on (0, 1). Moreover, lim x(s) = P R∗ ,

s→∞

(16.44)

where P R∗ is defined by (16.42). In addition, the values of pij , i = 0, 1, 2, j = 1, . . . , Mi , that result in P R∗ are p∗11

=

p∗21

=

p∗ij

=

p∗01

=

p∗0M0

=

³ N +1 ´ 11 P R∗ , N11 + P R∗ ³ N +1 ´ 21 P R∗ , N21 + P R∗ ´³ N + 1 ´ ³ N i,j−1 + 1 ij P R∗ , ∀ij 6= 11, 21, 01, 0M0 , Ni,j−1 + P R∗ Nij + P R∗ ´³ N ´³ N + 1 ´ ³ N 1M1 + 1 2M2 + 1 01 P R∗ , N1M1 + P R∗ N2M2 + P R∗ N01 + P R∗ ´ ³ N 0,M0 −1 + 1 P R∗ . (16.45) ∗ N0,M0 −1 + P R

Proof: Similar to the proof of Theorem 5.2. Corollary 16.2 If all buffers are of equal capacity, then p∗11

= p∗21 = p∗0M0 < p∗12 = . . . = p∗1M1

514

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS =

p∗22 = . . . = p∗2M2

= <

p∗02 = . . . = p∗0M0 −1 p∗01 .

Thus, when all buffers are of equal capacity, to optimize the production rate, the merge machine must have the largest efficiency, the first machines of the component lines and the last machine of the additional processing line must have the lowest efficiency, and the efficiency of all other machines must be in-between. Improvability with respect to WF and BC simultaneously: Theorem 16.7 A Bernoulli assembly system defined by assumptions (a)(f ) of Subsection 16.1 is unimprovable with respect to WF and BC re-allocation simultaneously if and only if (16.41) takes place and, in addition, puf i plf i

=

pub i ,

i = 1, . . . , M1 + M0 − 1,

=

plb i ,

i = 1, . . . , M2 + M0 − 1,

(16.46)

lf ub lb where puf i , pi , pi and pi are the steady states of the recursive aggregation procedure (16.18)-(16.25).

Proof: Similar to the proof of Theorem 5.4. The unimprovable allocation of N ’s and p’s for this case is given by Theorem 16.8 Assume that N ∗ is an integer multiple of M1 + M2 + M0 − 1 and denote ∗∗

d P R =

max P R(p1 , p2 , p0 , N1 , N2 , N0 ). p1 , p2 , p0 ; QM1 ∗ QM2 ∗ QM0 ∗ ∗ i=1 p1i i=1 p2i i=1 p0i = p N , N , N0 ; PM1 PM21 2 P M0 −1 N + N0i = N ∗ 1i i=1 i=1 N2i + i=1 (16.47) Then conditions (16.41) and (16.46) are satisfied if and only if, Ã ! N∗ + 1 ∗∗ M +M +M −1 1 2 0 d p∗11 = p∗21 = p∗0M0 = ∗∗ P R , N∗ d M1 +M2 +M0 −1 + P R Ã !2 ∗ N ∗∗ M1 +M2 +M0 −1 + 1 ∗ d pij = P R , ∀ij 6= 11, 21, 0M0 , 01, ∗∗ ∗ N d M1 +M2 +M0 −1 + P R Ã !3 N∗ ∗∗ M1 +M2 +M0 −1 + 1 ∗ d p01 = P R , (16.48) ∗∗ N∗ d + P R M1 +M2 +M0 −1 ∗ Nij

=

N∗ , M1 + M2 + M0 − 1

i = 0, 1, 2, ∀ij 6= 0M0 ,

16.2. IMPROVEMENT OF BERNOULLI ASSEMBLY SYSTEMS

515

∗∗

∗ d ’s defined by the last where P R is calculated according to (16.43) with Nij equation in (16.48).

Proof: Similar to the proof of Theorem 5.5. Clearly, this theorem supports the conclusion of the previous subsection that the merge machine must have the highest efficiency. Improvability with respect to BC: In agreement with the case of serial lines, we formulate BC-Improvability Indicator 16.1: A Bernoulli assembly system is practically unimprovable with respect to BC if the average occupancy of each buffer is as close to the average availability of its downstream buffer as possible (keeping in mind that the buffer immediately after the merge machine has two upstream buffers). This implies that the BC-Continuous Improvement Procedure 5.1 can be applied to assembly systems as well.

16.2.2

Unconstrained improvability

Definitions: The definitions of the bottleneck machine and bottleneck buffer in assembly systems remain the same as those in serial lines. Specifically: Definition 16.2 In a Bernoulli assembly system, mij , i ∈ {0, 1, 2}, j ∈ {1, . . . , Mi }, is the bottleneck machine (BN-m) if d d ∂P R ∂P R > , ∂pij ∂pmn

∀ij 6= mn.

Although, as in serial lines, the lowest efficiency machine is not necessarily the bottleneck, the true bottleneck can be identified using the arrow-based method of Section 5.2. Indeed, consider the upper and lower serial lines of Figure 16.5. In each of these lines, assign arrows directed from one machine to another according to the Arrow Assignment Rule 5.1 modified to account for the starvation of m01 as follows: if BLiMi > ST01 , i = 1, 2, direct the arrow from miMi to m01 ; if BLiMi < ST01 , i = 1, 2, direct the arrow from m01 to miMi . Bottleneck Indicator 16.1: The bottleneck machine of a Bernoulli assembly system can be identified as follows: • If there is a single machine with no emanating arrows, it is the BN-m. • If there are multiple machines with no emanating arrows, the one with the largest severity is the Primary BN-m (PBN-m), where the severity of each (local) bottleneck is defined by Si1 SiMi

=

|STi2 − BLi1 |,

=

|BLi,Mi −1 + ST01i | − |BLiMi + STiMi |,

i = 1, 2,

S01

=

|BL1M1 + BL2M2 + ST02 | − |BL01 + ST011 + ST012 |,

i = 1, 2,

516

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS S0M0

=

|BL0,M0 −1 − ST0M0 |,

Sij

=

|BLi,j−1 + STi,j+1 | − |BLij + STij |, ij 6= 11, 21, 1M1 , 2M2 , 01, 0M0 .

(16.49)

The accuracy of this bottleneck indicator is roughly the same as that for the serial Bernoulli lines (see Subsection 5.2.3). Several examples are given in Figures 16.11-16.14, where the quantities with the hat are calculated using recursive procedure (16.18)-(16.25) and those without the hat are evaluated by simulations. Note that in the first example (Figure 16.11), the merge machine, although of the highest efficiency, is the bottleneck. All other examples show the bottlenecks in other parts of the system. m 11

b 11

m 12

b 12

0.78

1

0.8

1

^ ST 1i

0

0.1004

^ BL 1i

0.335

0.2911

ST 1i

0

0.1045

BL 1i

0.3169

0.2758

∆ PR ∆ p ij

0.1447

0.2091

m 21

b 21

m 22

0.8

1

0.8

^ ST 2i

0

^ BL 2i

0.355

BN−m m 01 0.82

^ ST 0i b 22 1

^ BL 0i ^ ST 0i

(1)

(2)

b 01

m 02

b 02

m 03

1

0.8

1

0.8

0.1567

0.2993

0.355

0.1483

0.089

0

0.089

0.089

ST 0i

0.1575

0.2844

0.3369

0.2993

BL 0i

0.1488

0.0926

0

ST 0i

0.1488 0.202

0.1266

ST 2i

0

0.0926

BL 2i

0.3369

0.2844

∆ PR ∆ p ij

0.1268

0.202

∆ PR ∆ p ij

(1)

(2)

0.2576

Figure 16.11: Illustration of Bottleneck Indicator 16.1: Example 1 Based on the above, the BN-Continuous Improvement Procedure 5.1 can be used for assembly systems as well. Similarly, the notions of buffer potency and measurement-based management, discussed in Sections 5.2 and 5.3, are applicable to assembly systems. Assembly system with complex structure: Bottleneck Indicator 16.1 can be used for assembly systems with many component lines and merge operations. To accomplish this, using the arrow assignment rule in all constituent lines, the BN-m can be identified as the machine with no emanating arrows; if there are several of such machines, the PBN-m can be determined using an appropriately defined severity.

16.2. IMPROVEMENT OF BERNOULLI ASSEMBLY SYSTEMS m 11

b 11

m 12

b 12

0.8

1

0.8

1

^ ST 1i

0

0.0808

^ BL 1i

0.3958

0.3503

ST 1i

0

0.0844

BL 1i

0.3780

0.3366

∆ PR ∆ p ij

0.0840

0.1498

^ ST 0i

BN−m

^ BL 0i

m 21

b 21

0.8

1

m 22 0.65

m 01

b 22

^ ST 0i

(1)

ST 0i

0.0657

BL 0i

^ BL 2i

0.3958

0.2003

ST 0i

0

0.0686

0.378

0.1860

∆ PR ∆ p ij

0.1284

0.3085

b 02

m 03

0.8

1

0.8

0.1263

0.3503

0.3958

0.1263

0.0808

0

0.1259

0.3366

0.3780

0.1259

0.0844

0

0.1498

0.084

0.23

(2)

0

BL 2i

m 02

1

1

^ ST 2i

ST 2i

b 01

0.82

(1)

0.2335

(2)

∆ PR ∆ p ij

0.2321

Figure 16.12: Illustration of Bottleneck Indicator 16.1: Example 2 PBN−m

^ ST 1i ^ BL 1i ^ S

m 11

b 11

m 12

b 12

0.8

1

0.6

1

0

0.0562 0.1865

0.4253

BN−m m 01

0.4236

0.9

ST 1i

0

0.0584

BL 1i

0.4106

0.1753

^ ST 0i

0.2885

^ BL 0i

∆ PR ∆ p ij

0.0984 m 21

b 21

m 22

0.8

1

0.8

b 22 1

^ ST 0i

(1)

(2)

0

0.0749

ST 0i

^ BL 2i

0.4253

0.3865

BL 0i

0

0.0779

BL 2i

0.4106

0.377

∆ PR ∆ p ij

0.052

0.1011

∆ PR ∆ p ij

b 02

m 03

0.62

1

0.8

0.2065

0.4253

0.2256

0.0581

0

0.0877 0.4209

^ ST 2i

ST 2i

m 02

1

0.2427

^ S

ST 0i

b 01

(1)

(2)

0.2467

0.1953

0.4106

0.266

0.0604

0

0.2591

0.0921

0.0855 0.2321

Figure 16.13: Illustration of Bottleneck Indicator 16.1: Example 3

517

518

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

PBN−m m 11

b 11

m 12

b 12

0.8

1

0.6

1

^ ST 1i

0

0.0570

^ BL 1i

0.4202

0.1804

^ S ST 1i

0.4663 0

m 01 0.9

0.0586

BL 1i

0.4091

0.1730

∆ PR ∆ p ij

^ ST 0i

0.0927

0.2691

^ BL 0i

b 21

m 22

^ ST 0i

1

0.8

BN−m m 21 0.58

b 22

(1)

(2)

b 01

m 02

b 02

m 03

1

0.8

1

0.8

0.2876

0.3804

0.4202

0.1303

0.0760

0

0.2888

0.3745

0.4091

0.1268

0.0762

0

0.0919

0.0477

0.2474

1

ST 0i

^ ST 2i

0

0.22

^ BL 2i

BL 0i

0.2003

0.2763

ST 0i

^ S

0.0373

ST 2i

0

0.2264

BL 2i

0.1891

0.2607

∆ PR ∆ p ij

0.2468

0.1521

∆ PR ∆ p ij

(1)

(2)

0.2549 0.1362

Figure 16.14: Illustration of Bottleneck Indicator 16.1: Example 4

16.3. LEAN BUFFERING IN BERNOULLI ASSEMBLY SYSTEMS

16.3

519

Lean Buffering in Bernoulli Assembly Systems

Consider the assembly system defined by assumptions (a)-(f) of Section 16.1. Let E be its efficiency, i.e., PR , E= P R∞ where P R and P R∞ are the production rates of the system with finite and infinite buffers, respectively. As in Chapter 6, lean buffering of the assembly system is the smallest buffer capacity necessary and sufficient to attain the desired line efficiency E. Such a buffering is denoted as N11,E , . . . , N1M1 ,E , N21,E , . . . , N2M2 ,E , N01,E , . . . , N0(M0 −1),E . Based on the performance analysis techniques of Section 16.1, it is possible to show that the lean buffering for Bernoulli assembly systems can be evaluated using the same methods as those developed in Section 6 for serial lines. Specifically, • for the simplest, i.e., three-machine assembly system with identical machines, the lean buffering is given by expression (6.10); • for assembly systems with more than three identical machines, lean buffering can be evaluated using (6.12) and (6.13); • for assembly systems with non-identical machines, the six bounds derived in Section 6.3 can be used.

16.4

Customer Demand Satisfaction in Bernoulli Assembly Systems

The Production-Inventory-Customer (PIC) system considered here is shown in Figure 16.15, where the Finished Goods Buffer (FGB) is present and the customer, defined by the shipment size D and the shipping period T , is indicated. The rules of the system operation remain the same as in Chapter 9, i.e., defined by assumptions (i)-(v), except that the term “serial line” must be substituted by the term “assembly system”. The definition of the due-time performance, DT P , as well as the relative FGB capacity, ν, and the load factor, L, also remains the same as in (9.2)-(9.4). Given this scenario, all the results of Chapter 9 remain valid for the PIC system at hand. Specifically, • the DTP for one-machine production system can be evaluated using Theorem 9.1; • for multi-machine production system, the lower bound (9.12) holds; • the conservation of filtering law (9.15) also takes place; • DTP for random demand can be evaluated and has the same properties as described in Section 9.4.

520

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS Production

Inventory

Customer

Subsystem

Subsystem

Subsystem

p11

N11

p12

p1M

m11

b11

m12

m1M

p21

N21

p22

p2M

m21

b21

1

2

m2M

m22

N1M 1

1

2

b1M 1 N2M 2

b2M

p01

N01

p02

p0M

m01

b01

m02

m0M

0

D,T 0

N

FGB

2

Figure 16.15: PIC system with Bernoulli assembly subsystem

16.5

Case Studies

16.5.1

Automotive ignition module assembly system

The layout, structural modeling, and parameter identification of this system are described in Sections 3.2 and 3.10, respectively. Here we validate this model and develop a continuous improvement project. Model validation: Using iterative procedure (16.18)-(16.25), we calculate the production rate of the system for each of the consecutive six months and convert it into the throughput. The results are shown in Table 16.2 along with the actual throughput. Table 16.2: Model validation data Month d T P (parts/h) T P (parts/h) Error (%)

May 330 337 2.1

June 336 347 3.2

July 302 378 20.1

Aug. 335 340 1.6

Sept. 380 384 1

Oct. 337 383 11.9

As it follows from these data, the throughput estimates match closely the actual ones, with the exception of the months July and October. We rationalize the situation in July by the fact that during this month a two weeks shut-down period took place and, supposedly, some phenomena, are not reflected in the data identified. We do not have an explanation for the large error in the month of October. Analysis of MHS potency: Using Bottleneck Indicator 16.1, we determine that Operation 1 is the bottleneck based on May data (Figure 16.16). Similar analyses for five other months lead to the conclusions summarized in Table 16.3. In addition, Table 16.3 includes the machine with the smallest pij during the respective month. As it follows from these data, MHS is consistently not potent

16.5. CASE STUDIES

521

and an increase of the system T P may be obtained by improving MHS. The extent of the potential improvement has been characterized in Subsection 5.2.4. BN 1

2

3

4

^ ST:

0

0.2593

0.2656

0.2311

^ BL:

0.1110

0.1538

0.1731

0.1207

5

6

7

8

0.2737

0.3501

0.3136

0.2736

0.0814

0.1240

0.1569

0.1772

9

10

11

12

^ ST:

0

0.1520

0.1444

0.1576

^ BL:

0.2208

0.2723

0.2673

0.3356

13

14

15

16

17

18

0.2527 0.3084

0.2735

0.3355

0.2944

0.1975

0.1202 0.1353

0.0956

0.1434

0.1801

0

0.1064

Figure 16.16: Bottleneck identification for an automotive ignition module assembly system (based on May data)

Table 16.3: Bottleneck and the slowest machine of the systems Month Bottleneck Machines with the smallest isolation P R

May Op.1

June Op.13

July Op.13

Aug. Op.13

Sept. Op.13

Oct. Op.1

Op.4

Op.4

Op.4

Op.11

Op.4

Op.14

Designing a continuous improvement project: Three avenues for potential improvement have been investigated: increasing the capacity of all buffers, increasing the capacity of the buffer conveyor only, and eliminating the starvation of Operations 1 and 9 and the blockage of Operation 18. The results obtained are as follows: • Increasing capacity of all buffers: Assuming that the capacity of each buffer is increased from 1 to 5, and there is no starvation in Operations 1 and 9 and no blockage in Operation 18, the T P for each month has been calculated using the recursive procedure (16.18)-(16.25). The results are shown in Table 16.4. As it follows from these data, the MHS is potent: BNm is the least efficient machine and the T P of the system is almost equal to that of the slowest machine. However, due to practical considerations, all buffers cannot be increased without substantial capital investments (rebuilding the conveyor system). Therefore, two other avenues of continuous improvement have been investigated. • Increasing capacity of buffer conveyor: The buffering capacity of this conveyor can be increased by using additional pallets. At the time of the

522

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS

Table 16.4: Estimated production rates of the system with buffers of capacity 5 Month T P (parts/h) Isolation T P of the slowest machine BN machine Losses due to MHS (parts/h)

May 501

June 518

July 452

Aug. 490

Sept. 527

Oct. 491

522

534

468

498

540

492

Op.4

Op. 4

Op.4

Op.11

Op.4

Op.14

21

16

16

8

13

1

study, this conveyor contained 19 pallets. We have shown that 40 pallets would provide additional buffering capacity without creating congestion on the conveyor. With 40 pallets utilized, the capacity of the buffer b8 , i.e., the buffer in front of the bottleneck, becomes 29 parts. This results in T P = 368 parts/h, which is a 9.2% improvement (based on October data). • Eliminating starvations of first and blockages of last operations: This can be accomplished by either manual or robotic material handling. For instance, if Operation 8 is blocked, a part can be removed from the pallet, manually or by a robot, making this pallet available to Operation 1. If these modifications are put in place, the system performance is improved as shown in Table 16.5, i.e., a production rate increase of 7-17% is possible.

Table 16.5: Estimated production rates of the system with no starvations of Op.1 and 9 and no blockage of Op.18 Month T P (parts/h) Improvement (%) BN machine

May 372 12.7 Op.5

June 393 17 Op. 5

July 325 7.6 Op.5

Aug. 359 7.2 Op.13

Sept. 415 9.2 Op.12

Oct. 374 11 Op.5

The plant management accepted the above two recommendations, and they have been put in place.

16.5.2

Injection molding - assembly system

The layout and structural modeling of this system have been described in Section 3.2. The goal of this project was to select the capacity of the finished goods buffer so that DT P > 0.995. The shipping period and demand were defined as

16.6. SUMMARY

523

T = 12 and D = 9. Before this study, no strict limits on FGB was formally imposed, and the system was managed on the basis of an intuitive principle “the more finished goods at hand, the better.” Identification of virtual machine and buffers parameters: Based on two months performance data, the average up- and downtimes of each press were calculated. By assuming that the available presses were assigned uniformly to all molds, the parameters of the virtual model were identified as shown in Table 16.6, where N0 is the finished goods buffer capacity to be determined from the analysis. Table 16.6: Virtual machine and buffer parameters identified Machines pi Buffers Ni

m1 0.82 b1 26

m2 0.82 b2 27

m3 0.82 b3 30

m4 0.82 b4 15

m5 0.82 b5 18

m6 0.82 b6 45

m7 0.82 b7 45

m0 0.79 b0 N0

Performance evaluation: Using the parameters identified in Table 16.6, we first aggregate the assembly system without the finished goods buffer into one aggregated machine, with parameter pagg = 0.7872. This results in a moderate load factor L = 0.9527. The due-time performance analysis is carried out using \ this aggregated machine to obtain a lower bound of DT P 1 for various values of the capacity of the FGB, N0 . The results are shown in Table 16.7. Table 16.7: Lower bound of DTP of virtual model for various capacity of FGB N0 \ DT P 1 N0 \ DT P1

3 0.9508 8 0.9945

4 0.9691 9 0.9964

5 0.9802 10 0.9976

6 0.9872 11 0.9984

7 0.9916 12 0.9990

\ As it can be seen, DT P 1 > 0.995 for N0 = 9. Additional increases in N0 \ result in a very small improvement in DT P 1 . Therefore, the capacity of N0 has been recommended as one shipment (i.e., N0 = 9). This recommendation has been accepted by the plant management and implemented on the factory floor.

16.6

Summary

• Performance analysis of even the simplest, i.e., three-machine, assembly line requires a simplification based on iterations.

524

CHAPTER 16. BERNOULLI ASSEMBLY SYSTEMS • Using these iterations and the aggregation procedure (4.30), all performance measures of Bernoulli assembly systems can be evaluated. • The improvability properties of Bernoulli assembly systems are similar to those of Bernoulli serial lines. • If all buffers are of the same capacity, to ensure the largest throughput, the merge machine should have the largest efficiency, the first and the last machines the lowest efficiency, and the efficiency of all other machines should be in-between. • The bottleneck machines of Bernoulli assembly systems can be identified using the arrow-based method of Chapter 5 applied to all constituent serial lines. • The lean buffering for Bernoulli assembly systems can be calculated using the methods developed in Chapter 6 for serial lines. • Similarly, the due-time performance of Bernoulli assembly systems can be evaluated using the methods developed in Chapter 9 for serial lines.

16.7

Problems

Problem 16.1 Consider the Bernoulli assembly system defined by: M1 = 2, M2 = 4, M0 = 2, p1 = [0.85, 0.95], p2 = [0.9, 0.9, 0.9, 0.9], p0 = [0.95, 0.9], N1 = [2, 3], N2 = [1, 2, 2, 2], N0 = 2. (a) Is its buffering potent? (b) If not, design an improvement project that ensures strong potency and evaluate the resulting improvement of the production rate. Problem 16.2 Construct a recursive aggregation procedure for a Bernoulli assembly system with three component lines and one merge operation. Problem 16.3 Construct a recursive aggregation procedure for a Bernoulli assembly system with three component lines and two merge operations. Problem 16.4 Develop a rule-of-thumb for selecting lean buffering in Bernoulli assembly systems with identical machines and buffers (similar to that of Subsection 6.2.3). Problem 16.5 Develop a recursive aggregation procedure for the performance analysis of Bernoulli assembly systems under the symmetric blocking assumption.

16.8. ANNOTATED BIBLIOGRAPHY

16.8

525

Annotated Bibliography

The material of this chapter is based on [16.1] C.-T. Kuo, J.-T. Lim, S.M. Meerkov and E. Park, “Improvability Theory for Assembly Systems: Two Component - One Assembly Machine Case,” Mathematical Problems in Engineering, vol. 3, pp. 95-171, 1997. [16.2] S.-Y. Chiang, C.-T. Kuo, J.-T. Lim and S.M. Meerkov, “Improvability Theory of Assembly Systems I: Problem Formulation and Performance Evaluation,” Mathematical Problems in Engineering, vol. 6, pp. 321-357, 2000. [16.3] S.-Y. Chiang, C.-T. Kuo, J.-T. Lim and S.M. Meerkov, “Improvability Theory of Assembly Systems II: Improvability Indicators and Case Study,” Mathematical Problems in Engineering, vol. 6, pp. 359-393, 2000. More details can be found in [16.4] C.-T. Kuo, Bottlenecks in Production Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1996. Work allocation in assembly systems has been studied in [16.5] K.R. Baker, S.G. Powell and D.F. Pyke, “Optimal Allocation of Work in Assembly Systems,” Management Science, vol. 39, pp.101-106, 1993, where it has been shown that the merge operation must have a higher efficiency than other operations.

Chapter 17

Assembly Systems with Continuous Time Models of Machine Reliability Motivation: As it was pointed out on the number of occasions above, machines on the factory floor often obey continuous time reliability models. Hence, it is important to extend the results of Chapter 16 to both exponential and nonexponential machines. This is the goal of this chapter. Overview: Sections 17.1 and 17.2 address the problem of performance analysis for assembly systems with exponential and non-exponential machines, respectively. Section 17.3 is devoted to continuous improvement, while Section 17.4 discusses the issues of lean buffering and customer demand satisfaction.

17.1

Analysis of Assembly Systems with Exponential Machines

17.1.1

Systems addressed

The systems considered here are shown in Figures 17.1 and 17.2 for synchronous and asynchronous cases, respectively. The up- and downtime of the machines are distributed exponentially with parameters λij and µij , i = 0, 1, 2, j = 1, . . . , Mi , respectively, and the capacity of the machines is assumed to be cij = 1 for the synchronous case and arbitrary for the asynchronous one. The flow model is assumed. The states and the assumptions of system operation remain the same as in continuous serial lines, i.e., conventions (a)-(e) of Subsection 11.1 are satisfied. In addition, as in Chapter 16, it is assumed that (f) The merging machine m01 is starved when either b1M1 or b2M2 or both are empty. 527

528

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS

λ11 µ11

N11

λ12 µ12

λ1M µ1M 1

m11

b11

m12

m1M

λ21 µ 21

N21

λ22 µ 22

λ2M µ2M

m21

b21

m2M

N1M 1

b1M 1

1

2

m22

1

2

N2M 2

b2M

2

λ01 µ01

N01

m01

b01

λ02 µ02

λ0M µ 0 0M

m0M

m02

0

0

2

Figure 17.1: Exponential assembly system with synchronous machines λ11 µ11 τ11 N11

m11

b11

λ21 µ21 τ21

m21

λ12 µ12 τ12

1

1

m12

N21 c22 λ22 µ22

b21

λ1M µ1M τ1M

m22

m1M

N1M 1

1

b1M 1

1

λ2M µ2M τ2M N2M 2 2

2

m2M

2

2

b2M

τ 01 λ01 µ01 N01 τ02 λ02 µ02

m01

b01

m02

λ0M µ0M τ0M 0

0

m0M

0

0

2

Figure 17.2: Exponential assembly system with asynchronous machines Given this scenario, methods of performance analysis for both synchronous and asynchronous cases are described below. The approach is the same as in Chapter 16: It is based on the aggregation techniques for exponential lines developed in Subsections 11.1.2 and 11.2.2 applied to the constituent serial lines that comprise the assembly systems at hand.

17.1.2

Synchronous case

Recursive aggregation procedure: Consider the upper line {m11 , b11 , . . . , m1M1 , b1M1 , m001 , b01 , m02 , . . . , b0,M0 −1 , m0M0 }

(17.1)

and the lower line {m21 , b21 , . . . , m2M2 , b2M2 , m0001 , b01 , m02 , . . . , b0,M0 −1 , m0M0 }

(17.2)

where the virtual machines m001 and m0001 are modified versions of m01 to be defined so as to account for the starvation of m01 by b1M1 and b2M2 , respectively. For convenience, introduce the notations: ½ λ1i , i = 1, . . . , M1 , λui = λ0(i−M1 ) , i = M1 + 2, . . . , M1 + M0 , ½ µ1i , i = 1, . . . , M1 , µui = µ0(i−M1 ) , i = M1 + 2, . . . , M1 + M0 ,

17.1. ANALYSIS OF CONTINUOUS ASSEMBLY SYSTEMS ½ Niu

= ½

λli

= ½

µli

= ½

Nil

=

529

N1i , i = 1, . . . , M1 , N0(i−M1 ) , i = M1 + 1, . . . , M1 + M0 − 1, λ1i , i = 1, . . . , M2 , λ0(i−M2 ) , i = M2 + 2, . . . , M2 + M0 ,

(17.3)

µ1i , i = 1, . . . , M2 , µ0(i−M2 ) , i = M2 + 2, . . . , M2 + M0 , N2i , i = 1, . . . , M2 , N0(i−M2 ) , i = M2 + 1, . . . , M2 + M0 − 1,

where, as before, the superscripts u and l indicate the upper and lower lines, respectively. Then, the recursive aggregation procedure for analysis of synchronous exponential assembly systems can be introduced as follows: Equation for the λ- and µ-parameter of the virtual machine m001 : µuM1 +1 (s + 1) λuM1 +1 (s + 1)

=

µ01 [1 − Pb02M2 (s)],

=

λ01 + µ01 − µuM1 +1 (s + 1),

(17.4)

where the second equation is selected so as to euM1 +1 (s + 1) = eM1 +1 [1 − Pb02M2 (s)], and Pb02M2 (s) is the probability that buffer b2M2 is empty at step s; Equations for aggregation of the upper line: µub i (s + 1)

=

λub i (s + 1)

=

µuf i (s λuf i (s

+ 1) + 1)

uf uf ub u µui (1 − Q(λub i+1 (s + 1), µi+1 (s + 1), λi (s), µi (s), Ni )), i = 1, . . . , M0 + M1 − 1, uf ub u λui + µui Q(λi+1 (s + 1), µub (s + 1), λuf i+1 i (s), µi (s), Ni )), i = 1, . . . , M0 + M1 − 1,

Q(λuf i−1 (s

=

µui (1

−

=

λui

uf µui Q(λi−1 (s

+

+

1), µuf i−1 (s

+

1), µuf i−1 (s

+

1), λub i (s

+

1), µub i (s

(17.5)

u + 1), Ni−1 )),

i = 2, . . . , M0 + M1 , ub u + 1), λub i (s + 1), µi (s + 1), Ni−1 ), i = 2, . . . , M0 + M1 ;

Equation for the probability of buffer b1M1 being empty: uf ub ub u Pb01M1 = Q(λuf M1 (s + 1), µM1 (s + 1), λM1 +1 (s + 1), µM1 +1 (s + 1), NM1 ); (17.6)

Equation for the λ- and µ-parameter of the virtual machine m0001 : µlM2 +1 (s + 1)

=

µ01 [1 − Pb01M1 (s)],

λlM2 +1 (s + 1)

=

λ01 + µ01 − µlM2 +1 (s + 1);

Equations for aggregation of the lower line: µlb i (s + 1)

=

lf lf lb l µli (1 − Q(λlb i+1 (s + 1), µi+1 (s + 1), λi (s), µi (s), Ni )),

(17.7)

530

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS i = 1, . . . , M0 + M2 − 1,

λlb i (s + 1)

=

lf lf lb l λli + µli Q(λlb i+1 (s + 1), µi+1 (s + 1), λi (s), µi (s), Ni )), i = 1, . . . , M0 + M2 − 1,

µlf i (s + 1)

=

lf lb lb l µli (1 − Q(λlf i−1 (s + 1), µi−1 (s + 1), λi (s + 1), µi (s + 1), Ni−1 )), i = 2, . . . , M0 + M2 , (17.8)

λlf i (s + 1)

=

lf lb lb l λli + µli Q(λlf i−1 (s + 1), µi−1 (s + 1), λi (s + 1), µi (s + 1), Ni−1 ), i = 2, . . . , M0 + M2 ;

Equation for the probability of buffer b2M2 being empty: lf lb lb l Pb02M2 = Q(λlf M2 (s + 1), µM2 (s + 1), λM2 +1 (s + 1), µM2 +1 (s + 1), NM2 ). (17.9)

The boundary conditions for this aggregation procedure are: u λuf 1 (s) = λ1 ,

u λub M1 +M0 (s) = λM1 +M0 ,

s = 0, 1, . . . ,

u µuf 1 (s) = µ1 , l λlf 1 (s) = λ1 , l µlf 1 (s) = µ1 ,

µub M1 +M0 (s)

µuM1 +M0 ,

s = 0, 1, . . . ,

l λlb M2 +M0 (s) = λM2 +M0 ,

s = 0, 1, . . . ,

µlb M2 +M0 (s)

s = 0, 1, . . . ,

= =

µlM2 +M0 ,

(17.10)

and the initial conditions are: u λuf i (0) = λi ,

λlf i (0)

=

λli ,

u µuf i (0) = µi ,

i = 1, . . . , M1 + M0 ,

l µlf i (0) = µi , Pb02M2 (0) = 0.

i = 1, . . . , M2 + M0 ,

(17.11)

Theorem 17.1 Aggregation procedure (17.4)-(17.11) has the following properties: (i) It is convergent, i.e., the following limits exist: lim λuM1 +1 (s) =: λuM1 +1 ,

s→∞

uf lim λuf i (s) =: λi ,

s→∞

ub lim λub i (s) =: λi ,

s→∞

lim

s→∞

λuM2 +1 (s)

=:

lb lim λlb i (s) =: λi ,

s→∞

=: µuf i ,

ub lim µub i (s) =: µi ,

s→∞

λuM2 +1 ,

lf lim λlf i (s) =: λi ,

s→∞

lim µuM1 +1 (s) =: µuM1 +1 ,

s→∞ lim µuf (s) s→∞ i

lim

µuM2 +1 (s)

s→∞ lim µlf (s) =: s→∞ i

µlf i ,

lb lim µlb i (s) =: µi ,

s→∞

i = 1, . . . , M1 + M0 , i = 1, . . . , M1 + M0 ; =: µuM2 +1 ,

(17.12)

i = 1, . . . , M2 + M0 , i = 1, . . . , M2 + M0 .

(ii) These limits are unique solutions of the steady state equations corresponding to (17.4)-(17.9).

17.1. ANALYSIS OF CONTINUOUS ASSEMBLY SYSTEMS

531

(iii) In addition, these limits satisfy the relationships: euf M1 +M0

=

lf lb eub 1 = eM2 +M0 = e1

=

uf uf ub ub u euf i [1 − Q(λi+1 , µi+1 , λi , µi , Ni )]

=

uf uf ub ub u (17.13) eub i+1 [1 − Q(λi , µi , λi+1 , µi+1 , Ni )] i = 1, ..., M1 + M0 − 1,

=

lf lf lb lb l elf i [1 − Q(λi+1 , µi+1 , λi , µi , Ni )]

=

lf lf lb lb l elb i+1 [1 − Q(λi , µi , λi+1 , µi+1 , Ni )], i = 1, ..., M2 + M0 − 1.

where euf i =

µuf i λuf i

elf i =

, + µuf i

λlf i

, lf λlf i + µi

eub i =

µub i , ub λub i + µi

elb i =

µlb i , lb λlb i + µi

i = 1, . . . , M1 + M0 , i = 1, . . . , M2 + M0 .

Proof: Similar to those of Theorems 11.4 and 16.1. The limits (17.12)-(17.13) can be used to represent the assembly system at hand as shown in Figure 17.3. uf

λi

uf

µi

u

Ni

ub

ub

λi +1 µ i +1

i =1, ..., M 1+M 0 −1

lf

lf

λi µi

l

Ni

λ

lb

i +1

lb

µi

+1

i =1, ..., M 2+M 0 −1

Figure 17.3: Representation of the synchronous exponential assembly system through the aggregated machines

Formulas for the performance measures: Based on the above two-machine interpretation, the expressions for the performance measures are as follows: Production rate: d P R

= =

=

µuf i uf λuf i + µi

uf uf ub u [1 − Q(λub i+1 , µi+1 , λi , µi , Ni )]

µub uf i+1 ub ub u [1 − Q(λuf i , µi , λi+1 , µi+1 , Ni )] ub λub + µ i+1 i+1 i = 1, ..., M1 + M0 − 1, λlf i λlf i

+

µlf i

lf lf lb l [1 − Q(λlb i+1 , µi+1 , λi , µi , Ni )]

(17.14)

532

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS =

µlb lf i+1 lb lb l [1 − Q(λlf i , µi , λi+1 , µi+1 , Ni )], lb λlb + µ i+1 i+1 i = 1, ..., M2 + M0 − 1. u

(17.15)

l

\ \ IP i remain the same as Work-in-process: The expressions for W IP i and W f f b for the serial lines, i.e., (11.30), but with λi , µi , λi+1 and µbi+1 , i = 1, . . . , M − uf ub ub 1, being substituted by λuf i+1 , µi , λi+1 and µi+1 , i = 1, . . . , M0 + M1 − 1, lf lb lb respectively, for the upper line and by λlf i+1 , µi , λi+1 and µi+1 , i = 1, . . . , M0 + M2 − 1, respectively, for the lower line. Blockages and starvations: u

di BL

u

ci ST

l

d BL i u

c ST i

=

=

=

=

µui uf uf ub u Q(λub i+1 , µi+1 , λi , µi , Ni ), λui + µui i = 1, . . . , M1 + M0 − 1, µui uf uf ub u Q(λi−1 , µi−1 , λub i , µi , Ni−1 ), λui + µui i = 2, . . . , M1 + M0 , µli lf lf lb l Q(λlb i+1 , µi+1 , λi , µi , Ni ), λli + µli i = 1, . . . , M2 + M0 − 1, l µi lf lb lb l Q(λlf i−1 , µi−1 , λi , µi , Ni−1 ), λli + µli i = 2, . . . , M2 + M0 .

(17.16)

(17.17)

(17.18)

(17.19)

Accuracy: The accuracy of estimates (17.14)-(17.19) is the same as that for the exponential serial lines evaluated in Section 11.1.2. PSE Toolbox: Recursive procedure (17.4)-(17.11) and performance measures (17.14)-(17.19) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Section 19.3.

17.1.3

Asynchronous case

Recursive aggregation procedure: The asynchronous case is analyzed using the same approach as above but with two modifications. First, the aggregation of the constituent serial lines is based on iterating cij ’s, rather than λij ’s and µij ’s (using (11.54)) and, second, the effect of the other line is taken into account by modifying c01 , rather than λ01 and µ01 . Specifically, consider the upper and lower lines of Figure 17.2 and introduce the notations: ½ c1i , i = 1, . . . , M1 , cui = c0(i−M1 ) , i = M2 + 1, . . . , M1 + M0 ,

17.1. ANALYSIS OF CONTINUOUS ASSEMBLY SYSTEMS ½ λui

= ½

µui

=

eui

=

Niu

=

cli

=

λli

=

µli

=

eli

=

Nil

533

λ1i , i = 1, . . . , M1 , λ0(i−M1 ) , i = M1 + 1, . . . , M1 + M0 ,

µ1i , i = 1, . . . , M1 , µ0(i−M1 ) , i = M1 + 1, . . . , M1 + M0 , µui , i = 1, . . . , M1 + M0 , u λ + µui ½i N1i , i = 1, . . . , M1 , N0(i−M1 ) , i = M1 + 1, . . . , M1 + M0 − 1, ½ λ1i , i = 1, . . . , M2 , c0(i−M2 ) , i = M2 + 2, . . . , M2 + M0 , ½ λ1i , i = 1, . . . , M2 , λ0(i−M2 ) , i = M1 + 2, . . . , M2 + M0 , ½ µ1i , i = 1, . . . , M2 , µ0(i−M2 ) , i = M1 + 2, . . . , M2 + M0 ,

(17.20)

µli , i = 1, . . . , M2 + M0 , λl + µli ½i N2i , i = 1, . . . , M2 , = N0(i−M2 ) , i = M1 + 1, . . . , M2 + M0 − 1,

Then, the recursive aggregation procedure for analysis of asynchronous assembly systems can be defined as follows: Equation for the c-parameter of the virtual machine m001 : cuM1 +1 (s + 1) = c01 [1 − Pb02M2 (s)];

(17.21)

where, as before, Pb02M2 (s) is the probability that buffer b2M2 is empty at step s; Equations for aggregation of the upper line: bliu (s + 1)

=

cub i (s + 1)

=

stui (s + 1)

=

cuf i (s + 1)

=

u u uf u u ub u eui cuf i (s) − T P (λi , µi , ci (s), λi+1 , µi+1 , ci+1 (s + 1), Ni )

cui [1 − bliu (s + 1)], eui cub i (s + 1) −

eui cuf i (s) 1 ≤ i ≤ M0 + M1 − 1, 1 ≤ i ≤ M0 + M1 − 1,

u u ub T P (λui−1 , µui−1 , cuf i−1 (s + 1), λi , µi , ci (s u ub ei ci (s + 1)

,

(17.22) u + 1), Ni−1 )

2 ≤ i ≤ M0 + M1 , cui [1 − stui (s + 1)],

2 ≤ i ≤ M0 + M 1 ;

Equation for the probability of buffer b1M1 being empty: Pb01M1 = stuM1 +1 (s + 1); Equation for the c-parameter of the virtual machine clM2 +1 (s + 1) = c01 [1 − Pb01M1 (s)];

(17.23) m0001 : (17.24)

,

534

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS Equations for aggregation of the lower line:

blil (s + 1)

=

clb i (s + 1)

=

stli (s + 1)

=

l l lf l l lb l eli clf i (s) − T P (λi , µi , ci (s), λi+1 , µi+1 , ci+1 (s + 1), Ni )

eli clf i (s) 1 ≤ i ≤ M0 + M2 − 1, 1 ≤ i ≤ M0 + M2 − 1,

cli [1 − blil (s + 1)], eli clb i (s

+ 1) −

T P (λli−1 , µli−1 , clf i−1 (s + eli clb i (s + 1)

1), λli , µli , clb i (s

,

(17.25) l + 1), Ni−1 )

,

2 ≤ i ≤ M0 + M2 , clf i (s

+ 1)

=

cli [1

−

stli (s

+ 1)],

2 ≤ i ≤ M0 + M 2 ;

Equation for the probability of buffer b2M2 being empty: Pb02M2 = stlM2 +1 (s + 1);

(17.26)

The boundary conditions for this aggregation procedure are: u cuf 1 (s) = c1 ,

clf 1 (s)

=

cl1 ,

u cub M1 +M0 (s) = cM1 +M0 ,

s = 0, 1, . . . ,

clb M2 +M0 (s)

s = 0, 1, . . . ,

=

clM2 +M0 ,

(17.27)

and the initial conditions are: cuf i (0)

= cui ,

i = 1, . . . , M1 + M0 ,

clf i (0) 2M2 b P0 (0)

=

cli ,

i = 1, . . . , M2 + M0 ,

=

0.

(17.28)

Theorem 17.2 Aggregation procedure (17.21)-(17.28) has the following properties: (i) It is convergent, i.e., the following limits exists: lim cuM1 +1 (s) =: cuM1 +1 ,

s→∞

lim cuf (s) =: cuf i = 1, . . . , M1 + M0 , i , s→∞ i ub lim cub i = 1, . . . , M1 + M0 ; i (s) =: ci , s→∞ u u lim cM2 +1 (s) =: cM2 +1 , s→∞ lim clf (s) =: clf i = 1, . . . , M2 + M0 , i , s→∞ i lb lim clb i = 1, . . . , M2 + M0 . i (s) =: ci , s→∞

(17.29)

(ii) These limits are unique solutions of the steady state equations corresponding to (17.21)-(17.26).

17.1. ANALYSIS OF CONTINUOUS ASSEMBLY SYSTEMS

535

(iii) In addition, these limits satisfy the relationships: cuf M1 +M0 eM1 +M0

lf lb = cub 1 e1 = cM2 +M0 eM2 +M0 = c1 e1 u u ub u = T P (λui , µui , cuf i , λi+1 , µi+1 , ci+1 , Ni ) (17.30) i = 1, . . . , M1 + M0 − 1, l l lb l = T P (λli , µli , clf i , λi+1 , µi+1 , ci+1 , Ni ), i = 1, . . . , M2 + M0 − 1.

Proof: Similar to the proofs of Theorems 11.7 and 16.1. The limits (17.29) can be used to represent the assembly system at hand as shown in Figure 17.4. u

u

u

λi

µi c

u Ni

uf i

u

c

l

l

λi +1 µ i +1

λi µi

ub i +1

lf

ci

i =1, ..., M 1+M 0 −1

l Ni

λ

l

l

i +1

µi lb

+1

c i +1

i =1, ..., M 2+M 0 −1

Figure 17.4: Representation of the asynchronous exponential assembly system through the aggregated machines

Formulas for the performance measures: Based on the above two-machine interpretation, the expressions for the performance measures are as follows: Throughput: d T P

=

lf u l cuf M1 +M0 eM1 +M0 = cM2 +M0 eM2 +M0

(17.31)

=

u cub 1 e1

(17.32)

=

l clb 1 e1 . u

l

\ \ Work-in-process: The expressions for W IP i and W IP i remain the same as for the asynchronous serial lines (see Subsection 11.2.2), but with cfi and cbi+1 , ub i = 1, . . . , M − 1, being substituted by cuf i+1 and ci+1 , i = 1, . . . , M0 + M1 − 1, lf lb respectively, for upper line and by ci+1 and ci+1 , i = 1, . . . , M0 + M2 − 1, respectively, for lower line. Blockages and starvations: u

di BL u ci ST l di BL u ci ST

=

eui bliu ,

i = 1, . . . , M1 + M0 − 1,

(17.33)

=

eui stui ,

i = 2, . . . , M1 + M0 ,

(17.34)

=

eli blil , eli stli ,

=

i = 1, . . . , M2 + M0 − 1,

(17.35)

i = 2, . . . , M2 + M0 .

(17.36)

536

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS

Accuracy: The accuracy of estimates (17.31)-(17.36) is the same as that for the asynchronous exponential serial lines discussed in Subsection 11.2.2. PSE Toolbox: Recursive procedure (17.21)-(17.28) and performance measures (17.31)-(17.36) are implemented in the Performance Analysis function of the toolbox. For a description and illustration of this tool, see Section 19.3. Monotonicity and reversibility: Similar to the Bernoulli case, exponential assembly systems possess the properties of monotonicity with respect to all machine and buffer parameters and the property of reversibility. Generalizations: Exponential assembly systems with more than two component lines and more than one merging operation can be analyzed using the techniques described here and applied to all constituent serial lines, which comprise the assembly system at hand.

17.2

Analysis of Non-exponential Assembly Systems

The system considered here is shown in Figure 17.5. Each machine is defined f t up,1M 1 f t down,1M 1 N 1M1 τ1M1

f t up,11 f t down,11 τ11

N11

f t up,12 f t down,12 τ12

m11

b11

m12

m1M

f t up,22

f t up,2M

f t up,21

f t down,21 τ21

N21

f t down,22 τ22

m21

b21

m22

1

b1M

1

N1

f t up,02 f t down,02 τ02

f t up,0M 0 f t down,0M 0 τ0M

m02

m0M

0

2

f t down,2M 2 τ2M N 2M2 2

m2M2

f t up,01 f t down,01 τ01

m01

b01

0

b2M

2

Figure 17.5: Assembly system with non-exponential machines by its pdf’s of up- and downtime and by its cycle time. The non-exponential pdf’s considered are Weibull (3.4), gamma (3.5), and log-normal (3.6). The mathematical description of systems at hand and the approach to their analysis remain the same as in Chapter 12: numerical simulations followed by analytical approximations of the results obtained. Based on this approach, it is possible to show that, as in Chapter 12, the throughput of non-exponential assembly systems is practically insensitive to the nature of up- and downtime distributions and depends mainly on their coefficients of variation. Using this

17.2. NON-EXPONENTIAL CONTINUOUS ASSEMBLY SYSTEMS

537

observation, the formula for evaluating the throughput in the non-exponential assembly case is introduced as d T P = T P 0 − (T P 0 − T P exp )CVave , where

P CVave =

ij∈Im (CVup,ij

+ CVdown,ij )

2(M0 + M1 + M2 )

(17.37)

,

and Im is used to denote the set of all machines. Define the accuracy of this approximation as ∆T P =

d |T P − TP| . TP

(17.38)

Numerical Fact 17.1 Assume that CVup,ij ≤ 1 and CVdown,ij ≤ 1. Then ∆T P ¿ 1. Justification A set of 10,000 assembly systems has been generated with the parameters selected randomly and equiprobably from the sets M0 , M1 , M2 Tdown,ij eij cij Nij

∈ {1, 2, 3, 4, 5}, ∈ [1, 5], ∈ [0.75, 0.95], ∈ [0.8, 1.2], ∈ {2, 3, . . . , 10}.

(17.39) (17.40) (17.41) (17.42) (17.43)

For each of these systems, 50 additional assembly systems have been constructed by selecting the distributions of the machines’ up- and downtime and the coefficients of variation randomly and equiprobably from the following sets ftup ,ij , ftdown ,ij CVup,ij , CVdown,ij

∈ {exp, W, ga, LN }, ∈ [0.1, 1].

(17.44) (17.45)

d For each of the 500,000 assembly systems, thus constructed, T P and T P have been evaluated, respectively, by simulations and by calculations using (17.37). The accuracy of the approximation has been evaluated by (17.38). The simulations used a C++ code representing the system at hand and Simulation Procedure 4.1. This approach is illustrated by a system shown in Figure 17.6. For this system, the simulations result in T P 0 = 0.7066 and T P exp = 0.6284. Figure 17.7 shows the linear function representing expression (17.37). As mentioned before, to investigate the accuracy of this expression for the system at hand, 50 additional systems are generated and their T P s, obtained by simulations, are indicated in Figure 17.7. Clearly, these T P s are close to that provided by (17.37), with the largest error being less than 0.025.

538

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS

[Tup, Tdown]: [10.1, 1.49] c: 0.99

[10.0, 2.38] 0.97 7

[13.0, 2.82] 0.93 2

[4.07, 1.14] 0.90 9

10

[14.7, 1.00] 1.02

[39.8, 2.67] 0.96 9

[Tup, Tdown]: [16.7, 4.44] c: 0.95

[6.86, 1.96] 1.02 4

[10.3 2.48] 0.96 5

[16.9, 4.77] 0.92 8

[24. 4, 4.79] 0.93 10

[4.40, 1.43] 1.09 8

5

Figure 17.6: Illustration of systems analyzed

0.75 0.7 TP 0.65 0.6 0.55 0

0.2

0.4

0.6

0.8

1

CVave

Figure 17.7: Estimate of the throughput as a function of the average coefficient of variation

17.3. IMPROVEMENT OF CONTINUOUS ASSEMBLY SYSTEMS

539

Analyzing in a similar manner all 500,000 assembly systems under consideration, we determine that the average value of ∆T P is 0.0211 and the maximum one is 0.0676. Thus, we conclude that expression (17.37) is justified. As for the arbitrary continuous time model of machine reliability we conjecture that: Expression (17.37) can be used for evaluating throughput of assembly systems with machines having any unimodal distribution of up- and downtime.

17.3

Improvement of Assembly Systems with Continuous Time Models of Machine Reliability

17.3.1

Constrained improvability

The resource constraints for assembly systems under consideration are similar to those for serial lines with continuous models of machine reliability (see Chapter 13). Specifically, the buffer capacity (BC) constraint is M1 X i=1

N1i +

M2 X

N2i +

i=1

M 0 −1 X

N0i = N ∗ ,

(17.46)

i=1

while the cycle time (CT) constraint becomes Mi 2 X X

τij = τ ∗ .

(17.47)

i=0 j=1

The definitions of improvability with respect to BC and CT are similar to those of Chapter 13 (with obvious modifications to account for the parallel component lines). As a result, it is possible to show that the improvability indicators also remain the same. Specifically: CT-Improvability Indicator 17.1: An assembly system with machines having their reliability models in the set {exp, W , ga, LN } is unimprovable with respect to CT if each buffer is, on the average, close to being half full. BC-Improvability Indicator 17.1: An assembly system with machines having their reliability models in the set {exp, W , ga, LN } is unimprovable with respect to BC if the buffer occupancy allocation is such that max |W IPi(j−1) − (Nij − W IPij )|, ı = 0, 1, 2; j = 1, . . . , Mi ij

is minimized over all sequences Nij , i = 1, 2, j = 1, . . . , Mi and N0j , j = 1, . . . , M0 − 1, such that (17.46) is observed.

540

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS

As in Chapter 13, we conjecture that: These indicators hold for any unimodal continuous time model of machine reliability. Finally, we note that CT- and BC-Continuous Improvement Procedures of Chapter 13 can be used for assembly systems as well.

17.3.2

Unconstrained improvability

As it has been pointed out in Chapter 13, among all possible types of bottlenecks, the most useful one for practical application is the c-BN. Therefore, below this case is addressed. The definition of c-BN remains the same as in Section 13.2 with obvious modification to account for parallel component lines. The arrow assignment rule and the BN severity also remain the same as in Subsection 16.2.2. As a result, it is possible to show that c-BN Indicator 13.1 also holds. Namely, Bottleneck Indicator 17.1: The bottleneck machine and bottleneck buffer of an assembly system with machines having reliability models in the set {exp, W , ga, LN } can be identified as follows: • If there is a single machine with no emanating arrows, it is the c-BN. • If there are multiple machines with no emanating arrows, the one with the largest severity is the Primary c-BN (Pc-BN), where the severity is defined by (16.48). The accuracy of this bottleneck indicator is again similar to that of continuous serial lines. Specifically, a single machine with no emanating arrows is c-BN in 90.1% of cases; one of the multiple machines with no emanating arrows is Pc-BN in 96.6% of cases. Two examples of c-BN identification in continuous assembly systems are shown in Figures 17.8 and 17.9. To conclude this section, we point out that Bottleneck Indicator 17.1 can be used in assembly systems with more than two components lines and more than one merge machines as well. Also, we conjecture that: Bottleneck Indicator 17.1 is applicable to assembly systems with any unimodal continuous time model of machine reliability.

17.4

Design of Lean Buffering and Customer Demand Satisfaction in Assembly Systems with Continuous Machines

17.4.1

Lean buffering

The approach to lean buffering design in assembly systems with continuous time model of machine reliability remains the same as that of the continuous

17.4. DESIGN OF LEAN CONTINUOUS ASSEMBLY SYSTEMS

541

c-BN [ftup, ftdown]: [Tup, Tdown]: [CVup, CVdown]: c:

[ga, exp] [8.07, 2.21] [0.61, 1] 0.91

[W, exp] [13.6, 3.11] [0.23, 1] 0.89 9

2

STij:

0

0.010

BLij:

0.065

0.078

୼்௉ : ୼௖೔ೕ

0.045

0.305

[ga, ga] [13.7, 3.30] [0.48, 0.81] 0.95

[W, ga] [15.3, 3.06] [0.41, 0.53] 0.98 8

6

[ga, W] [30.9, 2.72] [0.80, 0.77] 1.03

[W, LN] [37.7, 2.13] [0.23, 0.23] 1.03

[ga, W] [15.2, 3.60] [0.63, 0.31] 0.85

[ftup, ftdown]: [Tup, Tdown]: [CVup, CVdown]: c:

8

5

0

0.056

0.016

BLij:

0.065

0.277

0.292

୼்௉ : ୼௖೔ೕ

0.021

0.000

0.007

STij:

0.132

0.187

0.211

BLij:

0.003

0.001

0

0.008

0.011

ST012: 0.006

୼்௉ : ୼௖೔ೕ

5

STij:

[ga, LN] [25.2, 3.89] [0.54, 0.42] 0.96

0.071

Figure 17.8: Single machine with no emanating arrows [ftup, ftdown]: [Tup, Tdown]: [CVup, CVdown]: c:

[ga, LN] [16.8, 1.71] [0.61, 0.74] 0.86

[W, ga] [17.3, 4.45] [0.61, 0.86] 0.93 4

8

STij:

0

0.011

BLij:

0.220

0.150

୼்௉ : ୼௖೔ೕ

0.014

0.032

c-BN candidate

6

8

c-BN candidate [ga, exp] [27.5, 3.51] [0.38, 1] 0.87

[exp, LN] [13.2, 3.88] [1, 0.26] 0.88

[exp, exp] [28.9, 3.96] [1, 1] 0.96

[ftup, ftdown]: [Tup, Tdown]: [CVup, CVdown]: c:

7

4

0

0.012

0.100

BLij:

0.266

0.091

0.110

୼்௉ : ୼௖೔ೕ

0.004

0.136

0.051

[LN, ga] [6.93, 1.54] [0.65, 0.94] 1.05 3

STij:

0.036

0.071

0.158

0.254

BLij:

0.099

0.025

0.019

0

0.127

0.035

0.001

ST012: 0.122

5

STij:

[exp, W] [21.1, 3.65] [1, 0.68] 1.04

[ga, exp] [14.4, 4.31] [0.46, 1] 0.88

[ga, exp] [13.0, 3.45] [0.69, 1] 1.11

୼்௉ : ୼௖೔ೕ

0.022

Figure 17.9: Multiple machines with no emanating arrows

542

CHAPTER 17. CONTINUOUS ASSEMBLY SYSTEMS

serial lines (see Chapter 14). Specifically, in the case of identical machines, normalizations (14.3) and (14.4) are used, and the lean level of buffering (LLB) is calculated based on expressions (14.9) and (14.20), (14.22) for exponential threeand M > 3-machine assembly systems, respectively. In the non-exponential case, the results of Section 14.4 can be used. In systems with non-identical machines, to evaluate LLB, the six bounds for lean level of buffering, discussed in Subsection 14.3.2, can be applied.

17.4.2

Customer demand satisfaction

Results of Chapter 15 are directly applicable to assembly systems since the Production Subsystem (PS) is modeled as a single machine. So, to evaluate DT P , the PS must be aggregated into a single machine (using expressions (11.25) or (11.46)) and then formulas (15.11) and (15.22) can be used. Keeping in mind (5.13), this provides a lower bound on DT P in Production-Inventory-Customer (PIC) systems with exponential machines. The lean capacity of the Inventory Subsystem, νDT P , in PIC systems with exponential assembly PS can be evaluated as in Section 15.4. For the case of non-exponential machines, the approach of Section 15.3 can be used.

17.5

Summary

• Aggregation procedures (11.25) and (11.54), applied to all constituent serial lines, can be used for performance analysis of synchronous and asynchronous assembly systems with exponential models of machine reliability. • Empirical fomulas can be used for T P evaluation in assembly systems with an arbitrary continuous time model of machine reliability. • The constrained improvability properties of assembly systems with continuous models of machine reliability are similar to those of exponential serial lines. • The c-BN and BN-b of assembly systems with continuous time models of machine reliability can be identified using the arrow-based method of Chapter 13 applied to all constituent serial lines.

17.6

Problems

Problem 17.1 Consider the synchronous exponential assembly system defined by M1 = 2, M2 = 4, M0 = 2, e1 = [0.85, 0.95]T , e2 = [0.9, 0.9, 0.9, 0.9]T , e0 = [0.95, 0.9]T , N1 = [2, 3]T , N2 = [1, 2, 2, 2]T , N0 = 2, and the average downtime of all machines equal to 2.

17.7. ANNOTATED BIBLIOGRAPHY

543

(a) Is its buffering potent? (b) If not, design an improvement project that ensures strong potency and evaluate the resulting improvement of the production rate. Problem 17.2 Construct a recursive aggregation procedure for a synchronous exponential assembly system with three component lines and one merge operation. Problem 17.3 Construct a recursive aggregation procedure for a synchronous exponential assembly system with three component lines and two merge operations.

17.7

Annotated Bibliography

Performance analysis of exponential assembly systems has been studied in numerous publications. Several examples can be given as follows: [17.1] S.B. Gershwin, “Assembly/disassembly Systems: An Efficient Decomposition Algorithm for Tree-structured Networks,” IIE Transactions, vol. 23, pp. 302–314, 1991. [17.2] M.D. Mascolo, R. David and Y. Dallery, “Modeling and Analysis of Assembly Systems with Unreliable Machines and Finite Buffers,” IIE Transactions, vol. 23, pp. 315–330, 1991. [17.3] S.B. Gershwin and M.H. Burman, “A Decomposition Method for Analyzing Inhomogeneous Assembly/disassembly Systems,” Annals of Operations Research, vol. 93, pp. 91–115, 2000. The material of this chapter is based on [17.4] S.-Y. Chiang, Bottlenecks in Production Systems with Markovian Machines: Theory and Applications, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1999. [17.5] J. Li, “Overlapping Decomposition: A System-Theoretic Method for Modeling and Analysis of Complex Manufacturing Systems,” IEEE Transactions on Automation Science and Engineering, vol. 2, pp. 40–53, 2005. [17.6] S. Ching, S.M. Meerkov and L. Zhang, “Assembly Systems with Nonexponential Machines: Throughput and Bottlenecks,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, pp. 911–917, 2008.

Chapter 18

Summary of Main Facts of Production Systems Engineering Motivation: All previous chapters described numerous facts concerning the behavior and properties of individual machines, serial lines, and assembly systems. In this chapter, these facts are summarized and their significance is commented upon. Sections or subsections, where these facts are derived, are indicated. Overview: The three sections of this chapter address the main facts pertaining to individual machines, serial lines, and assembly systems, respectively.

18.1

Individual Machines

IM 1 For any distribution of up- and downtime, the efficiency of a machine, i.e., the average number of parts produced per cycle time, is given by e=

Tup 1 = , Tup + Tdown 1 + TTdown up

(18.1)

where Tup and Tdown are average up- and downtimes of the machine (Subsection 2.2.6). The quantity e is referred to as machine efficiency. This fact implies that: • The efficiency of a machine in isolation (i.e., when it is neither starved nor blocked) can be calculated without knowing its reliability model. • The machine efficiency is, in fact, a function of one variable only: Tup /Tdown ; thus, if this ratio is constant, the machine efficiency remains the same, irrespective of how long or short Tup and Tdown are. 547

548

CHAPTER 18. SUMMARY OF MAIN FACTS • Increasing Tup by any factor leads to the same machine efficiency as decreasing Tdown by the same factor.

IM 2 If the breakdown and repair rates of a machine are increasing functions of time, the coefficients of variation of the resulting distributions are less than 1 (Subsection 2.2.4). This fact is of significance because CV < 1 leads to simple methods for the analysis of production systems at hand, e.g., to simple empirical formulas for the throughput and lean buffering of serial lines with any model of machine reliability (see Facts PA 6 and D 6).

18.2

Serial Lines

18.2.1

Performance analysis

PA 1 Unlike individual machines, the throughput of a serial line does depend on Tup and Tdown independently. In particular, shorter Tup and Tdown lead to Tup larger throughput than longer ones, even if Tdown (i.e., e) is fixed (Subsections 11.1.1, 11.1.2, 11.2.1, and 11.2.2). Thus, from the point of view of the overall system, it is better to have shorter up- and downtimes than longer ones. This is because shorter downtimes are easier to accommodate by finite buffers than longer ones. PA 2 Also unlike individual machines, decreasing Tdown by any factor leads to a larger increase in the serial line throughput than increasing Tup by the same factor (Subsections 11.1.1, 11.1.2, 11.2.1, and 11.2.2). Thus, management should strive more for decreasing Tdown than for increasing Tup . PA 3 Finally, unlike individual machines, decreasing coefficients of variation of up- and downtime leads to increased throughput of a serial line (Section 12.3). This implies that management should strive for decreasing variability of upand downtime, even if their average values remain the same. PA 4 The throughput of serial lines with Bernoulli, geometric, and exponential machines can be evaluated analytically using aggregation techniques. The same methods can be used to evaluate the work-in-process and the probabilities of blockages and starvations of the machines (Sections 4.1, 4.2, 11.1, and 11.2). Therefore, time consuming simulations, as a means for evaluating the performance measures, can be avoided.

18.2. SERIAL LINES

549

PA 5 The throughput of a serial line with machines having any reliability model is practically independent of the distributions of up- and downtime; it depends mostly on their coefficients of variation (Section 12.2). PA 6 If the coefficients of variation of machines’ up- and downtime are less than 1, the throughput of a serial line is almost a linear function with a negative slope of the average coefficient of variation (Section 12.3). The significance of the last two facts is that the throughput of a serial line can be evaluated without knowing detailed statistical properties of machines’ reliability but just based on the first two moments of up- and downtime distributions. PA 7 The throughput of serial lines is a saturating function, while the workin-process (for most machine efficiency allocations) is almost a linear function of buffer capacity (Subsections 4.2.3, 11.1.2, and 11.2.2). Thus, there is no reason for increasing buffer capacity beyond a certain level (see Fact D 4 for more details). PA 8 Serial lines possess the property of reversibility, i.e., the throughput remains the same if the flow of parts is (conceptually) reversed (Subsections 4.1.2, 4.2.2, 4.3.2, and 4.3.3). The significance of this fact can be illustrated as follows: • Some argue that to optimize the throughput of a line with identical machines, the largest buffers should be placed at the end of the line. Reversibility implies that one can equally argue the opposite, i.e., the largest buffers should be placed at the beginning of the line. Thus, both arguments are wrong. • Where should a single buffer be placed so that the throughput of a line with identical machines is maximized? Reversibility allows to establish that it should be in the middle of the line. • If all machines and buffers are identical and one machine can be improved, which one should it be, so that the throughput of a serial line is maximized? Similar to the above, the reversibility property allows to establish that it should be the machine in the middle of the line. • How should the total buffer capacity be allocated among all buffers in a line with identical machines so that the throughput is maximized? Reversibility implies that this allocation must be symmetric. Clearly, the uniform allocation meets this requirement as do the bowl and inverted bowl allocations. (As it is stated in Fact D 2, the optimal one is actually the “flat” inverted bowl pattern.) PA 9 Serial lines possess the property of monotonicity, i.e., the throughput of a serial line is a strictly monotonically increasing function of buffers capacity and machine repair rates and a strictly monotonically decreasing function of machine breakdown rates (Subsections 11.1.2 and 11.2.2).

550

CHAPTER 18. SUMMARY OF MAIN FACTS

Thus, one would always increase the throughput by improving any machine or buffer. One, however, must be careful in doing so: as Fact PA 7 states, increasing buffer capacity may lead to diminishing returns. PA 10 While it remains strictly monotonic with respect to buffers capacity and machine breakdown and repair rates, the throughput of closed serial lines (i.e., lines with a fixed number of carriers) is non-monotonic (concave) with respect to the number of carriers (Section 7.2). Thus, increasing (as well as decreasing) the number of carriers in closed lines may lead to throughput losses. PA 11 In serial lines with machines having reliability-capacity coupling, the throughput is a non-monotonic (concave) function of machine capacity (Problem 11.14). Thus, in such systems, the throughput may decrease when the machine operation is sped up. PA 12 In serial lines with machines having quality-quantity coupling, the production rate of non-defective parts is a non-monotonic (concave) function of machine efficiency (Subsection 8.2.2). Thus, increasing machine efficiency in serial lines with quality-quantity coupling may degrade the overall system performance. PA 13 Serial lines with rework may not possess the property of reversibility. As a result, to increase the throughput of serial lines with rework, more efficient machines should be placed toward the end of the line (Subsection 8.3.2). This fact is qualitatively different from that in lines with no rework, which do possess the property of reversibility and where the position of a machine does not indicate its importance for the overall system performance. PA 14 In serial lines, the transients (i.e., the process of reaching the steady states) of the throughput are typically orders of magnitude faster than the transients of work-in-process (Sections 10.3 and 10.4). PA 15 Larger buffers, as well as longer lines, lead to longer transients (Sections 10.2 – 10.4). PA 16 The throughput losses due to transients may be as high as 10%, if at the beginning of the shift all buffers were empty (Subsection 10.5.2). PA 17 To eliminate throughput losses due to transients, all buffers should be initially at least half full (Subsection 10.5.3). The last two facts are useful for managing daily shutdowns of production lines.

18.2. SERIAL LINES

18.2.2

551

Continuous improvement

CI 1 Serial lines are practically unimprovable with respect to workforce or cycle time re-allocation if each buffer is, on average, close to being half full (Subsections 5.1.2 and 13.1.2). This fact implies that if some buffers are mostly full or mostly empty, the throughput can be increased by workforce or cycle time re-allocation. This is a basis for a continuous improvement procedure. CI 2 A serial production line is practically unimprovable with respect to buffer capacity re-allocation if the average occupancy of each buffer is close to the average availability of the immediate downstream buffer (Subsections 5.1.4 and 13.1.3). Thus, if some buffers are mostly full and some are mostly empty, the throughput can be improved by buffer capacity re-allocation. This criterion can also be used as a basis of a continuous improvement procedure. CI 3 The bottleneck machine, i.e., the machine with the largest effect on the system throughput, is not necessarily the machine with the lowest efficiency (Subsections 5.2.1 and 13.2.1). This happens because the bottleneck property depends not only on the parameters of the machines in isolation but also on their position in the system, buffer capacity, and the overall topology of the production line. CI 4 The bottleneck machine can be identified using the following procedure: Let STi and BLi be the frequencies of starvation and blockage of machine i, respectively. Place arrows directed from mi to mi+1 if BLi > STi+1 and from mi+1 to mi if STi+1 > BLi (see Figure 18.1). Then, the machine with no emanating arrows is the bottleneck (Subsections 5.2.3, 13.2.4 and 13.2.5). BN−m BN−b

ST

i

BL i

m1

b1

m2

b2

m3

b3

m4

b4

m5

b5

m6

b6

m7

b7

m8

0.9

6

0.7

6

0.8

1

0.7

1

0.75

4

0.6

6

0.7

2

0.85

0

0

0

0.09

0.23

0.1

0.2

0.36

0.4

0.2

0.3

0.14

0.03

0

0.01

0

Figure 18.1: Arrow-based method for bottleneck identification Based on our practical experience, this rule is, perhaps, the most efficient way of day-to-day management of production systems. CI 5 The bottleneck buffer, i.e., the buffer with the largest effect on the overall throughput, is not necessarily the smallest buffer in the system (Subsections 5.2.1 and 13.2.1).

552

CHAPTER 18. SUMMARY OF MAIN FACTS

CI 6 The bottleneck buffer is either immediately upstream the bottleneck machine (if it is more often starved than blocked) or immediately downstream the bottleneck machine (if it is more often blocked than starved, see Figure 18.1) (Subsections 5.2.3, 13.2.4, and 13.2.5). The notion of bottleneck buffers is also useful for design and implementation of continuous improvement projects with predictable results. CI 7 The throughput of a closed line can be increased by adding a carrier if the sum of the frequencies of machine starvations is larger than the sum of the frequencies of machine blockages; if the inequality is reversed, the throughput can be increased by removing a carrier (Subsection 7.3.2). Again, this fact is also quite useful for day-to-day management of closed production systems. CI 8 The bottleneck machine of a closed line can be identified by the arrowbased method of CI 4 but using virtual, rather than real, frequencies of blockages and starvations of the first and the last machines (Subsection 7.4.2). CI 9 In general, the bottleneck machines in serial lines with inspection stations cannot be identified using the arrow-based method. However, if the only inspection station is the last machine, the arrow-based method works. In the general case, the BN-m can be identified using the estimates of throughput sensitivities with respect to machine parameters (Subsection 8.1.3). CI 10 In production lines with non-perfect quality machines and with inspection stations, the quality bottleneck (i.e., the machine, the quality of which has largest effect on the good parts throughput), is not necessarily the machine with the worst quality. However, if the only inspection machine is the last one, the machine with the worst quality is indeed the bottleneck (Subsection 8.1.3). Thus, the quality bottleneck in production lines with the inspection machine being the last one can be identified quite easily – by simply comparing the quality characteristics of the machines and selecting the worst one. CI 11 In production lines with non-perfect quality machines and only one inspection machine, which is the last one, the production rate bottleneck and the quality bottleneck are decoupled and each one can be identified as mentioned in CI 9 and CI 10 (Subsection 8.1.3). This implies, in particular, that eliminating the production rate bottleneck does not affect the position of the quality bottleneck, and vice versa. CI 12 In production lines with non-perfect quality machines and several inspection machines, or one inspection machine that is not the last one, the production rate bottleneck and the quality bottleneck are coupled (Subsection 8.1.3).

18.2. SERIAL LINES

553

This implies that, in general, eliminating the production rate bottleneck does affect the position of the quality bottleneck, and vice versa. CI 13 Bottlenecks in lines with rework may shift not only because of changes in machine efficiency but also due to changes in quality buy rates (Subsection 8.3.3). CI 14 In production lines with machines having reliability-capacity coupling, the bottleneck may be eliminated by increasing the machine cycle time (Problem 11.14). This implies that slowing down a machine may lead to a higher throughput if the reliability of this machine deteriorates at higher speeds of its operation. CI 15 In production lines with quality-quantity coupling machines, the bottleneck may be eliminated by decreasing its efficiency (Subsection 8.2.3). Here again, one may need to decrease the efficiency of the bottleneck machine in order to increase the production rate of good parts. CI 16 Serial lines with lean finished goods buffers obey a conservation law: the product of filtering in time and space, which are necessary and sufficient to ensure the desired level of customer demand satisfaction, is constant (Subsections 9.3.2 and 15.4.2). This implies that lean finished goods buffer capacity and the shipping period trade off one-to-one.

18.2.3

Design

D 1 In serial lines with buffers of identical capacity, the unimprovable workforce or machine efficiency allocation is a “flat” inverted bowl pattern (Subsection 5.1.2). Although the optimal inverted bowl workforce allocation provides the highest throughput, in most practical situations it is not much higher than the throughput ensured by the uniform allocation. D 2 In serial lines with identical machines, the unimprovable buffer capacity allocation is a “flat” inverted bowl pattern (Subsection 5.1.4). Although the inverted bowl pattern leads to the largest throughput, in most cases it is also not too different from that ensured by the uniform buffer capacity allocation. D 3 Workforce and buffer capacity in a serial line are allocated jointly optimally if all buffers are of identical capacity and work is allocated according to a “flat” inverted bowl pattern (Subsection 5.1.3).

554

CHAPTER 18. SUMMARY OF MAIN FACTS

D 4 If the desired efficiency of an exponential line, E, is equal to the machine efficiency, e, the buffer capable of accommodating 3-4 downtimes is lean, where, lean is understood as the smallest buffer necessary and sufficient to ensure the desired line efficiency (Subsection 14.2.3). D 5 If E > e, the lean buffer capacity is larger than 4 downtimes. If E < e, the lean buffer capacity is smaller than 3 downtimes (Subsection 14.2.3). D 6 In serial lines with non-exponential machines and the coefficients of variation of up- and downtime less than 1, the lean buffer capacity is a decreasing linear function of the average coefficient of variation (Subsection 14.4.3). Thus, decreasing variability of the up- and downtime leads to smaller lean buffering. D 7 An unimpeding number of carriers in closed lines is independent of the machines’ efficiency (Subsection 7.2.2). D 8 The lean number of carriers in closed lines can be selected as the number of machines plus half of the total buffer capacity (Subsection 7.3.3). D 9 The optimal position of an efficient inspection machine is either immediately downstream of the last non-perfect quality machines or around the downstream bottleneck (Subsection 8.1.4). Clearly, all these facts can be used for design of efficient production lines.

18.3

Assembly Systems

A 1 All performance measures of assembly systems with Bernoulli, geometric, and exponential machines can be evaluated analytically using aggregation techniques (Subsections 16.1.1, 16.1.2, 17.1.2, and 17.1.3). A 2 Assembly systems obey the properties of monotonicity and reversibility (Subsections 16.1.2 and 17.1.2). A 3 The bottleneck machines of assembly systems can be identified using the arrow-based method similar to that of serial lines (Subsections 16.2.2 and 17.3.2). A 4 An assembly system is practically unimprovable with respect to workforce or cycle time re-allocation if each buffer is, on average, close to being half full (Subsections 16.2.1 and 17.3.1). A 5 If all buffers are of equal capacity, to optimize the throughput, the merge machine must have the highest efficiency, the first machine of each component line and the last machine of the additional processing line must have the lowest efficiency, and the efficiency of all other machines must be in-between (Subsection 16.2.1).

Chapter 19

PSE Toolbox Motivation: Parts I - IV of this textbook presented various techniques for modeling, analysis, continuous improvement, and design of production systems. In this chapter, a toolbox, referred to as the PSE Toolbox (see Figure 19.1), which implements these techniques, is described. This toolbox can be used for both educational and industrial purposes. Overview: The architecture of the PSE Toolbox is characterized and a number of tools, included in it, are described. A demo of the PSE Toolbox can be found at: http://www.ProductionSystemsEngineering.com/PSE Toolbox

19.1

Architecture and Functions

The PSE Toolbox consists of a number of functions for modeling, analysis, design and continuous improvement of production systems. Each function consists of several tools, implementing various methods developed in this textbook. The functions included in the toolbox are: • • • • • • • •

Modeling Performance Analysis Continuous Improvement Bottleneck Identification Lean Buffer Design Product Quality Customer Demand Satisfaction Simulations

When a function is selected, the tools included appear on the right side of the window. Selecting and double-clicking a tool brings up a subwindow, and the 555

556

CHAPTER 19. PSE TOOLBOX

Figure 19.1: PSE Toolbox

19.2. MODELING FUNCTION

557

Figure 19.2: Functions of PSE Toolbox user can either input the appropriate parameters or load a text file containing the parameters. When entering machine and buffer parameters, each two numbers must be separated by a space or a comma. The results obtained by the tool appear in the lower part of the subwindow. In addition, the user can click the “View” button to represent the results in another popup window and save them into a text file. Below, the functions and some of the tools included in the PSE Toolbox to-date are described.

19.2

Modeling Function

The Modeling function, illustrated in Figure 19.2, includes the following tools: • • • • • •

aggregation of parallel machines; aggregation of consecutive dependent machines; exp-B transformation for serial lines; B-exp transformation for serial lines; exp-B transformation for assembly systems; B-exp transformation for assembly systems.

19.2.1

Aggregation of parallel machines

This tool, illustrated in Figure 19.3, is used to simplify several parallel machines into a single aggregated machine. The aggregation formulas are provided in Subsection 3.3.5. The input parameters of this tool are: • number of parallel machines S; • capacity of each parallel machine c; • average downtime of each parallel machine Tdown ;

558

CHAPTER 19. PSE TOOLBOX

Figure 19.3: Aggregation of parallel machines • average uptime of each parallel machine Tup . The outputs of this tool are: • capacity of the aggregated machine c; • average downtime of the aggregated machine Tdown ; • average uptime of the aggregated machine Tup .

19.2.2

Aggregation of consecutive dependent machines

This tool, illustrated in Figure 19.4, is used to simplify several consecutive dependent machines into a single aggregated machine. The aggregation formulas are provided in Subsection 3.3.5. The input parameters of this tool are: • • • •

number of consecutive dependent machines S; capacity of each consecutive dependent machine c; each consecutive dependent machine average downtime Tdown ; each consecutive dependent machine average uptime Tup .

The outputs of this tool are: • capacity of the aggregated machine c; • average downtime of the aggregated machine Tdown ; • average uptime of the aggregated machine Tup .

19.2. MODELING FUNCTION

559

Figure 19.4: Aggregation of consecutive dependent machines

19.2.3

Exp-B transformation for serial lines

This tool, illustrated Figure 19.5, is used to simplify an exponential model of a serial line to a Bernoulli line. The transformation formulas are presented in Subsection 3.9.3. The input parameters of this tool are: • • • • •

number of machines in serial line M ; failure rate of each machine λ; repair rate of each machine µ; capacity of each machine c; capacity of each buffer N .

The outputs are: • parameter p of each machine in the Bernoulli line; • buffer capacity N of each buffer in the Bernoulli line.

19.2.4

B-exp transformation for serial lines

This tool, illustrated Figure 19.6, is used to transform a Bernoulli line into the exponential description. The transformation formulas are presented in Subsection 3.9.4. The input parameters of this tool are:

560

CHAPTER 19. PSE TOOLBOX

Figure 19.5: Exp-B transformation for serial lines

19.2. MODELING FUNCTION

561

• number of machines M in the original exponential line; • failure rate λ of each machine in the original exponential line; • repair rate µ of each machine in the original exponential line; • capacity c of each machine in the original exponential line; • capacity N of each buffer in the original exponential line; • parameter p of each machine in the transformed (and, perhaps, modified) Bernoulli line; • capacity N of each buffer in the transformed (and, perhaps, modified) Bernoulli line.

Figure 19.6: B-exp transformation for serial lines The outputs are: • number of machines M in the transformed exponential line; • failure rate λ of each machine in the transformed exponential line; • repair rate µ of each machine in the transformed exponential line; • capacity c of each machine in the transformed exponential line; • capacity N of each buffer in the transformed exponential line.

562

CHAPTER 19. PSE TOOLBOX

19.3

Performance Analysis Function

The Performance Analysis function, illustrated in Figure 19.7, includes the following tools:

Figure 19.7: Performance Analysis function • • • • • • • •

analysis analysis analysis analysis analysis analysis analysis analysis

19.3.1

of of of of of of of of

serial lines with Bernoulli machines; synchronous serial lines with exponential machines; asynchronous serial lines with exponential machines; serial lines with general models of machine reliability; closed line with Bernoulli machines; closed line with exponential machines; assembly systems with Bernoulli machines; assembly systems with exponential machines.

Analysis of serial lines with Bernoulli machines

This tool, illustrated in Figure 19.8, is used to calculate the performance measures of Bernoulli serial lines. The calculation formulas are provided in Section 4.2. The input parameters of this tool are: • number of machines M ; • parameter of each machine p; • capacity of each buffer N . The outputs are: • line production rate P R; • work-in-process of each buffer W IP ; • probabilities of blockage BL;

19.3. PERFORMANCE ANALYSIS FUNCTION

563

Figure 19.8: Performance analysis of serial lines with Bernoulli machines • probabilities of starvation ST ; • parameters of forward aggregation pfi ; • parameters of backward aggregation pbi .

19.3.2

Analysis of synchronous serial lines with exponential machines

The calculation formulas for this tool are provided in Section 11.1. The input parameters are (Figure 19.9): • • • • •

number of machines M ; capacity of all machines c; failure rate of each machine λ; repair rate of each machine µ; capacity of each buffer N .

The outputs are: • • • • •

efficiency of each machine in isolation e; line production rate P R and throughput T P ; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

19.3.3

Analysis of asynchronous serial lines with exponential machines

The calculation formulas for this tool are provided in Section 11.2. The input parameters are (Figure 19.10):

564

CHAPTER 19. PSE TOOLBOX

Figure 19.9: Performance analysis of synchronous serial lines with exponential machines • • • • •

number of machines M ; capacity of each machine c; failure rate of each machine λ; repair rate of each machine µ; capacity of each buffer N .

The outputs are: • • • • •

efficiency of each machine in isolation e; line throughput T P ; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

19.3.4

Analysis of closed lines with Bernoulli machines

This tool is used for analysis of closed Bernoulli lines. Since no analytical formulas for performance measures of closed lines with M > 2 are available, this tool is based on simulations. The input parameters of this tool are (Figure 19.11): • • • •

number of machines M ; parameter of each machine p; capacity of each in-process buffer N ; capacity of empty carrier buffer N0 ;

19.3. PERFORMANCE ANALYSIS FUNCTION

565

Figure 19.10: Performance analysis of asynchronous serial lines with exponential machines • • • •

number of carriers S; simulation warm-up time; total simulation time (including both warm-up and results collection time); number of simulation iterations.

Figure 19.11: Performance analysis of closed line with Bernoulli machines The outputs are: • • • •

line production rate P R; work-in-process of in-process buffer W IP ; probabilities of blockage BL; probabilities of starvation ST ;

566

CHAPTER 19. PSE TOOLBOX • S-improvability indicator I.

19.3.5

Analysis of assembly systems with Bernoulli machines

The calculation formulas for this tool are provided in Section 16.1. The input parameters are (Figure 19.12): • number of machines M1 and M2 in the component lines 1 and 2 and number of machines M0 in the subsequent processing line; • parameters of machines in component lines 1, 2 and subsequent processing line, p1 , p2 , p0 , respectively; • capacity of buffers in component lines 1, 2 and subsequent processing line, N1 , N2 , N0 , respectively.

Figure 19.12: Performance analysis of Bernoulli assembly system The outputs are: • line production rate P R; • probabilities of blockage BL; • probabilities of starvation ST .

19.4

Continuous Improvement Function

The Continuous Improvement function, illustrated in Figure 19.13, includes the following tools: • unimprovable allocation of WF in Bernoulli serial lines;

19.4. CONTINUOUS IMPROVEMENT FUNCTION

567

Figure 19.13: Continuous improvement functions • unimprovable allocation of WF and BC simultaneously in Bernoulli serial lines; • WF-continuous improvement procedure for Bernoulli serial lines; • BC-continuous improvement procedure for Bernoulli serial lines; • CT-continuous improvement procedure for exponential serial lines; • BC-continuous improvement procedure for exponential serial lines; • CT-continuous improvement procedure for lines with general models of machine reliability; • BC-continuous improvement procedure for lines with general models of machine reliability; • S-continuous improvement procedure for closed Bernoulli lines; • N0 -continuous improvement procedure for closed Bernoulli lines; • S-continuous improvement procedure for closed exponential lines; • N0 -continuous improvement procedure for closed exponential lines; • S-continuous improvement procedure for closed lines with general machines; • N0 -continuous improvement procedure for closed lines with general machines.

19.4.1

Unimprovable allocation of WF in Bernoulli serial lines

The unimprovable allocation formulas for this tool are provided in Section 5.1. The input parameters are (Figure 19.14): • number of machines M ; • work force constraint p∗ ; • capacity of each buffer N .

568

CHAPTER 19. PSE TOOLBOX

Figure 19.14: Unimprovable allocation of WF in Bernoulli serial lines The outputs are: • • • • • • •

unimprovable allocation of machine reliability p; resulting line production rate P R; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST ; parameters of forward aggregation pfi ; parameters of backward aggregation pbi .

19.4.2

Unimprovable allocation of WF and BC simultaneously in Bernoulli serial lines

The allocation formulas for this tool are provided in Section 5.1. The input parameters are (Figure 19.15): • number of machines M ; • work force constraint p∗ ; • buffer capacity constraint N ∗ . The outputs are: • • • • • • • •

unimprovable allocation of machine reliability p; unimprovable allocation of buffer capacity N ; resulting line production rate P R; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST ; parameters of forward aggregation pfi ; parameters of backward aggregation pbi .

19.4. CONTINUOUS IMPROVEMENT FUNCTION

569

Figure 19.15: Unimprovable allocation of WF and BC for Bernoulli serial lines

19.4.3

WF-continuous improvement procedure for Bernoulli serial lines

This continuous improvement procedure is described in Section 5.1 The input parameters are (Figure 19.16): • • • •

number of machines M ; initial reliability of each machine p; capacity of each buffer N ; stopping criterion δ.

Figure 19.16: WF-continuous improvement procedure for Bernoulli serial lines The outputs are:

570

CHAPTER 19. PSE TOOLBOX • • • •

resulting allocation of machine reliability p; resulting line production rate P R; work-in-process of each buffer W IP ; differences W IP − N/2.

19.4.4

BC-continuous improvement procedure for Bernoulli serial lines

This continuous improvement procedure is described in Section 5.1. The input parameters are (Figure 19.17): • number of machines M ; • reliability of each machine p; • initial capacity of each buffer N .

Figure 19.17: BC-continuous improvement procedure for Bernoulli serial lines The outputs are: • • • •

resulting allocation of buffer capacity N ; resulting line production rate P R; work-in-process of each buffer W IP ; available (i.e., unoccupied) capacity of each buffer N − W IP .

19.4.5

S-continuous improvement procedure for closed Bernoulli lines

This continuous improvement procedure is described Section 7.3. The input parameters are (Figure 19.18): • number of machines M ; • reliability of each machine p;

19.5. BOTTLENECK IDENTIFICATION FUNCTION • • • • • •

571

capacity of each in-process buffer N ; capacity of empty carrier buffer N0 ; initial number of carriers S; simulation warm-up time; total simulation time; number of simulation iterations.

Figure 19.18: S-continuous improvement procedure for closed Bernoulli line The outputs are: • • • • • •

line production rate P R; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST ; resulting number of carriers S; S-improvability indicator I.

19.5

Bottleneck Identification Function

The Bottleneck Identification function, illustrated in Figure 19.19, includes the following tools: • • • •

BN-m BN-m BN-m BN-m

and BN-b in serial lines with Bernoulli machines; and BN-b in serial lines with exponential machines; and BN-b in serial lines with general models of machine reliability; and BN-b in closed lines with Bernoulli machines;

572

CHAPTER 19. PSE TOOLBOX

Figure 19.19: Bottleneck identification functions • • • • •

BN-m and BN-b in closed lines with exponential machines; BN-m and BN-b in closed lines with general models of machine reliability; BN-m and BN-b in assembly systems with Bernoulli machines; BN-m and BN-b in assembly systems with exponential machines; BN-m and BN-b in assembly systems with general models of machine reliability.

19.5.1

BN-m and BN-b in serial lines with Bernoulli machines

The identification methods are presented in Section 5.2. The input parameters are (Figure 19.20): • number of machines M ; • parameter of each machine p; • capacity of each buffer N . The outputs are: • • • • • •

BN-m and BN-b; severity of each local BN-m (for multiple BN case); line production rate P R; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

19.5.2

c-BN and BN-b in serial lines with exponential machines

The identification methods are presented in Section 13.2. The input parameters are (Figure 19.21):

19.5. BOTTLENECK IDENTIFICATION FUNCTION

573

Figure 19.20: BN-m and BN-b in serial lines with Bernoulli machines • • • • •

number of machines M ; failure rate of each machine λ; repair rate of each machine µ; capacity of each machine c; capacity of each buffer N .

The outputs are: • efficiency of each machine in isolation e; • bottleneck machine c-BN (primary bottleneck machine PcBN-m in multiple bottlenecks case) and bottleneck buffer BN-b; • line throughput T P ; • work-in-process of each buffer W IP ; • probabilities of blockage BL; • probabilities of starvation ST ; • severity of each local BN.

19.5.3

c-BN and BN-b in serial lines with general models of machine reliability

The input parameters of this tool are (Figure 19.22): • number of machines M ; • probabilities of blockage BL; • probabilities of starvation ST . The outputs are: • bottleneck machine c-BN (Pc-BN in multiple bottlenecks case) and bottleneck buffer BN-b; • severity of each local BN.

574

CHAPTER 19. PSE TOOLBOX

Figure 19.21: c-BN and BN-b in serial lines with exponential machines

Figure 19.22: c-BN and BN-b in serial lines with general models of machine reliability

19.5. BOTTLENECK IDENTIFICATION FUNCTION

19.5.4

575

BN-m and BN-b in closed lines with Bernoulli machines

This identification method is described in Section 7.4. The input parameters are (Figure 19.23): • number of machines M ; • reliability of each machine p; • capacity of each in-process buffer N ; • capacity of empty carrier buffer N0 ; • number of carriers S; • simulation warm-up time; • total simulation time; • number of simulation iterations.

Figure 19.23: BN-m and BN-b in closed lines with Bernoulli machines The outputs are: • bottleneck machine BN-m; • line production rate P R; • work-in-process of each buffer W IP ; • probabilities of virtual blockage BLv ; • probabilities of virtual starvation STv ; • severity of each local BN.

576

CHAPTER 19. PSE TOOLBOX

19.6

Lean Buffer Design Function

The Lean Buffer Design function, illustrated in Figure 19.24, includes the following tools:

Figure 19.24: Lean buffer design function

• lean buffering for serial lines with identical Bernoulli machines; • lean buffering for serial lines with non-identical Bernoulli machines; • lean buffering for serial lines with identical exponential machines; • lean buffering for serial lines with non-identical exponential machines; • lean buffering for assembly systems with identical Bernoulli machines; • lean buffering for assembly systems with non-identical Bernoulli machines; • lean buffering for assembly systems with identical exponential machines; • lean buffering for assembly systems with non-identical exponential machines.

19.6.1

Lean buffering for serial lines with identical Bernoulli machines

The calculation formulas for this tool are provided in Section 6.2. The input parameters are (Figure 19.25): • number of machines M ; • parameter of each machine p; • desired line efficiency E. The output is the capacity of lean buffer N .

19.6. LEAN BUFFER DESIGN FUNCTION

577

Figure 19.25: Lean buffering for serial lines with identical Bernoulli machines

19.6.2

Lean buffering for serial lines with non-identical Bernoulli machines

The calculation methods are introduced in Section 6.3. The input parameters are (Figure 19.26): • number of machines M ; • parameter of each machine p; • desired line efficiency E.

Figure 19.26: Lean buffering for serial lines with non-identical Bernoulli machines The outputs are the capacity of lean buffer N and resulting line efficiency E for the following six approaches: • local pair-wise approach; • global pair-wise approach; • local upper bound approach; • global upper bound approach; • full search approach; • bottleneck-based approach.

578

CHAPTER 19. PSE TOOLBOX

19.7

Product Quality Function

The Product Quality Function, illustrated in Figure 19.27, includes the following tools: • analysis of Bernoulli lines with non-perfect quality machines and inspection operations; • PR-BN in Bernoulli lines with non-perfect quality machines and inspection operations; • Q-BN in Bernoulli lines with non-perfect quality machines and inspection operations; • analysis of exponential lines with non-perfect quality machines and inspection operations; • PR-BN in exponential lines with non-perfect quality machines and inspection operations; • Q-BN in exponential lines with non-perfect quality machines and inspection operations; • analysis of Bernoulli lines with quality-quantity coupling machines.

Figure 19.27: Product quality tools

19.7.1

Analysis of Bernoulli lines with non-perfect quality machines and inspection operations

The calculation formulas for this tool are provided in Section 8.1. The input parameters are (Figure 19.28): • • • •

number of machines M ; reliability of each machine p; quality parameter of each machine g; capacity of each buffer N ;

19.8. CUSTOMER DEMAND SATISFACTION FUNCTION

579

Figure 19.28: Analysis of Bernoulli lines with non-perfect quality machines and inspection machines • positions of inspection machines. The outputs are: • production rate of good parts P R, consumption rate CR and scrap rate SR; • work-in-process of each buffer W IP ; • probabilities of blockage BL; • probabilities of starvation ST ; • parameters of forward aggregation pfi ; • parameters of backward aggregation pbi .

19.8

Customer Demand Satisfaction Function

The Customer Demand Satisfaction function, illustrated in Figure 19.29, includes the following tools: • • • • • •

DT P in PIC system with one Bernoulli machine; lower bound on DT P in PIC system with multiple Bernoulli machines; DT P in PIC system with one exponential machine; lower bound on DT P in PIC system with multiple exponential machines; DT P in PIC system with random demand; lower bound on DT P in assembly system with multiple Bernoulli machines; • lower bound on DT P in assembly system with multiple exponential machines.

580

CHAPTER 19. PSE TOOLBOX

Figure 19.29: Customer demand satisfaction tools

19.8.1

DT P in PIC system with one Bernoulli machine

The calculation formulas for this tool are provided in Section 9.2. The input parameters are (Figure 19.30): • • • •

number of cycles in an epoch T ; customer demand D; parameter of one-Bernoulli-machine PS p; capacity of finished goods buffer N .

Figure 19.30: DTP calculation in PIC system with one-Bernoulli-machine PS The output is DT P of this PIC system.

19.9

Simulation Function

The Simulation function, illustrated in Figure 19.31, includes the following tools:

19.9. SIMULATION FUNCTION

581

Figure 19.31: Simulation function • • • • • • • • •

simulation of serial lines with Bernoulli machines; simulation of serial lines with exponential machines; simulation of serial lines with general models of machine reliability; simulation of closed lines with Bernoulli machines; simulation of closed lines with exponential (synchronous or asynchronous) machines; simulation of closed lines with general models of machine reliability; simulation of assembly systems with Bernoulli machines; simulation of assembly systems with exponential machines; simulation of assembly systems with general models of machine reliability.

19.9.1

Simulation of serial lines with Bernoulli machines

The input parameters of this tool are (Figure 19.32): • • • • • •

number of machines M ; parameter of each machine p; capacity of each buffer N ; simulation warm-up time; total simulation time; number of simulation iterations.

The outputs are: • • • •

line production rate P R; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

582

CHAPTER 19. PSE TOOLBOX

Figure 19.32: Simulation of serial lines with Bernoulli machines

19.9.2

Simulation of serial lines with exponential machines

The input parameters of this tool are (Figure 19.33): • • • • • • • •

number of machines M ; failure rate of each machine λ; repair rate of each machine µ; capacity of each machine c; capacity of each buffer N ; simulation warm-up time; total simulation time; number of simulation iterations.

The outputs are: • • • • •

line throughput T P ; isolated efficiency of each machine e; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

19.9.3

Simulation of serial lines with general models of machine reliability

The input parameters of this tool are (Figure 19.34): • number of machines M ; • average uptime of each machine Tup ; • average downtime of each machine Tdown ;

19.9. SIMULATION FUNCTION

Figure 19.33: Simulation of serial lines with exponential machines • • • • • • • • •

coefficients of variation of uptime of each machine CVup ; coefficients of variation of downtime of each machine CVdown ; capacity of each machine c; capacity of each buffer N ; distribution of uptime of each machine fup ; distribution of downtime of each machine fdown ; simulation warm-up time; total simulation time; number of simulation iterations.

The distributions, which can be selected, are: • • • •

exp: exponential; LN: lognormal; W: Weibull; ga: gamma.

The outputs are: • • • • •

efficiency of each machine in isolation e; line throughput T P ; work-in-process of each buffer W IP ; probabilities of blockage BL; probabilities of starvation ST .

583

584

CHAPTER 19. PSE TOOLBOX

Figure 19.34: Simulation of serial lines with general models of machine reliability

19.9.4

Simulation of assembly systems with Bernoulli machines

The input parameters of this tool are (Figure 19.35): • number of machines in component lines 1, 2 and subsequent processing line, M1 , M2 , M0 , respectively; • parameters of machines in component lines 1, 2 and subsequent processing line, p1 , p2 , p0 , respectively; • capacity of buffers in component lines 1, 2 and subsequent processing line, N1 , N2 , N0 , respectively; • simulation warm-up time; • total simulation time; • number of simulation iterations. The outputs are: • line production rate P R; • probabilities of blockage BL; • probabilities of starvation ST .

19.9. SIMULATION FUNCTION

Figure 19.35: Simulation of assembly systems with Bernoulli machines

585

Chapter 20

Proofs Motivation: This chapter is intended to present proofs of theorems and other formal statements formulated throughout this volume. Overview: Sections 20.1-20.3 include proofs for Parts II, III, and IV, respectively.

20.1

Proofs for Part II

20.1.1

Proofs for Chapter 4

Proof of Lemma 4.1: For x 6= y, represent Q(x, y, N ) as follows: Q(x, y, N ) =

1−x N 1− x yα 1−α

1−x ´ ³

= 1−αN 1−α

+

1− x y

. αN

1−α

This expression can be rewritten as Q(x, y, N ) = = =

1−x 1 + α + α2 + . . . + αN −1 +

(y−x)y(1−x) N y(y(1−x)−x(1−y)) α

1−x (20.1) 1+α+ + . . . + αN −1 + (1 − x)αN 1−x ´ ³ . 2 N −1 α 1 + α + α + . . . + αN −2 + 1 + x(1−y) y α2

Since the numerator is in (0, 1) and the denominator is a positive number greater than 1, Q(x, y, N ) takes values on (0, 1). The monotonicity properties also follow from (20.1), since the numerator is monotonically decreasing in x (constant in y and N ), and the denominator is monotonically increasing in x (decreasing in y and increasing in N ). 587

588

CHAPTER 20. PROOFS

For x = y, expression (20.1) also holds (with α = 1). Therefore, all the above properties take place in this case as well. ¥ Proof of Theorem 4.1: From (4.14), • if p1 = p2 , it is clear that limN →∞ Q(p1 , p2 , N ) = 0; • if p1 < p2 , 1 − p2 p1 (1 − p2 ) < <1 α(p1 , p2 ) = p2 (1 − p1 ) 1 − p1 and, therefore, lim Q(p1 , p2 , N ) = (1 − p1 )(1 − α) =

N →∞

p2 − p1 ; p2

• if p1 > p2 , then α > 1, and lim Q(p1 , p2 , N ) = 0.

N →∞

Thus, from (4.17), • when p1 ≥ p2 , lim P R → p2 ;

N →∞

• when p1 < p2 , lim P R = p2 (1 −

N →∞

p2 − p1 ) = p1 . p2

Therefore, limN →∞ P R = min(p1 , p2 ). From (4.23), • when p1 = p2 , lim W IP = lim N/2 = ∞;

N →∞

N →∞

• when p1 < p2 , then αN → 0 and lim W IP = p1

N →∞

p1 1 − αN − (1 − α)N αN = ; (1 − α)(p2 − p1 αN ) (1 − α)p2

• when p1 > p2 , from the above equation we have lim W IP = lim N = ∞.

N →∞

N →∞

The asymptotic properties of BL1 (for p1 ≤ p2 ) and ST2 are proved analogously. For p1 > p2 , in the limit N → ∞, the system has no steady state, which is interpreted as BL1 = 0. ¥ The proof of statement (i) of Theorem 4.2 requires the following lemmas:

20.1. PROOFS FOR PART II

589

Lemma 20.1 Consider the two sequences pfi (s) and pbi (s), s = 1, 2, . . ., defined by recursive procedure (4.30). Then, pfi (s) < pfi (s − 1), i = 2, . . . , M , implies and is implied by pbj (s + 1) > pbj (s), j = 1, . . . , M − 1. Proof: The implication pfi (s) < pfi (s − 1) ⇒ pbj (s + 1) > pjb (s) is proved by induction: For the base case j = M − 1, using Lemma 4.1, from (4.30) and the assumption of Lemma 20.1, we obtain: pbM −1 (s + 1)

=

pM −1 [1 − Q(pM , pfM −1 (s), NM −1 )]

> pM −1 [1 − Q(pM , pfM −1 (s − 1), NM −1 )] = pbM −1 (s). For the general case j = M − 2, M − 3, . . . , 2, 1, we write pbj (s + 1)

=

pj [1 − Q(pbj+1 (s + 1), pbj (s), Nj )]

> pj [1 − Q(pbj+1 (s), pfj (s), Nj )] > pj [1 − Q(pbj+1 (s).pfj (s − 1), Nj )] = pbj (s). The implication pfi (s) < pfi (s − 1) ⇐ pbj (s + 1) > pbj (s) is proved analogously. ¥ Lemma 20.2 The sequences pfi (s) and pbj (s), s = 1, 2, . . .; i = 2, . . . , M ; j = 1, . . . , M − 1, defined by recursive procedure (4.30), are monotonically decreasing and increasing, respectively. Proof: By induction. For s = 1, due to Lemma 4.1, we have pfi (s) = pi [1 − Q(pfi−1 (s), pbi (s), Ni )] < pi = pfi (0),

2 ≤ i ≤ M.

Assume that for s ≥ 1, pbj (s + 1) > pbj (s),

1 ≤ j ≤ M − 1.

Then, due to the first implication of Lemma 20.1, pbj (s + 1) > pbj (s),

1 ≤ j ≤ M − 1.

By the second implication of Lemma 20.1 pfj (s + 1) < pfj (s),

2 ≤ j ≤ M. ¥

The proof of statements (ii) and (iii) of Theorem 4.2 requires the lemmas formulated next.

590

CHAPTER 20. PROOFS Consider the steady state equations of recursive procedure (4.30), i.e., pfi pbi

=

pi [1 − Q(pfi−1 , pbi , Ni−1 )],

=

pi [1 − Q(pbi+1 , pfi , Ni )], pbM = pM . pf1 = p1 ,

2 ≤ i ≤ M, 1 ≤ i ≤ M − 1,

(20.2)

Introduce (M − 1) two-machine serial production lines, Li , i = 1, . . . , M − 1, where the first machine has the isolation production rate pfi , the second pbi+1 , and the buffer capacity is Ni . The following properties hold: d Lemma 20.3 Let P Ri be the production rate of line Li , i = 1, . . . , M − 1, d d d Ri = pfi pbi /pi , i = 1, . . . , M . Moreover, P Ri = and let P RM = pfM . Then, P d P Rj , ∀i, j = 1, . . . , M . Proof: Using (4.19) and (20.2), we obtain: d P Ri

pfi pb pf = i i, pi pi i = 1, . . . , M − 1,

= pfi [1 − Q(pbi+1 , pfi , Ni )] = pi [1 − Q(pbi+1 , pfi , Ni )]

and

pf pM pb pf d P RM = pfM = M = M M, pM pM

which proves the first statement of the lemma. Moreover, d P Ri

= =

pbi pfi pb = i pi [1 − Q(pfi−1 , pbi , Ni−1 )] pi pi b d pi [1 − Q(pf , pbi , Ni−1 )] = P Ri−1 , i−1

i = 2, . . . , M,

which proves the second statement. ¥ Lemma 20.4 The production rate (4.19) is a monotonically increasing function of p1 , p2 , and N1 . Proof: Follows immediately from Lemma 4.1. ¥ Lemma 20.5 The equilibrium equation (20.2) of recursive procedure (4.30) has a unique solution. Proof: By contradiction: Assume that along with the solution Pagg = [pf1 , . . . , pfM , pb1 , . . . , pbM ], there exists another solution of equation (20.2) denoted

20.1. PROOFS FOR PART II

591

as P agg = [pf1 , . . . , pfM , pb1 , . . . , pbM ]. Suppose that pb1 > pb1 . Then, by Lemma 20.3, d d P Ri > P Ri , 1 ≤ i ≤ M. (20.3) d d R1 (p1 , pb2 , N1 ) > P R1 (p1 , pb2 , N1 ), by Lemma 20.4 pb2 > pb2 . Therefore, Since P by Lemma 4.1, pf2 = p2 [1 − Q(p1 , pb2 , N1 )] < p2 [1 − Q(p1 , pb2 , N1 )] = pf2 . Assume pbj > pbj and pfj < pfj . The base case d Rj (pfj , pbj+1 , Nj ) > (j = 2) has already been established. By equation (20.3), P d P Rj (pf , pb , Nj ). Since pf < pf , by Lemma 20.4, pb > pb . Using Lemma

Now proceed inductively.

j

j+1

j

j

j+1

j+1

4.1 and the assumption that pfj < pfj and pbj+1 > pbj+1 , pfj+1 = pj+1 [1 − Q(pfj , pbj+1 , Nj )] < pj+1 [1 − Q(pfj , pbj+1 , Nj )] = pfj+1 . Thus, the inductive hypothesis is established, and, therefore, pbj > pbj and pfj < d d RM < P RM , pfj , 2 ≤ j ≤ M . In particular, pfM < pfM , so by Lemma 20.3, P b b which contradicts equation (20.3). Therefore, we conclude that p1 ≤ p1 . Assuming that pb1 < pb1 , and proceeding analogously, yields pb1 ≥ pb1 . Therefore, pb1 = pb1 . The equality of the remaining components of P agg and Pagg is also shown by induction. ¥ Proof of Theorem 4.2: Since the sequences pfi (s) and pbj (s), s = 1, 2, . . .; i = 2, . . . , M ; j = 1, . . . , M − 1, are monotonic (Lemma 20.2) and bounded from above and below, they are convergent. This proves statement (i). Statement (ii) follows immediately from Lemma 20.5. Statement (iii) follows from Lemma 20.3 and the relationship pb pf pb pf d d RM . P R1 = 1 1 = pb1 = M M = pfM = P p1 pM ¥ The proof of Theorem 4.3 is based on three lemmas. To formulate them, introduce the following probabilities: p˜fi p˜bi

:=

Prob{mi produces|mi is not blocked},

:=

Prob{mi produces|mi is not starved}.

(20.4)

These probabilities play a crucial role in the proof of Theorem 4.3. Specifically, Lemma 20.6 associates p˜fi and p˜bi with the probabilities of buffers being empty and full, respectively. Lemma 20.7 states that if p˜fi and p˜bi+1 are known, then the

592

CHAPTER 20. PROOFS

stationary probability distribution of buffer occupancy, Pi [·], can be calculated with the error O(δ). Lemma 20.8 shows that p˜fi and p˜bi+1 can be calculated from the steady state of the recursive procedure with the accuracy O(δ). Therefore, since the production rate can be calculated by P R = (1 − PM −1 [0])pM , the claim of Theorem 4.3 will follow. Lemma 20.6 The conditional probabilities p˜fi , p˜bi can be expressed as (α) (β)

i = 2, . . . , M, p˜fi = pi [1 − Pi−1 [0]] + O(δ), j−1 M h ³ ´ i X Y p˜bi = pi 1 − pr (1 − pj )Pi,...,j−1 [Ni , . . . , Nj−1 ] + O(δ), j=i+1

r=i+1

i = 1, . . . , M − 1, where Pi,...,j−1 [hi , . . . , hj−1 ] is the steady state probability that consecutive buffers i, . . . , j contain hi , . . . , hj parts, respectively, and δ is defined in (4.44). Proof: The probability that machine i is blocked can be written as P {mi is blocked} =

M ³ j−1 X Y j=i+1

´ pr (1 − pj )Pi,...,j−1 [Ni , . . . , Nj−1 ).

(20.5)

r=i+1

Since machine i is not starved when buffer i − 1 contains one or more parts, using the conditional probability formula and the definition of δ, we write: P {mi is blocked|mi is not starved} PNi−1 M ³ j−1 ´ X Y Pi−1,...,j−1 [c, Ni , . . . , Nj−1 ] = pr (1 − pi ) c=1 1 − Pi−1 [0] j=i+1 r=i+1 =

M ³ j−1 X Y j=i+1

´ ³ pr (1 − pi ) Pi,...,j−1 [Ni , . . . , Nj−1 ]

r=i+1

´.³ ´ 1 − Pi−1 [0] −Pi−1,...,j−1 [0, Ni , . . . , Nj−1 ]

=

M ³ j−1 X Y j=i+1

´ ³ pr (1 − pi ) Pi,...,j−1 [Ni , . . . , Nj−1 ]

r=i+1

−Pi−1 [0]Pi,...,j−1 [Ni , . . . , Nj−1 ] =

M ³ j−1 X Y j=i+1

´.³ ´ 1 − Pi−1 [0] + O(δ)

´ pr (1 − pi )Pi,...,j−1 [Ni , . . . , Nj−1 ] + O(δ).

r=i+1

From here and equation (20.5) we obtain: P {mi is blocked|mi is not starved} = P {mi is blocked} + O(δ).

(20.6)

20.1. PROOFS FOR PART II

593

Using repeatedly the conditional probability formula, the definition of p˜fi , and equation (20.6), we have: p˜fi

= P {mi produces|mi is not blocked} = P {mi is up, not blocked, and not starved|mi is not blocked} P {mi is up, not blocked, and not starved} = P {mi is not blocked} P {mi is not blocked|mi is not starved} = P {mi is not starved} P {mi is not blocked} ·P {mi is up|mi is not blocked or starved} P {mi is not blocked|mi is not starved} = pi (1 − Pi−1 [0]) P {mi is not blocked} 1 − P {mi is blocked|mi is not starved} = pi (1 − Pi−1 [0]) 1 − P {mi is blocked} = pi (1 − Pi−1 [0]) + O(δ).

This proves statement (α) of the lemma. Statement (β) is proved analogously. ¥ ˜i, i = To formulate Lemma 20.7, consider (M − 1) two-machine lines, L f 1, . . . , M − 1, where the first machine is defined by p˜i , the second by p˜bi+1 , and the buffer is of capacity Ni . Let P˜i [·] be the equilibrium probability mass ˜ i . Along with these M − 1 lines, consider function of buffer occupancy of line L the line defined by assumptions (a)-(e) of Subsection 4.2.1 with M machines. Let Pi [·], as before, be the equilibrium probability mass function of buffer occupancy of buffer i. Then, Lemma 20.7 The following property holds: |P˜i [j] − Pi [j]| ∼ O(δ),

i = 1, . . . , M − 1, j = 0, . . . , Ni ,

where δ is defined by (4.44). Proof: Consider the line defined by assumptions (a)-(e) with M machines. Let Ki = [k1 , . . ., ki−1 , ki+1 , . . ., kM −1 ]T , 1 ≤ i ≤ M1 , 0 ≤ kj ≤ Nj , j 6= i, be an (M − 2)-dimensional vector. Let Yi (hi , Ki ), 1 ≤ i ≤ M − 1, denote the probability that there are hi parts in buffer i and kj parts in buffer j, ∀j 6= i. Since this line can be described by an ergodic Markov chain with states Yi (hi , Ki ), in the steady state we write: X Yi (0, Ki ) = Yi (0, Ki0 )P {mi does not produce|0, Ki0 } · P {Ki0 → Ki |0 → 0} Ki0

+

X

Yi (1, Ki0 )P {mi does not produce, mi+1 produces|1, Ki0 }

Ki0

·P {Ki0 → Ki |1 → 0},

594

CHAPTER 20. PROOFS

where P {mi does not produce|hi , Ki } denotes the conditional probability that machine i does not produce a part during a cycle, given that buffer i contains hi parts and buffer j contains kj parts, ∀j 6= i, and P {Ki0 → Ki |h0i → hi } denotes the conditional probability of the transition from the state where buffer j, j 6= i, contains kj0 parts to the state where buffer j contains kj parts, given that the number of parts in buffer i changes from h0i to hi . Summation over all Ki ∈ RM −2 yields X X Pi [0] = Yi (0, Ki0 )P {mi does not produce|0, Ki0 } P {Ki0 → Ki |0 → 0} Ki0

+

X

Ki0

Yi (1, Ki0 )P {mi does not produce, mi+1 produces|1, Ki0 }

Ki0

·

X

P {Ki0 → Ki |1 → 0}.

Ki0

Since

P

Pi [0]

Ki0

P {Ki0 → Ki |0 → 0} = 1,

=

X

Yi (0, Ki0 )P {mi does not produce|0, Ki0 }

(20.7)

Ki0

+

X

Yi (1, Ki0 )P {mi does not produce, mi+1 produces|1, Ki0 }.

Ki0

Consider now the first term on the right hand side of equation (20.7): X P {mi does not produce|0, Ki0 } Ki0

X

=

Yi (0, Ki0 )P {mi does not produce|0, Ki0 }

(20.8)

Ki0 +

such that ki−1 ≥ 1 X

Yi (0, Ki0 )P {mi does not produce|0, Ki0 }.

Ki0 such that ki−1 → 0 When buffer i − 1 contains at least one part, machine i is not starved, and when buffer i contains zero parts, machine i is not blocked. Therefore, the probability in the first term on the right hand side of equation (20.8) is equal to 1 − pi . When buffer i − 1 contains zero parts, machine i is starved, and the probability in the second term on the right hand side of equation (20.8) is equal to one. Consequently, X P {mi does not produce|0, Ki0 } Ki0

= (1 − pi )[Pi [0] − Pi−1,i [0, 0]] + Pi−1,i [0, 0]

20.1. PROOFS FOR PART II

595

= Pi [0](1 − pi ) + Pi−1,i [0, 0]pi .

(20.9)

Using (4.44), this can be rewritten as X P {mi does not produce|0, Ki0 } Ki0

= Pi [0](1 − pi ) + Pi−1 [0]Pi [0]pi + O(δ) = Pi [0][1 − pi (1 − Pi−1 [0])] + O(δ). By Lemma 20.6, we finally obtain: X P {mi does not produce|0, Ki0 } = Pi [0](1 − p˜fi + O(δ)). Ki0

Analysis of the second term on the right-hand side of equation (20.7) proceeds analogously and results in Pi [0] = Pi [0](1 − p˜fi + O(δ)) + Pi [1](1 − p˜fi )˜ pbi+1 . Similar arguments can be used to derive the equations for Pi [j], j = 1, . . . , Ni . As a result, we obtain the following set of equations: Pi [0] = Pi [0](1 − p˜fi + O(δ)) + Pi [1](1 − p˜fi )˜ pbi+1 , pfi p˜bi+1 + (1 − p˜fi )(1 − p˜bi+1 ) + O(δ)]Pi [1] Pi [1] = p˜fi Pi [0] + [˜ pbi+1 Pi [2], +(1 − p˜fi )˜ Pi [j]

=

+(1 Pi [Ni ]

(20.10)

p˜fi (1

− p˜bi+1 )Pi [j − 1] + [˜ pfi p˜bi+1 + (1 − p˜fi )(1 pbi+1 Pi [j + 1], 2 ≤ j ≤ Ni − 1, − p˜fi )˜

−

p˜bi+1 )

+ O(δ)]Pi [j]

= p˜fi (1 − p˜bi+1 )Pi [Ni − 1] + [˜ pfi p˜bi+1 + 1 − p˜bi+1 + O(δ)]Pi [Ni ].

These equations can be written in matrix form as Pi = (A + ∆A)Pi ,

Pi = [Pi [0], . . . , Pi [Ni ]]T

where        

A= 1 − p˜fi 0 ··· 0 ··· 0

(1 − p˜fi )p˜bi+1 p˜fi ··· p˜fi (1 − p˜bi+1 ) ··· 0

0 p˜fi p˜bi+1 + (1 − p˜fi )(1 − p˜bi+1 ) ··· p˜fi p˜bi+1 + (1 − p˜fi )(1 − p˜bi+1 ) ··· 0

0 (1 − p˜fi )p˜bi+1 ··· (1 − p˜fi )p˜bi+1 ··· p˜fi (1 − p˜bi+1 )

p˜fi p˜bi+1

0 0 ··· 0 ··· + 1 − p˜bi+1

       

(20.11)

and ∆A is a diagonal matrix with all diagonal elements of the order O(δ), and, therefore, ||∆A|| ∼ O(δ). As it follows from equation (4.6), the equilibrium distribution of parts P˜i [·] of ˜ i is described by P˜i = AP˜i , where A is given in equation (20.11). Since A line L

596

CHAPTER 20. PROOFS

is the state transition matrix of an ergodic Markov chain, λ = 1 is an eigenvalue of A with multiplicity 1. Therefore, using the perturbation theory, we obtain: |P˜i [j] − Pi [j]| ∼ O(δ),

1 ≤ i ≤ M − 1, 0 ≤ j ≤ Ni . ¥

Thus, Lemma 20.7 claims that if the conditional probabilities p˜fi and p˜bi , i = 1, . . . , M , are known, it is possible to determine, approximately, the steady state buffer occupancy probability mass functions Pi [·], i = 1, . . . , M − 1. The task of determining the values of these conditional probabilities, however, remains. Lemma 20.8 shows that they are given, approximately, by recursive procedure (4.30). Lemma 20.8 The following relationships hold: |˜ pfi − pfi | ∼ O(δ), |˜ pbi − pbi | ∼ O(δ), i = 1, . . . , M, where pfi and pbi are given in (4.35) and δ is defined in (4.44). Proof: Let P˜i [·] be the equilibrium probability distribution of buffer occu˜ i , i = 1, . . . , M − 1, as described earlier, and let Pi [·] be the pancy of line L equilibrium probability mass function of buffer occupancy for buffer i of line defined by assumptions (a)-(e). Let the conditional probabilities p˜fi and p˜bi , i = 1, . . . , M , be as defined in equation (20.4). Then by Lemma 20.6, p˜fi can be expressed in terms of Pi−1 [0] as p˜fi = pi (1 − Pi−1 [0]) + O(δ),

i = 2, . . . , M.

By Lemma 20.7, this can be approximated with the distribution of parts on line ˜ i by L i = 2, . . . , M. p˜fi = pi (1 − P˜i−1 [0]) + O(δ), Using Lemma 20.4, this can be rewritten as pfi−1 , p˜bi , Ni−1 )) + O(δ), p˜fi = pi (1 − Q(˜

i = 2, . . . , M.

(20.12)

Analogously, by Lemma 20.6, p˜bi

=

M ³ j−1 X Y j=i+1

´ pr (1 − pj )Pi,...,j−1 [Ni , . . . , Nj−1 ]] + O(δ)

r=i+1

= pi [1 − (1 − pi+1 )Pi [Ni ] M ³ j−1 ´ X Y − pr (1 − pj )Pi,...,j−1 [Ni , . . . , Nj−1 ]] + O(δ). j=i+2

r=i+1

Using (4.44), this can be approximated by p˜bi

=

pi [1 − (1 − pi+1 )P˜i [Ni ]

20.1. PROOFS FOR PART II

−Pi [Ni ]

M ³ j−1 X Y j=i+2

597 ´ pr (1 − pj )Pi+1,...,j−1 [Ni+1 , . . . , Nj−1 ]] + O(δ).

r=i+1

By Lemma 20.7, this may be rewritten as p˜bi

= pi [1 − (1 − pi+1 )P˜i [Ni ] M ³ j−1 ´ X Y −P˜i [Ni ] pr (1 − pj )Pi+1,...,j−1 [Ni+1 , . . . , Nj−1 ]] + O(δ). j=i+2

r=i+1

Rearranging and using Lemma 20.6, we obtain p˜bi

pi [1 − (1 − pi+1 )P˜i [Ni ] − P˜i [Ni ](pi+1 − p˜bi+1 )] + O(δ) = pi [1 − (1 − p˜i+1 )P˜i [Ni ] + O(δ).

=

Using Lemma 20.4, this may be rewritten as pbi+1 , p˜fi , Ni )] + O(δ). p˜bi = pi [1 − Q(˜

(20.13)

By Lemma 20.5, the equilibrium equation (20.2) has a unique solution pfi , pbi , i = 1, . . . , M . Equations (20.12) and (20.13) show that the conditional probabilities p˜fi , p˜bi , i = 1, . . . , M , solve equation (20.2) with error O(δ). Therefore, we conclude that |˜ pfi − pfi | ∼ O(δ), |˜ pbi − pbi | ∼ O(δ), i = 1, . . . , M. ¥ Proof of Theorem 4.3: Using Lemma 20.7, the production rate may be calculated as P R = (1 − PM −1 [0])pM = (1 − P˜M −1 [0])pM + O(δ). Using Lemma 20.4, this may be expressed as P R = [1 − Q(˜ pfM −1 , p˜M , NM −1 )]pM + O(δ). By Lemma 20.8, we obtain P R = [1 − Q(pfM −1 , pM , NM −1 )]pM + O(δ). By Lemma 20.5. we finally conclude that P R = pfM + O(δ). ¥

598

CHAPTER 20. PROOFS

Proof of Theorem 4.4: Let pfj and pbj , 1 ≤ j ≤ M , denote the steady states of recursive procedure (4.30) applied to the original line. Introduce the notations pfj = pbM −j and pbj = pfM −j . Observe that pfj and pbj solve the equilibrium equations of recursive procedure (4.30) for the reversed line. By Lemma 20.5, the equilibrium equations possess a unique solution, so pfj and pbj must be the limiting values obtained by recursive procedure (4.30) for the reversed line. Therefore, P R(pM , . . . , p1 , NM −1 , . . . , N1 ) = pfM = pb1 . By Lemma 20.3, pb1 = pfM . ¥ Proof of Theorem 4.5: Let P R(p1 , p2 , N1 ) denote the production rate of the serial production line defined by assumptions (a)-(e) with M = 2. Then the following three facts hold: (i) Function Q(x, y, N ) is monotonically decreasing in x and N , and monotonically increasing in y (Lemma 4.1). (ii) Function P R(p1 , p2 , N1 ) is monotonically increasing in p1 , p2 and N1 (Lemma 20.4). (iii) P R(pfi , pbi+1 , Ni ) = pfM , i = 1, . . . , M − 1, where pfi and pbi are defined in (4.35) (Lemma 20.3). Consider two serial production lines defined by assumptions (a)-(e) of Subsection 4.2.1, the first of which is described by parameters pi , i = 1, . . . , M , and Ni , i = 1, . . . , M − 1, and the second by parameters p˜i ≥ pi , i = 1, . . . , M , and ˜i ≥ Ni , i = 1, . . . , M − 1. Let pf , pb , p˜f , p˜b , i = 1, . . . , M , denote the steady N i i i i states of recursive procedure (4.30) for the first and second lines, respectively. We prove Theorem 4.5 by contradiction. Assume (20.14) p˜fM < pfM . Then, using (iii), ˜1 ) < P R(pf , pb2 , N1 ). P R(˜ pf1 , p˜b2 , N 1 Since, by (ii), P R(p1 , p2 , N1 ) is a monotonically increasing function of each of ˜1 ≥ N1 , it follows that p˜b < pb . its arguments, and by construction p˜b1 ≥ pf1 , N 2 2 Therefore, using (4.30) and monotonicity property (i), ˜1 )] > p2 [1 − Q(p1 , pb , N1 )] = pf . p˜f2 = p˜2 [1 − Q(˜ p1 , p˜b2 , N 2 2 Now proceed inductively. Assume p˜bi < pbi and p˜fi > pfi . The base case ˜i ) < (i = 2) has already been established. From (iii) and (20.14), P R(˜ pfi , p˜bi+1 , N f f f b b ˜ P R(pi , pi+1 , Ni ). Since p˜i > pI and Ni > Ni , it follows from (ii) that p˜i+1 < pbi+1 . Equation (4.30) and monotonicity property (i) then yield ˜i )] > pi+1 [1 − Q(pf , pbi+1 , Ni )] = pf . pfi , p˜bi+1 , N p˜fi+1 = p˜i+1 [1 − Q(˜ i i+1 The inductive hypothesis is, therefore, established, and p˜bi < pbi and p˜fi > pfi , i = 2, . . . , M . In particular, p˜fm > pfM , which contradicts the assumption

20.1. PROOFS FOR PART II

599

(20.14). Therefore, p˜fM ≥ pfM and, using (4.36), P R(p1 , . . . , pM , N1 , . . . , NM −1 ) is a monotonically increasing function of its arguments. ¥

20.1.2

Proofs for Chapter 5

Introduce fN (x, y) := [1 − Q(x, y, N )][1 − Q(y, x, N )],

x, y ∈ (0, 1),

where Q(x, y, N ) is given in (4.14). Lemma 20.9 Function fN (x, y) can be represented as follows: h g(1 − αN ) i2 fN (x, y) = , 1 − αN g 2 where

r g

=

α

=

1 , y x(1 − y) . y(1 − x)

Proof: From equation (4.14), we obtain 1 − Q(x, y, N ) = = =

1−

(1 − x)(1 − α) 1 − y1 αN

α + x(1 − α) − xy αN 1 − xy αN

³ x ´h 1 − αN i . y 1 − xy αN

By Lemma 20.3, x(1 − Q(y, x, N )) = y(1 − Q(x, y, N )), and, therefore, fN (x, y) =

[1 − Q(x, y, N )][1 − Q(y, x, N )] y = [1 − Q(x, y, N )]2 x ³ x ´h 1 − αN i2 = y 1 − xy αN h 1 − αN i2 = g2 . 1 − αN g 2 ¥

600

CHAPTER 20. PROOFS

Lemma 20.10 Under the constraint xy = p = const, function fN (x, y) achieves its maximum if and only if x = y. Furthermore, √ √ √ fN ( p, p) = [N/(N + 1 − p)]2 . p √ Proof: Define g = x/y, p = xy, and α = x(1−y) y(1−x) . Then, x = g p and √ y = p/y. Suppose x < y, which implies 0 < g < 1. Then √ ´ √ ³ p g p 1− g √ α= p √ . g (1 − g p) Observe that α is a monotonically increasing function of g. Using Lemma 20.9, fN (x, y) can be expressed as ³ √ √p ´ ³ g(1 − αN ) ´2 fN g p, = g 1 − αN g 2 = exp{2[ln(g) + ln(1 − αN ) − ln(1 − αN g 2 )]}. Using the series expansion ln(1 − x) = −

∞ X xi i=1

i

,

|x| < 1,

√ √p function fN (g p, g ) may be expressed, for 0 < g < 1, as (

) ∞ ∞ h X (αN )i X (αN g 2 )i i exp 2 ln(g) − + i i i=1 i=1 ) ( ∞ h X (αN g 2 )i − (αN )i i = exp 2 ln(g) − i i=1 ) ( ∞ i h X 1 N i 2i (α )(g − 1) . (20.15) = exp 2 ln(g) − i i=1

³ √ √p ´ fN g p, = g

Since each term in the exponent of equation (20.15) is a monotonically increas√ √ ing function of g, it can be concluded that fN (g p, p/g) is a monotonically increasing function of g for 0 < g < 1. Now, suppose y < x. Define g = 1/g and observe that 0 < g < 1. Because √ √ of the symmetry of its definition, fN (x, y) = fN (y, x). So fN (g p, p/g) − √ √ √ √ fN ( p/g, g p) = fN (g p, p/g). By the previous arguments, this function is monotonically increasing in g for 0 < g < 1, and, therefore, monotonically √ √ decreasing in g for 1 < g < ∞. Thus, fN (g p, p/g), 0 < g < ∞, attains its √ maximum at g = 1, which corresponds to x = y = p. Then, √ √ fN ( p, p, N ) =

√ √ [1 − Q( p, p, N )]2

20.1. PROOFS FOR PART II

601 √ 1 − p i2 = 1− √ N +1− p i2 h N = . √ N +1− p h

¥ Lemma 20.11 Define

q ci :=

Then,

pfi pbi+1 .

³ N +1 ´ i d P R, d Ni + P R

ci ≥

and the equality takes place if and only if pfi = pbi+1 . Proof: Using Lemma 20.3, c2i

= = = =

pfi pbi+1 ³p P d R ´³ p i

pbi

d´

i+1 P R pfi+1

d pi pi+1 P R

2

pi (1 − Q(pbi+1 , pfi , Ni ))pi+1 (1 − Q(pfi , pbi+1 , Ni )) d P R

2

fNi (pfi , pbi+1 )

.

By Lemma 20.10 and the definition of ci , i2 h Ni , fNi (pfi , pbi+1 ) ≤ Ni + 1 − ci with the equality taking place if and only if pfi = pbi+1 , so c2i

≥

ci

≥

³ d P R´ ≥ ci 1 + Ni ci

≥

h

d P R

2

Ni Ni +1−ci

i2 ,

d P R(Ni + 1 − ci ) , Ni d P R(Ni + 1) , Ni d P R(Ni + 1) . d Ni + P R ¥

602

CHAPTER 20. PROOFS

Lemma 20.12 The total workforce, p∗ , necessary to achieve the production d rate value P R, is bounded by p∗ ≥

M −1 ³ Y j=1

Nj + 1 ´2 d M PR . d Nj + P R

The equality holds if and only if pfi = pbi+1 , i = 1, . . . , M − 1. Proof: By Lemma 20.3, d P R d P R

M

M

· p∗

³ pf pb ´ 1 1

=

p1

···

³ pf pb ´ i

i

pi

···

³ pf pb ´ M M , pM

pb1 (pf1 pb2 ) · · · (pfM −1 pbM )pfM .

=

d Since, by Lemma 20.3, pb1 = pfM = P R, we have p∗ =

c21 c22 · · · c2M −1 d P R

M −2

.

Using Lemma 20.11 we obtain, with equality if and only if pfi = pbi+1 , i = 1, . . . , M − 1, QM −1 ³ d Ni +1 ´2 PR i=1 d Ni +P R p∗ ≥ M −2 d PR M −1 ³ Y Ni + 1 ´2 d M PR . = d Ni + P R i=1 ¥ Lemma 20.13 The condition pfi = pbi+1 , i = 1, . . . , M − 1, is achieved if and only if the workforce is distributed as ³ N +1 ´ 1 , p1 = d N1 + P R ´³ N + 1 ´ ³ N j−1 + 1 j d pj = P R, 2 ≤ j ≤ M − 1, (20.16) d d Nj−1 + P R Nj + P R ´ ³ N M −1 + 1 , pM = d NM −1 + P R d where P R is the production rate of the line. Proof: Suppose pfi = pbi+1 , i = 1. . . . , M . Then, by Lemmas 20.3 and 20.4, d P R

=

pfi (1 − Q(pbi+1 , pfi , Ni ))

20.1. PROOFS FOR PART II

603

= pfi (1 − Q(pfi , pfi , Ni )) ³ ´ 1 − pfi = pfi 1 − f Ni + 1 − pi =

pfi Ni Ni + 1 − pfi

.

Solving this equation for pfi and recalling the assumption that pbi = pbi+1 , we obtain ³ N +1 ´ i d P R. (20.17) pfi = pbi+1 = d Ni + P R Using Lemma 20.3, for i = 2, . . . , M − 1, 2 ´³ N + 1 ´ P ³ N d pfi pbi R i−1 + 1 i d PR = = , d d pi Ni−1 + P R Ni + P R pi

which can be rearranged into ´³ N + 1 ´ ³ N i−1 + 1 i d P R, pi = d d Ni−1 + P R Ni + P R

i = 2, . . . , M − 1.

(20.18)

The expressions for p1 = pf1 and pM = pbM are obtained from equation (20.15). Now, suppose that the workforce is distributed as in equation (20.16). We next show that this implies that pfi = pbi+1 , i = 1, . . . , M − 1. By Lemma 20.5, there exists a unique solution to the equilibrium equations (20.2) of recursive procedure (4.30). We claim that this solution is ³ N +1 ´ i d , i = 1, . . . , M − 1, (20.19) R, pfi = pbi+1 = pb1 = pfM = P d Ni + P R Since equation (20.2) has a unique solution, we only need to show that (20.19) is indeed a solution. Using equation (20.18), for i = 2, . . . , M , we have: pi (1 − Q(pfi−1 , pbi , Ni−1 )) ´³ N + 1 ´³ ´ ³ N 1 − pbi i−1 + 1 i d 1− =P R d d Ni−1 + 1 − pbi Ni−1 + P R Ni + P R ´ ´³ ´³ ³ N Ni + 1 Ni−1 i−1 + 1 d =P R b d d Ni−1 + P R Ni + P R Ni−1 + 1 − pi ³ ´³ ´ ³ N Ni−1 ´ Ni + 1 1 i−1 + 1 d =P R b Ni−1 +1 d d −1 Ni−1 + P R Ni + P R pi pbi ´ ³ N ³ ´³ ´ Ni−1 Ni + 1 1 i−1 + 1 d =P R +1 d NNi−1 d d −1 Ni−1 + P R Ni + P R Ni−1 +1 i−1 +1 d dPR Ni−1 +P R

³ N +1 ´ i d P R = d Ni + P R

d Ni−1 +P R

PR

604

CHAPTER 20. PROOFS = pfi .

The proof for pbi , i = 1, . . . , M − 1, is similar.

¥

Lemma 20.14 The minimum workforce p∗min required to achieve production d rate P R is given by p∗min =

M −1 ³ Y j=1

Nj + 1 ´2 d M PR . d Nj + P R

Moreover, this rate is achieved if and only if p∗ is distributed among Q production ∗ p1 , . . . , pM , i pi = p , so that pfi = pbi+1 , i = 1, . . . , M − 1. Proof: By Lemma 20.12, p∗min ≥

M −1 ³ Y j=1

Nj + 1 ´2 d M PR , d Nj + P R

and equality is achieved if and only if pfi = pbi+1 , i = 1, . . . , M − 1. By Lemma 20.13, the lower bound is attainable with the workforce distribution as specified in equation (20.16). ¥ Lemma 20.15 The minimum workforce p∗min , necessary to achieve the prod d duction rate P R, is a monotonically increasing function of P R. Proof: From Lemma 20.14, p∗min is given by p∗min =

M −1 h³ Y j=1

Nj + 1 ´2 d M i PR . d Nj + P R

Consider one term of the product, i.e., Tj =

³ N + 1 ´2 j d P R, d Nj + P R

j = 1, . . . , M − 1.

d R yields Differentiating Tj with respect to P ³ ´ (N + 1)2 (N − P d d ∂Tj 2P R R) 1 j j = = (Nj + 1)2 − > 0. d d d d ∂P R (Nj + P R)2 (Nj + P R)3 (Nj + P R)3 d R, Since p∗min is the product of positive, monotonically increasing function of P the lemma is proved. ¥

20.1. PROOFS FOR PART II

605

Proof of Theorem 5.1: By contradiction: Suppose the line is unimprovable, but that there exists an i such that pfi 6= pbi+1 . Then, by Lemma 20.14, p∗ > p∗min and, by Lemma 20.15, the workforce p∗ , being optimally distributed, can achieve a larger production rate, which is a contradiction. ¥ Proof of Corollary 5.1: From equation (4.8), when applied to line Li = with pfi = pbi+1 , we have:

{pfi , Ni , pbi+1 }

Pi [j] =

Pi [0] 1 − pfi

,

1 ≤ j ≤ Ni .

So, using equation (4.10), \ W IP i

=

Ni X

jPi [j]

j=0

=

Ni ³ X j j=0

=

´³

1 1−

pfi

Ni (Ni + 1) 2(Ni + 1 − pfi )

´

1 − pfi Ni + 1 −

pfi

. ¥

Proof of Theorem 5.2: It was established in Lemma 20.13 that (5.9) is satisfied if and only if ³ N +1 ´ ∗ 1 d ∗ PR , d N1 + P R ´³ N + 1 ´ ³ N ∗ i−1 + 1 i d = ∗ ∗ PR , d d Ni−1 + P R Ni + P R ´ ³ N ∗ + 1 M −1 d = ∗ PR d NM −1 + P R =

p1 pi pM

i = 2, . . . , M − 1, (20.20)

∗

d where P R = pfM is defined in (5.12). Therefore, for a line satisfying (5.9), p∗ =

M Y

∗

d pi = (P R )M

i=1

M −1 ³ Y i=1

Ni + 1 ´2 . ∗ d Ni + P R

This may be rearranged as ∗

1 d P R = (p∗ ) M

M −1 ³ Y i=1

∗ d R ´ M2 Ni + P . Ni + 1

(20.21)

606

CHAPTER 20. PROOFS

Define function f (·) by 1

f (x) := (p∗ ) M

M −1 ³ Y i=1

and observe that

Ni + x ´ M2 , Ni + 1

∗

∗

d d R ). P R = f (P

(20.22)

We next show that x(n + 1) = f (x(n)) is a contraction on [0, 1], from which it follows that (20.22) has exactly one solution and that recursive procedure (5.13) converging to this solution. Using the relationship QN N N ih X d[ i=1 yi (x)] h Y yi0 (x) i = , yi (x) dx y (x) i=1 i=1 i where yi0 (x) = dyi (x)/dx, we calculate df (x) dx

1

= (p∗ ) M 1

= (p∗ ) M

−1 ³ M −1 M −1 ³ 2 ´h MY 1 i Ni + x ´i M2 −1 h Y ³ Nx + x ´ih X M Ni + 1 Ni + 1 Ni + x i=1 i=1 i=1 −1 ³ M −1 ³ 2 ´h MY 1 i Ni + x ´i M2 h X . M Ni + 1 Ni + x i=1 i=1

Since p∗ < 1 and (Ni + x)/(Ni + 1) ≤ 1 for x ∈ [0, 1], using the assumption PM −1 i=1 1/Ni ≤ M/2 we obtain ¯ df (x) ¯ ¯ ¯ ¯ ¯ < 1, dx

x ∈ [0, 1].

The Mean Value Theorem guarantees that for all x, y ∈ [0, 1] there exists a c ∈ [x, y] such that df (c) (x − y), f (x) − f (y) = dx and, therefore, |f (x) − f (y)| < |x − y|. This implies that x(n + 1) = f (x(n)) is a contraction on [0, 1], which establishes the theorem. ¥ Proof of Theorem 5.3: Equations (5.15)-(5.17) follow directly from the proof of Theorem 5.2. ¥ Proof of Theorem 5.4: Define 1

f (N1 , . . . , NM −1 , x) := (p∗ ) M

M −1 ³ Y i=1

Ni + x ´ M2 Ni + 1

(20.23)

20.1. PROOFS FOR PART II

607

and f ∗ (x) =

max P

N1 ,...,NM −1 ,

M −1 i=1

Ni =N ∗

f (N1 , . . . , NM −1 , x).

(20.24)

∗ The values N1∗ , . . . , NM −1 which solve (20.24) can be determined by the Lagrange multiplier technique. The Lagrange function is:

F (N1 , . . . , NM −1 , λ) = =

f (N1 , . . . , NM −1 , x) + λ(N1 + . . . , NM −1 − N ∗ ) M −1 ³ Y 1 Ni + x ´ M2 + λ(N1 + . . . , NM −1 − N ∗ ). (p∗ ) M N + 1 i i=1

Therefore, the optimality condition ∂F (N1 , . . . , NM −1 , λ) ∂Ni −1 ³ ³ 2 ´h MY i 1 Ni + x ´ M2 ih 1−x ∗ M + λ = 0, = (p ) M Ni + 1 (Ni + 1)(Ni + x) i=1 is satisfied if and only if Ni = Nj , ∀i, j. Thus, (20.24) is solved by Ni∗ =

N∗ , M −1

i1, . . . , M − 1.

(20.25)

Consider now recursive procedure (5.13): x(n + 1) = f (N1 , . . . , NM −1 , x(n)), where f (·) is defined by (20.23). Recall that, according to Theorem 5.3, limn→∞ x(n) = P R(p∗1 , . . ., p∗M , N1 , . . ., NM −1 ), where p∗i , i = 1, . . . , M , are defined by (5.15)(5.17). Define the recursive procedure (5.13) for two sequences of Ni ’s, the optimal one and any other: x∗ (n + 1) x0 (n + 1)

= =

∗ ∗ f (N1∗ , . . . , NM −1 , x (n)), 0 0 0 f (N1 , . . . , NM −1 , x (n)),

(20.26) (20.27)

where Ni∗ is defined by (20.25) and Ni0 , i = 1, . . . , M − 1, is any sequence PM −1 satisfying i=1 Ni0 = N ∗ . Assume that the initial conditions for (20.26) and (20.27) are the same: x∗ (0) = x0 (0) ∈ [0, 1]. We show below that (i). x∗ (n) ≥ x0 (n), ∀n > 0, i.e., ∗ 0 0 0 0 P R(p∗1 , . . . , p∗M , N1∗ , . . . , NM −1 ) ≥ P R(p1 , . . . , pM , N1 , . . . , NM −1 )

(ii). pfi = pbi , i = 2, . . . , M − 1.

608

CHAPTER 20. PROOFS

Since pf1 = p1 and pbM = pM , this would prove Theorem 5.4. Fact (i) is proved by induction. For n = 1, the result x∗ (1) ≥ x0 (1) follows immediately from the fact that the sequence Ni∗ , i = 1, . . . , M −1, solves (20.24). Now, assume that x∗ (n) ≥ x0 (n). Because −1 ³ M −1 ³ 2 ´h MY 1 1 i df (N1 , . . . , NM −1 , x) Ni + x ´ M2 ih X ∗ M = (p ) > 0, dx M Ni + 1 Ni + 1 i=1 i=1 (20.28) that is, f (N1 , . . . , NM −1 , x) is a monotonically increasing function of x, and since the sequence Ni∗ , i = 1, . . . , M − 1, solves (20.24),

x∗ (n + 1)

∗ ∗ f (N1∗ , . . . , NM −1 , x (n))

=

∗ 0 ≥ f (N1∗ , . . . , NM −1 , x (n)) 0 0 ≥ f (N10 , . . . , NM −1 , x (n))

=

x(n + 1).

Statement (ii) follows from (20.25) and the fact that the unique solution to the equilibrium equation (20.2) of recursive procedure (4.30), when the workforce is distributed according to (5.15)-(5.17)), is given in equation (20.19) by d R, pb1 = pfM = P i.e.,

pfi = pbi+1 =

³ N +1 ´ i d P R, d Ni + P R

p1 = pM = pfi = pbi ,

i = 1, . . . , M − 1, (20.29)

i = 2, . . . , M − 1. ¥

Proof of Corollary 5.3: Equations (5.20)-(5.23) follow directly from Theorem 5.4, equation (20.25), and Corollary 5.1. ¥ Proof of Theorem 5.5: Similar to the proof of Theorem 5.4. Equations (5.25)-(5.27) follow directly from (20.25) and (5.15)-(5.17). ¥ Proof of Theorem 5.6: Using the identity nr a r b o √ , , min{a, b} = ab min b a in

0 < a, 0 < b,

d P R = min{pfi , pbi }, i

we obtain: d P R = min i

q pfi pbi

n min

s

pfi , pbi

s

pbi o pfi

.

20.1. PROOFS FOR PART II

609

By Lemma 20.3, this can be written as q d P R = min i

which implies that

d P Rpi min

n

s

pfi , pbi

s

pbi o pfi

,

(

) n p f pb o i i d P R = min pi min , . pbi pfi ¥

Proof of Theorem 5.7: Consider a WF-unimprovable line defined by assumptions (a)-(e) of Subsection 4.2.1. Let the workforce distribution be denoted by pi = p∗i , i = 1, . . . , M . Suppose that the workforce distribution is modified by pi = gp∗i and pj = (1/g)p∗i , 1 ≤ i ≤ M and 1 ≤ j ≤ M . Observe that the total workforce does not depend on g, but that the line is unimprovable when g = 1. Therefore, the production rate achieves its maximum when g = 1. Letting P R = P R(g), we observe ∂P R(1) = 0. ∂g

(20.30)

Using the chain rule, ∂P R(1) ∂g

pi ∂P R(1) ∂(gpi ) ∂P R(1) ∂( g ) + · · ∂pi ∂g ∂pj ∂g ∂P R(1) ∂P R(1) = pi − pj . ∂pi ∂pj

=

(20.31)

Since i and j were chosen arbitrarily, we therefore conclude, from (20.30) and (20.31), that ∂P R(1) ∂P R(1) = pj , ∀i, j. pi ∂pi ∂pj ¥ To prove Theorem 5.8, we need the following three lemmas: d R with respect to pi , Lemma 20.16 Let fi (p1 , p2 , N1 ) be the sensitivity of P i.e., fi (p1 , p2 , N1 ) = ∂P R(p1 , p2 , N1 )/∂pi , i = 1, 2. Then functions f1 (p1 , p2 , N1 ) and f2 (p1 , p2 , N1 ) are monotonically decreasing and increasing in p1 , respectively. Proof: Recall that P R(p1 , p2 , N1 ) = p2 [1 − Q(p1 , p2 , N1 )] = p1 [1 − Q(p2 , p1 , N1 )].

(20.32)

610

CHAPTER 20. PROOFS

Hence, f1 (p1 , p2 , N1 ) =

∂P R(p1 , p2 , N1 ) ∂Q(p1 , p2 , N1 ) = −p2 . ∂p1 ∂p1

(20.33)

From equation (4.14), function Q(p1 , p2 , N1 ) can be re-written as follows: Q(p(p1 , p2 , N1 ) =

1 − p2 , α + . . . + αN1 + 1 − p2

where α is given in (4.7). Then, ´ ³ (1−p2 )2 N1 −1 α − 1 + 2α + . . . + N 1 p2 (1−p1 )2 ∂Q(p1 , p2 , N1 ) = . ∂p1 [α + . . . + αN1 + (1 − p2 )]2 After some algebra, this simplifies to   

∂Q(p1 , p2 , N1 ) =  ∂p1 

−1+αN1 +N1 (1−α)αN1 p p2 (1− p1 αN1 )2

, p1 6= p2 ,

2

−N1 (N1 +1) 2p1 (N1 +1−p1 )2 ,

(20.34) p1 = p2 .

Therefore, from (20.33) and (20.34),  (1−αN1 )−N1 (1−α)αN1  , p1 6= p2 , p   p2 (1− p1 αN1 )2 2 f1 (p1 , p2 , N1 ) =   N12 +N1 )  p1 = p 2 . 2(N1 +1−p1 )2 , Analogously, it can be shown that  (1−α−N1 )−N1 (1−1/α)α−N1  , p1 6= p2 , p   p2 (1− p2 α−N1 )2 1 f2 (p1 , p2 , N1 ) =   N12 +N1 )  p 1 = p2 . 2(N1 +1−p1 )2 ,

(20.35)

(20.36)

Consider now the two cases, p1 6= p2 and p1 = p2 , separately. For the case p1 6= p2 , we write f1 (p1 , p2 , N1 ) (1 − α)(1 + α + . . . + αN1 −1 − N1 αN1 ) = (1 − pp12 αN1 )2 = =

1 p

1− p1 αN1 2 1−α

p

·

1− p1 αN1 2

1+α+...+αN1 −1 −N1 αN1

1 p

1−αN1 +(1− p1 )αN1 2 1−α

·

p 1−αN1 +(1− p1 )αN1 2 N N (1−α 1 )+(α−α 1 )+...+(αN1 −1 −αN1 )

20.1. PROOFS FOR PART II

611 1

=

1 + α + . . . + αN1 −1 +

p2 −p1 p2 p2 (1−p1 )−p1 (1−p2 ) p2 (1−p1 )

αN −1

(1 − α)(1 + α + . . . + αN1 −1 ) + . . . + (1 − α)αN1 −1 1 − αN1 + (1 − pp12 )αN1 i.h h 1 + 2α + . . . + (N1 − 1)αN1 −2 = 1 + 2α + 3α2 + . . . + N1 αN1 −1 i +(N1 + 2A)αN1 −1 + . . . + (2 + 2A)α2N1 −3 + (1 + A)2 α2N1 −2 ·

=

k(α) , k(α) + l(α, A)

(20.37)

where A

=

k(α) = l(α, A) −

p1 (1 − p2 ) , p2 1 + 2α + 3α2 + . . . + N1 αN1 −1 ,   2AαN1 −1 + . . . + (2 + 2A)α2N1 −3 + (1 + A)2 α2N1 −2 , N1 ≥ 2, 

2A + A2 ,

N1 = 1.

For an arbitrary p1 ∈ (0, 1), choose p1 so that p1 > p1 . Then, α = and A =

p1 (1−p2 ) p2

p1 (1−p2 ) p2 (1−p1 )

>α

> A. Therefore, for N1 = 1,

f1 (p1 , p2 , N1 ) − f1 (p1 , p2 , N1 ) =

1 2

1+A+A

−

1 < 0. 1 + A + A2

For N1 ≥ 2, f1 (p1 , p2 , N1 )

− = =

f1 (p1 , p2 , N1 ) k(α) k(α) − k(α) + l(α, A) k(α) + l(α, A) k(α)l(α, A) − k(α)l(α, A) . [k(α) + l(α, A)][k(α) + l(α, A)]

(20.38)

The numerator of (20.38) can be re-written as k(α)l(α, A) − k(α)l(α, A) = (1 + 2α + . . . + N1 αN1 −1 )[2AαN1 −1 + . . . + (1 + A)2 α2N1 −2 ] N1 −1

N1 −1

(20.39)

2 2N1 −2

−(1 + 2α + . . . + N1 α )[2Aα + . . . + (1 + A) α ] = (αN1 −1 + 2ααN1 −1 + . . . + N1 αN1 −1 αN1 −1 )[2A + . . . + (1 + A)2 αN1 −1 ] −(αN1 −1 + 2αN1 −1 α + . . . + N1 αN1 −1 αN1 −1 )[2A + . . . + (1 + A)2 αN1 −1 ]. Since α > α and A > A, (αN1 −1 + 2ααN1 −1 + . . . + N1 N1 − 1αN1 −1 )

612

CHAPTER 20. PROOFS < (αN1 −1 + 2αN1 −1 α + . . . + N1 N1 − 1αN1 −1 )

and

[2A + . . . + (1 + A)2 αN1 −1 ] < [2A + . . . + (1 + A)2 αN1 −1 ].

Hence, k(α)l(α, A) − k(α)l(α, A) < 0, and, therefore, f1 (p1 , p2 , N1 ) − f1 (p1 , p2 , N1 ) < 0, i.e., f1 (p1 , p2 , N1 ) is monotonically decreasing in p1 if p1 6= p2 . For the case p1 = p2 , expression (20.38) holds as well, since in this case α = 1. Hence, f (p1 , p2 , N1 ) is again monotonically decreasing in p1 . The monotonicity property of f2 (p1 , p2 , N1 ) with respect to p1 is proved analogously. ¥ Lemma 20.17 Functions BL1 and ST2 defined by (4.24) are monotonically increasing and decreasing in p1 , respectively. Proof: By (4.24), BL1 = p1 Q(p2 , p1 , N1 ). Since Q(p2 , p1 , N1 ) is monotonically increasing in p1 (Lemma 4.1), function BL1 is also monotonically in p1 . Similar argument is used to prove that ST2 is monotonically decreasing in p1 . ¥ Lemma 20.18 For the case p1 = p2 , BL1 = ST2 , and f1 (p2 , p1 , N1 ) = f2 (p1 , p2 , N1 ). Proof: Follows directly from (4.24), (20.35) and (20.36). ¥ Proof of Theorem 5.8: Denote BL1 = BL1 (p1 , p2 , N1 ) and ST2 = ST2 (p1 , p2 , N1 ). Suppose BL1 < ST2 . Then we show, by contradiction, that p1 < p2 . Indeed, assume p1 ≥ p2 . Then, by Lemmas 20.17 and 20.18, BL1 (p1 , p2 , N1 ) ≥ BL1 (p2 , p2 , N1 ) = ST2 (p2 , p2 , N1 ) ≥ ST2 (p1 , p2 , N1 ), which contradicts the assumption. Therefore, p1 < p2 . Since p1 < p2 , using Lemmas 20.16 and 20.18, we have: f1 (p1 , p2 , N1 ) > f1 (p2 , p2 , N1 ) = f2 (p2 , p2 , N1 ) > f2 (p1 , p2 , N1 ),

20.1. PROOFS FOR PART II

613

i.e., ∂P R(p1 , p2 , N1 )/∂p1 > ∂(p1 , p2 , N1 )/∂p2 . This proves the “if” part. The “only if” part is proved by assuming ∂P R(p1 , p2 , N1 )/∂p1 > ∂(p1 , p2 , N1 )/∂p2 and using arguments analogous to the above. ¥ Proof of Lemma 5.1: Under Hypothesis 5.1, c j, dj−1 > ST BL

∀j ≤ i.

Under this condition, we show, by contradiction, that there exists N1∗ < ∞ such that pfj−1 > pjb , for all Nj−1 > N1∗ , ∀j ≤ i. Assume that, for all N1∗ < ∞, there exists Nj−1 > N1∗ , ∀j ≤ i, such that pfj−1 ≤ pbj . From (4.14) and (4.24), lim

Nj−1 →∞

dj−1 BL

=

=

lim

Nj−1 →∞

pj−1 Q(pbj , pfj−1 , Nj−1 )

 1 (1−pbj )(1− α )  lim p ,  N →∞ j−1 b j−1 p  j  1 Nj−1 1− f ( α )  p j−1

  b    limN →∞ pj−1 1−pj b , j−1 Nj−1 +1−p j

lim

Nj−1 →∞

cj ST

= =

=

=

where α=

pbj > pfj−1

pbj = pfj−1

0, lim

Nj−1 →∞

pj Q(pfj−1 , pbj , Nj−1 )

 (1−α)(1−pfj−1 )  limNj−1 →∞ pj pf , pbj > pfj−1    Nj−1  1− j−1 α pb j

  b    limN →∞ pj 1−pj b , j−1 Nj−1 +1−pj  pbj −pf f   pj pbj−1 , pbj > pj−1 ,

pbj = pfj−1

j

 

pbj = pfj−1 ,

0,

pfj−1 (1 − pbj ) pbj (1

−

pfj−1 )

< 1, if pbj > pfj−1 .

Therefore, there exists Nj−1 > N1∗ , ∀j ≤ i, such that c j, dj−1 ≤ ST BL which contradicts the assumption. Choose N1∗ large enough so that pj−1 > pbj , for all Nj−1 > N1∗ , ∀j ≤ i. Then, (1 − pfj−1 )(1 − α) f b lim Q(pj−1 , pj , Nj−1 ) = = 0, pf Nj−1 →∞ Nj−1 1 − j−1 α b p j

614

CHAPTER 20. PROOFS

where α=

pfj−1 (1 − pbj ) pbj (1 − pfj−1 )

> 1.

Thus, for any 0 < ² ¿ 1, there exists N1∗∗ > N1∗ , such that, for all Nj−1 > N1∗∗ , ∀j ≤ i, Q(pfj−1 , pbj , Nj−1 ) = ²j1 < ²10 . Similarly, for j ≥ i, we can conclude that, for any 0 < ²20 ¿ 1, there exists N2∗∗ > N2∗ such that, for all Nj > N2∗∗ , Q(pbj+1 , pfj , Nj ) = ²j2 < ²20 . Therefore, for given ²0 , choose N ∗ = max(N1∗∗ , N2∗∗ ) so that ² = max(²j1 , ²s2 ) < ²0 . j,s

¥ Proof of Theorem 5.9: The proof is based on evaluating the sensitivities of • Q(x, y, N ) to x and y, • pfi and pbi to pi , and d • P R to pi . These evaluations include straightforward but tedious and lengthy calculations, which are not included here and can be found in [20.3] and [20.4]. ¥

20.1.3

Proofs for Chapter 6

It is convenient to prove Theorem 6.3 first and then specialize it to Theorem 6.2. Proof of Theorem 6.3: From (4.36) with i = M − 1, we obtain P R = pbM [1 − Q(pfM −1 , pbM , NM −1 )] = p[1 − Q(pfM −1 , p, NE )]. Using (6.1), (4.33), and (4.34), we find 1 − E = Q(pfM −1 , p, NE ) =

α=

(1 − pfM −1 )(1 − α) 1−

pfM −1 (1 − p) p(1 − pfM −1 )

.

pfM −1 N E p α

,

(20.40)

20.1. PROOFS FOR PART II

615

Solving (20.40) for NE yields · ln NE =

µ p pfM −1

E−pfM −1 −α+αpfM −1 E−1

¶¸ .

ln α

(20.41)

The steady states of (4.30) give pfM −1 = p[1 − Q(pfM −2 , pbM −1 , NE )] = p[1 − Q],

(20.42)

Q = Q(pfM −2 , pbM −1 , N ).

(20.43)

where

Substituting (20.42) into (20.41) finally yields 

1−E−Q ln( (1−E)(1−Q) )



. NE =   (1−p)(1−Q)   ln( 1−p(1−Q) ) 

(20.44)

¥ Proof of Theorem 6.2: In three-machine lines, (20.42) becomes pf2 = p[1 − Q(pf1 , pb2 , NE )] = p[1 − Q(p, pb2 , NE )].

(20.45)

and, in addition, pf2 = pb2 . Therefore, P R = pf2 [1 − Q(p, pf2 , NE )] = p[1 − Q(p, pf2 , N )]2 = p[1 − Q]2 ,

(20.46)

where Q was defined in (20.43). From (20.46) and (6.1) we find Q=1−

√ E.

Substituting this into (20.42), and then substituting the resulting expression for pfM −1 into (20.41), we obtain (6.10). ¥ Proof of Theorem 6.5: Following immediately from (6.18) by observing that α(p1 , p2 ) is equal to 1/α(p2 , p1 ) and P R(p1 , p2 ) is equal to P R(p2 , p1 ). ¥

616

CHAPTER 20. PROOFS

20.1.4

Proofs for Chapter 7

Proof of Theorem 7.1: Under assumptions (a)-(h) of Subsection 7.1.1, the system under consideration is described by an ergodic Markov chain with states being the probability of occupancy of buffer b1 (since, given the probability of occupancy of b1 , the probability of occupancy of b0 can be immediately calculated). Let Pj be the stationary probability that b1 contains j parts, i.e., Pj = P {h1 = j},

j = 0, 1, . . . , N1 .

(20.47)

Then, the performance analysis of the system at hand amounts to evaluating Pj ’s, and then evaluating P R, STi and BLi , i = 1, 2. It turns out that it is convenient to calculate Pj separately for three cases of relationships among S, N0 , and N1 . This is carried out below. Case 1: S < min(N1 , N0 ). The balance equations for this case are: P0 P1 Pi PS Pj N1 X

Pi

=

(1 − p1 )P0 + (1 − p1 )p2 P1 ,

= =

p1 P0 + [p1 p2 + (1 − p1 )(1 − p2 )]P1 + (1 − p1 )p2 P2 , p1 (1 − p2 )Pi−1 + [p1 p2 + (1 − p1 )(1 − p2 )]Pi + i = 2, . . . , S − 1, (1 − p1 )p2 Pi+1 , = p1 (1 − p2 )PS−1 + (1 − p2 )PS , = 0, j = S + 1, . . . , N1 , =

1.

i=0

Their solution is: Pi

=

PS

=

Pj

=

αi P0 , i = 1, ..., S − 1, 1 − p2 αS (1 − p1 )P0 , 1 − p2 0, j = S + 1, . . . , N1 ,

(20.48)

where α(p1 , p2 )

P0

=

=

p1 (1 − p2 ) , (1 − p1 )p2

Qcl 1 (p1 , p2 , N1 , N0 , S)

(20.49)        =

     

1−p S+1−2p ,

if p1 = p2 = p,

(1−p1 )[1−α(p1 ,p2 )] , p (1−p ) 1− p1 (1−p1 ) α(p1 ,p2 )S 2

(20.50)

2

if p1 6= p2 .

Case 2a: N1 < S ≤ N0 . In this case, m1 is never starved and m2 is never blocked. In other words, the closed loop does not impede the open system

20.1. PROOFS FOR PART II

617

performance. Thus, the stationary probability mass function is the same as the corresponding open line. Therefore, it follows that, for N1 < S ≤ N0 ,  1−p  min(N1 ,N0 )+1−p ,     if p1 = p2 = p,  cl P0 = Q2 (p1 , p2 , N1 , N0 , S) = (20.51)  (1−p1 )[1−α(p1 ,p2 )]  p1  min(N ,N ) 1 0  1− p2 α(p1 ,p2 )   if p1 6= p2 . Case 2b: N0 < S ≤ N1 . Here, in the reversed flow scenario, the first machine, m2 , is never starved and the second machine, m1 , is never blocked. Thus, the line again is equivalent to an open line with the same machines but with the in-process buffer of capacity N0 . Therefore, in this case, P0

=

Qcl 2 (p2 , p1 , N0 , N1 , S).

(20.52)

Case 3: S > max(N1 , N0 ). The balance equations in this case are: Pj

= 0, j = 0, 1, . . . , S − N0 − 1, = [(1 − p1 ) + p1 p2 ] PS−N0 + p2 (1 − p1 )PS−N0 +1 , = Pi−1 p1 (1 − p2 ) + [p1 p2 + (1 − p1 )(1 − p2 )]Pi + (1 − p1 )p2 Pi+1 , i = S − N0 + 1, . . . , N1 − 1,

PS−N0 Pi PN1 N1 X

= PN1 −1 p1 (1 − p2 ) + (p1 p2 + 1 − p2 )PN1 ,

Pi

= 1.

i=0

Their solution is: Pi

=

α(p1 , p2 )i−(S−N0 ) PS−N0 ,

i = S − N0 , ..., N1 ,

(20.53)

.

(20.54)

where α(p1 , p2 ) is given in (20.49) and PS−N0

=

1 PN1 +N0 −S i=0

α(p1 , p2 )i

Given the above, the probability that b0 is full and m1 is down can be expressed as  1−p   N1 +N0 −S+1 ,    if p1 = p2 = p,  cl (20.55) Q3 (p1 , p2 , N1 , N0 , S) =  (1−p1 )[1−α(p1 ,p2 )]   1−α(p , N +N −S+1  1 0 1 ,p2 )   if p1 6= p2 . Using the three probability mass functions derived above and following the same arguments as in open line, we obtain the expressions for the performance measures (7.3)-(7.8) given in Theorem 7.1. ¥

618

as

CHAPTER 20. PROOFS Proof of Theorem 7.2: In this proof, we again consider three cases: Case 1: S < min(N1 , N0 ). For p1 6= p2 , function Qcl 1 (p1 , p2 , N1 , N0 , S) given in (20.50) can be rewritten Qcl 1 (p1 , p2 , N1 , N0 , S) = (1 − p1 ) i h . 1 +p2 ) αS−2 1 + α + · · · + αS−3 + 1 + p1 (1−p2p)(p 2 2

Clearly, it is strictly decreasing in p1 and S. Similarly, Qcl 1 (p2 , p1 , N0 , N1 , S) is strictly decreasing in p2 and S. Thus, P Rcl

= p2 [1 − Qcl 1 (p1 , p2 , N1 , N0 , S)] = p1 [1 − Qcl 1 (p2 , p1 , N0 , N1 , S)],

is strictly increasing in p1 , p2 , and S, and is independent of, i.e. constant in, N1 and N0 . For p1 = p2 = p, it is easy to show that P Rcl =

p(1 − p) p(S − p) =p− , S + 1 − 2p S + 1 − 2p

S ≥ 2,

which again implies that P Rcl is strictly increasing in p and S, and is independent of, i.e. constant in, N1 and N0 . Case 2: min(N1 , N0 ) < S ≤ max(N1 , N0 ). In this situation, the closed line is exactly equivalent to an open line. Thus, P Rcl is strictly increasing in p1 and p2 , monotonically increasing in N1 and N0 and independent of, i.e. constant in, S. Case 3: S > max(N1 , N0 ). For p1 6= p2 , Qcl 3 (p1 , p2 , N1 , N0 , S) =

(1 − p1 ) , 1 + α + · · · + αN1 +N0 −S

which implies that this function is strictly decreasing in p1 , strictly decreasing in N1 and N0 , and strictly increasing in S. Therefore, P Rcl

= p2 [1 − Qcl 3 (p1 , p2 , N1 , N0 , S)] = p1 [1 − Qcl 3 (p2 , p1 , N0 , N1 , S)],

is strictly increasing in p1 and p2 , strictly decreasing in S, and strictly increasing in N1 and N0 . For p1 = p2 = p, P Rcl =

p(N1 + N0 − S − p) , N1 + N0 − S + 1

S ≥ 2,

20.1. PROOFS FOR PART II

619

and the same conclusions hold. Thus, P Rcl is strictly increasing in p1 and p2 , non-strictly increasing in N1 and N0 , and non-monotonic concave in S. ¥ Proof of Theorem 7.3: It has been shown that in open lines P Ro (p1 , p2 , N1 ) = p2 [1 − Qo (p1 , p2 , N1 )],

(20.56)

where ( Qo (p1 , p2 , N1 ) =

1−p N1 +1−p , (1−p1 )(1−α) p 1− p1 αN1

if p1 = p2 = p, if p1 = 6 p2 .

(20.57)

2

It can be shown that < Qcl 1 (p1 , p2 , N1 , N0 , S),

Qo (p1 , p2 , N1 ) Qo (p1 , p2 , N1 ) Qo (p1 , p2 , N1 )

< =

Qo (p1 , p2 , N1 )

<

Qcl 2 (p1 , p2 , N1 , N0 , S), Qcl 2 (p1 , p2 , N1 , N0 , S), Qcl 3 (p1 , p2 , N1 , N0 , S),

S ≤ min(N1 , N0 ), N0 < S ≤ N1 , N1 < S ≤ N0 , S > max(N1 , N0 ).

Therefore, the pair (N0 , S) is unimpeding, i.e. P Rcl (p1 , p2 , N1 , N0 , S) = P Ro (p1 , p2 , N1 ), if and only if N1 < S ≤ N0 . ¥ Proof of Theorem 7.4: When the machines are asymptotically reliable in the sense that pi = 1 − ²ki ,

i = 1, 2,

where 0 < ² ¿ 1 and ki > 0 is independent of ², it is easy to show that Qcl 1 (p1 , p2 , N1 , N0 , S) = Qcl 2 (p1 , p2 , N1 , N0 , S) = Qcl 3 (p1 , p2 , N1 , N0 , S) =

Qeo (p1 , p2 , S − 1) + O(²2 ), eo

(20.58) 2

Q (p1 , p2 , min(N1 , N0 )) + O(² ), Qeo (p1 , p2 , N1 + N0 − S + 1) + O(²2 ),

where ( eo

Q (p1 , p2 , Neo ) =

1−p Neo +1−p , (1−p1 )(1−α) p 1− p1 αNeo

if p1 = p2 = p, if p1 = 6 p2 .

(20.59)

2

Substituting (20.58) and (20.59) into (7.3)-(7.8), respectively, proves Theorem 7.4. ¥

620

CHAPTER 20. PROOFS

Proof of Theorem 7.5 When (7.24) occurs, m1 is never starved and mM is never blocked. Hence, the closed nature of the line does not impact the open line performance. ¥ Proof of Theorem 7.8: Two cases are considered: Case 1: N1 6= N0 . Under this condition, it follows from Theorem 7.1 that a closed line with two machines is S + -improvable if S < min(N1 , N0 ) + 1, i.e. ∗ = min(N1 , N0 ) + 1, and Smin ST1cl

=

BLcl 1

=

ST2cl

=

BLcl 2

=

p1 Qcl (p2 , p1 , N0 , N1 , S),   p1 (1 − p2 )Qcl (p2 , p1 , N0 , N1 , S), if S = N1 < N0 ,  0, if S < N1 , p2 Qcl (p1 , p2 , N1 , N0 , S),   (1 − p1 )p2 Qcl (p1 , p2 , N1 , N0 , S), if S = N0 < N1 ,  0, if S < N0 .

(20.60) (20.61) (20.62) (20.63)

cl cl + Clearly, ST1cl > BLcl 1 and ST2 > BL2 . Therefore, for S -improvable situation, cl ST1cl + ST2cl > BLcl 1 + BL2 .

(20.64)

Similarly, a closed line with two machines is S − -improvable if S > max(N1 , N0 ), ∗ = max(N1 , N0 ), and i.e. Smax ST1cl BLcl 1

= 0, = p1 (1 − p2 )Qcl (p2 , p1 , N0 , N1 , S),

(20.65) (20.66)

ST2cl BLcl 2

= 0, = (1 − p1 )p2 Qcl (p1 , p2 , N1 , N0 , S).

(20.67) (20.68)

Therefore, for S − -improvable situation, cl ST1cl + ST2cl < BLcl 1 + BL2 .

Case 2: N1 = N0 = N . Under this condition, Qcl 1 (p1 , p2 , N1 = N, N0 = N, S = N ) < Qcl 3 (p1 , p2 , N1 = N, N0 = N, S = N + 1), and therefore, P Rcl (p1 , p2 , N1 = N, N0 = N, S = N ) > P Rcl (p1 , p2 , N1 = N, N0 = N, S = N + 1).

(20.69)

20.1. PROOFS FOR PART II

621

In other words, the line is S + -improvable if S < min(N1 , N0 ) = N and S − ∗ ∗ = Smax = N . Then, using again improvable if S > min(N1 , N0 ) = N , i.e. Smin (20.62) and (20.67), we obtain that ST1cl + ST2cl ST1cl + ST2cl

cl > BLcl 1 + BL2 , cl < BL1 + BLcl 2,

if S < N, if S > N,

(20.70) (20.71)

which completes the proof. ¥

20.1.5

Proofs for Chapter 8

d c and CR d Proof for Theorem 8.2: The proof of the monotonicity of P R, SR, with respect to pi and Ni is similar to the proof of Theorem 4.5 of Chapter 4. As for the monotonicity with respect to gi , it follows from (8.7) and the properties of function Q that pfi (s) and pbi (s), s = 1, 2, . . . , are monotonically increasing and d R = pfM = lims→∞ pfM (s) and decreasing in gj , j ∈ Inp , respectively. Since P d and P d d = pb = lims→∞ pb (s), the above implies that CR R are monotonically CR 1 1 c = CR− d P d decreasing and increasing in gi , i ∈ Inp , respectively. Also, since SR R, the scrap rate is monotonically decreasing in gi . Finally, if the only inspection d is equal to the production rate of the corresponding line machine is mM , CR d is independent of gi . without scrap and, therefore, CR ¥ Proof for Theorem 8.3: As it follows from (8.5), the consumption rate d d CR(1) of L1, is equal to P R(2). Therefore, d d ∂ CR(1) ∂ CR(1) > , ∀ j 6= i ∂pi ∂pj

⇐⇒

d d ∂P R(2) ∂P R(2) > , ∀ j 6= i. (20.72) ∂pi ∂pj

Due to (8.3), Y d d ∂P R(1) ∂ CR(1) = gi , ∂pl ∂pl

l = 1, . . . , M.

(20.73)

i∈Inp

Hence, d d d d d d ∂ CR(1) ∂ CR(1) ∂P R(1) ∂P R(1) ∂P R(2) ∂P R(2) > ⇐⇒ > ⇐⇒ > , ∀j 6= i. ∂pi ∂pj ∂pi ∂pj ∂pi ∂pj ¥ Proof of Theorem 8.5: Based on recursive procedure (8.7) and using the chain rule, we write d d d ∂P R ∂q l ∂P R ql ∂P R = · = , ∂q l ∂gil ∂q l gil ∂gil

622

CHAPTER 20. PROOFS

where q l is the quality buy rate of the l-th inspection machine and gil is the quality parameter of the machines in the l-th Q-segment. Therefore, the LQBN of each segment is the machine with the smallest gi . Further, we prove by contradiction that one of the LQ-BNs is the Q-BN. Indeed, assume that none of the LQ-BNs is the Q-BN. Then, a non-perfect quality machine, other than a LQ-BN, is the Q-BN. This implies that in one d of the Q-segments, this machine has a larger effect on the P R than its LQ-BN, which is a contradiction. ¥ Proof of Theorem 8.6: As it follows from (4.19), P R in two-machine lines defined by assumptions (a)-(i) can be expressed in the closed form as follows: PR

= p2 [1 − Q(p1 , p2 , N1 )]g1 (p1 )g2 (p2 ) = p1 [1 − Q(p2 , p1 , N1 )]g1 (p1 )g2 (p2 ),

(20.74)

where Q(·) is defined by (4.14). Therefore, ¶ µ ∂Q(p1 , p2 , N1 ) ∂P R dg1 (p1 ) = p2 (1 − Q(p1 , p2 , N1 )) − g1 (p1 ) g2 (p2 ), ∂p1 dp1 ∂p1 ¶ µ ∂P R ∂Q(p2 , p1 , N1 ) dg2 (p2 ) = p1 (1 − Q(p2 , p1 , N1 )) − g2 (p2 ) g1 (p1 ), ∂p2 dp2 ∂p2 where

 y(1−x)(αN (x,y)−1)+(y−x)N αN (x,y) ∂Q(x, y, N )  , (y−xαN (x,y))2 (1−x) = N (N +1)  − ∂x 2p(N +1−p)2 ,

if x 6= y, if x = y = p.

R ∂P R Clearly, ∂P ∂p1 and ∂p2 are continuous with respect to p1 and p2 , respectively. Assume that {1} ⊆ IQQC , then

lim

p1 →0

∂P R ∂p1

= g1 (0)g2 (p2 ) > 0.

(20.75)

In addition, for N1 ≥ 2, it is possible to show that ∂P R lim p1 →1 ∂p1

=

¯ dg(p1 ) ¯¯ p2 g2 (p2 ) < 0. dp1 ¯p1 =1

Therefore, there exist p∗1 , p∗∗ 1 ∈ (0, 1), such that ∂P R ∂p1 ∂P R ∂p1

>

0,

∀0 < p1 < p∗1 ,

<

0,

∀p∗∗ 1 < p1 < 1.

Similar arguments in the case of {2} ⊆ IQQC lead to ∂P R ∂p2

>

0,

∀0 < p2 < p∗2 ,

(20.76)

20.1. PROOFS FOR PART II ∂P R ∂p2

623 <

0,

∀p∗∗ 2 < p2 < 1. ¥

Proof of Theorem 8.7: The production rates of Lines 2, 3, and 4 can be expressed as: d P R2 d P R3

b k )(1 − bl b k ), b k ) = pk (1 − st b k )(1 − bl = pk2 (1 − st 1 2 b k ) = pk q(1 − st b k )(1 − bl b k ), b = pk3 (1 − bl )(1 − bl k 1 1 2

d P R4

b k ) = pk (1 − q)(1 − st b k )(1 − bl b k ). b k )(1 − bl = pk4 (1 − bl 2 1 2

This leads to d d R2 , P R3 = q P

d d P R4 = (1 − q)P R2 .

On the other hand, d P R1

=

b j )(1 − st b j1 ) = pj (1 − bl b j1 )st b j2 , p1j (1 − st

d P R4

=

b j )(1 − st b j2 ) = pj (1 − bl b j2 ). p4j (1 − st

Therefore, b j )(1 − st d d d b j1 st b j2 ) = P R4 = pj (1 − bl R2 , P R1 + P and, d d R3 . P R1 = P ¥

20.1.6

Proofs for Chapter 9

The proofs of Theorems 9.1 and 9.2 are similar to more general proofs of Theorems 15.1 and 15.2. Therefore, we postpone the proofs of Theorems 9.1 and 9.2 until Theorems 15.1 and 15.2 are proven (see Section 20.2).

20.1.7

Proofs for Chapter 10

Proof of Theorem 10.1: The similarity transformation for (10.1) with N = 1 is   1 1 . (20.77) Q =  √ 1−p − √ 12 2 2+p −2 p

This leads to C˜

=

CQ

−1

1 = 2−p

"

2+p −2 p

p 1

p # p 2 + p2 − 2 p , p 2 + p2 − 2 p

(20.78)

624

CHAPTER 20. PROOFS P R(n)

=

W IP (n)

=

p p 2 + p2 − 2 p p + x ˜1 (0)λn1 , 2−p 2−p p 2 + p2 − 2 p 1 + x ˜1 (0)λn1 2−p 2−p

(20.79) (20.80)

and, therefore, to (10.30). ¥ Proof of Theorem 10.2: Expression (10.45), (10.46) follow directly from d \ (10.35), (10.41) by evaluating the time necessary for P R(n) and W IP (n) to reach 95% of their steady state values. To prove (10.47) and (10.48), we observe that tˆsW IP − tˆsP R

= 1−

ln(20β) ln(20γ) ln(γ/β) −1+ = , ln λ1 ln λ1 − ln λ1

(20.81)

and · N2+1 1 − p(N +1−p) γ N ≥ 1, = β 1 − p(N +1−p) ·1 N

0 < N < ∞, 0 < p < 1.

(20.82) ¥

20.2

Proofs for Part III

20.2.1

Proofs for Chapter 11

Proof of Lemma 11.1:

For e1 6= e2 ,

Q(λ1 , µ1 , λ2 , µ2 ) =

1+

1 − e1 , − e−βN )

φ 1−φ (1

(20.83)

where φ = β

=

e1 (1 − e2 ) , e2 (1 − e1 ) (λ1 + µ1 + λ2 + µ2 )(λ1 µ2 − λ2 µ1 ) . (λ1 + λ2 )(µ1 + µ2 )

Then, n −µ1 λ2 µ2 µ1 λ2 µ2 −βN + e 2 ∂λ1 (λ1 µ2 − λ2 µ1 ) (λ1 µ2 − λ2 µ1 )2 µ1 λ2 [(λ1 + λ2 )(λ1 + µ1 + λ2 + µ2 )µ2 − (µ1 + µ2 )(λ1 µ2 − λ2 µ1 )] o . +N (λ1 + λ2 )2 (µ1 + µ2 )(λ1 µ2 − λ2 µ1 ) (20.84)

∂[1 +

φ 1−φ (1

− e−βN )]

=

20.2. PROOFS FOR PART III

625

If e1 < e2 , then it is easy to show that β < 0. Let ³ µ1 λ2 µ2 f (N ) = e−βN (λ1 µ2 − λ2 µ1 )2 µ1 λ2 [(λ1 + λ2 )(λ1 + µ1 + λ2 + µ2 )µ2 − (µ1 + µ2 )(λ1 µ2 − λ2 µ1 )] ´ . +N (λ1 + λ2 )2 (µ1 + µ2 )(λ1 µ2 − λ2 µ1 ) (20.85) Since f (0) =

µ1 λ1 µ2 (λ1 µ2 −λ2 µ1 )2 ,

f (∞) = 0, and

³ −µ λ µ1 λ2 [(λ1 + λ2 )2 µ2 + (µ1 + µ2 )2 λ2 ] ´ 1 2 − βN 2 (λ1 + λ2 ) (λ1 + λ2 )2 < 0, (20.86) i h λ1 λ2 function f (N ) takes values in 0, (λ1 µµ21−λ 2 . Substituting the value of f (N ) 2 µ1 ) into (20.84), we obtain h i φ ∂ 1 + 1−φ (1 − e−βN ) ≤ 0, (20.87) ∂λ1 ∂f (N ) ∂N

e−βN

=

i.e., the denominator of (20.83) is monotonically decreasing in λ1 . Therefore, using (20.87) and the fact that the numerator of (20.83) is monotonically increasing in λ1 , ∂Q(λ1 , µ1 , λ2 , µ2 , N ) > 0. ∂λ1

(20.88)

A similar conclusion can be derived for e1 > e2 as well. If e1 = e2 , then λ1 +∆λ ∆λ λ1 +∆λ+µ1 λ1 +∆λ

Q(λ1 + ∆λ, µ1 , λ2 , µ2 , N ) = 1− λ1 +∆λ

= =

− µ1 λ2 µ2 (λ1 +∆λ) e

(λ1 +∆λ+µ1 +λ2 +µ2 )µ2 ∆λ N (λ1 +∆λ+λ2 )(µ1 +µ2 )

∆λ

hλ1 +∆λ+µ1 λ1 +∆λ i + O(∆λ2 ) +∆λ+µ1 +λ2 +µ2 )µ2 ∆λ µ1 λ2 1 − µ2 (λ1 +∆λ) 1 − (λ1(λ N 1 +∆λ+λ2 )(µ1 +µ2 ) (λ1 + λ2 )(µ1 + µ2 )(λ1 + ∆λ) (λ1 + ∆λ + µ1 )[(λ1 + λ2 )(µ1 + µ2 ) + µ1 λ2 (λ1 + µ1 + λ2 + µ2 )N ] +O(∆λ2 ).

Therefore, Q(λ1 + ∆λ, µ1 , λ2 , µ2 , N ) − Q(λ1 , µ1 , λ2 , µ2 , N ) ∆λ  (λ 1 +λ2 )(µ1 +µ2 )(λ1 +∆λ)  (λ1 +∆λ+µ1 )[(λ1 +λ2 )(µ1 +µ2 )+µ1 λ2 (λ1 +µ1 +λ2 +µ2 )N ] = lim ∆λ→0  ∆λ

lim

∆λ1 →0

626

CHAPTER 20. PROOFS

−

(λ1 +λ2 )(µ1 +µ2 )λ1 (λ1 +µ1 )[(λ1 +λ2 )(µ1 +µ2 )+µ1 λ2 (λ1 +µ1 +λ2 +µ2 )N ]

∆λ

  

(λ1 + λ2 )(µ1 + µ2 ) (λ1 + µ1 )[(λ1 + λ2 )(µ1 + µ2 ) + µ1 λ2 (λ1 + µ1 + λ2 + µ2 )] > 0, =

i.e., Q(λ1 , µ1 , λ2 , µ2 ) is monotonically increasing in λ1 for the case e1 = e2 . Thus, Q(λ1 , µ1 , λ2 , µ2 ) is monotonically increasing in λ1 . Similarly, the monotonicity properties of Q(λ1 , µ1 , λ2 , µ2 ) with respect to λ2 , µ1 , µ2 can be proved. For the monotonicity of function Q with respect to N , if e1 < e2 , then 0 < φ < 1,

β > 0.

Thus, the denominator of (20.83) is positive and monotonically increasing in N . Therefore, function Q is monotonically decreasing in N . Also, under the above condition, 0 < 1 − φ < 1 − φe−βN < 1, which implies that 0 < Q(λ1 , µ1 , λ2 , µ2 , N ) < 1. Similar results can be obtained for e1 ≥ e2 as well. ¥

Derivations of marginal pdf of buffer occupancy for synchronous exponential two-machine lines: A synchronous exponential two-machine line is characterized by a continuous time, mixed space Markov process. The transition rates of this process from state (s1 = i, s2 = j) to state (s1 = k, s2 = l), i.e., νkl,ij , are as follows: ν11,00 = 0, ν10,00 = µ1 , ν01,00 = µ2 , ν11,01 = µ1 , ν10,01 = 0, ν00,01 = λ2 , ν11,10 = µ2 , ν01,10 = 0, ν00,10 = λ1 , ν10,11 = λ2 , ν01,11 = λ1 , ν00,11 = 0. While the discrete part of the Markov process at hand is described by the above transition rates, the continuous part, i.e., buffer occupancy, is characterized by   1, if s1 (t) = 1, s2 (t) = 0, dh(t) ¯¯ 0, if s1 (t) = s2 (t), = ¯  dt 0
20.2. PROOFS FOR PART III

627

Using these data in the evolution equations (2.65) derived in Chapter 2, we obtain the following equations for the steady state pdf for the the internal states h ∈ (0, N ): µ2 µ1 + fH,I,J (h, 0, 1) , fH,I,J (h, 1, 1) = fH,I,J (h, 1, 0) λ 1 + λ2 λ1 + λ2 ∂fH,I,J (h, 1, 0) = fH,I,J (h, 1, 1)λ2 − fH,I,J (h, 1, 0)(λ1 + µ2 ) + fH,I,J (h, 0, 0)µ1 , ∂h ∂fH,I,J (h, 0, 1) = fH,I,J (h, 0, 1)(λ2 + µ1 ) − fH,I,J (h, 1, 1)λ1 − fH,I,J (h, 0, 0)µ2 , ∂h λ1 λ2 + fH,I,J (h, 0, 1) . (20.89) fH,I,J (h, 0, 0) = fH,I,J (h, 1, 0) µ1 + µ2 µ1 + µ2 Since h(t) is continuous in t, events A1 = {h(t) = h and h(t) is increasing in t} and A2 = {h(t) = h and h(t) is decreasing in t} occur alternately with the equal frequencies. Therefore, fH,I,J (h, 1, 0) = fH,I,J (h, 0, 1).

(20.90)

The stationary probabilities of the boundary states, h = 0 and h = N , are derived as follows: The evolution equations for these probabilities are: P0,11 (t + δt) = P0,10 (t + δt) = P0,01 (t + δt) = P0,00 (t + δt) = PN,11 (t + δt) = PN,10 (t + δt) = PN,01 (t + δt) = PN,00 (t + δt) =

P0,11 (t)[1 − (λ1 + λ2 )δt] + P0,01 (t)µ1 δt + o(δt), 0, P0,01 (t)[1 − (µ1 + λ2 )δt] + P0,11 (t)λ1 δt + P0,00 (t)µ2 δt +fH,I,J (0, 0, 1, t)δt + o(δt), P0,00 (t)[1 − (µ1 + µ2 )δt] + P0,01 (t)λ2 δt + o(δt), PN,11 (t)[1 − (λ1 + λ2 )δt] + PN,10 (t)µ2 δt + o(δt), PN,10 (t)[1 − (λ1 + µ2 )δt] + PN,11 (t)λ2 δt + PN,00 (t)µ1 δt +fH,I,J (N, 1, 0, t)δt + o(δt), 0, PN,00 (t)[1 − (µ1 + µ2 )δt] + PN,10 (t)λ1 δt + o(δt).

In the limit δt → 0, these equations lead to the following steady state relationships: P0,11 P0,10 P0,01 P0,00 PN,11 PN,10 PN,01 PN,00

= = = = = = = =

µ1 λ1 +λ2 P0,01 ,

0,

1 µ1 +λ2 [fH,I,J (0, 0, 1) + λ1 P0,11 + µ2 P0,00 ] λ2 µ1 +µ2 P0,01 , µ2 λ1 +λ2 PN,10 , 1 λ1 +µ2 [fH,I,J (N, 1, 0) + λ2 PN,11 + µ1 PN,00 ]

0,

λ1 µ1 +µ2 PN,10 .

(20.91)

628

CHAPTER 20. PROOFS Solving (20.89) and (20.90), we obtain fH,I,J (h, 1, 0)

=

fH,I,J (h, 0, 1) = C0 eKh ,

fH,I,J (h, 1, 1)

=

fH,I,J (h, 0, 0)

=

C0 D1 eKh , C0 Kh e , D1

(20.92)

where D1 is defined in (11.11) and C0 is a free constant. Substituting (20.92) into (20.91) results in P0,11

=

P0,10

=

P0,01

=

P0,00

=

PN,11

=

PN,10

=

PN,01

=

PN,00

=

(µ1 + µ2 )C0 , λ2 (λ1 + µ1 + λ2 + µ2 ) 0, (λ1 + λ2 )(µ1 + µ2 )C0 , λ2 µ1 (λ1 + µ1 + λ2 + µ2 ) (λ1 + λ2 )C0 , µ1 (λ1 + µ1 + λ2 + µ2 ) µ1 + µ2 C0 eKN , λ1 (λ1 + µ1 + λ2 + µ2 ) (λ1 + λ2 )(µ1 + µ2 ) C0 eKN , λ1 µ2 (λ1 + µ1 + λ2 + µ2 ) 0, λ 1 + λ2 C0 eKN , µ2 (λ1 + µ1 + λ2 + µ2 )

(20.93)

To evaluate the free constant, substitute (20.92) and (20.93) into the total probability formula, 1 X 1 Z X i=0 j=0

N−

0+

fH,I,J (h, i, j)dh +

1 X 1 X

P0,ij +

i=0 j=0

1 X 1 X

PN,ij = 1,

(20.94)

i=0 j=0

leading to C0 =

1 . D2 + D3 + D4

(20.95)

Thus, the steady state distributions of buffer occupancy are as follows: fH,I,J (h, 1, 1)

=

fH,I,J (h, 1, 0)

=

fH,I,J (h, 0, 1)

=

fH,I,J (h, 0, 0)

=

D1 eKh , D2 + D3 + D4 1 eKh , D2 + D3 + D4 1 eKh , D2 + D3 + D4 1 eKh , D1 (D2 + D3 + D4 )

20.2. PROOFS FOR PART III P0,11

=

P0,10

=

P0,01

=

P0,00

=

PN,11

=

PN,10

=

PN,01

=

PN,00

=

629

µ1 + µ2 , λ2 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 ) 0, (λ1 + λ2 )(µ1 + µ2 ) , (20.96) λ2 µ1 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 ) (λ1 + λ2 ) , µ1 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 ) µ1 + µ2 eKN , λ1 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 ) (λ1 + λ2 )(µ1 + µ2 ) eKN , λ1 µ2 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 ) 0, λ 1 + λ2 eKN . µ2 (λ1 + µ1 + λ2 + µ2 )(D2 + D3 + D4 )

The marginal steady state pdf of buffer occupancy is given by fH (h) =

1 X 1 X

fH,I,J (h, i, j) +

i=0 j=0

+

1 X 1 X

1 X 1 X

P0,ij δ(h)

i=0 j=0

PN,ij δ(h − N ).

(20.97)

i=0 j=0

Substituting (20.96) into (20.97), we obtain (11.10). Formulas for the performance measures in synchronous exponential two-machine lines: Using the above distributions, P R, W IP , BL1 and ST2 can be expressed as follows: PR

W IP

= P [s2 = 1](1 − P [h = 0, s1 = 0|s2 = 1]) = P [s1 = 1](1 − P [h = N, s2 = 0|s1 = 1]) µ ¶ µ ¶ P0,01 PN,10 = e2 1 − = e1 1 − , e2 e1 Z N 1 X 1 X 1 Z N− 1 X X fH (h)hdh = fH,I,J (h, i, j)hdh + PN,ij N , = 0

i=0 j=0

0+

BL1

= P [h = N, s1 = 1, s2 = 0] = PN,10 ,

ST2

= P [h = 0, s1 = 0, s2 = 1] = P0,01 .

Using (20.96), it can be shown that P0,01 e2 PN,10 e1

= Q(λ1 , µ1 , λ2 , µ2 , N ), = Q(λ2 , µ2 , λ1 , µ1 , N ).

i=0 j=0

630

CHAPTER 20. PROOFS

where function Q is defined in (11.4). Therefore, PR

W IP

=

e2 [1 − Q(λ1 , µ1 , λ2 , µ2 , N )]

=

e1 [1 − Q(λ2 , µ2 , λ1 , µ1 , N )],  D5   D2 +D3 +D4 , for e1 6= e2 ,

=

 

D

( 22 +D4 )N D2 +D3 +D4 ,

for e1 = e2 ,

BL1

=

e1 Q(λ2 , µ2 , λ1 , µ1 , N ),

ST2

=

e2 Q(λ1 , µ1 , λ2 , µ2 , N ). ¥

Proof of Theorem 11.1: Let λli1 , µli1 , λli2 , µli2 , i = 1, 2, be the failure and repair rates of machine mi of lines l1 and l2 , respectively. Assume µl11 = a1 µl12 ,

µl21 = a2 µl22 .

Then, from (11.19), a1 > 1, In addition, from

eli1

=

eli2 ,

i = 1, 2, it follows that

λl11 = a1 λl12 , When Q l1

= =

Ql2

=

l

λ1i l

µ1i

=

l

λ2i l

µ2i

a2 > 1.

λl21 = a2 λl22 .

, i = 1, 2, we obtain:

λl11 (λl11 + λl21 )(µl11 + µl21 ) (λl11 + µl11 )[(λl11 + λl21 )(µl11 + µl21 ) + λl21 µl11 (λl11 + λl21 + µl11 + µl21 )N ] [(1 − el1 )(a1 λl12 + a2 λl22 )(a1 µl12 + a2 µl22 )]/[(a1 λl12 + a2 λl22 )(a1 µl12 +a2 µl22 ) + a1 a2 λl22 µl12 (a1 λl12 + a2 λl22 + a1 µl12 + a2 µl22 )N ], (1 − el2 )(λl12 + λl22 )(µl12 + µl22 ) . (λl12 + λl22 )(µl12 + µl22 ) + λl22 µl12 (λl12 + λl22 + µl12 + µl22 )N

Thus, 1 Ql1

=

[(a1 λl12 + a2 λl22 )(a1 µl12 + a2 µl22 ) + a1 a2 λl22 µl12 (a1 λl12 + a2 λl22

+a1 µl12 + a2 µl22 )N ]/[(1 − el1 )(a1 λl12 + a2 λl22 )(a1 µl12 + a2 µl22 )N ] ³ ´ i a1 a2 1 h a1 a2 l2 l2 1 + λ N µ + = 2 1 l l l l l 1−e1 a1 µ12 + a2 µ22 a1 λ12 + a2 λ22 ³ ´ i 1 1 h 1 l2 l2 1 + λ N µ + = 2 1 l l l l l 1 1 1 1 2 2 2 2 1−e1 a2 µ1 + a1 µ2 a2 λ1 + a1 λ2 ³ ´ i 1 h 1 1 l2 l2 1 + λ N > µ + 2 1 1 − el1 µl12 + µl22 λl12 + λl22

20.2. PROOFS FOR PART III

631

1 . Q l2

=

Therefore, Ql1 < Ql2 and l

l

1 2 d d R . P R = el21 (1 − Ql1 ) > el22 (1 − Ql2 ) = P

When β l1

l

λ1i

l µ1i

6=

l

λ2i l

µ2i

, since eli1 = eli2 , i = 1, 2, it follows that φl1 = φl2 . Then,

=

(λl11 + λl21 + µl11 + µl21 )(λl11 µl21 − λl21 µl11 ) (λl11 + λl21 )(µl11 + µl21 )

=

(a1 λl12 + a2 λl22 + a1 µl12 + a2 µl22 )(a1 λl12 a2 µl22 − a2 λl22 a1 µl12 ) (a1 λl12 + a2 λl22 )(a1 µl12 + a2 µl22 )

=

a1 a2 (λl12 µl22 − λl22 µl12 ) a1 a2 (λl12 µl22 − λl22 µl12 ) + (a1 µl12 + a2 µl22 ) (a1 λl12 + a2 λl22 )

=

λl12 µl22 − λl22 µl12 λl12 µl22 − λl22 µl12 + . 1 l2 1 l2 1 l2 1 l2 a2 µ1 + a1 µ2 a 2 λ 1 + a 1 λ2

If λl12 µl22 > λl22 µl12 , β l1

>

λl12 µl22 − λl22 µl12 λl12 µl22 − λl22 µl12 + µl12 + µl22 λl12 + λl22

=

β l2 > 0.

If λl12 µl22 < λl22 µl12 , then

0 > β l1 > β l2 .

Hence, in both cases, e−β

l1

l2

< e−β ,

Thus, Ql1 < Ql2 , i.e., l

l

1 2 d d R . P R >P

¥ Proof of Theorem 11.2: Consider a two-machine synchronous exponential line defined by parameters λi , µi , i = 1, 2, and N . Let l1 denote the line with Tdown,1 decreased by a factor 1 + α, α > 0, and l2 denote the line with Tup,1 increased by the same factor. Then, Q l1

=

Q l2

=

Q(λ1 , µ1 (1 + α), λ2 , µ2 , N ), λ1 , µ1 , λ2 , µ2 , N ). Q( 1+α

632

CHAPTER 20. PROOFS When β l1

λ1 µ1 (1+α)

= =

β l2

= = =

6=

λ2 µ2 ,

(λ1 + λ2 + µ1 (1 + α) + µ2 )(λ1 µ2 − λ2 µ1 (1 + α)) (λ1 + λ2 )(µ1 (1 + α) + µ2 ) λ1 µ2 − λ2 µ1 (1 + α) λ1 µ2 − λ2 µ1 (1 + α) + , µ1 (1 + α) + µ2 λ 1 + λ2 λ1 λ1 + λ2 + µ1 + µ2 )( 1+α µ2 − λ2 µ1 ) ( 1+α λ1 ( 1+α + λ2 )(µ1 + µ2 ) 1 (λ1 + (1 + α)λ2 + (1 + α)(µ1 + µ2 ))(λ1 µ2 − λ2 µ1 (1 + α)) 1+α

(λ1 + (1 + α)λ2 )(µ1 + µ2 ) λ1 µ2 − λ2 µ1 (1 + α) λ1 µ2 − λ2 µ1 (1 + α) + . (1 + α)(µ1 + µ2 ) (1 + α)(λ1 + (1 + α)λ2 )

If λ1 µ2 < λ2 µ1 (1 + α), it follows that β l1 > β l2 > 0. If λ1 µ2 > λ2 µ1 (1 + α), it implies that 0 > β l1 > β l2 . In both cases, e−β Therefore, Ql1 < Ql2 and

l1 N

l

< e−β

l2

N

.

l

1 2 d d P R >P R .

When Q l1

=

Ql2

=

λ1 µ1 (1+α)

=

λ2 µ2 ,

³ ´.³ λ1 (λ1 + λ2 )(µ1 (1 + α) + µ2 ) (λ1 + µ1 (1 + α))[(λ1 + λ2 )(µ1 (1 + α) ´ +µ2 ) + λ2 µ1 (λ1 + λ2 + µ1 (1 + α) + µ2 )] λ1 λ1 1+α ( 1+α λ1 ( 1+α

³ =

λ1 + λ2 )(µ1 + µ2 ) + λ2 µ1 ( 1+α + λ2 + µ1 + µ2 )] ´.³ λ1 (λ1 + (1 + α)λ2 )(µ1 + µ2 ) (λ1 + (1 + α)µ1 )[(λ1

+

1 1+α

+ λ2 )(µ1 + µ2 )

λ1 µ1 )[( 1+α

´ +(1 + α)λ2 )(µ1 + µ2 ) + λ2 µ1 (λ1 + (1 + α)λ2 + (1 + α)(µ1 + µ2 ))] . Then, 1 Q l1 1 Q l2

1 λ2 µ1 1 λ2 µ1 1 + · (1 + α) , µ2 + λ1 λ1 µ1 + 1+α λ1 λ 1 + λ2 i h1 λ2 µ1 1 λ2 µ1 1+α + = (1 + α) · + · λ λ1 µ1 + µ2 λ1 λ1 + (1 + α)λ2 =

and

which implies that

1 1 > l2 , l 1 Q Q Ql1 < Ql2 .

20.2. PROOFS FOR PART III

633

Therefore, again

l

l

1 2 d d P R >P R .

¥ Proof of Theorem 11.3: If µλ11 = µλ22 , from (11.4) and (11.13), it is easy to show that Q → 0 when N → ∞. If µλ11 6= µλ22 and e2 < e1 , then β < 0, which leads to Q = 0 and P R = e2 when N → ∞. If µλ11 6= µλ22 and e2 > e1 , it follows that β > 0, and Q = (1 − e1 )(1 − φ) when N → ∞. Therefore, lim P R

=

e2 (1 − (1 − e1 )(1 − φ)) ³ e1 (1 − e2 ) e1 (1 − e2 ) ´ = e2 e 1 + − e1 e2 (1 − e1 ) e2 (1 − e1 ) ³ e1 (1 − e2 ) ´ = e2 e1 + e2 = e1 .

N →∞

Thus, limN →∞ P R = min(e1 , e2 ). If e1 = e2 , from (11.11) and (11.14), N = ∞. N →∞ 2

lim W IP = lim

N →∞

If e1 > e2 , we have K > 0 and from (11.15), lim D5 = lim D2 N + D4 N,

N →∞

N →∞

which leads to lim W IP = lim N = ∞.

N →∞

N →∞

If e1 < e2 , then K < 0, and limN →∞ eKN = 0, which implies that lim D3

N →∞

lim D4 N

N →∞

1 eKN − 1 ) N →∞ D1 K 1 (2 + D1 + D1 ) , = − K (λ1 + λ2 + µ1 + µ2 )(λ1 + µ2 ) + λ2 µ1 − λ1 µ2 N eKN = lim N →∞ λ1 µ2 (λ1 + λ2 + µ1 + µ2 ) = 0 =

lim (2 + D1 +

and, after some algebra, D

lim W IP

N →∞

2 D5 −K = lim = lim N →∞ D2 + D3 N →∞ D2 + D3 (λ1 + λ2 )(µ1 + µ2 ) = . (λ2 µ1 − λ1 µ2 )2

634

CHAPTER 20. PROOFS

The asymptotic properties of BL1 (for e1 ≤ e2 ) and ST2 are proved analogously. For e1 > e2 , in the limit N → ∞, the system has no steady state, which is interpreted as BL1 = 0. ¥ Similar to the Bernoulli case, the proof of Theorem 11.4 requires, along with Lemma 11.1, the following facts: Lemma 20.19 Consider the four sequences λfi (s), µfi (s), λbi (s) and µbi (s), i = 1, · · · , M , defined by recursive procedure (11.25). Then µfj (s) < µfj (s − 1) and λfj (s) > λfj (s−1), j = 2, · · · , M , imply and are implied by µbj (s) > µbj (s−1) and λbj (s) < λbj (s − 1), j = 1, · · · , M − 1. Lemma 20.20 The sequences λfj (s) and λbj (s), s = 1, 2, . . . , are monotonically increasing and the sequences µfj (s) and µbj (s), s = 1, 2, . . . , are monotonically decreasing. To formulate the next lemma, consider the steady state equations of the recursive procedure (11.25), i.e., µbi λbi µfi λfi

=

µi − µi Q(λbi+1 , µbi+1 , λfi , µfi , Ni ),

=

λi + µi Q(λbi+1 , µbi+1 , λfi , µfi , Ni ), i µi − µi Q(λfi−1 , µfi−1 , λbi , µfi , Ni−1 ), λi + µi Q(λfi−1 , µfi−1 , λbi , µfi , Ni−1 ),

= =

i = 1, · · · , M − 1, = 1, · · · , M − 1, i = 2, · · · , M,

(20.98)

i = 2, · · · , M.

Introduce (M − 1) two-machine lines, Li , i = 1, . . . , M − 1, where the first machine is defined by λfi , µfi , the second machine is defined by λbi+1 , µbi+1 , and the buffer capacity is Ni . d Lemma 20.21 Let P Ri be the production rate of line Li , i = 1, . . . , M − 1, f d d d Ri = efi ebi /ei , i = 1, . . . , M . Moreover, P Ri = and let P RM = eM . Then, P const, ∀i = 1, . . . , M . Lemma 20.22 The production rate (11.13) is a monotonically decreasing function of λ1 , λ2 , and monotonically increasing function of µ1 , µ2 , and N1 . Lemma 20.23 The equilibrium equation (20.98) of recursive procedure (11.25) has a unique solution. The proofs of these lemmas are similar to those of Lemmas 20.1-20.5 and are omitted here; they can be found in [20.15]. Proof of Theorem 11.4: Since the sequences λfi (s), µfi (s), λbi (s) and µbi (s), i ≤ j ≤ M , are monotonic (Lemma 20.20) and bounded from above and below, they are convergent. This proves statement (i). Statements (ii) and (iii) follow directly from Lemmas 20.23 and 20.21, respectively. ¥

20.2. PROOFS FOR PART III

635

Derivations of marginal pdf of buffer occupancy for asynchronous exponential two-machine lines: In the asynchronous case, the transition rates, νkl,ij , remain the same as in synchronous lines. The dynamics of the buffer occupancy become  c1 − c2 , if s1 (t) = 1, s2 (t) = 1,    dh(t) ¯¯ if s1 (t) = 1, s2 (t) = 0, c1 , = ¯ , if s1 (t) = 0, s2 (t) = 1, −c  dt 0
=

∂fH,I,J (h, 1, 0) ∂h

=

∂fH,I,J (h, 0, 1) ∂h

=

fH,I,J (h, 0, 0) =

λ1 + λ2 µ2 fH,I,J (h, 1, 1) + fH,I,J (h, 1, 0) c1 − c2 c1 − c2 µ1 + fH,I,J (h, 0, 1) c1 − c2 λ1 + µ2 λ2 − fH,I,J (h, 1, 0) + fH,I,J (h, 1, 1) c1 c1 µ1 + fH,I,J (h, 0, 0), c1 λ2 + µ1 λ1 fH,I,J (h, 0, 1) − fH,I,J (h, 1, 1) c2 c2 µ2 − fH,I,J (h, 0, 0), (20.99) c2 λ1 λ2 fH,I,H (h, 1, 0) + fH,I,J (h, 0, 1). µ1 + µ2 µ1 + µ2 −

In addition, since as in the synchronous case, events A1 and A2 occur with equal frequencies, (c1 − c2 )fH,I,J (h, 1, 1) + c1 fH,I,J (h, 1, 0) = c2 fH,I,J (h, 0, 1).

(20.100)

Using (20.100), (20.99) can be simplified to fH,I,J (h, 1, 1) = ∂fH,I,J (h, 1, 0) ∂h ∂fH,I,J (h, 0, 1) ∂h

c1 c2 fH,I,J (h, 1, 0) − fH,I,J (h, 0, 1), c2 − c1 c2 − c1

=

E1 fH,I,J (h, 1, 0) + E2 fH,I,J (h, 0, 1),

=

E3 fH,I,H (h, 1, 0) + E4 fH,I,J (h, 0, 1),

636

CHAPTER 20. PROOFS λ1 λ2 fH,I,H (h, 1, 0) + fH,I,J (h, 0, 1), µ1 + µ2 µ1 + µ2

fH,I,J (h, 0, 0) =

where Ei , i = 1, . . . , 4, are defined in (11.36). The probabilities of the boundary states for asynchronous lines can be characterized as follows: • If c1 < c2 , P0,11

=

P0,10

=

P0,01

=

P0,00

=

PN,11

=

PN,10

=

PN,01

=

PN,00

=

PN,00

=

1 [µ1 P0,01 + (c2 − c1 )fH,I,J (0, 1, 1)], λ1 + λ 2 0, 1 [λ1 P0,11 + µ2 P0,00 + c2 fH,I,J (0, 0, 1)], µ1 + λ2 λ2 P0,01 , µ1 + µ2 0, (20.101) c2 − c1 fH,I,J (N, 1, 1), µ2 0, λ1 PN,10 , µ1 + µ2 c2 fH,I,J (N, 0, 1). µ2

• If c1 > c2 , P0,11

=

0,

P0,10

=

P0,01

=

P0,00

=

P0,00

=

PN,11

=

PN,10

=

PN,01

=

PN,00

=

0, c1 − c2 fH,I,J (0, 1, 1), µ1 c1 fH,I,J (0, 1, 0), µ1 λ2 P0,01 , µ1 + µ2 1 [µ2 PN,10 + (c1 − c2 )fH,I,J (N, 1, 1)], (20.102) λ 1 + λ2 1 [λ2 PN,11 + µ1 PN,00 + c1 fH,I,J (N, 1, 0)], λ1 + µ 2 0, λ1 fH,I,J (N, 0, 1). µ1 + µ2

Finally, the total probability formula takes the form: 1 X 1 X 1 X 1 Z N− 1 1 X X X fH,I,J (h, i, j)dh + P0,ij + PN,ij = 1. i=0 j=0

0+

i=0 j=0

i=0 j=0

20.2. PROOFS FOR PART III

637

Using the same procedure as in the synchronous case, the solutions to the above equations can be obtained as follows: • If c1 < c2 ,

h i C0 eK1 h − F1 e(K1 −K2 )N · eK2 h , · ¸ F2 K1 h F1 F3 (K1 −K2 )N K2 h fH,I,J (h, 0, 1) = C0 − e + e ·e , 2E2 2E2 c1 c2 fH,I,J (h, 1, 0) − fH,I,J (h, 0, 1), fH,I,J (h, 1, 1) = c2 − c1 c2 − c1 λ1 λ2 fH,I,J (h, 0, 0) = fH,I,H (h, 1, 0) + fH,I,J (h, 0, 1), µ1 + µ2 µ1 + µ2 c1 (λ2 + µ1 + µ2 )fH,I,J (0, 1, 0) − c2 λ2 fH,I,J (0, 0, 1) , P0,11 = λ2 (λ1 + µ1 + λ2 + µ2 ) P0,10 = 0, (µ1 + µ2 )[c1 λ1 fH,I,J (0, 1, 0) + c2 λ2 fH,I,J (0, 0, 1)] P0,01 = , λ2 µ1 (λ1 + µ1 + λ2 + µ2 ) c1 λ1 fH,I,J (0, 1, 0) + c2 λ2 fH,I,J (0, 0, 1) P0,00 = , µ1 (λ1 + µ1 + λ2 + µ2 ) PN,11 = 0, (20.103) c1 fH,I,J (N, 1, 0) − c2 fH,I,J (N, 0, 1) PN,10 = , µ2 PN,01 = 0, λ1 [c1 fH,I,J (N, 1, 0) − c2 fH,I,J (N, 0, 1)] PN,00 = . µ2 (µ1 + µ2 )

fH,I,J (h, 1, 0)

=

• If c1 > c2 , fH,I,J (h, 1, 0) = fH,I,J (h, 0, 1) = fH,I,J (h, 1, 1) = fH,I,J (h, 0, 0) = P0,11 P0,10 P0,01 P0,00 PN,11

=

h i C0 eK1 h − F1 e(K1 −K2 )N · eK2 h , · ¸ F2 K1 h F1 F3 (K1 −K2 )N K2 h C0 − e + e ·e , 2E2 2E2 c1 c2 fH,I,J (h, 1, 0) − fH,I,J (h, 0, 1), c2 − c1 c2 − c1 λ1 λ2 fH,I,H (h, 1, 0) + fH,I,J (h, 0, 1), µ1 + µ2 µ1 + µ2 0,

= 0, (20.104) −c1 fH,I,J (0, 1, 0) + c2 fH,I,J (0, 0, 1) = , µ1 c1 = fH,I,J (0, 1, 0), µ1 c2 (λ1 + µ1 + µ2 )fH,I,J (N, 0, 1) − c1 λ1 fH,I,J (N, 1, 0) , = λ1 (λ1 + µ1 + λ2 + µ2 )

638

CHAPTER 20. PROOFS PN,10

=

PN,01

=

PN,00

=

(µ1 + µ2 )[c1 λ1 fH,I,J (N, 1, 0) + c2 λ2 fH,I,J (N, 0, 1)] , λ1 µ2 (λ1 + µ1 + λ2 + µ2 ) 0, c1 λ1 fH,I,J (N, 1, 0) + c2 λ2 fH,I,J (N, 0, 1) . µ2 (λ1 + µ1 + λ2 + µ2 )

where C0 , K1 , K2 , Fi , i = 1, 2, 3, are defined in (11.35)-(11.38). The marginal pdf of buffer occupancy can be expressed as fH (h) =

1 X 1 X

1 X 1 X 1 1 X X fH,I,J (h,i,j) + P0,ij δ(h) + PN,ij δ(h−N ). (20.105)

i=0 j=0

i=0 j=0

i=0 j=0

Substituting (20.103) and (20.104) into (20.105) leads to (11.34). Formulas for the performance measures in asynchronous exponential two-machine lines: The performance measures of interest can be expressed as follows: (Z − ) 1 N X fH,I,J (h, i, 1)dh + PN,i1 + min(c1 , c2 )P0,11, (20.106) T P = c2 i=0

W IP

=

0+

1 X 1 Z X i=0 j=0

N−

0+

fH,I,J (h, i, j)hdh +

1 X 1 X

PN,ij N .

(20.107)

i=0 j=0

Substituting (20.103) and (20.104) into (20.106) and (20.107) leads to (11.40) and (11.41). ¥ Proof of Theorem 11.7: This proof is similar to the proofs of Theorems 4.2 and 11.4, and, therefore, is omitted here; details can be found in [20.14]. ¥

20.2.2

Proofs for Chapter 12

The proof of Theorem 12.1 is based on the following two lemmas: Lemma 20.24 Consider a two-machine line defined by assumption (12.6) with CVup,i = CVdown,i = 0, i = 1, 2, and assume that Tup,1 = Tup,2 = Tup , Tdown,1 = Tdown,2 = Tdown . Then its production rate is given by PR = e =

Tup . Tup + Tdown

Proof: Let si (t), i = 1, 2, denote the states of machine mi , i = 1, 2, at time t, i.e., ½ 0, mi is down, si (t) = 1, mi is up,

20.2. PROOFS FOR PART III

639

and h(t) be the buffer occupancy at time t. Without loss of generality, h(0) = 0. Assume that si (0) = 0, i = 1, 2, and mi changes its state at time ti ≤ Tdown , i = 1, 2, i.e.,  0, t ∈ [0, ti ),    1, t ∈ [ti + lT, ti + lT + Tup ), si (t) = 0, t ∈ [ti + lT + Tup , ti + (l + 1)T ),    i = 1, 2, l = 0, 1, 2, . . . , where T = Tup + Tdown . Then, the following three cases are possible: • Case 1: t1 = t2 . Obviously, in this case machine m1 is never blocked and m2 is never starved, and therefore, P R = e. • Case 2: t1 > t2 . In this case, (α). If t2 + Tup ≥ t1 , the following hold: s1 (t) = 0, s1 (t) = 0, s1 (t) = 1, s1 (t) = 1, s1 (t) = 0, s1 (t) = 0, s1 (t) = 1, s( t) = 1, s1 (t) = 0, ...,

s2 (t) = 0, h(t) = 0, t ∈ [0, t2 ), s2 (t) = 1, h(t) = 0, t ∈ [t2 , t1 ), s2 (t) = 1, h(t) = 0, t ∈ [t1 , t2 + Tup ), s2 (t) = 0, h(t) = t − Tup − t2 , t ∈ [t2 + Tup , t1 + Tup ), s2 (t) = 0, h(t) = t1 − t2 , t ∈ [t1 + Tup , t2 + T ), s2 (t) = 1, h(t) = t1 + T − t, t ∈ [t2 + T, t1 + T ), s2 (t) = 1, h(t) = 0, t ∈ [t1 + T, t2 + T + Tup ), s2 (t) = 0, h(t) = t − t2 − T − Tup , t ∈ [t2 + T + Tup , t1 + T + Tup ), s2 (t) = 0, h(t) = t1 − t2 , t ∈ [t1 + T + Tup , t2 + 2T ), ..., ....

By induction, we obtain s1 (t) = 0, s2 (t) = 0,

h(t) = t1 − t2 , t ∈ [t1 + (l − 1)T + Tup , t2 + lT ), s1 (t) = 0, s2 (t) = 1, h(t) = t1 − t + lT, t ∈ [t2 + lT, t1 + lT ), s1 (t) = 1, s2 (t) = 1, h(t) = 0, t ∈ [t1 + lT, t2 + lT + Tup ), s1 (t) = 1, s2 (t) = 0, h(t) = t − t2 − lT − Tup , t ∈ [t2 + lT + Tup , t1 + lT + Tup ), l = 1, 2, . . . .

640

CHAPTER 20. PROOFS Since k ≥ 1, i.e., N ≥ Tdown , machine m1 is never blocked and machine m2 is not starved (except for the initial period t ∈ [t2 , t1 )). Thus, after the period [0, t2 + Tup ), the line is producing during the interval of length Tup and is not producing during the interval of length Tdown . (β). If t2 + Tup < t1 , which implies that Tup < Tdown , the following hold: s1 (t) = 0, s2 (t) = 0, s1 (t) = 0, s2 (t) = 1, s1 (t) = 0, s2 (t) = 0, s1 (t) = 1, s2 (t) = 0, s1 (t) = 0, s2 (t) = 0, s1 (t) = 0, s2 (t) = 1, s1 (t) = 0, s2 (t) = 0, s1 (t) = 1, s2 (t) = 1, ...,

...,

h(t) = 0, t ∈ [0, t2 ), h(t) = 0, t ∈ [t2 , t2 + Tup ), h(t) = 0, t ∈ [t2 + Tup , t1 ), h(t) = t − t1 , t ∈ [t1 , t1 + Tup ), h(t) = Tup , t ∈ [t1 + Tup , t2 + T ), h(t) = Tup − t + t2 + T, t ∈ [t2 + T, t2 + T + Tup ), h(t) = 0, t ∈ [t2 + T + Tup , t1 + T ), h(t) = t − t1 − T, t ∈ [t1 + T, t1 + T + Tup ), ....

Clearly, in this case, s1 (t) = 0, s2 (t) = 0, s1 (t) = 1, s2 (t) = 0, s1 (t) = 0, s2 (t) = 0, s1 (t) = 0, s2 (t) = 1,

h(t) = 0, t ∈ [t2 + lT + Tup , t1 + lT ), h(t) = t − t1 − kT, t ∈ [t1 + lT, t1 + lT + Tup ), h(t) = Tup , t ∈ [t1 + lT + Tup , t2 + (l + 1)T ), h(t) = t2 + (l + 1)T + Tup − t, t ∈ [t2 + (l + 1)T, t2 + (l + 1)T + Tup ), l = 1, 2, . . . .

Thus, m2 is not starved (except for the initial interval t ∈ [t2 , t2 + Tup )) and m1 is never blocked. Therefore, in both (a) and (b), t2 + Tup ≥ t1 or t2 + Tup < t1 , and, hence, P R = e. • Case 3: t1 < t2 . Similar arguments can be used. If t1 + Tup ≥ t2 , we

20.2. PROOFS FOR PART III

641

show that: s1 (t) = 0, s2 (t) = 0,

h(t) = 0, t ∈ [t2 + (l − 1)T + Tup , t1 + lT ), s1 (t) = 1, s2 (t) = 0, h(t) = t − t1 − lT, t ∈ [t1 + lT, t2 + lT ), s1 (t) = 1, s2 (t) = 1, h(t) = t2 − t1 , t ∈ [t2 + lT, t1 + lT + Tup ), s1 (t) = 0, s2 (t) = 1, h(t) = t2 − t + lT + Tup , t ∈ [t1 + lT + Tup , t2 + lT + Tup ), l = 1, 2, . . . .

Again, due to N ≥ Tdown , machine m1 is never blocked and m2 is never starved. If t1 + Tup < t2 , the following hold: s1 (t) = 0, s2 (t) = 0, s1 (t) = 1, s2 (t) = 0, s1 (t) = 0, s2 (t) = 0, s1 (t) = 0, s2 (t) = 1,

h(t) = 0, t ∈ [t2 + (l − 1)T + Tup , t1 + lT ), h(t) = t − t1 − lT, t ∈ [t1 + lT, t1 + lT + Tup ), h(t) = Tup , t ∈ [t1 + lT + Tup , t2 + lT ), h(t) = Tup − t + lT + t2 , t ∈ [t2 + lT, t2 + lT + Tup ), l = 1, 2, . . . .

Therefore, in both cases, P R = e. Next, we repeat this analysis under the assumptions (α) (β) (γ)

si (0) = 1,

i = 1, 2, s1 (0) = 1, s2 (0) = 0, s1 (0) = 0, s2 (0) = 1,

and show that after the initial period t ∈ [0, Tup + Tdown ), the line produces during the interval of the length Tup and does not produce during the interval of the length Tdown . Therefore, in all cases, P R = e. ¥

Lemma 20.25 Consider an M -machine line defined by assumption (12.6) with CVup,i = CVdown,i = 0, i = 1, . . . , M , and assume that Tup,i = Tup and Tdown,i = Tdown , i = 1, . . . , M . Then its production rate is given by PR = e =

Tup . Tup + Tdown

642

CHAPTER 20. PROOFS

Proof: From Lemma 20.24, machine m2 is not starved except for a subinterval of [0, Tup + Tdown ). Thus, in the steady state, the two-machine line is up (producing) for Tup time units and down (not producing) for Tdown , which is equivalent to a single machine. Since N ≥ Tdown , no blockage of m2 can occur. Therefore, aggregating these two machines with the third one, we conclude that the three machines are again equivalent to a single machine. Continuing this process until all machines are aggregated, we obtain P R = e for M > 2-machine lines when CV = 0 and N ≥ Tdown . ¥ Proof of Theorem 12.1: Consider the production line, L, defined by assumptions (a)-(e) of Subsection 11.1.1 with arbitrary values of Tup,i and Tdown,i , but with CVup,i = CVdown,i =0, i = 1, . . . , M . Along with it, consider the production line, L0 , also defined by assumptions (a)-(e) but with identical machines and buffers given by: 0 Tdown

=

e0

=

0 Tup

=

N0

=

max Tdown,i ,

i=1,...,M

min

i=1,...,M

Tup,i = min ei = emin , i=1,...,M Tup + Tdown,i

(20.108)

e0 T0 , 1 − e0 down 0 Tdown .

As it follows from Lemma 20.25, the production rate, P R0 , of line L0 is emin . Due to (20.108) and the monotonicity property of production rate of serial lines with respect to machine and buffer parameters, the production rate, P R, of line L satisfies the inequality P R ≥ P R0 . However, P R is limited by the least efficient machine in the system. Therefore, P R = P R0 = emin . ¥

20.2.3

Proofs for Chapter 13

Proof of Theorem 13.1: Since for a single machine with any reliability model TP = c

Tup , Tup + Tdown

the quantities involved in Definition 13.3 can be easily calculated: ¯ ∂T P ¯ ¯ ¯ ¯ ¯ = ∂Tdown

c

Tup , (Tup + Tdown )2

20.2. PROOFS FOR PART III ∂T P ∂Tup

643 = c

Tdown , (Tup + Tdown )2

leading to the conclusion of Theorem 13.1. ¥ Proof of Theorems 13.2-13.4: These proofs are based on evaluating the derivatives of P R with respect to Tup and Tdown . These evaluations are based on straightforward but tedious and lengthy calculations; the details can be found in [20.14] and [20.25]. ¥

20.2.4

Proofs for Chapter 14

It is convenient first to prove Theorem 14.3 and then specialize it to Theorem 14.2. Proof of Theorem 14.3: The steady states of the recursive procedure (11.25) are given in (20.98). When N is selected so that the efficiency of the production line is E, it follows from (11.29) that Ee = e[1 − Q(λfM −1 , µfM −1 , λ, µ, NE )], which implies that Q(λfM −1 , µfM −1 , λ, µ, NE ) = 1 − E. This, along with (11.4), provides an equation for NE from which it follows that (1 − efM −1 )(1 − φ) = 1 − E, 1 − φ exp{−βNE } i.e., 1 NE = − ln β

Ã

E − φ − efM −1 + efM −1 φ φ(E − 1)

! .

Therefore, the lean level of buffering, kE = NE /Tdown , is given by Ã ! E − φ − efM −1 + efM −1 φ µ kE = − ln . β φ(E − 1) In this formula, as it follows from (11.7), (11.29), and (20.98), efM −1 φ

= e(1 − Q), =

(1 − Q)(1 − e) , 1 − e + eQ

(20.109)

644

CHAPTER 20. PROOFS β

=

(µfM −1 + µ + λfM −1 + λ)(λfM −1 µ − λµfM −1 ) (µfM −1 + µ)(λfM −1 + λ)

,

where λfM −1 =

λ λµ , µfM −1 = , 1+Q Qµ + λ

Q = Q(λfM −2 , µfM −2 , λbM −1 , µbM −1 , NE ).

(20.110)

Using these expressions and the identities λ = λ+µ =

µ(1 − e) , e µ , e

equation (20.109) can be re-written as kE

=

e(1 − Q)(eQ + 1 − e)(eQ + 2 − 2e)(2 − Q) Q(2e − 2eQ + eQ2 + Q − 2) µ ¶ E − eE + eEQ − 1 + e − 2eQ + eQ2 + Q · ln . (20.111) (1 − e − Q + eQ)(E − 1)

This proves the top expression of (14.10). To prove the bottom one, we observe that if N = 0, recursive procedure (11.25) results in P R = eM = eE. 1

This implies that if e ≥ E M −1 , M ≥ 2, line efficiency E is attained with kE = 0. ¥ Proof of Theorem 14.2: In the case of M = 3, function Q defined by (20.110) can be evaluated explicitly. Indeed, for M = 3, expression (20.110) can be written as follows: Q = Q(λ, µ, λb2 , µb2 , NE ). To evaluate this function, we observe that, as it follows from (11.29), Q(p, r, λb2 , µb2 , NE ) = Q(λ, µ, λf2 , µf2 , NE ) and, moreover, Ee = e(1 − Q)2 . This implies that Q=1−

√

E.

20.2. PROOFS FOR PART III

645

Substituting this expression in (20.111) results in (14.9). ¥ Proof of Theorem 14.5: Follows immediately from (14.26) by observing that β(λ1 , µ1 , λ2 , µ2 ) = −β(λ2 , µ2 , λ1 , µ1 ) and φ(e1 , e2 ) = 1/φ(e2 , e1 ). ¥

20.2.5

Proofs for Chapter 15

It is convenient to prove Theorem 15.2 first and then specialize it to Theorem 15.1. Proof of Theorem 15.2: The steps of this proof are as follows: Step 1: Derive the characterization of the probability mass function, P [H(i) = k], for k = 0, 1, . . .,N , where H(i) is the number of parts in FGB at the end of the i-th epoch. Step 2: Represent the results of Step 1 in a matrix-vector form and solve for the probability mass function. Step 3: Express the DT P in terms of P [H(i) = k] and, thus, obtain the claim of the theorem. Below these three steps are carried out. Step 1: For k = 1, 2, . . . , N − 1, introduce zk

:= = =

P [(H(i) = k] P [k ≤ tˆ(i) + H(i − 1) − D(i) < k + 1] N X

P [k ≤ tˆ(i) + H(i − 1) − D(i) < k + 1|H(i − 1) = l]P [H(i − 1) = l]

l=0

=

N X

P [D(i) + k − l ≤ tˆ(i) < D(i) + k + 1 − l]P [H(i − 1) = l]

l=0

=

N ³ ´ X P [tˆ(i) ≥ D(i) + k − l] − P [tˆ(i) ≥ D(i) + k − l + 1] P [H(i − 1) = l]

=

N ³ ´ X P [t(i) ≥ D(i) + k − l] − P [t(i) ≥ D(i) + k − l + 1] P [H(i − 1) = l].

l=0

l=0

In the steady state of PIC system operation, zk

=

N ³ ´ X P [t(i) ≥ D(i) + k − l] − P [t(i) ≥ D(i) + k − l + 1] P [(H(i) = l], l=0

646

CHAPTER 20. PROOFS N ³X J X

=

l=0

−

¯ ¯ P [t(i) ≥ D(i) + k − l¯D(i) = Dj ]P [D(i) = Dj ]

j=1

J X

´ P [t(i) ≥ D(i) + k − l + 1|D(i) = Dj ]P [D(i) = Dj ] P [H(i) = l]

j=1 N X J ³ X

´ P [t(i) ≥ Dj + k − l] − P [t(i) ≥ Dj + k − l + 1] Pj zl

=

l=0 j=1 N X J X

=

[P(Dj + k − l) − P(Dj + k − l + 1)]Pj zl

l=0 j=1 N X J X

=

rk,l,j Pj zl ,

(20.112)

l=0 j=1

where P(Dj + k − l) and P(Dj + k − l + 1) are defined in (15.6). For k = N , introduce zN

= P [tˆ(i) + H(i − 1) − D(i) = N ] = P [t(i) + H(i − 1) − D(i) ≥ N ] =

N X

P [t(i) + H(i − 1) − D(i) ≥ N |H(i − 1) = l]P [H(i − 1) = l]

l=0

=

N X

P [t(i) ≥ D(i) + N − l]P [H(i) = l]

l=0

=

N X J X

P [t(i) ≥ D(i) + N − l|D(i) = Dj ]P [D(i) = Dj ]P [H(i) = l]

l=0 j=1

=

N X J X

P [t(i) ≥ Dj + N − l]Pj zl

l=0 j=1

=

N X J X

P(Dj + N − l)Pj zl

l=0 j=1

=

N X J X

rbN,l,j zl .

(20.113)

l=0 j=1

For k = 0, introduce z0 = P [H(i) = 0]. Clearly z0 = 1 −

N X k=1

P [H(i) = k] = 1 −

N X k=1

zk .

(20.114)

20.2. PROOFS FOR PART III

647

Step 2: Substituting z0 into (20.112) and (20.113), we obtain: J J hX i X (r1,1,j − r1,0,j )Pj − 1 z1 + (r1,2,j − r1,0,j )Pj z2 + . . . j=1

j=1

+ J X

J X

J X

j=1

j=1

(r1,N,j − r1,0,j )Pj zN = − J hX

(r2,1,j − r2,0,j )Pj z1 +

j=1

i (r2,2,j − r2,0,j )Pj − 1 z2 + . . .

j=1

+ ... J X

r1,0,j Pj ,

J X

J X

j=1

j=1

(r2,N,j − r2,0,j )Pj zN = − ...

(b rN,1,j − rbN,0,j )Pj z1 +

j=1

J X

...

r2,0,j Pj ,

(b rN,2,j − rbN,0,j )Pj z2 + · · ·

j=1

+

J hX

J i X (b rN,N,j − rbN,0,j )Pj − 1 zN = − rbN,0,j Pj ,

j=1

j=1

or, in the matrix-vector form, RZ = −Z0 ,

(20.115)

where R and Z0 are defined in (15.25) and (15.26), respectively, and R is nonsingular. Thus, (20.116) Z = −R−1 Z0 . Step 3: From the definition of DT P , using the total probability formula, we have: DT P

= =

P [b t(i) + H(i − 1) ≥ D(i)] N X

P [tˆ(i) + H(i − 1) ≥ D(i)|H(i − 1) = k]P [H(i − 1) = k]

k=0

=

N X

P [tˆ(i) ≥ D(i) − k]P [H(i − 1) = k]

k=0

=

N X

P [t(i) ≥ D(i) − k]P [H(i − 1) = k],

k=0

=

N X

P [t(i) ≥ D(i) − k]P [H(i) = k]

k=0

=

N X J X k=0 j=1

P [t(i) ≥ D(i) − k]P [D(i) = Dj ]P [H(i) = k]

648

CHAPTER 20. PROOFS

=

N X J X

P(Dj − k)Pj zk .

(20.117)

k=0 j=1

Theorem 15.2 is proved. ¥ Proof of Theorem 15.1: In the case of constant demand, J = 1 and Pj = 1, Dj = D, which reduces (15.27) to (15.11). ¥ Proof of Theorem 9.2: In the case of a Bernoulli line, P [k ≤ tˆ(i) + H(i − 1) − D(i) < k + 1] is simplified to P [tˆ(i) + H(i − 1) − D(i) = k]. Thus, zk is representedµby P¶[t(i) = D(i) + k − l]P [H(i − 1) = l]. In addition, P(x) is T expressed by px (1 − p)T −x . This leads to equation (9.20). x ¥ Proof of Theorem 9.1: In the case of constant demand, J = 1 and Pj = 1, Dj = D. This reduces equation (9.20) to (9.10). ¥

20.3

Proofs for Part IV

Proof of Theorem 16.1: By induction. Since function Q(x, y, N ) takes values in (0, 1) and is monotonically increasing in y, from (16.5), for s = 0, we have Pb01 (1)

= Q(p1 , p0 [1 − Pb02 (0)], N1 ) = Q(p1 , p0 , N1 ) > Q(p1 , p0 [1 − Q(p2 , p0 [1 − Pb01 (1)], N2 )], N1 ) = Q(p1 , p0 [1 − Pb2 (1)], N1 ) 0

= Pb02 (1)

(20.118)

Pb01 (2),

Q(p2 , p0 [1 − Pb01 (1)], N2 ) < Q(p2 , p0 [1 − Pb01 (2)], N2 ) = Pb2 (2).

=

(20.119)

0

Assume that for s > 0, Pb01 (s) < Pb01 (s − 1), Pb02 (s) > Pb02 (s − 1). Then, due to the monotonicity of Q(x, y, N ), Pb01 (s + 1)

=

Q(p1 , p0 [1 − Pb02 (s)], N1 )

(20.120)

20.4. ANNOTATED BIBLIOGRAPHY Q(p1 , p0 [1 − Pb02 (s − 1)], N1 ) Pb01 (s),

(20.121)

= Q(p2 , p0 [1 − Pb01 (s + 1)], N2 ) > Q(p2 , p0 [1 − Pb01 (s)], N2 ) = Pb02 (s).

(20.122)

< = Pb02 (s + 1)

649

Thus, Pb01 (s) and Pb02 (s) are monotonically decreasing and increasing in s, respectively. Since the sequences Pb01 (s) and Pb02 (s) are monotonic and bounded from above and below (function Q takes values in (0, 1)), they are convergent, i.e., lim Pb01 (s)

=: Pb01 ,

lim Pb02 (s)

=: Pb02 .

s→∞ s→∞

¥ Proofs of Theorems 16.2 and 16.3: These proofs are similar to the proof of Theorem 4.3; details are available in [20.22]. ¥ Proof of Theorem 16.4: This proof is based on the combination of the proofs of Theorems 4.2 and 16.1; details are available in [20.23]. ¥

20.4

Annotated Bibliography

The proofs for Chapter 4 originally appeared in [20.1 ] D.A. Jacobs and S.M. Meerkov, “A System-Theoretic Property of Serial Production Lines: Improvability,” International Journal of System Science, vol. 26, pp. 95-137, 1995. [20.2 ] D.A. Jacobs and S.M. Meerkov, “Mathematical Theory of Improvability for Production Systems,” Mathematical Problems in Engineering, vol. 1, pp. 95-137, 1995. Details of the proofs for Chapter 5 can be found in [20.1], [20.2], and [20.3 ] C.-T. Kuo, J.-T. Lim and S. M. Meerkov, “Bottlenecks in Serial Production Lines: A System-theoretic Approach,” Mathematical Problems in Engineering, vol. 2, pp. 233-276, 1996. [20.4 ] C.-T. Kuo, Bottlenecks in Production Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1996.

650

CHAPTER 20. PROOFS The proofs for Chapter 6 are similar to those published in

[20.5 ] A.B. Hu and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Bernoulli Machines,” Mathematical Problems in Engineering, vol. 2006, Article ID 17105, 2006. The proofs for Chapter 7 originally appeared in [20.6 ] S. Biller, S.P. Marin, S.M. Meerkov and L. Zhang, “Closed Bernoulli Lines: Analysis, Continuous Improvement, and Leanness,” IEEE Transactions on Automation Science and Engineering, vol. 5, 2008. Details of the proofs for Chapter 8 are given in [20.7 ] S.M. Meerkov and L. Zhang, “Product Quality Inspection in Bernoulli Lines: Analysis, Bottlenecks, and Design”, Control Group Report No. CGR-07-10, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007. [20.8 ] S.M. Meerkov and L. Zhang, “Bernoulli Production Lines with QualityQuantity Coupling Machines: Monotonicity Properties and Bottlenecks”, Control Group Report No. CGR-07-11, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007. [20.9 ] S. Biller, J. Li, S.P. Marin, S.M. Meerkov and L. Zhang, “Bottlenecks in Bernoulli Serial Lines with Rework”, Control Group Report No. CGR-0706, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, 2007. The proofs for Chapter 9 can be found in [20.10 ] J. Li and S.M. Meerkov, “Production Variability in Manufacturing Systems: Bernoulli Reliability Case,” Annals of Operations Research, vol. 93, pp. 299-324, 2000. [20.11 ] J. Li, Production Variability in Manufacturing Systems: A Systems Approach, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 2000. The proofs for Chapter 10 originally appeared in [20.12 ] S.M. Meerkov and L. Zhang, “Transient Behavior of Serial Production Lines with Bernoulli Machines,” IIE Transactions, vol. 40, pp. 297-312, 2008. The proofs for Chapter 11 are modifications of those published in [20.13 ] D.A. Jacobs, Improvability in Production Systems: Theory and Case Studies, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1993.

20.4. ANNOTATED BIBLIOGRAPHY

651

[20.14 ] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “c-Bottleneck in Serial Production Lines”, Mathematical Problems in Engineering, vol. 7, pp. 543-578, 2001. [20.15] J. Li, “Performance Analysis of Production Systems with Rework Loops,” IIE Transactions, vol. 36, pp. 755-765, 2004. Details of the proofs for Chapter 12 can be found in [20.16 ] J. Li and S.M. Meerkov, “Evaluation of Throughput in Serial Production Lines with Non-exponential Machines”, in Analysis, Control and Optimization of Complex Dynamic Systems, ed. E.K. Boukas and R. Malhame, pp. 55-82, Kluwer Academic Publishers, 2005. The proofs for Chapter 13 originally appeared in [20.14], and [20.17 ] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “Bottlenecks in Markovian Production Lines: A Systems Approach,” IEEE Transactions on Robotics and Automation, vol. 14, pp. 352-359, 1998. [20.18 ] S.-Y. Chiang, C.-T. Kuo and S.M. Meerkov, “DT-Bottleneck in Serial Production Lines: Theory and Application”, IEEE Transactions on Robotics and Automation, vol. 16, pp. 567-580, 2000. The proofs for Chapter 14 originally appeared in [20.19 ] E. Enginarlar, J. Li and S.M. Meerkov, “How Lean Can Lean Buffers Be?” IIE Transactions, vol. 37, pp. 333-342, 2005. [20.20 ] S.-Y. Chiang, A.B. Hu and S.M. Meerkov, “Lean Buffering in Serial Production Lines with Non-identical Exponential Machines,” IEEE Transactions on Automation Science and Engineering, vol. 5, pp. 298-306, 2008. The proofs for Chapter 15 are taken from [20.21 ] J. Li, E. Enginarlar and S.M. Meerkov, “Random Demand Satisfaction by Unreliable Production-Inventory-Customer Systems,” Annals of Operations Research, vol. 126, pp. 159-175, 2004. The proofs for Chapter 16 have been initially published in [20.3] and [20.22 ] C.-T. Kuo, J.-T. Lim, S.M. Meerkov and E. Park, “Improvability Theory for Assembly Systems: Two Component - One Assembly Machine Case,” Mathematical Problems in Engineering, vol. 3, pp. 95-171, 1997. [20.23 ] S.-Y. Chiang, C.-T. Kuo, J.-T. Lim and S.M. Meerkov, “Improvability Theory of Assembly Systems I: Problem Formulation and Performance Evaluation,” Mathematical Problems in Engineering, vol. 6, pp. 321-357, 2000.

652

CHAPTER 20. PROOFS

[20.24 ] S.-Y. Chiang, C.-T. Kuo, J.-T. Lim and S.M. Meerkov, “Improvability Theory of Assembly Systems II: Improvability Indicators and Case Study,” Mathematical Problems in Engineering, vol. 6, pp. 359-393, 2000. The original proofs for Chapter 17 can be found in [20.25 ] S.-Y. Chiang, Bottlenecks in Production Systems with Markovian Machines: Theory and Applications, Ph.D Thesis, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 1999.

Epilogue This book presented numerous facts of Production Systems Engineering. Some of them are simple and, from our point of view, elegant. Others are more involved and aesthetically less attractive. All of them, we believe, are practical. What is it that makes them possible? What are the common themes in the foundation of these facts? Three answers can be given: 1. The possibility of exact analysis of two-machine systems and a certain level of machine decoupling, afforded by buffering, allow for aggregation procedures, which, in the final account, describe the systems at hand in the Markovian case. 2. Filtering properties, also afforded by buffering, eliminate the dependence of various performance measures on higher order moments of random phenomena that affect system behavior. Only the first two moments are of importance. Moreover, these two moments affect the behavior not independently but in conjunction with each other – in the form of the coefficient of variation. This allows for the analysis of systems in the non-Markovian case. 3. The sensitivity of the throughput to machine and buffer parameters is closely related to machines’ blockages and starvations. This places the probabilities of blockages and starvations at the center of continuous improvement of both Markovian and non-Markovian systems, which is the foundation of the main practical outcome of this work – Measurement-Based Management of production systems. These simplifying phenomena allow us to conclude this volume with a paraphrase of the well-known maxim attributed to Albert Einstein, nature is complex but not evil:

Production systems are complex but not evil.

653

Abbreviations and Notations Abbreviations B

BAS BBS BC BN BN-b BN-m c-BN cdf CS CT DT-BN DTPM-BN

exp

FGB

G

Bernoulli model of machine reliability blocked after service blocked before service buffer capacity bottleneck bottleneck buffer bottleneck machine cycle time bottleneck cumulative distribution function customer subsystem cycle time downtime bottleneck downtime preventive maintenance bottleneck exponential model of machine reliability finished goods buffer

ga

GBN Geo

IS JIT JR LBC LBN LF LLB LN

LPR-BN LQ-BN M

655

general model of machine reliability gamma model of machine reliability reliability global bottleneck geometric model of machine reliability inventory subsystem Just-in-Time “Just Right” operation Lean buffer capacity local bottleneck load factor lean level of buffering log-normal model of machine reliability local production rate bottleneck local quality bottleneck mixed model of machine reliability

656 MBM

MHS P-BN Pc-BN pdf PEF PIC

pmf PR-BN PS

MeasurementBased Management material handling system primary bottleneck primary cycle time bottleneck probability density function pre-exponential function productioninventorycustomer system probability mass function production rate bottleneck production subsystem

PSE Q-BN QQC Ra

RCC SLE UT-BN UPPM-BN

W

WF

Production Systems Engineering quality bottleneck quality-quantity coupling Rayleigh model of machine reliability reliabilitycapacity coupling second largest eigenvalue uptime bottleneck uptime preventive maintenance bottleneck Weibull model of machine reliability work force

657

Notations A, B, C b BL d BL c cbi

cfi

Cov(X, Y ) CR d CR CV CVave CVdown CVef f CVup ∆ or ² δ D DT P \ DT P E E(X) or x e ebi

efi events buffer probability of blockage estimate of BL capacity of a machine capacity of exponential machine mi in backward aggregation capacity of exponential machine mi in forward aggregation covariance of X and Y consumption rate estimate of CR coefficient of variation average coefficient of variation coefficient of variation of downtime effective coefficient of variation coefficient of variation of uptime various errors accuracy shipment size due-time performance estimate of DT P line efficiency expected value of X machine efficiency efficiency of exponential machine mi in backward aggregation

FX (x) fX (x) ftdown ftup g

H

h I k kE exp kE

ΛT λ

λ1 λbi

λfi

L LT

efficiency of exponential machine mi in forward aggregation cdf of X pdf of X pdf of downtime pdf of uptime probability to produce a good part number of parts in FGB at the end of shipping period number of parts in the buffer S-improvability indicator level of buffering lean level of buffering lean level of buffering for exponential lines percent of production losses due to transients repair rate of an exponential machine second largest eigenvalue of a transition matrix breakdown rate of exponential machine mi in backward aggregation breakdown rate of exponential machine mi in forward aggregation load factor production losses due to transients

658 µ µbi

µfi

M

m ν

νij νDT P N NDT P NE Ψij P Pi ∅ P [·] P

P Pij p

breakdown rate of an exponential machine repair rate of exponential main chine mi backward aggregation repair rate of exponential main chine mi forward aggregation number of machines in the system machine relative finished goods buffer capacity transition rates lean level of FGB buffer capacity lean FGB capacity lean buffer capacity pre-exponential factors probability probability that a buffer contains i parts empty set probability of an event probability of breakdown of geometric machine transition matrix transition probabilities efficiency of a Bernoulli machine

pbi

pfi

PR d P R PX q ρ R

RT d RT σ S Si SX SL SR c SR ST si τ τi T Tdown Tup TP d T P tdown,i

efficiency of Bernoulli main chine mi backward aggregation efficiency of Bernoulli main chine mi forward aggregation production rate estimate of P R probability mass function quality buy rate correlation coefficient probability of repair of geometric machine residence time estimate of RT standard deviation number of carriers severity of bottleneck machine mi sample space service level scrap rate estimate of SR probability of starvation machine state relative shipping period cycle time of machine mi shipping period average downtime average uptime throughput estimate of T P i-th downtime

659 ts,P R ts,W IP tup,i V ar(X) or σ2 W IP \ W IP ζ

settling time of PR settling time of W IP i-th uptime variance of X work-in-process estimate of W IP outcomes

Index Aggregated machines, 77 consecutive, 80 parallel, 77 Aggregation procedures, 135, 251, 274, 352, 374, 498, 528 Bernoulli assembly systems, 498– 509 Bernoulli serial lines, 135–149 exponential assembly systems, 528–536 exponential serial lines, 352, 374 asynchronous, 374–381 synchronous, 352–361 serial lines with non-perfect machines, 251 serial lines with rework, 274 Areas of manufacturing, 3 machine tools and material handling devices, 3 production planning and scheduling, 4 production systems, 4 quality assurance, 4 work systems, 4 Arrow-based method for bottleneck identification, 180, 432, 433, 515, 540 Bernoulli assembly systems, 515 Bernoulli serial lines, 180 continuous assembly systems, 540 exponential serial lines, 432 non-exponential serial lines, 433 Assembly systems, 64, 495, 527, 536 Bernoulli, 495, 511, 519 continuous improvement, 511 lean buffering, 519

performance analysis, 495 with finished goods buffer, 519 exponential, 527, 539, 540, 542 continuous improvement, 539 lean buffering, 540 performance analysis, 527 with finished goods buffer, 542 non-exponential, 536, 539, 540, 542 continuous improvement, 539 lean buffering, 540 performance analysis, 536 with finished goods buffer, 542 Asymptotic properties, 133, 143, 350, 356, 374, 377, 502 Bernoulli assembly systems, 502 Bernoulli serial lines, 133, 143 Exponential serial lines, 350, 374, 377 exponential serial lines, 356 asynchronous, 374, 377 synchronous, 350, 356 Axioms of probability, 14 Bayes’s formula, 20, 28 Blockages-equal-starvations criterion, 173 Blocking conventions, 85 block after service, 85 block before service, 85 Bottleneck indicators, 180, 237, 254, 259, 279, 280, 431–433, 540 Bottleneck machines and buffers, 176, 236, 254, 267, 278, 424, 515, 540 p+ and p− , 267–269 661

662 Bernoulli assembly systems, 515– 516 Bernoulli serial lines, 176–189 closed serial lines, 236–239 continuous assembly systems, 540 lines with continuous time models of machine reliability, 424 c-BN, 426, 432, 433, 540 DT-BN, 426, 429–431 UT-BN, 426, 429–431 serial lines with non-perfect quality machines, 254–259 PR-BN, 254 Q-BN, 254 serial lines with QQC machines, 267–269 serial lines with rework, 278– 287 Buffer half-full criterion, 170, 414, 513, 539 Bernoulli assembly systems, 513 Bernoulli serial lines, 170 continuous assembly systems, 539 continuous serial lines, 414 Buffering potency, 189, 434 Case studies, 102, 156, 194, 216, 241, 308, 381, 434, 520 crankshaft line, 383 ignition assembly, 110, 520 ignition coil processing, 102, 156, 194, 216, 308, 381, 434 injection - modeling assembly, 522 paint shop, 106, 159, 195, 217, 241 Central limit theorem, 42 Closed serial lines, 221 Coefficient of variation, 36 Conditional probability, 16, 28 Conservation of filtering, 302, 479 Continuous improvement, 167, 230, 236, 411, 511, 515, 539

INDEX constrained improvability, 168, 230, 411, 511, 539 Bernoulli assembly systems, 511–515 Bernoulli serial lines, 168–176 closed serial lines, 230–235 continuous assembly systems, 539 continuous serial lines, 411– 424 unconstrained improvability, 176, 236, 424, 515, 540 Bernoulli assembly systems, 515–516 Bernoulli serial lines, 176–193 closed serial lines, 236–239 continuous assembly systems, 540 exponential serial lines, 424– 433 non-exponential serial lines, 433 Customer demand satisfaction, 293, 471, 519, 542 Bernoulli assembly systems, 519 Bernoulli serial lines, 296–311 continuous assembly systems, 542 exponential serial lines, 474–490 non-exponential serial lines, 477 Design, 170, 173, 201, 259, 302, 442, 479, 513, 514, 540 lean buffering, 201, 442, 540 lean finished goods buffering, 302, 479, 542 quality inspection system, 259 WF and BC unimprovable system, 173, 514 WF unimprovable system, 170, 513 Distribution of buffer occupancy, 126, 345, 369, 497 Bernoulli assembly systems, 497 Bernoulli serial lines, 126 exponential serial lines, 345, 369

INDEX asynchronous, 369 synchronous, 345 Due-time performance evaluation, 298, 303, 474, 482 fixed demand, 298, 474 random demand, 303, 482 Effects of up- and downtime, 349, 356, 374, 377 asynchronous exponential lines, 374, 377 synchronous exponential lines, 349, 356 Empirical design rule for inspection position, 261 Empirical formulas, 404, 464 lean buffering, 464–467 throughput, 404 Empty carrier buffer, 63, 223 Equivalent representations of M -machine lines, 141, 354, 376, 508, 531, 535 Bernoulli assembly systems, 508 Bernoulli serial lines, 141 exponential assembly systems, 531, 535 asynchronous, 535 synchronous, 531 exponential serial lines, 354, 376 asynchronous, 376 synchronous, 354 Expected value, 36 Failures, 73 operation-dependent, 74 time-dependent, 74 Flat inverted bowl phenomenon, 171, 174 Formulas for performance measures, 131, 141, 251, 276, 347, 355, 371, 376, 404, 501, 508, 531, 535, 537 Bernoulli assembly systems, 501, 508 Bernoulli serial lines, 131, 141

663 exponential assembly systems, 531, 535 asynchronous, 535 synchronous, 531 exponential serial lines, 347, 355, 371, 376 asynchronous, 371, 376 synchronous, 347, 355 non-exponential assembly systems, 537 non-exponential serial lines, 404 serial lines with non-perfect quality machines, 251 serial lines with rework, 276 Improvability, 169, 231, 413, 511 S + and S − , 231–235 buffer capacity, 174–176, 418– 424, 515 cycle time, 414–418, 539 work force, 512–515 workforce, 169–174 Improvability indicators, 170, 174, 231, 414, 418, 424, 513, 515, 539 Independence, 18 Law of large numbers, 41 Lean buffering, 201, 239, 302, 443, 456, 479, 519 Bernoulli assembly systems, 519 Bernoulli serial lines, 201 identical machines, 202–205 non-identical machines, 207– 216 closed serial lines, 239–240 exponential serial lines, 443 identical machines, 443–451 non-identical machines, 451– 456 finished goods, 302, 479 non-exponential serial lines, 456– 467 Machine model identification, 75 Machine quality models, 82

664 Bernoulli, 82 exponential, 82 general, 82 Machine reliability models, 70 continuous time, 71 exponential, 71 gamma, 72 general, 73 log-normal, 73 mixed, 73 Rayleight, 72 Weibull, 72 slotted time, 71 Bernoulli, 71 geometric, 71 Machine timing issues, 68 capacity, 70 continuous time, 70 cycle time, 68 discrete event, 70 flow system, 70 slotted time, 70 Main facts of PSE, 547 assembly systems, 554 individual machines, 547 serial lines, 548 continuous improvement, 551 design, 553 performance analysis, 548 Measurement-based management, 190, 434 Model validation, 91 Modeling steps, 92 Models of buffers, 83 Models of machines, 68 Performance analysis, 123, 225, 250, 272, 298, 303, 315, 343, 404, 474, 484, 495, 527 Bernoulli assembly system, 495– 511 Bernoulli serial lines, 124–149 closed serial lines, 225–230 due-time performance, 298, 303, 474, 484

INDEX exponential assembly systems, 527 asynchronous, 532–536 synchronous, 528–532 exponential serial lines, 343 asynchronous, 361–381 synchronous, 344–361 non-exponential assembly systems, 536 non-exponential serial lines, 404– 405 serial lines with non-perfect quality machines, 250–253 serial lines with rework, 272– 277 transient characteristics, 315– 336 Performance measures, 86 due-time performance, 90 probability of blockages, 88 probability of starvations, 88 production rate, 86 residence time, 89 throughput, 86 transient characteristics, 90 work-in-process, 87 Probability Theory, 13 Procedures for continuous improvement, 172, 175, 186, 232, 233, 415, 418, 419 Product quality, 247 lines with non-perfect quality machines, 248–263 lines with QQC machines, 263– 269 lines with rework, 269–287 Production-inventory-customer system, 296 PSE problems, 4 analysis, 6 continuous improvement, 6 constrained improvability, 6 unconstrained improvability, 6 design, 6 PSE Toolbox, 555

INDEX bottleneck identification function, 571 continuous improvement function, 566 customer demand satisfaction function, 579 lean buffer design function, 576 modeling function, 557 performance analysis function, 562 product quality function, 578 simulation function, 580 QQC machines, 263 Random events, 14 Random processes, 43 continuous time, continuous spaces, 44 ergodic, 46 Markov, 46 chains, 47 continuous time, continuous space, 53 continuous time, discrete space, 50 continuous time, mixed state, 54 wide sense stationary, 45 Random variables, 21 continuous, 21, 26 decreasing transition rates, 32 Erlang, 32 exponential, 28 gamma, 32 Gaussian, 34 log-normal, 34 Rayleigh, 30 Weibull, 33 discrete, 21, 22 Bernoulli, 22 binomial, 22 geometric, 24 Poisson, 24 mixed, 21, 36 vector, 38

665 Rework, 63, 269 Rule-of-thumb for selecting lean buffering, 206, 451 Bernoulli serial lines, 206 exponential serial lines, 451 Sensitivity to machine reliability model, 394, 458, 477 lean buffering, 458–462 throughput, 394–404 Serial lines, 62, 123, 167, 201, 221, 247, 293, 315, 343, 391, 411, 441, 471 Bernoulli, 123, 167, 201, 293 continuous improvement, 167 lean buffering, 201 performance analysis, 123 with finished goods buffer, 293 closed lines, 63, 221 exponential, 343, 411, 441, 471 continuous improvement, 411 lean buffering, 441 performance analysis, 343 with finished goods buffer, 471 non-exponential, 391, 411, 456, 477 continuous improvement, 411 lean buffering, 456 performance analysis, 391 with finished goods buffer, 477 transient behavior, 315 with quality inspection, 63, 247 with quality-quantity coupling, 263 with rework, 63, 269 Structural modeling, 66 System-theoretic properties, 154, 208, 253, 264, 277, 361, 381, 536 monotonicity, 156, 253, 264, 278, 361, 509, 536 reversibility, 154, 208, 253, 277, 361, 509, 536 Total probability formula, 19, 28 Transformations, 93

666 B-exp, 100 exp-B, 94 Transient behavior, 315 pre-exponential factors, 318, 323– 326 production losses, 319, 332–336 second largest eigenvalue, 317, 320–323 settling time, 319, 326–332 Types of production systems, 62 assembly systems, 64 multiple merge, 65 single merge, 65 serial lines, 62 closed, 63 re-entrant, 63 with finished goods buffer, 62 with quality inspection, 63 with rework, 63 Variance, 36

INDEX