Rapid Modelling and Quick Response: Intersection of Theory and Practice

Rapid Modelling and Quick Response

Gerald Reiner Editor

Rapid Modelling and Quick Response Intersection of Theory and Practice

123

Editor Prof. Dr. Gerald Reiner Université de Neuchâtel Faculté des Sciences Économiques Institut de l’Entreprise (IENE) rue A.-L. Breguet 1 2000 Neuchâtel Switzerland [email protected]

ISBN 978-1-84996-524-8

e-ISBN 978-1-84996-525-5

DOI 10.1007/978-1-84996-525-5 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010932606 Ó Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudio Calamar S.L. Printed on acid-free paper. Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Rapid Modelling and Quick Response - Intersection of Theory and Practice This volume is a sequel of the 1st Rapid Modelling Conference proceedings volume that focused on Rapid Modelling for increasing competitiveness. The main focus of the 2nd Rapid Modelling Conference proceedings volume “Rapid Modelling and Quick Response - Intersection of Theory and Practice” is the transfer of knowledge from theory to practice, providing the theoretical foundations for successful performance improvement (based on lead time reduction, etc. as well as financial performance measures). Furthermore illustrations will be given by teaching/business cases as well as success stories on new software tools in this field as well as new approaches. In general, Rapid Modelling is based on queueing theory but other mathematical modelling techniques as well as simulation models which facilitate the transfer of knowledge from theory to application are of interest as well. Together with the proceedings volume of selected papers presented a the 1st Rapid Modelling Conference “Increasing Competitiveness - Tools and Mindset” the interested reader should have a good overview on what is going on in this field. The objective of this conference series is to provide an international, multidisciplinary platform for researchers and practitioners to create and exchange knowledge on increasing competitiveness through Rapid Modelling. In this volume, we demonstrate that lead time reduction (through techniques ranging from quick response manufacturing to lean production) is very important but not enough. Additional factors such as risk, costs, revenues, environment, etc. have to be considered as well. We accepted papers that contribute to these themes in the form of: • • • •

Rapid Modelling Case study research, survey research, action research, longitudinal research Theoretical papers Teaching/business case studies

v

vi

Preface

Relevant topics are: • • • • • •

Queueing Theory Rapid Modelling in Manufacturing and Logistics Rapid Modelling in Services Rapid Modelling and Financial Performance Measurement Product and Process Development Supply Chain Management

Based on these categories, the proceedings volume has been divided into six chapters and brings together selected papers which present different aspects of the 2nd Rapid Modelling Conference. These papers are allocated based on their main contribution. All papers passed through a double-blind referee process to ensure their quality. While the RMC10 (2nd Rapid Modelling Conference “Rapid Modelling and Quick Response - Intersection of Theory and Practice”) takes place at the University of Neuchˆatel, located in the heart of the city of Neuchˆatel, Switzerland, it is based on a collaboration with the project partners within our IAPP Project (No. 217891, see also http://www.unine.ch/iene-kje). We are happy to have brought together authors from Algeria, Austria, Belgium, United Kingdom, Finland, Germany, Hungary, Italy, Sweden, Switzerland, Turkey and the United States of America.

Acknowledgement We would like to thank all those who contributed to the conference and this proceedings volume. First, we wish to thank all authors and presenters for their contribution. Furthermore, we appreciate the valuable help from the members of the international scientific board, the referees and our sponsors (see the Appendix for the appropriate lists). In particular, our gratitude goes to our team at Enterprise Institute at the University of Neuchˆatel, Gina Fiore Walder, Reinhold Schodl, Boualem Rabta, Arda Alp, Gil Gomes dos Santos, Yvan Nieto, who supported this conference project and handled the majority of the text reviews as well as the formating work with LaTex. Ronald Kurz created the logo of our conference and he took over the development of the conference homepage http://www.unine.ch/rmc10. Finally, it has to be mentioned that the conference as well as the book are supported by the EU SEVENTH FRAMEWORK PROGRAMME - THE PEOPLE PROGRAMME - Industry-Academia Partnerships and Pathways Project (No. 217891) “How revolutionary queuing based modelling software helps keeping jobs in Europe. The creation of a lead time reduction software that increases industry competitiveness and supports academic research.” Neuchˆatel, June 2010

Gerald Reiner

Contents

Part I Queueing Theory Perturbation Analysis of M/M/1 Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karim Abbas and Djamil A¨ıssani

3

Series Expansions in Queues with Server Vacation . . . . . . . . . . . . . . . . . . . . 17 Fazia Rahmoune and Djamil A¨ıssani Part II Rapid Modelling in Manufacturing and Logistics Optimal Management of Equipments of the BMT Containers Terminal (Bejaia’s Harbor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Djamil A¨ıssani, Mouloud Cherfaoui, Sma¨ıl Adjabi, S. Hocine and N. Zareb Production Inventory Models for a Multi-product Batch Production System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Ananth Krishnamurthy and Divya Seethapathy Dependency Between Performance of Production Processes and Variability – an Analysis Based on Empirical Data . . . . . . . . . . . . . . . . . . . . 61 Martin Poiger, Gerald Reiner and Werner Jammernegg Improving Business Processes with Rapid Modeling: the Case of Digger . 77 Reinhold Schodl, Nathan Kunz, Gerald Reiner and Gil Gomes dos Santos Part III Rapid Modelling in Services Quick Response Service: The Case of a Non-Profit Humanitarian Service Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Arda Alp, Gerald Reiner and Jeffrey S. Petty

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Contents

Applying Operations Management Principles on Optimisation of Scientific Computing Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Ari-Pekka Hameri and Tapio Niemi Increasing Customer Satisfaction in Queuing Systems with Rapid Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 No´emi Kall´o and Tam´as Koltai Rapid Modelling of Patient Flow in a Health Care Setting: Integrating Simulation with Lean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Claire Worthington, Stewart Robinson, Nicola Burgess and Zoe Radnor Part IV Rapid Modelling and Financial Performance Measurement Evaluation of the Dynamic Impacts of Lead Time Reduction on Finance Based on Open Queueing Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Dominik Gl¨aßer, Boualem Rabta, Gerald Reiner and Arda Alp The Financial Impact of a Rapid Modeling Issue: the Case of Lot Sizing . 163 Lien G. Perdu and Nico J. Vandaele Part V Product and Process Development A Flexibility Based Rapid Response Model in Ready to Wear Sector, in Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 M¨ujde Erol Genevois and Deniz Yensarfati Modular Product Architecture: The Role of Information Exchange for Customization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 AHM Shamsuzzoha and Petri T. Helo Part VI Supply Chain Management The Impact of Technological Change and OIPs on Lead Time Reduction . 215 Krisztina Demeter and Zsolt Matyusz Global Supply Chain Management and Delivery Performance: a Contingent Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Ruggero Golini and Matteo Kalchschmidt In-Transit Distribution Strategy: Hope for European Factories? . . . . . . . . 249 Per Hilletofth, Frida Claesson and Olli-Pekka Hilmola Effect of component interdependency on inventory allocation . . . . . . . . . . 263 Yohanes Kristianto Nugroho, AHM Shamsuzzoha and Petri T. Helo

Contents

ix

Dynamic Nature and Long-Term Effect of Events on Supply Chain Confidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Harri Lorentz and Olli-Pekka Hilmola Evaluation of Supply Process Improvements Illustrated by Means of a JIS Supply Process from the Automotive Industry . . . . . . . . . . . . . . . . . . . . 289 Gerald Reiner and Martin Poiger Information Needs for Decisions on Supply Chain Design . . . . . . . . . . . . . . 303 Stefan Seuring and Tino Bauer A Conceptual Framework for the Integration of Transportation Management Systems and Carbon Calculators . . . . . . . . . . . . . . . . . . . . . . . 317 Stefan Treitl, Heidrun Rosiˇc and Werner Jammernegg A Conceptual Framework for the Analysis of Supply Chain Risk . . . . . . . 331 Monika Weish¨aupl and Werner Jammernegg A

International Scientific Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

B

Sponsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

List of Contributors

Karim Abbas Laboratory LAMOS, University of B´ejaia, Compus of Targa Ouzemour, 06000 B´ejaia, Algeria, e-mail: [email protected] Sma¨ıl Adjabi Laboratory LAMOS, University of B´ejaia, Targa Ouzemour, 6000 B´ejaia, Algeria e-mail: [email protected] Djamil A¨ıssani Laboratory LAMOS, University of B´ejaia, Targa Ouzemour, 6000 B´ejaia, Algeria e-mail: lamos bejaia.hotmail.com Arda ALP Enterprise Institute, University of Neuchˆatel, Rue A.L. Breguet 1, CH-2000 Neuchˆatel, Switzerland e-mail: [email protected] Tino Bauer FTI Consulting Deutschland GmbH, Maximilianstrasse 54, 80538 Muenchen, Germany e-mail: [email protected] Nicola Burgess Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK, e-mail: [email protected] Mouloud Cherfaoui Laboratory LAMOS, University of B´ejaia, Targa Ouzemour, 6000 B´ejaia, Algeria e-mail: [email protected] Frida Claesson School of Technology and Society, University of Sk¨ovde, 541 28 Sk¨ovde, Sweden e-mail: [email protected] xi

xii

List of Contributors

Suzanne de Treville University of Lausanne, Faculty of Business and Economics, Internef 315, CH-1015 Lausanne, Switzerland e-mail: [email protected] Krisztina Demeter Department of Logistics and Supply Chain Management, Corvinus University of Budapest, Fovam ter 8, H-1093 Budapest, Hungary e-mail: [email protected] M¨ujde Erol Genevois Industrial Engineering Department, Galatasaray University, Ciragan Cad. No: 36 Ortakoy, Istanbul, Turkey e-mail: [email protected] Dominik Gl¨aßer Institut de l’entreprise, Universit´e de Neuchˆatel, Rue A.-L. Breguet 1, CH-2000 Neuchˆatel, Switzerland e-mail: [email protected] Ruggero Golini Department of Economics and Technology Management, Universit`a degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy e-mail: [email protected] Gil Gomes dos Santos Enterprise Institute, Faculty of Economics, University of Neuchˆatel, Avenue A.-L. Breguet 1, 2000 Neuchˆatel, Switzerland, e-mail: [email protected] Ari-Pekka Hameri Ecole des HEC, University of Lausanne, Internef, Lausanne 1015, Switzerland e-mail: [email protected] Petri T. Helo Department of Production, University of Vaasa, Finland e-mail: [email protected] Per Hilletofth Logistic Research Group, University of Sk¨ovde, 541 28 Sk¨ovde, Sweden e-mail: [email protected] Olli-Pekka Hilmola Lappeenranta Univ. of Tech., Kouvola Unit, Prikaatintie 9, 45100 Kouvola, Finland Safia Hocine Department Operational Research, University of B´ejaia, Targa Ouzemour, 6000 B´ejaia, Algeria e-mail: lamos [email protected]

List of Contributors

xiii

Werner Jammernegg Vienna University of Economics and Business, Nordbergstraße 15, 1090 Vienna, Austria e-mail: [email protected] Matteo Kalchschmidt Department of Economics and Technology Management, Universit`a di Bergamo, Viale Marconi 5, 24044 Dalmine, Italy e-mail: [email protected] No´emi Kall´o Department of Management and Corporate Economics, Budapest University of Technology and Economics, M¨uegyetem rkp. 9. T. e´ p. IV. em., 1111 Budapest, Hungary e-mail: [email protected] Tam´as Koltai Department of Management and Corporate Economics, Budapest University of Technology and Economics, M¨uegyetem rkp. 9. T. e´ p. IV. em., 1111 Budapest, Hungary e-mail: [email protected] Ananth Krishnamurthy University of Wisconsin-Madison, Department of Industrial and Systems Engineering, 1513 University Avenue, Madison, WI 53706, USA e-mail: [email protected] Yohanes Kristianto Nugroho Department of Production, University of Vaasa, Finland e-mail: [email protected] Nathan Kunz Enterprise Institute, Faculty of Economics, University of Neuchˆatel, Avenue A.-L. Breguet 1, 2000 Neuchˆatel, Switzerland, e-mail: [email protected] Harri Lorentz Turku School of Economics, Finland e-mail: [email protected] D´avid Losonci Department of Logistics and Supply Chain Management, Corvinus University of Budapest, Fovam ter 8, H-1093 Budapest, Hungary e-mail: ([email protected] Zsolt Matyusz Department of Logistics and Supply Chain Management, Corvinus University of Budapest, Fovam ter 8, H-1093 Budapest, Hungary e-mail: [email protected]

xiv

List of Contributors

Tapio Niemi Helsinki Institute of Physics, CERN, CH-1211 Geneva, Switzerland e-mail: [email protected] Lien G. Perdu Dept of Business and Economics, K.U. Leuven, Naamsestraat 69, BE-3000 Leuven, Belgium e-mail: [email protected] Jeffrey S. Petty Lancer Callon Ltd., Suite 298, 56 Gloucester Road, UK-SW7 4UB London, United Kingdom e-mail: [email protected] Martin Poiger University of Applied Sciences BFI Vienna, Wohlmutstrasse 22, A-1020 Wien, Austria e-mail: [email protected] Boualem Rabta Enterprise Institute, University of Neuchatel, Rue A.-L. Breguet 1, CH-2000 Neuchatel, Switzerland e-mail: [email protected] Zoe Radnor Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] Fazia Rahmoune LAMOS Laboratory of Modelling and Optimization of Systems - University of Bejaia 06000, Algeria e-mail: [email protected] Gerald Reiner Institut de l’entreprise, Universit´e de Neuchˆatel, Rue A.-L. Breguet 1, CH-2000 Neuchˆatel, Switzerland e-mail: [email protected] Stewart Robinson Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] Heidrun Rosiˇc Vienna University of Economics and Business, Nordbergstraße 15, 1090 Vienna, Austria e-mail: [email protected]

List of Contributors

xv

Reinhold Schodl Institut de l’entreprise, Universit´e de Neuchˆatel, Rue A.-L. Breguet 1, CH-2000 Neuchˆatel, Switzerland e-mail: [email protected] Divya Seethapathy Department of Industrial and Systems Engineering, University of Wisconsin, Madison, WI 53706, USA e-mail: [email protected] Stefan Seuring University of Kassel, Department of International Management, Steinstr. 19, 37213 Witzenhausen, Germany e-mail: [email protected] AHM Shamsuzzoha Department of Production, University of Vaasa, Finland e-mail: [email protected] Stefan Treitl WU Vienna University of Economics and Business, Nordbergstraße 15, 1090 Vienna, Austria e-mail: [email protected] Nico J. Vandaele Research Center for Operations Management, Department of Decision Sciences and Information Management, K.U. 3000 Leuven, Belgium e-mail: [email protected] Monika Weish¨aupl WU Vienna University of Economics and Business, Nordbergstr. 15, 1090 Wien, Austria e-mail: [email protected] Claire Worthington Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] Deniz Yensarfati Industrial Engineering Department, Galatasaray University, Ciragan Cad. No: 36 Ortakoy, Istanbul, Turkey e-mail: [email protected] Nadira Zareb ˜ Targa Ouzemour, 6000 Department Operational Research, University of BejaA¯a, B´ejaia, Algeria e-mail: lamos [email protected]

Part I

Queueing Theory

Perturbation Analysis of M/M/1 Queue Karim Abbas and Djamil A¨ıssani

Abstract This paper treats the problem of evaluating the sensitivity of performance measures to changes in system parameters for a specific class of stochastic models. Motivated by the problem of the coexistence on transmission links of telecommunication networks of elastic and unresponsive traffic, we study in this paper the impact on the stationary characteristics of an M/M/1 queue of a small perturbation in the server rate. For this model we obtain a new perturbation bound by using the Strong Stability Approach. Our analysis is based on bounding the distance of stationary distributions in a suitable functional space.

1 Introduction A manufacturing process or a telecommunication network is a dynamical system which can, in principle, be described by using a rapid modelling technique such as queueing theory. In particular, the usefulness of the M/M/1 queueing model is multiplied many times if the model approximates the behaviour of queues that deviate slightly from the assumptions in the model. An analysis of the sensitivity of performance measures to changes in model parameters is an issue of practical importance. The study of this queueing model is motivated by the following engineering problem: Consider a transmission link of a telecommunication network carrying elastic traffic, able to adapt to the congestion level of the network, and a small proportion of traffic, which is unresponsive to congestion. The problem addressed in this paper

Karim Abbas (B) and Djamil A¨ıssani Laboratory LAMOS, University of B´ejaia, Compus of Targa Ouzemour, 06000 B´ejaia, Algeria, e-mail: [email protected] Djamil A¨ıssani e-mail: lamos [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 1, 

3

4

Karim Abbas and Djamil A¨ıssani

is to derive quantitative results for estimating the influence of unresponsive traffic on elastic traffic. Suppose, for example, we wish to model the behaviour of a single server queue with an infinite waiting room and a first-in first-out discipline. Let P denote the transition kernel of the imbedded jump chain of the M/M/1 queue with arrival rate λ and service rate μ . This Markov chain, defined on some (denumerable) state-space S, has unique stationary distribution π . What would be the effect on the stationary performance of the queue if we increased the service rate of the server by ε ? Let P denote the Markov transition kernel of the Markov chain modeling the alternative system, in our example the M/M/1 queue with service rate μ + ε , and assume that P has unique stationary distribution π. The question about the effect of switching from P to P on the stationary behavior is expressed by π − π, the difference between the stationary distributions. Obviously, a bound on the effect of the perturbation is of great interest. More specifically, let  · tv denote the total variation norm, then the above problem can be phrased as follows: Can π − πtv be approximated or  tv ? This is known as ”perturbation analysis” or ”stabounded in terms of P − P bility problem” of Markov chains in the literature. However, convergence w.r.t. to the total variational norm allows for only bounding the effect of switching from P to P for bounded performance measures only. There exists numerous results on perturbation bounds of Markov chains. General results are summarized by Heidergott and Hordijk (2003). One group of results concerns the sensitivity of the stationary distribution of a finite, homogeneous Markov chain (see Heidergott et al, 2007), and the bounds are derived using methods of matrix analysis; see the review of Cho and Meyer (2001) and recent papers of Kirkland (2002) and Neumann and Xu (2004). Another group includes perturbation bounds for finite-time and invariant distributions of Markov chains with general state space; see Anisimov (1988), Rachev (1989), A¨ıssani and Kartashov (1983), Kartashov (1996) and Mitrophanov (2005). In these works, the bounds for general Markov chains are expressed in terms of ergodicity coefficients of the iterated transition kernel, which are difficult to compute for infinite state spaces. These results were obtained using operator-theoretic and probabilistic methods. Recent work examines the robustness of queueing models in a general framework. The sensitivity of the queue to a perturbation is measured by various functions (metrics) of the probability distributions associated with the perturbed and nominal queueing processes. These analyses can be found in Kotzurek and Stoyan (1976), Whitt (1981), Zolotarev (1977), Fricker et al (2009) and Altman et al (2004). Related work on the robustness of statistical models can be found in Albin (1982). In Albin (1984), the author has examined the robustness of the M/M/1 queueing model to several specific perturbations in the arrival process. The author has used the Taylor series expansion to predict accurately the operating characteristics of queues with arrival processes that are slightly different from the Poisson process. More recently, the M/M/1 queueing model with perturbations in the service process, has been studied via a perturbation analysis of a Markov chain by Heidergott (2008). In this paper, we consider the same perturbation introduced in Heidergott (2008), and obtain a new results by applying an other approach. Therefore, we set out to

Perturbation Analysis of M/M/1 Queue

5

explain our approach with the M/M/1 queue with service rate μ as P and the M/M/1 queue with service rate μ + ε as the P system, for ε sufficiently small. For these systems π and π are known and everything can be computed explicitly. This allows for evaluating the potential of our approach. The paper is organized as follows. Section 2 presents the Strong Stability Approach. Section 3 is devoted to establish the bound on the perturbation. Numerical examples are presented in Section 4. Eventually, we will point out directions of further research.

2 Strong Stability Approach The main tool for our analysis is the weighted supremum norm, also called υ -norm, denoted by  · υ , where υ is some vector with elements υ (s) > 1 for all s ∈ S, and for any w ∈ RS |w(i)| wυ = sup . i∈S υ (i) Let μ be a probability measure on S, then the υ -norm of μ is defined as μ υ = ∑ υ (s)μ (ds). s∈S

The υ -norm is extended to operators on S in the following way: let A ∈ RS×S then Aυ =

∑Sj=1 |A(i, j)w( j)| . υ (i) i,wυ ≤1 sup

Note that υ -norm convergence to 0 implies elementwise convergence to 0. Suppose that π and π have finite υ -norm, then |π f − π f | ≤ π − π υ  f υ inf υ (s), s∈S

for all f with  f υ finite. For our analysis, we will have S ⊂ N and we will assume that υ (s) is of particular form υβ (s) = β s , for β > 1. Hence, the bound becomes |π f − π f | ≤ π − π υβ  f υβ ,

(1)

for all f such that | f (s)| ≤ cβ s for some finite number c. Denote the stationary distribution of P by π (and the stationary projector of P by Π ) and denote the stationary distribution of P by π (and the stationary operator of  ). Let P by Π D = ∑ (Pn − Π ). n≥0

Elementary calculation shows

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Karim Abbas and Djamil A¨ıssani

(I − P)D = I − Π .  and using the fact that it holds that Π  Π = Π yields Multiplying this equation by Π  −Π.  (I − P)D = Π Π  =Π  P,  we obtain Using that Π  = Π +Π  (P − P)D. Π

(2)

 yields Inserting the righthand side of the above expression repeatedly for Π  =Π Π

∞

∑ ((P − P)D)n.

(3)

n≥0

Switching from matrix to vector notation we arrive at the following basic series expansion

π = π

∞

∑ ((P − P)D)n .

(4)

n≥0

We say that the Markov chain X with transition kernel P verifying Pυ < ∞ and invariant measure π is strongly υ -stable, if every stochastic transition kernel P in some neighborhood {P : P − Pυ < ε } admits a unique invariant measure π such that π − π υ tends to zero as P − Pυ tends to zero. The key criterion of strong stability of a Markov chain X is the existence of a deficient version of P defined in the following. Let X be a Markov chain with the transition kernel P and invariant measure π . We call a deficient Markov kernel T a residual for P with respect to  · υ if there exists a measure σ and a nonnegative measurable function h on N satisfying the following conditions: a. b. c. d.

π h > 0, σ 1 = 1, σ h > 0, and the operator T = P − h ◦ σ is nonnegative, the norm of the operator T is strictly less than one, i.e., T υ < 1, Pυ < ∞,

where ◦ denotes the convolution between a measure and a function and 1 is the vector having all the components equal to 1. It has been shown in A¨ıssani and Kartashov (1983) that a Markov chain X with the transition kernel P is strongly stable with respect to υ if and only if a residual for P with respect to υ exists. Although the strong stability approach originates from stability theory of Markov chains, the techniques developed for the strong stability approach allow to establish numerical algorithms for bounding π − π υ (Abbas and A¨ıssani, 2010a,b; Bouallouche-Medjkoune and A¨ıssani, 2006; Rabta and A¨ıssani, 2005). To see this, revisit the series expansion in (4). The following equality has been established independently by Kartashov (1986) and Hordijk and Spieksma (1994):

Perturbation Analysis of M/M/1 Queue

7

D = (I − Π ) ∑ T n (I − Π ).

(5)

n≥0

It is worth noting that for the above relation to hold it is necessary that T is a residual    with respect to P. Note that (P−P)(I − Π ) = (P−P) and multiplying (5) by (P−P) yields (P − P)D = (P − P) ∑ T n (I − Π ). n≥0

Inserting the above into (3) yields

π − π = π

∞

n



∑

(P − P)

n≥1

∑T

m

(I − Π )

.

(6)

m≥0

Based on this series expansion, a bound on π − π υ is established in the following theorem. Theorem 0.1. (Kartashov, 1986) Let P be strongly stable. If P − Pυ <

1 − T υ I − Π υ

then, we the following bound holds π − π υ ≤ π υ

I − Π υ P − Pυ . 1 − T υ − I − Π υ P − Pυ

Proof. Under the assumptions of the theorem, it holds that             m n   (P − P) ∑ T (I − Π ) ≤ (P − P)υ  ∑ T  I − Π υ   n≥0  m≥0 υ

υ

I − Π υ ≤ (P − P)υ , 1−ρ

where we use the fact that strong stability implies that T υ < 1. Provided that (P − P)υ <

1 − T υ , I − Π υ

the series in (6) converges in υ -norm sense and we obtain with the help of the above inequality that   n   ∞  1 − T υ   m ,  ∑ (P − P) ∑ T (I − Π )  ≤ n≥0  1 − T υ − (P − P)υ I − Π υ m≥0 υ which proves the claim.

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Karim Abbas and Djamil A¨ıssani

Note that the term I − Π υ in the bound provided in Theorem 0.1 can be bounded by I − Π υ ≤ 1 + 1υ π υ .

3 Analysis of the Model We first consider the M/M/1 queue with arrival rate λ and service rate μ . Let ρ = λ /μ . Markov kernel P is then given by ⎧ λ ⎪ λ +μ j = i + 1, ⎪ ⎪ ⎪ ⎨ μ Pi j = λ + μ j = i − 1, i > 0, ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise, and P00 = 1 − λ /(λ + μ ). The kernel P of the M/M/1 queue with the service rate μ + ε , such that μ + ε > λ , is given by ⎧ λ ⎪ ⎪ λ +μ +ε j = i + 1, ⎪ ⎪ ⎪ ⎨ μ +ε Pi j = λ + μ +ε j = i − 1, i > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 otherwise, and P00 = 1 − λ /(λ + μ + ε ). In the following we will derive bounds for the effect on the stationary distribution of the queue length in the M/M/1 queue when we increased the service rate. For our bounds, we require bounds on the basic input entities such as π and T . In order to establish those bounds, we have to specify υ . Specifically, for β > 1, we will choose υβ (s) = β s , s ∈ S, as our norm-defining mapping. For ease of reference, we introduce the following condition: (C)

ρ <1

and

1<β <

1 . ρ

Essential for our numerical bound on the deviation between stationary distributions π and π is a bound on the deviation of the transition kernel P from P. This bound is provided in the following lemma. Lemma 0.1. If condition (C) is satisfied, then

Perturbation Analysis of M/M/1 Queue

P − Pυβ ≤

9

λ (1 + β )ε = Δ (β ). (λ + μ )(λ + μ + ε )

Proof. By definition, we have 1 k≥0 υ (k)

P − Pυβ = sup

∑ υ ( j)|Pk j − Pk j |.

j≥0

For k = 0:

Δ0 =

∑ υ ( j)|Pk j − Pk j |

j≥0

= υ (0)|P00 − P00 | + υ (1)|P01 − P01 | λ (1 + β )ε . = (λ + μ )(λ + μ + ε ) For k ≥ 1:

Δ1 = sup k≥1

1 βk

∑ β j |Pk j − Pk j |

j≥0

    λ λ  μ  k−1  μ + ε +β − λ + μ +ε − λ + μ  λ +μ +ε λ +μ  

ε λ = λβ + (λ + μ )(λ + μ + ε ) β

1 = sup k k≥1 β



 k+1  β 

Note that P − Pυβ = max{Δ0 , Δ1 }, and from Δ1 < Δ0 , it follows that P − Pυβ ≤ Δ0 = Δ (β ). In the following lemma we will identify the range for β that leads to finite υβ norm of π . Lemma 0.2. Provided that (C) holds, the υβ -norm of π is bounded by π υβ =

1−ρ = c0 (β ) < ∞. 1 − ρβ

Proof. The stationary distribution of P is known to be equal to π (i) = ρ i (1 − ρ ). Hence, ∞

π υβ = (1 − ρ ) ∑ β i ρ i , i=0

which is finite if β ρ < 1. Note that ρ < 1 is assumed for stability and β > 1 is assumed in the definition of υβ .

10

Karim Abbas and Djamil A¨ıssani

Lemma 0.3. If condition (C) holds, then I − Π υ ≤ Proof. We have

2 − ρ (β + 1) = c(β ). 1 − ρβ

I − Π υ ≤ 1 + 1υ π υ .

By definition, we obtain 1υ = 1, and 1υ π v = c0 (β ). Therefore, I − Π υ ≤ 1 + c0 (β ) = c(β ). Let T denote the taboo Markov kernel for taboo state zero, that is, T is a deficient Markov kernel that avoids jumps to state zero; more specifically, for i, j let ⎧ ⎨ 0, if i = 0, (7) Ti j = ⎩ Pi j , otherwise. Lemma 0.4. Provided that condition (C) holds, it hold that T υβ = τ (β ) < 1. Proof. For k = 0: T υ (0) =

∑ υ ( j)T0 j = ∑ β j × 0 = 0.

j≥0

j≥0

For k ≥ 1: T υ (k) =

∑ υ ( j)Pk j

j≥0



 λ μ υ (k + 1) × + υ (k − 1) × λ +μ λ +μ

 μ λ 1 + = βk β λ +μ β λ +μ = τ (β ) × υ (k), =

(8) (9)

with τ (β ) = (λ β /(λ + μ )) + (μ /β (λ + μ )). Thus, T υ = τ (β ), and it follows that the υ -norm of T is equal to τ (β ). Provided that ρβ < 1, it holds that

τ (β ) =

1 + ρβ 2 , (1 + ρ )β

Perturbation Analysis of M/M/1 Queue

11

which yields τ (β ) < 1. To summarize,

ρβ < 1 ⇒ T υ = τ (β ) < 1, for υ (k) = β k . which proves the claim. Lemma 0.5. Provided that condition (C) holds, Pυβ is finite. Proof. Let H denote the matrix with row zero equal to P and all other rows equal to zero. Then, T = P − H ⇒ P = T + H ⇒Pυβ ≤ T υβ + Hυβ . By Lemma 0.4, it holds that T υβ < ∞ and for the proof of the claim it suffices to show that Hυβ < ∞, which follows from Hυβ = Let

μ

λ

∑ β j P0 j = λ + μ + β λ + μ < ∞.

j≥0

B = sup {β > 1 : τ (β ) < 1} .

Theorem 0.2. Let ρ < 1. For β ∈ B, the deficient Markov kernel T defined in (7) is a residual of the Markov chain P with respect to υβ for provided that τ (β ) < 1. Proof. Let h(i) = 1i=0 = choose σ to be

⎧ ⎨ 1, if i = 0; ⎩

0, otherwise,

⎧ μ ⎪ λ +μ , if j = 0; ⎪ ⎪ ⎪ ⎨ λ σ j = P0 j = λ + μ , if j = 1; ⎪ ⎪ ⎪ ⎪ ⎩ 0, otherwise.

In the following we will show that conditions a, b, c, and d hold for T defined in (7). We start by verifying condition a and b. With the above definitions it holds that

π h = π0 = (1 − ρ ) > 0, and μ λ σ1 = λ+ μ + λ +μ + 0 = 1, here 1 is the vector having all the components equal to 1,

μ μ λ σh = λ+ + 0 × μ λ +μ + 0 = λ +μ > 0. In the same way, Ti j = Pi j − h(i)σ j =

⎧ ⎨ P0 j − σ j = P0 j − P0 j = 0, if i = 0; ⎩

Pi j − (0 × σ j ) = Pi j ,

otherwise,

12

Karim Abbas and Djamil A¨ıssani

and from the definition of the kernel (Pi j )i j , it is obvious that (Ti j )i j ≥ 0. We not turn to condition c. In the following we will provide a sufficient condition such that T υ < 1 for υ (k) = β k , for all k ≥ 0, with β > 1. By Theorem 0.2, the general bound provided Theorem 0.1 can be applied to the Markov kernels P and P for our M/M/1 queue. Specifically, we will insert the individual bounds provided in Lemma 0.1, Lemma 0.2, Lemma 0.3 and Lemma 0.4, which yields the following result. Theorem 0.3. Let ρ < 1. For β ∈ B such that

Δ (β ) < it holds that π − π υβ ≤

1 − τ (β ) , c(β )

c0 (β )c(β ) Δ (β ) = SSB(β ). 1 − τ (β ) − c(β ) Δ (β )

Proof. Note that β ∈ B already implies c0 (β ) < ∞ and τ (β ) < 1. Hence, Lemma 0.2 and Lemma 0.4 apply. Following the line of thought put forward in Section 2, see (1), we will translate the norm bound in Theorem 0.3 to bounds for individual performance measures f . Corollary 0.1. Under the conditions put forward in Theorem 0.3, it holds for any f such that  f υβ < ∞ that |π f − π f | ≤  f υβ × SSB(β ) = h(ε , β ). Note that the bound in Corollary 0.1 has β as a free parameter. This give the opportunity to minimize the right hand side of the inequality in Corollary 0.1 with respect to β . For given ρ , this leads to the following optimization problem. min h(ε , β ) β ∈B

s.t.

Δ (β ) <

1 − τ (β ) . c(β )

By inserting ε > 0 small, all inequalities can be made strict and in the above optimization problem can be solved using any standard technique.

4 Numerical Examples In this section we will apply our bounds put forward in Theorem 0.3 and Corollary 0.1. For the numerical examples we set μ = 1. We will discuss the following three cases in detail: the light traffic case ρ = 0.1, the medium traffic case ρ = 0.6,

Perturbation Analysis of M/M/1 Queue

13

and the heavy traffic case ρ = 0.9. To illustrate the application of Corollary 0.1 to a particular performance function, we take f (s) = s the identical mapping. In words, we are interested in the effect of perturbing the service rate by ε on the mean queue length. It is worth noting that in this case  f υβ =

1 − 1 β ln(β ) . ln(β )

The light traffic case: We let ρ = 0.1. For applying our bounds we compute the value for βopt that minimizes h(ε , β ). Then, we can compute the bounds put forward in Theorem 0.3 and Corollary 0.1 for various values for ε . The numerical results are presented in Table 1 Table 1 Errors comparative table for ρ = 0.1

ε

βopt

π − πβ |π f − π f | Bound True Bound True 0.01 2.6500 0.0021 5.0481e − 004 7.7964e − 004 1.9056e − 004 0.1 2.6000 0.0204 0.0049 0.0079 0.0019 1 3 0.2641 0.0539 0.0884 0.0180

The medium traffic case: We let ρ = 0.6. In a similar manner, the mapping h(ε , β ) is minimized at βopt . The perturbation ε = 1 is excluded for the medium traffic case. The numerical results are presented in Table 2, where the symbol ”x” indicates that our bounds are not applicable. Table 2 Errors comparative table for ρ = 0.6

ε

βopt

π − πβ |π f − π f | Bound True Bound True 0.01 1.2600 0.0945 0.0041 0.1504 0.0065 0.1 1.2000 1.6245 0.0294 3.2778 0.0594 1 x x x x x

The heavy traffic case: We let ρ = 0.9. The mapping h(ε , β ) is minimized at βopt . Like for the medium traffic case, ε = 1 is excluded. The numerical results are presented in Table 3, where the symbol ”x” indicates that our bounds are not applicable. Table 3 Errors comparative table for ρ = 0.9

ε

βopt

π − πβ |π f − π f | Bound True Bound True 0.01 1.0480 0.2783 0.0021 2.1837 0.0166 0.1 1.0500 2.2340 0.0090 16.8448 0.0680 1 x x x x x

14

Karim Abbas and Djamil A¨ıssani

From these numerical results, it is easy to see that, the values of our bounds increase as the value of perturbation’ parameter ε increases. Indeed, it is completely logical that the M/M/1 queue with service rate μ + ε is close to the M/M/1 queue with the same arrival flux and distribution of service time when ε tends to zero. Besides, we can notice the remarkable sensitivity of the Strong Stability Approach in the variation of the perturbation’ parameter ε with regard to the value of the true distance. The bound obtained by the Strong Stability Approach is much more refined than that obtained by the true distance when the parameter ε tends to zero.

5 Concluding Remarks The only input required from the perturbed queue is distance in υ -norm between P  In this paper, we discussed an example, where P and P are Markov chains and P. and a bound on P − Pυ can be computed. On the other hand, as π˜ is known in closed form, one can compute π˜ − π υ directly and there is no imminent reason for applying our approach, other than the tutorial value of explaining the method for a simple example. As further research, we will show how to estimate P − Pυ from a single sample path of the G/G/1 queue. This is topic of further research.

References Abbas K, A¨ıssani D (2010a) Strong stability of the embedded Markov chain in an GI/M/1 queue with negative customers. Applied Mathematical Modelling 34(10):2806–2812 Abbas K, A¨ıssani D (2010b) Structural perturbation analysis of a single server queue with breakdowns. Stochastic Models 26(1):78–97 Albin S (1982) On Poisson approximations for superposition arrival processes in queues. Management Science 28(2):126–137 Albin SL (1984) Analyzing M/M/1 queues with perturbations in the arrival process. The Journal of the Operational Research Society 35(4):303–309 Altman E, Avrachenkov KE, N´un˜ ez-Queija R (2004) Perturbation analysis for denumerable Markov chains with application to queueing models. Advances in Applied Probability 36(3):839–853 Anisimov V (1988) Estimates for the deviations of the transition characteristics of nonhomogeneous Markov processes. Ukrainian Mathematical Journal 40(6):588–592 A¨ıssani D, Kartashov N (1983) Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels. In: Doklady Akademii Nauk Ukrainskoi SSR 11, (seriya A), vol 11, pp 3–5 Bouallouche-Medjkoune L, A¨ıssani D (2006) Performance analysis approximation in a queueing system of type M/G/1. Mathematical Methods of Operations Research 63(2):341–356

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Cho G, Meyer C (2001) Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra and its Applications 335(1–3):137–150 Fricker C, Guillemin F, Robert P (2009) Perturbation analysis of an M/M/1 queue in a diffusion random environment. Queueing Systems: Theory and Applications 61(1):1–35 Heidergott B (2008) Perturbation analysis of Markov chains. In: Proceedings of the International Workshop on DES, G¨oteborg, Sweden Heidergott B, Hordijk A (2003) Taylor series expansions for stationary Markov chains. Advances in Applied Probability 35(4):1046–1070 Heidergott B, Hordijk A, van Uitert M (2007) Series expansions for finite-state Markov chains. Probability in the Engineering and Informational Sciences 21(03):381–400 Hordijk A, Spieksma F (1994) A new formula for the deviation matrix. In: Kelly F (ed) Probability, Statistics and Optimization, Wiley Kartashov N (1986) Strongly stable Markov chains. Journal of Soviet Mathematics 34:1493–1498 Kartashov N (1996) Strong stable Markov chains. VSP/TbiMC, Utrecht/Kiev, Utrecht/Kiev Kirkland S (2002) On a question concerning condition numbers for Markov chains. SIAM Journal on Matrix Analysis & Applications 23(4):1109–1119 Kotzurek M, Stoyan D (1976) A quantitative continuity theorem for mean stationary waiting time in GI/G/I. Math Operations forsch Statist 7:595–599 Mitrophanov AY (2005) Sensitivity and convergence of uniformly ergodic Markov chains. Journal of Applied Probability 42(4):1003–1014 Neumann M, Xu J (2004) Improved bounds for a condition number for Markov chains. Linear Algebra and its Applications 386:225–241, special Issue on the Conference on the Numerical Solution of Markov Chains 2003 Rabta B, A¨ıssani D (2005) Strong stability in an (R,s,S) inventory model. International Journal of Production Economics 97(2):159–171 Rachev S (1989) The problem of stability in queueing theory. Queueing Systems 4:287–318 Whitt W (1981) Quantitative continuity results for the GI/G/1 queue. Tech. rep., Bell Laboratories Report Zolotarev V (1977) General problems of the stability of mathematical models. Proceedings of the session the 41st International Statistical Institute: New Delhi

Series Expansions in Queues with Server Vacation Fazia Rahmoune and Djamil A¨ıssani

Abstract This paper provides series expansions of the stationary distribution of finite Markov chains. The work presented is a part of research project on numerical algorithms based on series expansions of Markov chains with finite state-space S. We are interested in the performance of a stochastic system when some of its parameters or characteristics are changed. This leads to an efficient numerical algorithm for computing the stationary distribution. Numerical examples are given to illustrate the performance of the algorithm, while numerical bounds are provided for quantities from some models like manufacturing systems to optimize the requirement policy or reliability models to optimize the preventive maintenance policy after modelling by vacation queuing systems.

1 Introduction Let P denote the transition kernel of a Markov chain defined on a finite state-space S having unique stationary distribution πP . Let Q denote the Markov transition kernel of the Markov chain modeling the alternative system and assume that Q has unique stationary distribution πQ . The question about the effect of switching from P to Q on the stationary behavior is expressed by πP − πQ , the difference between the stationary distributions (Heidergott and Hordijk, 2003). In this work, we show that the performance measure of some stochastic models, which are gouverned by a finite Markov chain, can be obtained from other performance of more simple models, via series expansion method. Let  . tv denote the total variation norm, then the above problem can be phrased as follows: Can  πP − πQ tv be approximated or bounded Fazia Rahmoune (B) and Djamil A¨ıssani LAMOS Laboratory of Modelling and Optimization of Systems - University of Bejaia 06000, Algeria, e-mail: [email protected] Djamil A¨ıssani e-mail: lamos [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 2, 

17

18

Fazia Rahmoune and Djamil A¨ıssani

in terms of  P − Q tv ? This is known as perturbation analysis of Markov chains (PAMC) in the literature. This paper is considered as a continuity of the work (Rahmoune and A¨ıssani, 2008), where quantitative estimate of performance measure has been established via strong stability method for some vacation queueing models. In this work, we will show that πP − πQ can be arbitrarily closely approximated by a polynomial in (Q − P)DP , where DP denotes the deviation matrix associated with P. A precise definitions and notations will be given later. Starting point is the representation of πQ given by:

πQ =

k

∑ πP ((Q − P)DP )n + πQ ((Q − P)DP )k+1 ;

(1)

n=0

for any k ≥ 0. This series expansion of πQ provides the means of approximating πQ by Q and entities given via the P Markov chain only. We obtain a bound for the remainder term working with the weighted supremum norm, denoted by  . v , where v is some vector with positive non-zero elements, and for any w ∈ RS  w v = sup i∈S

w(i) , v(i)

(2)

see, for example (Meyn and Tweedie, 1993). We will show that for our models  

πQ (s) −

k

∑ πP ((Q − P)DP)n

(s) ≤ d  ((Q − P)DP ) k+1 v

n=0

for any k ∈ N and any s ∈ S, where v can be any vector satisfying v(s) ≥ 1 for s ∈ S, and d is some finite computable constant. In particular, the above error bound can be computed without knowledge of πQ . The key idea of the approach is to solve for all k the optimization problem

min  ((Q − P)DP )k v , subject to v(s) ≥ 1, for s ∈ S.

(3)

The vector v thus yields the optimal measure of the rate of convergence of the series in (1). Moreover, the series in (1) tends to converge extremely fast which is due to the fact that in many examples v be found such that  ((Q−P)DP )k v << 1. The limit of the series (1) first appeared in (Cao, 1998), however, neither upper bounds for the remainder term nor numerical examples were given there. The derivation of this has been done in (Heidergott and Hordijk, 2003), which is a generalization of (Cao, 1998). The use of series expansion for computational purposes is not new. It has been used in the field of linear algebra (Cho and Meyer, 2001). The work presented in this paper is part of research project on numerical algorithms based on series expansions of Markov chains as it was in Heidergott and Hordijk

Series Expansions in Queues with Server Vacation

19

(2003). The present paper establishes the main theoretical results. In particular, numerical examples are provided for vacation queueing systems.

2 Preliminaries on Finite Markov chains Let S denote a finite set {1, · · · , S}, with 0 < S < ∞ elements. We consider Markov kernels on state space S, where the Markov kernel Pn is simply obtained from taking the nth power of P. Provided it exists, we denote the unique stationary distribution of P by πP and its ergodic projector by ΠP . For simplicity, we identify πP and πQ with ΠP and ΠP , respectively. Throughout the paper, we assume that P is aperiodic and unichain, which means that there is one closed irreducible set of states and a set of transient states. Let |A|(i; j) denote the (i; j)th element of the matrix of absolute values of A ∈ RS×S , and additionally we use the notation |A| for the matrix of absolute values of A. The main tool for this analysis is the v-norm, as defined in (2). For a matrix A ∈ RS×S the v-norm is given by S

de f

 A v =

∑ |A(i, j)w( j)|

sup i,wv ≤1

j=1

v(i)

Next we introduce v-geometric ergodicity of P, see Meyn and Tweedie (1993) for details. Definition 0.1. A Markov chain P is v-geometric ergodic if c < ∞, β < 1 and N < ∞ exist such that  Pn − ΠP v ≤ cβ n , for all n ≥ N. The following lemma shows that any finite-state aperiodic Markov chain is vgeometric ergodic. Lemma 0.1. For finite-state and aperiodic P a finite number N exists such that  Pn − ΠP v ≤ cβ n , for all n ≥ N; where c < ∞ and β < 1. Proof. Because of the finite state space and aperiodicity.

3 Series Expansions in Queues with Server Vacation We are interested in the performance of a queuing system with single vacation of the server when some of its parameters or characteristics are changed. The system as

20

Fazia Rahmoune and Djamil A¨ıssani

given is modeled as a Markov chain with kernel P, the changed system with kernel Q. We assume that both Markov chains have a common finite state space S. We assume too, as indicated earlier, that both Markov kernels are aperiodic and unichain. The goal of this section is to obtain the stationary distribution of Q, denoted by πQ , via a series expansion in P. In the next section, we comment on the speed of convergence of this series. We summarize our results in an algorithm, presented in an other section. Finally, we illustrate our approach with numerical examples.

3.1 Description of the models Let us consider the M/G/1//N queueing systems with multiples vacations of the server modelling the reliability system with multiple preventives maintenances. We suppose that there is on the whole N machines in the system. Our system consists of a source and a waiting system (queue + service). Each machine is either in source or in the waiting system at any time. A machine in source arrives at the waiting system precisely, with the durations of inter-failure exponentially distributed with parameter λ ) for repairment (corrective maintenance). The distribution of the service time is general with a distribution function B(.) and mean b. The repairmen take maintenance period at each time the system is empty. If the server returns from maintenance finding the queue impty, he takes an other maintenance period (mulitiple maintenance). In addition, let us consider the M/G/1//N queuing system with unique vacation of the server modelling the reliability system with periodic prenventives maintenances, having the same distributions of the inter-arrivals and the repair time previously described. In this model, the server (repairman) will wait until the end of the next activity period during which at least a customer will be served, before beginning another maintenance period. In other words, there is exactly only one maintenance at the end of each activity period at each time when the queue becomes empty (exhaustive service). If the server returns from maintenane finding the queue nonempty, then the maintenance period finishes for beginning another activity period. We also suppose that the maintenance times V of the server are independent and iid, with general distribution function noted V (x).

3.2 Transition Kernels Let Xn (resp. X¯n ) the imbedded Markov chains at the end moments of repair tn for the nth machine associated with the M/G/1//N system with multiple maintenance (resp.to the system with the unique maintenance). In the same way, we define the following probabilities: fk = P[k broken down machines at the end of the preventive maintenance period]

Series Expansions in Queues with Server Vacation

= CNk

 ∞ 0

(1 − e−λ t )k e−(N−k)λ t dV (t), k = 0, N.

for witch

α¯ k = and

αk =

21

(4)

f0 + f1 , for k = 1 for 2 ≤ k ≤ N. fk , fk for k = 1, N. 1 − f0

The one stage transition probabilities of the imbedded Markov chains Xn and X¯n allow us to describe the general expression of the transition kernels P = (Pi j )i j and Q = (Qi j )i j summarized below respectively. ⎧ j+1 ⎪ ⎪ ∑ Pj−k+1 αk , if i = 0, j = 0, N − 1, k = 1, N, ⎪ ⎨ k=1 Pi j = if 1 ≤ i ≤ j + 1 ≤ N − 1, Pj−i+1 ⎪ ⎪ ⎪ ⎩ 0 else.

Qi j =

⎧ j+1 ⎪ ⎪ ⎪ ⎨ ∑ Pj−k+1 α¯ k , if i = 0, j = 0, N − 1, k = 1, N, k=1

if 1 ≤ i ≤ j + 1 ≤ N − 1, else.

Pj−i+1 ⎪ ⎪ ⎪ ⎩ 0

Clearly, the Markov chain {X¯n }n∈N is irreducible, aperiodic with finite state space S = {0, 1, · · · , N − 1}. So, we can applied the main theorical results established in this paper to this model, in order to approach another Markov chain whose transition kernel is neighborhood of its transition kernel Q.

3.3 Series Development for πQ We write DP for the deviation matrix associated with P; in symbols: ∞

DP =

∑ (Pm − ΠP )

(5)

m=0

Note that DP is finite for any aperiodic finite-state Markov chain. Moreover, the deviation can be rewritten as DP =

∞

∑ (P − ΠP )m − ΠP ,

m=0 ∞

where ∑ (P − ΠP )m is often referred to as the group inverse, see for instance Cao m=0

(1998) or Coolen-Schrijner and van Doorn (2002). A general definition which is

22

Fazia Rahmoune and Djamil A¨ıssani

valid for periodic Markov chain, can be found in, e.g., Puterman (1994). Let P be unichain. Using the definition of DP , we obtain: (I − P)DP = I − ΠP . This is the Poisson equation in matrix format. Let the following equation:

ΠQ = ΠP

k

∑ ((Q − P)DP )n + ΠQ ((Q − P)DP)k+1 .

(6)

n=0

for k ≥ 0, where: de f

H(k) = ΠP

k

∑ ((Q − P)DP )n,

n=0

is called a series approximation of degree k for ΠQ , T (k), with de f

T (k) = ΠP ((Q − P)DP )k ,

(7)

denotes the kth element of H(k), and de f

R(k) = ΠQ ((Q − P)DP )k+1 ,

(8)

is called the remainder term (see Heidergott et al, 2007, for details). The quality of the approximation provided by H(k) is given through the remainder term R(k).

3.4 Series Convergence In this section we investigate the limiting behavior of H(k) as k tends to ∞. We first establish sufficient conditions for the existence of the series. Lemma 0.2. (Heidergott and Hordijk, 2003) The following assertions are equivalent: ∞

(i) The series ∑ ((Q − P)DP )k is convergent. k=0

(ii) There are N and δN ∈ (0, 1) such that  ((Q − P)DP )N v < δN . (iii) There are κ and δ < 1 such that  ((Q − P)DP )k v < κδ k for any k. (iv) There are N and δ ∈ (0; 1) such that ((Q − P)DP )k v < δ k for any k ≥ N. Proof. See Heidergott and Hordijk (2003). The fact that the maximal eigenvalue of |(Q − P)DP | is smaller than 1 is necessary ∞

for the convergence of the series ∑ ((Q − P)DP )k . k=0

Series Expansions in Queues with Server Vacation

23

Remark 0.1. Existence of the limit of H(k), see (i) in Lemma 0.3, is equivalent to an exponential decay in the v-norm of the elements of the series, see (iv) in Lemma 0.2. For practical purposes, one needs to identify the decay rate δ and the threshold value N after which the exponential decay occurs. The numerical experiments have shown that the condition (ii) in Lemma 0.2 is the most convenient to work with. More specifically, we work with the following condition (C) as in Heidergott and Hordijk (2003), which is similar to the geometric series convergence criterion. The Condition(C): There exists a finite number N such that we can find δN ∈ (0; 1) which satisfies:  ((Q − P)DP )N v < δ N ; and we set de f

cvδN =

N−1 1  ∑ ((Q − P)DP )k v 1 − δN k=0

As shown in the following lemma, the factor cvδN in condition (C) allows to establish an upper bound for the remainder term that is independent of ΠQ . Lemma 0.3. Under (C) it holds that: (i)  R(k − 1) v ≤ cvδN  T (k) v for all k, ∞

(ii) lim H(k) = ΠP ∑ ((Q − P)DP )n = ΠQ k→∞

n=0

Proof. To proof the lemma it is sufficient to use the definition of the norm  . v and the remainder term R(k − 1), using the condition (iv) of Lemma 0.2. Remark 0.2. An example where the series H(k) fails to converge is illustrated in Heidergott and Hordijk (2003). Remark 0.3. The series expansion for ΠQ put forward in the assertion (ii) in Lemma 1 is well known; see Cao (1998) and Kirkland (2003) for the case of finite Markov chains and Heidergott and Hordijk (2003) for the general case. It is however worth noting that in the aforementioned papers, the series was obtained via a differentiation approach, whereas the representation is derived in this paper from the elementary equation 6. Remark 0.4. Provided that det(I − (Q − P)DP ) = 0, one can obtain πQ from

πQ = ΠP (I − (Q − P)DP )−1

(9)

Moreover, provided that the limit lim H(k) = lim πP

k→∞

k→∞

∞

∑ ((Q − P)DP )n

n=0

exists (see Lemma 0.3 for sufficient conditions), it yields πQ as πP ∑∞ n=0 ((Q − P)DP )n .

24

Fazia Rahmoune and Djamil A¨ıssani

Remark 0.5. Note that a sufficient condition for (C) is  (Q − P)DP v < δ , δ < 1.

(10)

In Altman et al (2004); Cho and Meyer (2001) it is even assumed that  (Q − P)DP v < g1 ,

(11)

with g1 > 0 a finite constant, and  D P v <

c , 1−β

(12)

with c > 0 and 0 < β < 1 finite constants. If g1 c < 1, 1−β

(13)

then (10) and hence (C) is clearly fulfilled. Hence, for numerical purposes these conditions are too strong.

3.5 The remainder term Bounds The quality of approximation by H(k − 1) is given by the remainder term R(k − 1) and in applications v should be chosen such that it minimizes cvδN  T (k) v , thus minimizing our upper bound for the remainder term. For finding an optimal upper bound, since cvδN is independent of k, we focus on T (k). Specifically, we have to find a bounding vector v that minimizes  T (k) v uniformly w.r.t. k. As the following theorem shows, the unit vector, denoted by 1, with all components equal to one, yields the minimal value for  T (k) v for any k. Theorem 0.1. (Heidergott and Hordijk, 2003) The unit vector 1 minimizes  T (k) v uniformly over k, i.e., ∀k ≥ 1 : in fv  T (k) v = T (k) 1

(14)

Remark 0.6. It can be shown as for the results in Altman et al (2004) and Cho and cg1 Meyer (2001), that the smallest 1− β is precisely the maximal eigenvalue of |DP |. Again we note that often the product of these maximal eigenvalues is not smaller than 1. If this is the case, then according to Altman et al (2004) and Cho and Meyer (2001) we cannot decide whether the series H(k) converges to ΠQ . Hence, their condition is too restrictive for numerical purposes.

Series Expansions in Queues with Server Vacation

25

3.6 Algorithm In this section we describe a numerical approach to computing our upper bound for de f

the remainder term R(k). We search for N such that 1 > δN =  ((Q − P)DP )N 1 , which implies that for N and δN , the condition (C) holds. Then the upper bound for R(k) is obtained from c1δN  ((Q − P)DP )k+1 1 . Based on the above, the algorithm that yields an approximation for πQ with ε precision can be described, with two main parts. First c1δN is computed. Then, the series can be computed in an iterative way until a predefined level of precision is reached.

The Algorithm Chose precision ε > 0. Set k = 1, T (1) = ΠP (Q − P)DP and H(0) = ΠP . Step 1: Find N such that  ((Q − P)DP )N 1 < 1. Set δN = ((Q − P)DP )N 1 and compute N−1 1 c1δN =  ∑ ((Q − P)DP )k 1 . 1 − δN k=0 Step 2: If

cδ  T (k) 1 < ε ,

the algorithm terminates and H(k − 1) yields the desired approximation. Otherwise, go to step 3. Step 3: Set H(k) = H(k − 1) + T (k). Set k := k + 1 and T (k) = T (k − 1)(Q − P)DP . Go to step 2. ∞

Remark 0.7. Algorithm 1 terminates in a finite number of steps, since ∑ ((Q − P)DP )k is finite, .

k=0

3.7 Numerical Application The present paper established the main theoretical results, and the analysis provided applies to the case of optimization of preventive maintenance in reparable reliability models. The application of this algorithm step by step gives us the following results. This part of the paper is reserved for theoretical and numerical results obtained via series expansion method to obtain the development of the stationary distribution of the M/G/1//N queueing models with single server vacation, witch modeless reliability system with preventive maintenance.

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Fazia Rahmoune and Djamil A¨ıssani

Let S the state space of the imbedded Markov chains Xn and X¯n of the both considered queueing systems. Note that the both chains are irreducible and aperiodic, with finite state space S, so they are v-geometric ergodic. We note by DP the deviation matrix associated to X¯n chain, and by πP its stationary distribution, with stationary projector ΠP . In the same time, πQ is the stationary distribution of Xn , with the projector ΠQ . We want to express πQ in terms of puissance series on (P − Q)DP and πP as follows: ∞

πQ =

∑ πP ((Q − P)DP )n ;

(15)

n=0

We show that this series is convergent. In fact, since the state space of the both chains is finite, so we can give the first following elementary result: Lemma 0.4. Let Xn and X¯n the imbedded Markov chains of the M/G/1//N queueing system with server vacation and the classical M/G/1//Nsystem respectively. Then, the finite number N exist and verified the following:  Pn − ΠP v ≤ cβ n , for all n ≥ N;

(16)

where c < ∞, β < 1. For the same precedent raisons we give the most important result about the deviation matrix DP associated to the imbedded Markov chain X¯n . Lemma 0.5. Let X¯n the imbedded Markov of the classical M/G/1//N queueing system and DP its deviation matrix. Then, DP is finite. Using Lemma 0.2, we obtain the following result about the required series expansion: Lemma 0.6. Let πP (resp. πQ ) the stationary distribution of the M/G/1//N classical system, (resp. M/G/1//N system with unique vacation), and DP the associated deviation matrix. Then, the series ∞

∑ πP ((Q − P)DP )n ;

(17)

n=0

converge normally then uniformly. This result is equivalent to say that the reminder term R(k) is uniformly convergent to zero. From the condition (C) and the Lemma 0.3, the sum function of the series 15 is the stationary vector πQ . Lemma 0.7. Let πP (resp. πQ ) the stationary distribution of the M/G/1//N classical system, (resp. M/G/1//N with vacation of the server), and DP the associated deviation matrix. Then, the series

Series Expansions in Queues with Server Vacation

πQ =

27

∞

∑ πP ((Q − P)DP )n ;

(18)

n=0

converge uniformly to the stationary vector πQ . From the work of Heidergot, we describe in this section a numerical approach to compute the supremum borne of the reminder term R(k). We ask about the number N as: δN = ((Q − P)DP )N 1 < 1, witch implies that the condition (C) is verified for N and δn . Then the limit of R(k) is obtained from:  ((Q − P)DP )k+1 1 < c1δN . The performance measure for witch we are interesting is the mean number of costumers at the stationary state in the system. The considered entries parameters are: N¯ = 5, λ = 2, service rate → Exp(μs = 5), vacation rate → Exp(μv = 300). Our goal is to compute approximatively the quantities π  w. The error to predict the stationary queue length via the quantities H(n) is then given and illustrated in the Figure1.

Fig. 1 Error in Average queue length

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Fazia Rahmoune and Djamil A¨ıssani

The figure show that

π  w − H(n)w | (19) π w is a graph on n. The numerical value of π w is 2.4956. For this example, we have obtained N = 14, δN = 0.9197 and c1δN = 201.2313. |

The algorithm terminates when the upper bound for  R 1 given by c1δN  R 1 is under the value ε . By taking ε = 10−2 , the algorithm compute π  w juste to the precision 10−2  w 1 .

Fig. 2 Relative error of the upper bound of the remainder term

13

From this figure we conclude that π ∑ ((Q − Q)D)k w approximates π  w with k=0

a maximal absolute error ε  w 1 = 3 ∗ 10−2 .

4 Conclusion In this work, we have presented a part of research project on numerical algorithms based on series expansions of finite Markov chains. We are interested in the performance of a stochastic system when some of its parameters or characteristics are perturbed. This leads to an efficient numerical algorithm for computing the stationary distribution. We have shown theoretically and numerically that introducing a small

Series Expansions in Queues with Server Vacation

29

disturbance on the structure of maintenance policy in M/G/1//N system with multiples maintenances after modelling by queues with server vacation, we obtain the M/G/1//N system with single maintenance policy (periodic maintenance). Then characteristics of this system can be approximated by those of the M/G/1//N system with periodic maintenance, with a precision which depends on the disturbance, in other words on the maintenance parameter value.

References Altman E, Avrachenkov KE, N´un˜ ez-Queija R (2004) Perturbation analysis for denumerable Markov chains with application to queueing models. Advances in Applied Probability 36(3):839–853 Cao XR (1998) The Maclaurin series for performance functions of Markov chains. Advances in Applied Probability 30(3):676–692 Cho G, Meyer C (2001) Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra and its Applications 335(1–3):137–150 Coolen-Schrijner P, van Doorn EA (2002) The deviation matrix of a continuoustime Markov chain. Probability in the Engineering and Informational Sciences 16(03):351–366 Heidergott B, Hordijk A (2003) Taylor series sxpansions for stationary Markov chains. Advances in Applied Probability 35(4):1046–1070 Heidergott B, Hordijk A, van Uitert M (2007) Series expansions for finite-state Markov chains. Probability in the Engineering and Informational Sciences 21(03):381–400 Kirkland S (2003) Conditioning properties of the stationary distribution for a Markov chain. Electronic Journal of Linear Algebra 10:1–15 Meyn S, Tweedie R (1993) Markov chains and stochastic stability. Springer, London Puterman M (1994) Markov decision processes: discrete stochastic dynamic programming. John Wiley and sons Rahmoune F, A¨ıssani D (2008) Quantitative stability estimates in queues with server vacation. Journal of Stochastic Analysis and Applications 26(3):665–678

Part II

Rapid Modelling in Manufacturing and Logistics

Optimal Management of Equipments of the BMT Containers Terminal (Bejaia’s Harbor) Djamil A¨ıssani, Mouloud Cherfaoui, Sma¨ıl Adjabi, S. Hocine and N. Zareb

Abstract The BMT (Bejaia Mediterranean Terminal) Company of Bejaia’s harbor became aware that the performances of the terminal with containers is measured by the time of stopover, the speed of the operations, the quality of the service and the cost of container’s transit. For this end, the company has devoted several studies to analyze the performance of its terminal: elaboration of a global model for the ”loaded / unloaded” process, modeling of the system by another approach which consists of the decomposition of the system into four independent subsystems (namely: the ”loading” process, the ”unloading” process, the ”full-stock” process and the ”empty-stock” process - (see A¨ıssani et al, 2009; A¨ıssani et al, 2009). The models used in this last study describe in detail the comportment of the real systems and the obtained results given by the simulators corresponding to each model are approximately the same as the real values. It is the reason for which the company wants to exploit these models in order to determine an optimal management of its equipments. Indeed, this work consists, more specifically, in determining the optimal number of trucks to be used in each process that minimizes the waiting time of trucks and GQ (Gantry of Quay). This is a multi-objectives optimization problem, exactly a stochastic bi-objectives optimization problem. For that, we have modeled the problem by an open network which is the most suitable for this situation. After the identification of the process parameters, we conclude that the model is an open network of unspecified queues (G[X] /G/1, M/G/1, G/G/N/0, ...). In the literature, there is no exact method for analyzing this kind of networks. For this, we have established a simulation model that can imitate the functioning of each system. Djamil A¨ıssani (B), Mouloud Cherfaoui and Sma¨ıl Adjabi Laboratory LAMOS, University of Bejaia, e-mail: lamos [email protected] Mouloud Cherfaoui e-mail: [email protected] Sma¨ıl Adjabi e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 3, 

33

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The simulations allowed us to evaluate the performances of the park with containers according to the number of the trucks used, on the basis of current conditions and in the case of variation of the flow of arrivals of ships and the service rate of the trucks. This allowed us to determine the optimal number of trucks to be used in the loading process and unloading process. We have also determined the performance of the stock, on basis of current conditions.

1 Introduction The performance of a Terminal with containers is measured by the time of stopover, the speed of the operations, the quality of the service and the cost of container’s transit. For that, in order to ensure the best functioning of the Terminal with containers at the level of the BMT Company (Bejaia Mediterranean Terminal - Bejaia’s Harbor), studies of performance’s evaluation were initiated. The first study was carried out in 2007 (Sait et al, 2007). It had for objective the global modeling of the unloading/loading process and had shown that if the number of ships [having a mean size of 170 ETU (Equivalent Twenty Units)], which was of 0.83 ships/day, increases to 1.4 ships/day, the full park will undergo a saturation of 94%. The second study was carried out in 2009 (A¨ıssani et al, 2009; A¨ıssani et al, 2009). It was intended to suggest an alternative approach for modeling the system. The authors proposed an approach which consists of decomposing the system into four independent subsystems, namely: the loading process, the unloading process, the full stock process and the empty stock process. The study showed that the park with containers has a possibility to handle 116226 ETU for an entry rate of 0.6104 ships / day for the loading process and 0.7761 ships/day for the unloading process. The study showed also that for a 30% increase in the number of ships arriving at the port of Bejaia, we note a small increase in the average number of ships in roads and in the quays. On the other hand, there is a clear increasing in the total number of treated containers which passes from 116226 ETU to 148996 ETU. We note also an increase in the average number of containers in the full park which passes from 3372 to 4874 ETU. As for the number of treated ships, it passes from 240 to 305 ships at the loading and from 296 to 382 ships at the unloading. In the present work, we propose to supplement this last study where we try to minimize the number of trucks to use in the treatment of the ships. The interest of this analysis comes from the fact that the permanent increase of traffic constrains the BMT Company to exploit other quays. To this end, in order to optimize the existing equipments, the problem will be modeled by an open network, which belongs to the multi-objectives optimization problems. Because of the complexity and the unavailability of analytical methods for analyzing this type of models, we apply the simulation approach.

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2 Park with containers and motion of the containers In this section, we are going to give a brief description of the Terminal of BMT Company, where we will present different operations and movements of a container and its capacities and equipments.

2.1 Motions of the containers Any container (ship) arriving at the Terminal of BMT Company passes by the following steps: • The step of anchorage: Any ship arriving at the Bejaia harbor’s is put on standby in the anchorage (roads) for a duration of time which varies from one ship to another, because of the occupation of the quay stations or unavailability of pilot or tug boats. • The step of service: • Service of accosting: The accosting of the ships is ensured by the operational section of the Harbor Company of Bejaia, such as the section of piloting and towing. • Vessel handling: the treatment of a ship is done mainly in three sub-steps: 1. Service before operations: It is the preparatory step of the ship for the handling (Loading/Unloading). 2. Step of Unloading/Loading: It consists of the unloading/loading of the containers. This is carried out with the two gantries of quay which have carriages being able to raise the containers from the container ships, to put them on trucks and to raise the container from the trucks and put them on board the container ship, if it’s the loading process. 3. Service after operations: It is the preparatory step of the ship for accosting towards outside. • Deliveries: The delivery concerns the full containers or discharged goods. The means used to perform this operation are: RTG (Rubber Tyre Gantry), trucks, stacker and forklifts if it’s necessary. • Restitution of the containers: At the restitution of the containers (empty containers), two zones are intended for the storage, one for empty containers of 20 units and the other for empty containers of 40 units.

2.2 The BMT Park with containers: capacity and equipments The Terminal of the BMT Company is provided with four quays of 500 m (currently only two are in the exploitation), a draught of 12 m starting from the channel, and a storage capacity of 10300 ETU, the Terminal with containers of Bejaia offers

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specialized installations for the refrigerating containers and the dangerous products. Moreover, this Terminal is the only Terminal with containers in Algeria, sufficiently equipped and has specialized equipments (Gantry of Quay, RTG . . .), handling and lifting, which can reduce the times of stopover, making it possible to fulfill waiting and the requirements of the operators (See Table 1). Table 1 Characteristics and equipments of the Terminal of BMT Company Quay /Anchorage Length: Depth: Basin surface: Quay: Utilisation rate of the quay Full Park Capacity: Area: Empty Park Capacity: Area: Refrigerating Park Capacity: Area: Zone for Capacity: Discharge / Potting Area:

500 m 12 m 60 h 4 70% 8300 ETU 68500 m2 900 ETU 15200 m2 500 Catches 2800 m2 600 ETU 3500 m2

Numbers: Tonnage: Type: Numbers: RTG Tonnage: (Rubber Tyre Gantry) Stacking: Stakers Numbers: Tonnage: Spreaders Numbers: Tonnage: Lifting trucks Numbers: Gantry of Quay

Truck-Tug

2 40 Tons Post Panamax 5 36 Tons 6+1 on the ground and 4+1 in Height 4 36 Tons 4 10 Tons 02 of 03 Tons, 02 of 05 Tons 02 of 10 Tons and 02 of 28 Tons Numbers: 8 of 60 Tons and 4 of 32 Tons

3 Mathematical Models After analyzing the main movements of a container at the level of BMT’s Terminal, we chose to model the problem by network, which is most suitable for this type of situation. To this effect, we obtained four models, namely: the empty stock, the full stock, the loading and the unloading processes which are given respectively by Figures 1 and 2.

Fig. 1 Diagram of the models of the storage processes

Optimal Management of Equipments of the BMT Containers Terminal (Bejaia’s Harbor)

37

Fig. 2 Diagram of the models of the ship’s treatments (unloaded/loaded process)

4 Calculation of the Forecasts In February 2009, a calculation of forecast had been carried out. The designed series is the number of containers treated (loaded/unloaded) in ETU. The data used were collected monthly and were held forth over a period of two years (from January 2006 to February 2009). The method used for calculation of the forecasts is the exponential smoothing method (David and Michaud, 1983). Figure 3 and Table 2 represent the original series of the number of containers in ETU, as well as the forecasts (from March to December 2009). Thus, it is noted that the objective that BMT Company had fixed at the beginning of the year was likely to be achieved.

5 Performance Evaluation of the BMT Terminal First, we will conduct a statistical analysis to identify the network corresponding to our system.

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Table 2 Original series and forecasts of the number of containers to be treated (ETU) in the year 2009 Months Historic Forecast 2006 2007 2008 2009 January 4938 6102 9695 10066 February 6006 10083 9928 11448 March 6445 8565 9882 11579.74 April 5604 9535 8791 11941.29 12314.13 May 6519 8938 10155 June 5909 8337 8799 12698.61 13095.09 July 6041 7582 9338 August 7552 7245 9304 13503.96 September 5915 8135 9171 13925.59 14360.38 October 5938 7982 8779 November 7858 7579 10984 14808.75 December 7636 9971 11596 15271.12 Total 133498.7

Fig. 3 Graph of original series and forecasts of the number of containers to be treated (ETU) in the year 2009

5.1 Statistical Analysis and Identification of Models The results of the preliminary statistical analysis (estimate and test adjustment) on the collected data for the identification of process parameters are summarized in Table 3. According to this preliminary analysis, we conclude that the performance evaluation of the Terminal of Bejaia is really a complex problem. Indeed, the system is modeled by an opened network of unspecified queues, because it consists of queues of type (G[X] /G/1, M/G/1, G/G/N/0, with blocking,...). Therefore, we cannot use analytical methods (as for the Jackson networks or BCMP) to obtain the characteristics of the system. This is why we will call upon the simulation approach to solve the problem.

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Table 3 Results of the statistical analysis on the collected data Process

Loading

Variable

Distribution Parameters of the distribution

Inter-arrivals of the ships to be loaded (minutes)

Exponential

service duration of the anchorage (minutes)

Normal

service duration of before operations (minutes)

Normal

Size of groups to be loaded

Geometric

μ = 99.175 et σ 2 = 38.678 p = 0.0059

service duration of the gantries of quay (minutes)

Normal

Service duration of the trucks (minutes)

Normal

μ = 2.944 et σ 2 = 1.097 μ = 8.823 et σ 2 = 5.359

service duration of the after operations (minutes)

Normal

μ = 99.175 et σ 2 = 38.678

Inter-arrivals of the ships to be unloaded (minutes) Exponential service duration of the anchorage (minutes)

Normal

service duration of before operations (minutes)

Normal

Unloading Size of groups to be loaded

Storage

λ = 2710 μ = 57.595 et σ 2 = 18.174

Geometric

λ = 2710 μ = 57.595 et σ 2 = 18.174 μ = 99.175 et σ 2 = 38.678 p = 0.007

Service duration of the gantries of quay (minutes)

Normal

Service duration of the trucks (minutes)

Normal

μ = 2.947 et σ 2 = 1.072 μ = 9.228 et σ 2 = 4.994

service duration of the after operations (minutes)

Normal

μ = 99.175 et σ 2 = 38.678

Size of groups of delivered containers/day

Uniform

Mean=145

Storage Size of groups of restored containers/day

Uniform

Mean=140

5.2 Determination of the optimal number of trucks by simulation In this section, the aim is to determine by simulation approach the optimal number of trucks to use during the loading process and unloading process. For that, we propose two approaches. 5.2.1 First approach We designed a simulator for each model under the Matlab environment. After the validation tests of each simulator, their executions provided the results summarized in Table 4. Where: • The 3rd column represents the mean number of ships in roads to be loaded (respectively to be unloaded) during one year. • The 4th column represents the mean number of ships loaded (respectively unloaded) during one year. • The 5th column represents the mean number of containers loaded (respectively unloaded) during one year. • The 6th column represents the mean number of blocking of the server ”GQ” according to the number of trucks used during the loading (respectively the unloading) on one year. • The 7th column represents the mean time of blocking of the server ”GQ” during the loading (respectively the unloading) on one year. • The 8th column represents the mean number of blocking of the server ”trucks” in the loading process (respectively the unloading) on one year.

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Table 4 Some performances of the processes obtained by simulation approach Process

N-trucks 1 2 3 4 5 Loading 6 7 8 9 10 11 12 1 2 3 4 5 Unloading 6 7 8 9 10 11 12

N-ship 1.20 0.87 0.88 0.79 0.78 0.85 0.83 0.89 0.84 0.93 0.84 0.84 0.90 0.90 0.73 0.80 0.87 0.73 0.90 0.77 0.77 0.83 0.70 0.73

D-ship N-Cts (ETU) 191.09 49020 193.88 49397 193.74 49681 196.08 50204 195.63 50298 196.59 50753 196.00 50090 194.59 49577 195.64 50233 194.34 49717 193.90 49260 192.93 49187 192.13 41689 191.57 41523 193.00 42321 197.20 42093 196.70 41971 191.33 41087 199.60 42798 195.13 42548 192.97 40966 194.90 43800 194.40 43401 192.97 41572

N-GQ 27495 18804 11775 7150.8 4749.0 3442.5 2577.0 2079.3 1725.9 1548.9 1316.9 1268.9 24112 15930 9035 3731.5 1025.2 166.83 19.80 1.50 0.10 0 0 0

W-GQ 3349.5 1118.1 449.2 196.7 103.8 66.3 47.5 38.1 32.2 29.3 25.6 25 2955.5 947.7 342.8 102.4 21.6 2.9 0.3 0 0 0 0 0

N-trucks W- trucks Proportions 2927 72.6 0.0909 11878 342.1 0.3642 18797 588.3 0.5757 22876 761.4 0.7094 25723 899.6 0.7818 27543 1001.7 0.8190 27453 1030.7 0.8365 27699 1068.1 0.8463 27428 1080.7 0.8496 28240 1133.8 0.8517 26870 1095.6 0.8509 28594 1181.4 0.8512 2099 50.40 0.0770 9289 259.90 0.3418 13713 406.95 0.4952 15794 435.37 0.5734 16544 516.73 0.6023 16386 548.40 0.6095 17103 537.75 0.6103 17002 545.60 0.6106 16378 549.78 0.6109 17525 551.85 0.6115 17359 546.22 0.6111 16607 539.27 0.6104

• The 9th column represents the total mean time of blocking of the server ”trucks” in the loading (respectively the unloading) on one year. • The 10th column represents the probabilities of the blocking of the servers ”trucks” in the loading process (respectively the unloading process); for example: the value 0.5757 of the third row represents the blocking probability in the case of three servers ”trucks” in the loading process, which is the sum of the probabilities of blocking of one server, two servers and three servers. These probabilities is distributed as following: P(X = 0) = 0.4276, P(X = 1) = 0.4117, P(X = 2) = 0.1492, P(X = 3) = 0.0148, where X: ’number of servers ”trucks” blocked and P(X = 1) + P(X = 2) + P(X = 3) = 0.5757. This distribution is illustrated by the figure (left).

Interpretation and discussion of the results • Loading process • From the obtained results, we note that the variation of the mean number of the loaded containers in ETU during one year is independent of the number of trucks used . Indeed, the mean number of the loaded containers varies only between

Optimal Management of Equipments of the BMT Containers Terminal (Bejaia’s Harbor)

41

Fig. 4 Probabilities of blocking of the servers ”trucks”: case of three trucks (loading process on the left and unloading process on the right)

49019.5401 ETU and 50753.3083 ETU which is practically the same. This independence can be explained by the fact that the inter-arrivals of ships to be loaded are very large compared to the time spent by ship at the quay. Similarly, for the loading process, the mean number of ships on the roads is also independent of the number of trucks used, except in the case of one truck, where the mean number of ships in roads is a bit high (1.2000 ships). But, according to the dashed curve, we note that the mean waiting time of the trucks (blocking of servers ”trucks”) is proportional to the number of the used servers ”trucks”. So, to minimize the blocking duration of the servers ”trucks”, we must use the minimum possible of servers ”trucks”. • This problem can be formulated (written) mathematically as follows: min ←− T1 , min ←− T2 ,

min ←− (T1 , T2 )

⎧ ⎧ or ⎨ Capacities of the company, ⎨ Capacities of the company, S.C. Available equipments, S.C. Available equipments, ⎩ Processing time. ⎩ Processing time. (1) Where T1 and T2 represent respectively the mean waiting time of the server ”GQ” and the mean waiting time of the servers ”trucks”. Here, we note that we are facing a stochastic multi-objectives optimization problem, specifically a stochastic bi-objectives problem. So, to determine the optimal number of trucks to use in the loading process, it is necessary to find a compromise between the blocking time of the server ”GQ” and the blocking time of the servers ”trucks”. It is thus necessary to find the number of servers ”trucks” which minimizes the blocking time of the server ”GQ” and minimizes the blocking time of the servers ”trucks” at the same time. So, we transform the problem (1) to the following form Weighted Sum Scalarization ( see Ehrgott, 2005; Bot et al, 2009):

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min ←− α T1 + (1 − α )T2 , ⎧ ⎨ Capacities of the company, S.C. Available equipments, ⎩ Processing time.

(2)

Where: α is weight reflecting the preference of the waiting time of the server ”GQ” and 1 − α the weight reflecting the preference the waiting time of the servers ”trucks”. In this work, we assume that there is not a preference between the waiting of the servers ”trucks” and the waiting of server ”GQ” i.e α = 0.5. In this case, we determine the minimum of the sum of the blocking times of the server ”GQ” and servers ”trucks” which is represented by the solid curve on the Figure 5 (left). According to this curve, we note that the optimal number of trucks to be used (which minimize the sum of the blocking times for loading process) is four (04) trucks. • Unloading process: With the same manner and same reasoning as in the loading process, we can determine the optimal number of trucks that will be used in the unloading process. In this case, the result is also four (04) trucks.

5.2.2 Second approach In this part, we propose another approach (reasoning) to determine the optimal number of trucks to be used. This method consists of determining the number of trucks to use in order to minimize the mean time of loading or unloading of a ship (beginning operations - end operations). The obtained results for different number of servers ”trucks” used are summarized in the table 5. Table 5 The Variation of the mean time (hours) of the loading/unloading service, according to the number of servers ”trucks” Number of trucks 1 2 3 4 5 6 Loading service 26.9163 13.9531 10.3773 8.2871 8.2671 8.4758 Unloading service 23.8718 13.4766 10.2773 9.1891 9.0743 9.3003 Number of trucks 7 8 9 10 11 12 Loading service 8.3842 8.4986 8.4049 8.4796 8.4955 8.3235 Unloading service 9.1870 8.9528 8.6114 8.9666 9.0482 8.6716

Interpretation and discussion of results Loading process: Figure 5 (right) and the second row of the Table 5 show that the mean time of loading service decreases with the number of servers ”trucks” from (01) to four (04) servers ”trucks”, and from four (04) trucks, the mean time of loading service remains almost constant, which means that beyond four (04) servers

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”trucks”, the mean time of loading service depends only on the ability of the server ”GQ”. To this end, we conclude that we no interest to use more than four (04) servers ”trucks” in the loading process. So the optimal number of trucks towing in this case is four (04) trucks. Unloading process : for the same arguments as the loading process, the optimal number for the unloading process is four (04) trucks.

Fig. 5 The mean waiting time of the servers ”trucks” and ”GQ” on one year (left) and the variation of the mean time of loading service (right) according to the number of servers ”trucks”

6 Performance study of storage process After the validation tests of empty stock and full stock simulators, their executions provided the results summarized in Table 6. Where the 2nd, 3rd and 4th column repTable 6 Storage performances Parameters Number of trucks ETU saturation (%) Full stock 4 4570.9995 55.0722 5 4440.8319 53.5044 Empty stock

4 5

1157.5036 1192.9385

128.6115 132.5487

resents respectively the number of servers ”trucks” used, the total mean number of containers (ETU) in the full stock and empty stock on one year and their saturation rate expressed as a percentage.

Interpretation and discussion of the results The simulation results show that:

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• With the current parameters, the average number of containers in the full park over a period of one year is 4570.9995 ETU in the case of four (04) ”trucks”, and 3610.8734 ETU in the case of five (05) ”trucks” and the mean number of containers in the empty park over a period of one year is 1157.5036 ETU in the case of four (04) ”trucks” and 1192.9385 ETU in the case of five (05) servers ”trucks”.

7 Conclusion The objective of this work is to determine an optimal management of the equipments of the Terminal with containers of the BMT Company, more specifically the optimal number of trucks to use in the loading process and unloading process. For this, we developed a mathematical model for each process (the ”loading”, the ”unloading”, the ”full stock” and the ”empty stock” process). Indeed, in order to analyze the different processes and determine the optimal number of trucks to use, each system (process) is modeled by an open network. We have also established a simulation model of each system, where the goal of each simulator is to reproduce the functioning of the park with containers. The study shows that: • For the loading process: For an arrival rate of 0.5317 ships/day, a mean service trucks of 8.8234 minutes and a mean service GQ of 2.9440 minutes, the optimal number of trucks is four (04) trucks. This mean that the BMT Company can recover a truck from each GQ, i.e in total two (02) trucks. • For the unloading process: For an arrival rate of 0.5317 ships/day, a mean service trucks of 9.2281 minutes and a mean service GQ of 2.9473 minutes, the optimal number of trucks is four (04). This mean that the BMT Company can recover a truck from each GQ, i.e in total two (02) trucks. • Regarding the stock: The study shows that for the current settings at the end of the year 2009 it will undergo a saturation of 55% for the full stock and 130% for the empty stock, hence the need for expanding the capacity of empty stock. It would be interesting to achieve this work, by discussing the following items: • An analytical resolution of the problem. • Determination of an optimal management of the others equipments of the BMT Company. • Take account the variation of the parameters of the system.

References A¨ıssani D, Adjabi S, Cherfaoui M, Benkhellat T, Medjkoune N (2009) Evaluation des performances du terminal a´ conteneurs B.M.T. (Port de B´ejaia).

Optimal Management of Equipments of the BMT Containers Terminal (Bejaia’s Harbor)

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In: Actes du S´eminaire International ”Terminal Development and Management”, vol PORTEK Company and B.M.T. (Bejaia Mediterranean Terminal) A¨ıssani D, Adjabi S, Cherfaoui M, Benkhellat T, Medjkoune N (2009) Forecast of the traffic and performance evaluation of the BMT container terminal (Bejaia’s harbor). In: Reiner G (ed) Rapid Modelling for Increasing Competitiveness: Tools and Mindset, Springer Verlag Ed. (Germany), pp 53–64 Bot R, Grad S, Wanka G (2009) Duality in Vector Optimization. Springer Verlag David M, Michaud J (1983) La pr´evision: approche empirique d’une m´ethode statistique. Masson Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer Verlag Sait R, Zerrougui N, Adjabi S, A¨ıssani D (2007) Evaluation des performances du parc a´ conteneurs de l’Entreprise Portuaire de B´ejaia. In: Proceedings of an International Conference Sada’07 (Applied Statistics for Development in Africa), Cotounou (Benin)

Production Inventory Models for a Multi-product Batch Production System Ananth Krishnamurthy and Divya Seethapathy

Abstract This paper presents an analytical model for capacitated batch production systems with the objective of investigating the effect of system utilization, production batch size, and production lead time on performance measures such as backorders and on-hand inventory levels. First, an exact analysis of a single product system and two product system is presented using the Markov chain approach. To overcome the computation complexity resulting from the explosion of state space, an alternative approach based on decomposition is proposed to analyze multi-product systems with more than two products. Numerical results indicate that the decomposition approach provides fairly accurate estimates of the performance measures. The analytical model also provides some useful insights on the effect of the production batch size and utilization on system performance. Key words: Batch production systems, Markov chain, multi-product, inventory control

1 Background and Literature Review Batch production systems find many applications in manufacturing. Typical applications include heat treat operations, annealing furnaces, and various forms of plating and coating operations. The production batch size used in these operations is typically governed by the physical constraints of the operation, namely size of furnace, capacity of tanks holding the liquid required for the coating operations. However,

Ananth Krishnamurthy (B) and Divya Seethapathy Department of Industrial and Systems Engineering, University of Wisconsin, Madison, WI 53706, USA, e-mail: [email protected] Divya Seethapathy e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 4, 

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these production batch sizes in many applications are much larger than the typical demands for the products from the customer. Consequently, managing production inventory decisions under these constrained environments poses an interesting analytical problem. Figure 1 describes a typical batch production environment operating in a make to stock environment. Fig. 1a provides a schematic description while Fig 1b provides a queuing model representation of the same system. External demands follow a random process and customer demands are satisfied from available finished goods inventory. Customer demands that cannot be satisfied immediately from stock are backordered. Assume that the system starts with an initial finished goods inventory level corresponding to Q + r, where Q represents the production batch size and r represents the re-order point of the system. As external demands arrive (one at a time), they are satisfied from available stock. When the level of finished goods inventory reaches the re-order point, an order for Q units is placed with the supplier. As additional demands arrive, inventory depletes further and might trigger additional orders, each of size Q units. At the manufacturing station (for instance, a heat treat furnace), batches of Q units undergo processing simultaneously for a random service time. Upon service completion, a full batch of Q units is delivered to finished goods inventory. Customer demands that are not satisfied from existing inventory are backordered and satisfied first, when inventory becomes available. In the queuing model described in Fig. 1b, the production facility is assumed to have a lead time, L. Finished goods inventory and backordered demands are represented by buffers FG and BO in a synchronization station and a batch queue, BQ models the batch formation process prior to release of a production order of size, Q.

Fig. 1 Schematic (a) and queuing model (b) of a batch production system

In terms of associated literature, the production inventory system being analyzed has strong relationship with the classical Q, r inventory model under continuous review. However, the model discussed in this paper corresponds to a capacitated system (see Benjaafar et al (2004); Buzacott and Shanthikumar (1993); George and Tsikis (2003) and the references therein), while a majority of prior studies have

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focused on un-capacitated systems (see Hopp and Spearman (2000); Thoneman and Bradley (2002); Nahmias (1993); Zipkin (1995) and the references therein). In the studies involving un-capacitated systems, production lead times at the manufacturing facility have either been assumed to be deterministic or assumed to follow a specific probability distribution. Typically, this distribution has been is independent of the load on the system. However, the distribution of the production lead time depends on several parameters such as demand and service rates, and more importantly the batch size, Q. This explicit dependency between the production inventory policy parameter, Q and the production lead time L, has received limited attention in the literature. The objective of this paper is to build analytical models for multi-product batch production systems that could be used to investigate how performance measures such as average backorders and average finished goods inventory levels vary with the parameters of the production inventory control policy, namely, Q and r. These models would also help to study the effect of product variety on system performance. The analysis approach and outline followed in this paper is as follows. First, in Section 2, an exact analysis of a single product batch production system is carried out using a Markov chain analysis. Next, the approach is extended to analyze a batch production system with two products in Section 3.1. Although a two-product batch production system can be analyzed exactly using Markov chains, the approach is not scalable to systems with large a number of products. Therefore, a decomposition based approach is developed in Section 3.2. The approach uses the notion of “machine availability” to decompose a two-product system into two separate oneproduct systems. The accuracy of the approach is validated using numerical experiments and the analytical model is used to carry out studies that provide insights in the behavior of various performance measures. In Section 4, an extension of this approach to general systems with multiple products is presented. Section 5 presents the summary and conclusions of this research.

2 Batch Production System with a Single Product This section presents the analytical model for a capacitated batch production system with a single product. Figure 2 describes the system. In the figure, MFG represents the manufacturing station and FG, BO, and BQ represent the buffers of finished goods inventory, backordered demands, and batch formation queue, respectively. The system is assumed to operate under a continuous review policy. External demands from customers are assumed to arrive for single units according to a Poisson process with parameter, λ . Orders to suppliers are placed for fixed quantities, Q, when the on-hand inventory in the finished goods buffer reaches the re-order point, r. The manufacturing facility, MFG is assumed to be composed of a single server with service times that have an exponential distribution with mean, μ −1 . The three performance measures of interest are on-hand inventory (OH), average backorders (BO), and on-order inventory (OO). On hand inventory represents the

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Fig. 2 Analytical model for a capacitated batch production system with a single product

number of finished products in the FG buffer. Backorders represent the number of orders in BO and on-order inventory being or waiting to be processed at MFG. Since orders are placed in batches of Q units, for this batch production system, the inventory position R satisfies the following equality: Inventory Position, R = On-Hand Inventory (OH) +On Order (OO) − Backorders (BO) . Under these assumptions stated above, the dynamics of the system can be modeled precisely as a Markov chain. The state of the system at any time can be represented using the state variable i = iFG − iBO , where iFG denotes the number of finished goods at FG, and iBO denotes the number of backordered demands waiting at BO. The state space of the system, S is defined as S = {r + Q, r + Q − 1, r + Q − 2...r. . .1, 0, −1, −2, . . ., −∞}. Note that since, orders that cannot be satisfied from stock are backordered, a statei¡0 implies that a stock-out has occurred and all future orders would be backordered till subsequent supplier deliveries increase on-hand inventory to positive levels. Figure 3 describes the Markov chain for this system.

Fig. 3 Markov chain for a capacitated batch production system with a single product

Let πi represent the steady state probability of the system being in state i. Then the Chapman- Kolmogorov equations for steady state can be written as follows:

λ πQ+r = μπr for i = r + Q λ πi = μπi−Q + λ πi+1 for i = r + 1 to r + Q − 1 (μ + λ )πi = μπi−Q + λ πi+1 for i = r to − ∞ .

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These equations are solved using the additional relation involving summation of probabilities,

∏e = 1 . The solution of the state equations gives the value of the steady state probabilities, πi . The performance measures are subsequently expressed in terms of πi as shown below: ∞

Expected On − hand Inventory, E(I) = ∑ iπi i=1

Expected Backorders, E(B) =

−1

∑

| − i|πi

i=−∞

Expected On-orders, E(O) =

∞

∑

r− jQ

jQ

j=1

∑

πi .

i=r−( j+1)Q−1

Closed form expressions for the steady state probabilities and performance measures can be derived in terms of the r0 the unique root of the equation (in operator notation)

μ DQ+1 − (μ + λ )D + λ = 0 . where r0 is such that

λ = μ



r0 1 − r0Q 1 − r0

.

Then the key performance measures are given by the following equations, namely:   r0r+1 r0 Q+1 − +r+ρ E(I) = 2 1 − r0 1 − r0   r0r+1 E(B) = ρ 1 − r0 r0 E(O) = . 1 − r0 Note that in the above equations, the utilization of the manufacturing station, ρ = λ /(μ Q) is a function of the batch size, Q. In order to verify the analysis, numerical experiments are conducted by varying the input parameters and examining the values of the performance measures under different settings. In all experiments, the service rate, μ is set to be equal to 1. Further, the batch size, Q is varied so that it takes the values: 1, 5 and 10, respectively. The demand rate λ is varied so that in

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all cases, the server utilization ρ takes values 60%, 70%, 80%, and 90%. Sample results from these experiments are displayed in Fig. 4.

(a)

(b) Fig. 4 Performance tradeoffs for a batch production system with a single product

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From the figure it can be observed that the expected number of backorders increase with utilization, i.e. E(B) increases with utilization, ρ . Further, the expected number of backorders increases with the batch size, Q, i.e. E(B) increases with batch size, Q. The graphs also indicate that the expected value of inventory that is on-order at MFG increases with utilization and the batch size, Q. This increase correlates with the increase in lead time for these parameters, suggesting that performance measures in a capacitated system show obvious relationships with the capacity utilization of the manufacturing station. The next section describes the analysis of a system with two products.

3 Batch Production System with Two Products This section presents the analysis of a batch production system with two products. An exact analysis is described in Section 3.1 while a decomposition based approximation is discussed in Section 3.2.

3.1 Exact Analysis Figure 5 describes the analytical model for a capacitated batch production system with N = 2 products. As observed in the figure, the manufacturing station represented by MFG, is shared by the two products. However FGn , BOn and BQn (n = 1, 2) represent the buffers of finished goods inventory, backordered demands, and batch formation queue, for each of the two products respectively. The system is assumed to operate under a continuous review policy. External demands for each product n arrive for single units at a time according to a Poisson process with parameter, λn . Orders for product n are placed for fixed quantities, Qn , when the on-hand inventory in the finished goods buffer for each reaches the re-order point, rn . The manufacturing facility, MFG is assumed to be composed of a single server with service times that has an exponential distribution with mean, μ −1 , independent of the product being processed at the station. It is assumed that the station does not process mixed batches of different products in the furnace. However, the station does use a priority rule while choosing batches for production. When the there are multiple batches waiting in queue for processing, the server chooses the next batch to be processed based on the current inventory level of the different products. In particular, if at the instant of service completion, BO1 > BO2 , then a batch of product 1 is released to potentially raise the inventory level of product 1. If there are no backorders for either product (i.e. when BO1 = BO2 =0) and if FG1 < FG2 , then a batch of product 1 is released to potentially raise the inventory level of product 1. Finally, if finished goods inventory levels are also equal (i.e. when FG1 = FG2 ), then a batch of product 1 is released if λ1 > λ2 . Otherwise, a batch of product 2 is released.

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Fig. 5 Analytical model for a capacitated batch production system with two products

Based on these assumptions, the dynamics of the system can be modeled precisely as a Markov chain. However, unlike the single product system, in order to represent the state of the system at any time, three parameters (i, j, k) are required; where i = iFG1 − iBO1 , j = iFG2 − iBO2 , and k = 0, 1, or 2 depending on whether the server is idle or processing product 1 or 2 respectively. The Markov chain can be divided into 4 regions and Fig. 6 shows the transitions in the different regions.

Fig. 6 Regions in the Markov chain for batch production system with two products

As with the case of the single product, the performance of a system with two products is analyzed by solving the balance equations for the underlying Markov chain. The solution for the balance equations yields the steady state probabilities for the system with two products. Expressions for performance measures such as average on-hand inventory E(In ), average backorders, E(Bn ), and average on-orders E(On ); for each product n = 1, 2 are written in terms of these steady state probabilities. Numerical results obtained from these expressions are discussed in the subsequent sections. However, the approach of determining the performance measures through the exact solution of the underlying Markov chain poses computational

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difficulties for systems with a large number of products, due to the explosion of the size of the underlying state space. There an alternative approach based on decomposition is proposed. This is discussed next.

3.2 Decomposition Based Approximation The main motivation behind the approximations is the observation that when the manufacturing station is processing product 1, the station is “unavailable” to process product 2. Similarly, when the manufacturing station is processing product 2, the station is “unavailable” to process product 1. Let the availability of the manufacturing station to serve product n be defined as An . Then the effect of unavailability is modeled by defining an effective service time τ˜n for product n at the manufacturing station. For the two product system, these effective service times are written as: 1 = τ˜1 = μ˜ 1 1 For product 2 : = τ˜2 = μ˜ 2

For product 1 :

1/μ1 where A1 = 1 − ρ2 A1 1/μ2 where A2 = 1 − ρ1 . A2

Then the performance of the two product system can be analyzed by solving two separate single product system with these effective service times.

Fig. 7 Decomposition based approach for analysis of two product systems

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Each single product system is then analyzed using the approach described in Section 2 to obtain the values of r0n for each product, n = 1, 2 from the following equation:

Q r 1 − r 0n 0n λn = for n = 1, 2. μ˜ n 1 − r0n Then, the performance measures E(Bn ), E(In ), and E(On ) for n = 1, 2 are obtained using r0n , n = 1, 2 using the following equations:   rn +1 r0n r0n Q+1 − E(In ) = + rn1 + ρn for n = 1, 2 2 1 − r0n 1 − r0n   rn +1 r0n for n = 1, 2 E(Bn ) = ρn 1 − r0n r0n E(On ) = for n = 1, 2. 1 − r0n In order to verify the analysis and the accuracy of the approximation approach, numerical experiments were conducted by varying the input parameters and examining the values of the performance measures under different settings. In all experiments, the service rate, μ was set to be equal to 1. Further, the batch size, Q, was varied so that it took the values: 1, 5 and 10, and the demand rate λn was varied so that in all cases, the value of the server utilization ρ took values 60%, 70%, 80%, 90%. The results of the analytical model were obtained using a Matlab program. These results were further validated against detailed simulation done using Arena simulation software. The simulation model was run for 100 000 h and the number of replications chosen for each scenario was 12. A sample of the results for expected backorders and expected on-hand inventory for product 1 are displayed in Fig. 8. From Fig. 8 it can be observed that the decomposition based approximation approach provides reasonably accurate estimates of the performance measures. This implies that fairly accurate estimates of performance measures can be obtained through the decomposition approach. Next, the trends related to the performance measures are discussed. It is observed that, as with the single product system, the expected number of backorders increase with the utilization of the manufacturing station (i.e. E(B1 ) increases with ρ ). Further, the expected number of backorders increases with the batch size, i.e. E(B1 ) increases with batch size, Q. Correspondingly, it is also noted that the expected on-hand inventory decreases with increase in utilization of the manufacturing station. Although larger batches decrease the utilization of the manufacturing station, larger batches also reduce the frequency of orders placed to the supplier which in turn causes increase in the backordered demands. These results suggest that the performance measures of a capacitated batch production system are significantly influenced by capacity utilization of the manufacturing station and its effect on the replenishment lead time for finished goods

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Fig. 8 Performance tradeoffs for a batch production system with two products

inventory. The next section describes the extension of the approximation approach to analyze a system with N > 2 products.

4 Batch Production Systems with Multiple Products This section provides a brief description of the extension of the approximation approach to a system with N > 2 products. The basic idea is to decompose the N-product system into N single product systems that each models the performance of a single product in the N-product system (see Fig. 9). As with the two product system, the main idea is that, when the manufacturing station is processing product n, the station is “unavailable” to process all the other products. If the availability of the manufacturing station to serve product n, is defined as An , then, the effect of this unavailability can be incorporated into each single product model by defining the effective service time τ˜n for product n at the manufacturing station. For the two product system, these effective service times are written as follows: 1 1/μn = τ˜n = where An = 1 − ∑ ρi for n = 1, . . ., N. μ˜ n An i =n

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Fig. 9 Decomposition based approach for analysis of multi-product systems

Having defined the effective service times, each single product system can analyzed using the approach described in Section 2 to obtain the values of r0n for each product, n = 1, . . . , N using the following equation:

Q r0n 1 − r0n λn = for n = 1, . . ., N. μ˜ n 1 − r0n Finally, the performance measures E(Bn ), E(In ), and E(On ) for n = 1, . . . , N can obtained using r0n using methods described in Section 3.2.

5 Summary and Conclusions This paper presents an analytical model for capacitated batch production systems with the objective of investigating the effect of system utilization, production batch size, and production lead time on performance measures such as backorders and onhand inventory levels. First, an exact analysis of a single product system is presented using the Markov chain approach. Next, the approach is extended to analyze systems with two products. Due to the computation complexity resulting from the explosion of state space, an alternative approach based on decomposition is proposed to analyze the two product systems using the notion of “server unavailability”. Numerical results indicate that the decomposition approach provides fairly accurate estimates

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of the performance measures. Finally the paper presents a framework that demonstrates how this approach could be extended to analyze systems with N > 2 products. The analytical model provides some useful insights on the effect of the production batch size and utilization on system performance. It is observed that expected number of backorders, E(Bn ) and expected number of orders on order, E(On ) increase with batch size, Q, and utilization, ρ . In contrast, expected on hand inventories, E(In ) decrease with increase in batch size, Q, and utilization, ρ . The ongoing work includes extensive numerical validation of the approximation approach for various N-product systems, and estimation of optimal production parameters that minimize overall system costs.

References Benjaafar S, Kim JS, Vishwanadham N (2004) On the Effect of Product Variety in Production Inventory Systems. Annals of Operations Research 126:71–101 Buzacott J, Shanthikumar J (1993) Stochastic Models of Manufacturing Systems. Prentice Hall George, Tsikis I (2003) George and isidoros tsikis, 2003, “unified modelling framework of multistage production-inventory control policies with lot sizing and advance demand information. In: Shanthikumar J, Yao D, Zijm We (eds) Stochastic Modeling and Optimization of Manufacturing Systems and Supply Chain, Chapter (11), Kluwer Academic Publishers, pp 271–297 Hopp W, Spearman M (2000) Factory Physics: Foundations of Factory Management. Irwin/McGraw Hill Nahmias S (1993) Production and Operations Analysis, 4th edn. McGraw Hill Irwin, Chicago, IL. Thoneman U, Bradley J (2002) The Effect of Product Variety on Supply Chain Performance. European Journal of Operations Research 143(3):548–569 Zipkin P (1995) Performance Analysis of Multi-item Production Inventory System under Alternative Policies. Management Science 41:690–703

Dependency Between Performance of Production Processes and Variability – an Analysis Based on Empirical Data Martin Poiger, Gerald Reiner and Werner Jammernegg

Abstract It is commonly accepted that variability is one of the main challenges in designing and managing manufacturing processes. Many process improvement concepts that focus primarily on communication and information exchange, flow time reduction, etc., finally influence variability. In particular, they reduce variability or mitigate the operational effects of variability. Vendor managed inventory or collaborative planning, forecasting, and replenishment are just two examples of such concepts. The effect on selected performance measures is mostly shown by idealized quantitative models, but there are only few results from real-world processes. In our study we want to illustrate and quantify the impact of variability on the performance of production processes by means of two real manufacturing processes. Case process one is an assembly process of frequency inverters and case process two is the assembly process of sliding glass top systems. For the inverter assembly process we want to show the operational impact of reduced demand variability (reduced forecast error), achieved by implementing Vendor managed inventory as well as collaborative planning. In the glass top assembly process internal variability is addressed by assessing the impact of the production lot size. Both processes are mainly evaluated by using WIP and flow time as key performance measures. The analysis is conducted with rapid modeling software based on open queuing networks. Our results show that a reasonable decrease in inventory and flow time can be achieved without any decline of customer service. Martin Poiger (B) University of Applied Sciences BFI Vienna, Wohlmutstrasse 22, A-1020 Wien, Austria, e-mail: [email protected] Gerald Reiner Enterprise Institute, Faculty of Economics, University of Neuchˆatel, Rue de la Maladi`ere 23, CH-2000 Neuchˆatel, Switzerland, e-mail: [email protected] Werner Jammernegg Institute for Production Management, Vienna University of Economics and Business Administration, Nordbergstrasse 15, A-1090 Wien, Austria, e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 5, 

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1 Introduction It is commonly accepted that variability is one of the main challenges in designing and managing manufacturing processes. Increasing variability always deteriorates process performance (Hopp and Spearman, 2006). Dealing with variability, respectively removing variability from one’s processes significantly differentiates high-performing plants from low-performing plants (Mapes et al, 2000). Klassen and Menor (2007) provide a useful classification of variability using the dimensions source and form, where source is divided into internal and external variability, and form into random and predictable variability. Examples for internal random variability are quality defects, equipment breakdown, or worker absenteeism and for internal predictable variability preventative maintenance, setup time, or product mix. Examples for external random variability are arrival of individual customers, transit time for local delivery, or quality of incoming supplies, and for external predictable variability daily or seasonal cycles of demand, technical support following new product launch, or supplier quality improvements based on learning curve.For each of these types there exist many strategies to decrease variability or mitigate its effects. Concerning internal variability the Toyota Production System is probably the most famous concept to remove variability in processes. Regarding external variability supply chain management with its collaboration concepts has to be mentioned. Vendor managed inventory (VMI) or collaborative forecasting, planning and replenishment (CFPR) are just two of plenty strategies, which primarily focus on communication and information exchange. Generally there are two major streams in operations management research. On the one hand, there are plenty of surveys based on perceptions of managers or decision makers (Rungtusanatham et al, 2003). Such studies usually do not address process physics based on real process data such as cycle time or on-hand inventory. On the other hand, there are analytical models dealing with idealized problems, which are far abstracted from reality in order to make them tractable for mathematical analysis (Bertrand and Fransoo, 2002). Due to this abstraction it is often difficult to transfer solutions from such models to real-life problems. Therefore, Bertrand and Fransoo (2002) suggest quantitative model-driven research based on empirical data. This is in line with Silver (2004), who promotes a holistic approach of operations management, concentrating on improving system performance under consideration of uncertainty and continual change, instead of optimization under existing deterministic conditions. The research methodology suggested by Bertrand and Fransoo can be regarded as empirical normative research, conducting all four steps of the so-called Mitroff-Cycle (Mitroff et al, 1974), which consists of conceptualization, modeling, model solving and implementation. Especially the implementation is difficult to conduct and observe. Therefore the empirical normative research is not found very often in the scientific literature. In our study we analyze ex ante and ex post two real-world production processes. For the ex ante analysis, mainly consisting of modeling and evaluation of process alternatives, we used rapid modeling (Suri et al, 1995) based on open queueing networks. The ex post analysis was conducted after finalized implementation of the

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suggested process improvements and includes the observation of the impact on the process performance. Case process one is an assembly process of frequency inverters and case process two is the assembly process of sliding glass top systems. For the inverter assembly process we show the operational impact of reduced demand variability (reduced forecast error), achieved by implementing Vendor-managedinventory as well as collaborative planning. In the glass top assembly process internal variability is addressed by assessing the impact of the production lot size. Both processes are mainly evaluated by using work-in-process (WIP) and flow time as key performance measures. Our paper is organized as follows. Section 2 provides an overview of the theoretical background including relevant literature. In this section we present the necessary basics from queuing, bring up the operations management triangle (Schmidt, 2005), and introduce the software we use for our analysis. In Section 3 we show the impact of demand variability on process performance by means of the production process of a frequency inverter. In Section 4 the model is presented, which shows the impact of batching. Finally we conclude our paper in Section 5, by summarizing the most important results and deriving some managerial implications.

2 Theoretical Background 2.1 Queuing Theory For analyzing manufacturing systems queuing theory plays an important role. A queueing system usually consists of an arrival process, a production or service process, and a queue (Hopp and Spearman, 2006). The expected time spent by a job in such a system is called flow time T , defined by the sum of the mean effective process time te plus the expected waiting time tq (1). T = te + tq .

(1)

The mean effective process time te is the average time required to process one job and includes setups, downtime, etc. It determines the capacity of such a system, as the possible average flow rate re is the reciprocal value of te . The waiting time tq depends on te , on the utilization ρ and on the variability of both process and interarrival time. The utilization ρ is the probability that a station is busy, and is calculated by dividing mean arrival rate ra by mean flow rate re (2). The mean arrival rate ra is the average number of jobs coming to the station per time unit (reciprocal value of mean interarrival time ta ). ra ρ= . (2) re Variability in queueing systems is usually measured by the coefficient of variation. The coefficients of variation of the process time (ce ) and of the interarrival time (ca ) are defined by (3), in which σe denotes the standard deviation of the effective

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process time te , and σa the standard deviation of the interarrival time ta . ce =

σe te

ca =

σa . ta

(3)

Using the parameters above waiting time tq can be calculated by (4). tq = t e

ρ (c2e + c2a ) . (1 − ρ ) 2

(4)

In general, this is a good approximation for waiting time in the so-called G/G/1 queueing model, if process and interarrival times are “generally” distributed, i.e. they are specified by mean and standard deviation without any specific information about the distribution. For further explanation we refer the reader to Hopp and Spearman (2006). This model clearly shows that waiting times rise with process time, with average utilization and with variability. For managers it follows that in case of high variability it is not reasonable to maximize average utilization, as flow time will rise exponentially, and customers may have to wait too long. This is somehow counterintuitive and goes against standard management intuition, which emphasizes increasing resource utilization (de Treville and van Ackere, 2006). Figure 1 graphically shows the nonlinear relationship of waiting time and average utilization for different coefficients of variation. In case of very low variability (ca = ce = 0.5) the system can be highly utilized, as flow time remains low also with utilization of over 0.9. In case of moderate variability (ca = ce = 1) reasonable waiting time occurs already with utilization of 0.8, and in systems with high variability (ca = ce = 2) even a utilization of 0.6 results in very long waiting times.

Fig. 1 Waiting time depending on utilization

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2.2 OM-Triangle Usually uncertainty in operations is addressed by installing buffers, like excess inventory or excess capacity. This idea is somehow extended by the so-called OMTriangle, derived from the G/G/1 queueing model stating that capacity, inventory and information are substitutes in providing customer service (Lovejoy, 1998; Schmidt, 2005). In particular, the OM-Triangle is obtained by fixing three extreme points on the waiting time curve in Figure 1. Figure 2 shows three points called inventory point, capacity point and information point. As inventory is direct proportional to flow time (Little’s Law) we can show inventory instead of flow time on the y-axis.

Fig. 2 OM triangle

Companies running their operations at the capacity point operate at low average utilization to be able to quickly respond to volatile demand. This is especially necessary in service industries such as emergency service, where waiting times are critical. Companies with high fixed capacity costs try to run their operations at nearly 100% utilization, and demand variability is buffered by high inventory. This makes sense for industries with durable products, meaning commodities such as paper, steel or petrochemicals. Finally, companies capable of reducing variability in their operations operate at the information point. The assumption behind this proposition is that by having better information it might be possible to significantly decrease variability, which means that the process can be run at higher utilization without increasing flow time (inventory). The most famous example for this point is Toyota. Where to operate on such curves is also the question in the study of Bitran and Morabito (1999). They underline the usefulness of these trade-off curves for the evaluation of strategic options as a function of resources required to provide customer service. Klassen and Menor (2007) provide empirical evidence for the ex-

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istence of this trade-off between capacity, inventory and variability by examining archival economic data on industry level.

2.3 Rapid Modeling Software – MPX Rapid modeling is a technique that is based on the idea that a process is made up of a network of queues (Enns, 1996). Furthermore it applies queuing relationships to come up with analytical performance results. On the one hand, demand and process variability are incorporated into the model and on the other hand, based on the used queuing models the degree of control over the probability distributions is limited. For our analysis we use the Rapid Modeling Software MPX. This software uses the node-decomposition approach, where a network is broken down in its workstations, each regarded as a G/G/m queue (Suri et al, 1995). Compared to simulation it is easy (without any programming expertise), quick to use, and generates quite good results for processes in steady state. Suri et al (1995) underline the advantage of using open queuing networks by stating that it is better to find approximate solutions to models close to reality, than finding exact solutions for models that are only rough approximations of reality (e.g., linear programming). MPX is successfully used in academic teaching (de Treville and van Ackere, 2006) and consulting, mainly for analyzing manufacturing systems. MPX guides the user intuitively through a complete process analysis, by requesting inputs of labor, equipment, products, and lot sizes, bill of materials, demand, variability, operations and routings. If the manufacturing process is modeled, performance measures like WIP and flow time can be computed almost instantaneous. Process alternatives can be evaluated by incorporating “what-if” scenarios to the validated model.

3 Empirical Analysis of the Impact of Demand Variability 3.1 Demand Variability By means of the following production processes we want to show the impact of reduced demand variability on process performance. This is highly relevant as generally speaking supply chain management tries to reduce harmful variability through information sharing and cooperation. It is commonly accepted that information sharing has a positive impact on supply chain performance. In particular, information sharing is subsumed normally by improved or enriched forecasting methods (reduced forecast error) which finally reduce demand variability. Overviews of models dealing with the value of information in supply chains are provided by Chen et al (2000) and Li et al (2005).

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Particularly interesting in this context is the bullwhip effect, based on de de Kok et al (2005): The bullwhip is the metaphor for the phenomenon that variability increases as one move up a supply chain.

It is obvious that the bullwhip effect has a disturbing influence on performance upand downstream the supply chain process. Chen et al (2000) investigate the dependencies between forecasting and lead times in a simple supply chain. Boute et al (2006) show for a two-echelon supply chain how the bullwhip effect could be dampened. Hosoda and Disney (2006) analyze the variance amplification in a three-echelon supply chain model under the assumption of a first-order autoregressive consumer demand. Motivated by this research Reiner and Fichtinger (2009) investigate the influence of demand modeling on supply chain process performance by comparing different demand forecasting models. As already mentioned there are not too many empirical studies on process level, providing results on the impact of reduced demand variability on process performance. Therefore we want to analyze and quantitatively evaluate this impact by means of a real-world production process modeled in MPX.

3.2 Initial Situation of Frequency Inverter Assembling The process we use to illustrate the impact of demand variability is the production process of frequency inverters, produced in 9 different sizes: 9, 10, 11A, 11B, 12, 13, 14A, 14B, 15. These sizes are further differentiated by software and labeling, which has no impact on process times. Therefore, we analyze the production on product size level, and not on final brand level (overall 48 different final products are produced). The production process, which is mainly an assembly process, consists of the activities (1) assembly, (2) pre-inspection, (3) inspection and (4) finalizing (see Fig. 3). In the current situation the assembling is done on five parallel roller conveyors (RC0 – RC4), each for particular sizes. After the assembling a pre-inspection is conducted. This is done by one person for all conveyors with a mobile inspection device at the very end of each roller conveyor. After pre-inspection the inverter is transferred to one of two inspection stations (IN) by crane. After passing inspection the inverter is finalized at station MPFZ. Seven per cent of the products do not pass inspection and have to be repaired on the roller conveyor. The MPX analysis shows that some of the stations are highly utilized (above 90%), which lead to very long flow times, mainly driven by long waiting time for equipment. The highest utilized stations are the inspection stations and the roller conveyor RC0 and RC4. Table 1 shows flow time and WIP for the various sizes of the initial situation as well as of the improved process described in Section 3.3.

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Fig. 3 Process flow chart of Inverter production Table 1 WIP and flow time for various for the initial situation (IN) and for the improved process (IP) product sizes 9 10 11A 11B 12 13 14A 14B 15

waiting time waiting time process flow for equipment for labor time time IS IP IS IP IS IP IS IP IS IP [pieces] [pieces] [days] [days] [days] [days] [days] [days] [days] [days] 9.669 4.045 1.081 0.146 0.01 0.01 0.583 0.544 1.83 0.7 3.836 1.602 1.096 0.152 0.01 0.01 0.586 0.547 1.854 0.71 3.967 2.807 0.564 0.174 0.058 0.058 0.709 0.709 1.563 0.942 0.311 0.252 0.394 0.174 0.058 0.058 0.709 0.709 1.393 0.942 1.551 2.524 0.365 0.148 0.057 0.057 0.743 0.743 1.37 0.948 5.415 4.543 0.531 0.305 0.009 0.009 0.859 0.859 1.713 1.172 9.077 6.141 3.723 2.149 0.009 0.009 1.244 1.209 7.134 3.367 5.599 3.85 3.723 2.149 0.009 0.009 1.396 1.367 7.286 3.525 0.843 0.581 3.905 2.252 0.009 0.009 1.489 1.464 7.664 3.725 WIP

3.3 Process Improvements and Impact of Demand Variability Various process alternatives were evaluated to identify an appropriate design of the process after the analysis of the current situation. It turned out that two main improvements should be made. First, some activities of the roller conveyor should be transferred to a new station, and second, an additional inspection station should be

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installed. Flow times and WIP of the improved process are shown in Table 1. The improved process performs much better in terms of WIP and flow time. Finally, we want to show the impact of different coefficients of variation on process performance. Varying is done by using a scaling factor in MPX to multiply the basic model coefficient of variation. Figure 4 shows that an increase as well as a decrease of the variability factor heavily impacts flow time of some of the products. Especially sizes 14A, 14B, and 15 (size 15 is shown in Fig. 4) are very sensitive to changed variability. This sensitivity is caused by high utilization, which means there is no safety capacity to address increased demand volatility.

Fig. 4 Flow time depending on variability factor

For the improved process variability sensitivity is much lower. Figure 5 shows that flow times are generally on a lower level, but further they nearly does not react in increased (scaled) variability.

Fig. 5 Flow time depending on different variability factors for the improved process

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These results show also that a comprehensive process evaluation has to take into consideration “robustness” of the results and not only the “ideal” solution based on average values. In the presented example the variability sensitivity has been applied. Based on this variation of the variability further evaluation model extensions in terms of different risk level analysis, e.g., process risks and disruption risks can be easily implemented.

4 Empirical Analysis of the Impact of Variability Caused by Batching 4.1 Theoretical Impact of Batching With the following production process we want to show explicitly the impact of internal variability induced by batch production. Producing in batches is necessary if different products are produced on the same work station and a setup is necessary to switch between the products. Batching is an important cause of variability and obviously influences flow time through a process. In processes using batching in a way that only complete batches are passed on to the next station, calculating flow time per part does not make sense. To know when a part is ready for the next station the flow time TB of a batch has to be calculated, because every part of a batch has to wait until the last part of the batch is finished. Flow time of a batch TB is the sum of waiting time tq , the setup time ts , the wait-in-batch time tqB and the process time per part t0 (5)(Hopp and Spearman, 2006). TB = tq + ts + tqB + t0 .

(5)

Setup time ts and process time per part t0 are independent from batch size, but waiting time tq and wait-in-batch time tqB depend on batch size. According to Section 2, tq mainly depends on utilization and variability. Batching influences utilization by its impact on average capacity of a station re . Equations (6) and (7) (Hopp and Spearman, 2006) show for given setup time ts that the lower the batch size Ns the higher mean effective process time te per part, and consequently, the lower average capacity re . te = t0 + re =

1 . te

ts Ns

(6) (7)

For given arrival rate ra decreased capacity re leads to higher utilization ρ and by that to longer waiting time tq (4). Utilization ρ for a batch process is defined by (8).

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  ts ra ρ = rate = ra t0 + = (Nst0 + ts ). Ns Ns

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(8)

Batching also influences tq because it induces process variability. Equations (9) and (10) (Hopp and Spearman, 2006) show the impact of batch size Ns and setup time ts on the variance of mean effective process time σe2 , and on squared coefficient of variation c2e . Clearly, in case of high setup times a rising batch size decreases squared coefficient of variation c2e , as less setups are executed.

σe2 = σ02 + c2e =

σs2 Ns − 1 2 + t Ns Ns2 s

σe2 . te2

(9) (10)

Consequently, from a utilization point of view and also from a variability point of view, batch sizes should be as high as possible to minimize average waiting time tq . Unfortunately the second parameter of average flow time, affected by batch size, i.e. wait-in-batch time tqB , reacts the other way around, as it rises with increasing batch size (11). (11) tqB = (Ns − 1)t0 . Therefore it is necessary to calculate a somehow “optimal” batch size, which minimizes average flow time TB . Figure 6 shows the factors described above. Setup time and process time are independent from batch size and remain constant. Waiting time (for resource) first sharply decreases with batch size and then slightly increases (due to the impact of

Fig. 6 Process performance measures depending on batch size

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batch size on the coefficient of variation – see (9)), and wait-in-batch time linearly increases with batch size. These factors lead to average flow time having this particular shape, with first a sharp decrease and afterwards a nearly linear increase. From that follows that first a minimum batch size is necessary to be able to produce the requested demand at all, i.e. having utilization below 1. As soon as there is enough capacity flow time rises nearly proportionally with batch size. Consequently, it is not easy to intuitively generate an appropriate batch size for a longer production process or a production network, consisting of several work stations with various routings. The following case shows how MPX can be used to analyze a real world production process by varying the batch size and by showing the impact with respect to average flow time.

4.2 Initial Situation of Glass Top Assembling The considered process is the assembly process of sliding glass top systems. This process is executed in an Austrian subsidiary, which belongs to a multinational company, dealing primarily with polymer solutions. The glass top is produced for another Austrian company, manufacturing industrial refrigerators. The top is delivered in six different variants (A1, A2, A3, B1, B2, and B3) and in sets of one upper top and one lower top. Both the upper and the lower top consist of a glass pane and a plastic frame. Figure 7 shows the simplified process flow diagram of the assembly procedure. The important point to mention is that for a particular batch of glass top sets first all upper tops are assembled and packed into a box. Then after a setup all lower tops are assembled and packed into the same box with the upper tops to complete the sets.

Fig. 7 Assembly process of glass tops

The process was again modeled and analyzed by MPX. Table 2 shows the quarterly production, the used batch size and the flow time calculated by MPX. Batch sizes were calculated and fixed by the company using the standard EOQ-model, but without taking into account the model’s restrictions and assumptions. Especially because of consideration of very high and questionable fixed costs the batch size is very high and of course responsible for the long flow time (along some other factors). During the process analysis many alternatives were evaluated and compared. It turned out that the process could be really enhanced by adding a second assembling table enabling parallel assembling of upper and lower tops. By that, one part of the long waiting time can be cancelled as upper and lower tops have their own

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Table 2 Production quantities, batch sizes and flow time per batch for the six different glass top sets Product name Demand per quarter Batch size Flow time [h] A1 2921 288 153.7 A2 13696 720 198.5 A3 6051 277 160.9 B1 2232 156 154.8 B2 3303 98 112.5 B3 594 24 72.5

equipment and do not have to wait for each other. The analysis of course brought some further measures to improve the process which are not discussed. In fact we want to focus on the impact of batching and the batch size on process performance.

4.3 Process Improvements and Impact of Batching To explicitly illustrate the impact of batch size we use the model of the improved process, in which the additional assemble table and some other measures are implemented. Figure 8 shows the flow time for a batch of A1 depending on the batch size.

Fig. 8 Flow time for a batch of A1 depending on batch size

The minimum batch size which is necessary to be able to produce at all is 37. Batch sizes below 37 lead to an over utilized process. From batch sizes 37 to 40 flow time sharply decreases with batch size. With batch sizes between 43 and 48 the flow time is at its minimum of 7.5 hours. From batch size 49 the flow time increases

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nearly linearly with batch size. From that follows that the batch size used in the initial situation is much too high in terms of flow time. This example clearly shows that it really makes sense to search for the flow time minimal batch size. In this case using inappropriately the EOQ-model as well as state of the art extensions with at least questionable cost information resulted in much too high flow times. Up to now the process has been improved also in reality. The company has installed the second assemble table, reduced the setup time from 30 minutes to 18 minutes and uses now much lower batch sizes. The sum of process improvements doubled the output of the assembly station.

5 Conclusions In this study, two real-world production processes are analyzed, using empirical data from process level. By means of these processes we want to show the impact of variability on process performance. In particular we use one process to show the impact of external variability in terms of demand variability, and the other process to show the impact of internal variability induced by batching. For the analysis we use MPX, a rapid modeling software based on open queueing networks. The process improvements concerning demand variability show two things. First, a process properly designed, i.e. with sufficiently high capacity, is much more robust against demand variability. Second, decreasing demand variability, achieved by VMI or other information sharing concepts, leads to an improved process performance in terms of WIP and flow time. The presented approach to identify the “ideal”, i.e. flow time minimal batch size is an interesting alternative compared to existing approaches based on mathematical optimization in a deterministic environment. The advantages of the presented approach are on the one hand, low hardware and time requirements for solution finding, compared to the classical approaches. On the other hand, it is possible to take the variability of the input-parameters into account. Finally, it has to be stated that with our approach it is not possible to find the mathematically optimal solution. Rather, we find an ideal solution in terms of robustness. We assume, following Hopp and Spearman (2006) that it is better to find a solution which works just fine in many cases than an optimal solution which might be very sensitive to any variation of the input-parameters.

References Bertrand J, Fransoo J (2002) Operations management research methodologies using quantitative modeling. International Journal of operations and production management 22(2):241–264

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Bitran G, Morabito R (1999) An overview of tradeoff curves in manufacturing system design. Production and operations management 8(1):56–75 Boute R, Disney S, Lambrecht M, Van Houdt B (2006) An integrated production and inventory model to dampen upstream demand variability in the supply chain. European Journal of Operational Research 178(1):121–142 Chen F, Drezner Z, Ryan J, Simchi-Levi D (2000) Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information. Management science 46(3):436–443 Enns S (1996) Analysis of a process improvement path using rapid modeling. Total Quality Management & Business Excellence 7(3):283–292 Hopp W, Spearman M (2006) Factory physics: foundations of manufacturing management. McGraw-Hill/Irwin Hosoda T, Disney S (2006) On variance amplification in a three-echelon supply chain with minimum mean square error forecasting. Omega 34(4):344–358 Klassen R, Menor L (2007) The process management triangle: An empirical investigation of process trade-offs. Journal of Operations Management 25(5):1015–1034 de Kok T, Janssen F, van Doremalen J, van Wachem E, Clerkx M, Peeters W (2005) Philips electronics synchronizes its supply chain to end the bullwhip effect. Interfaces 35(1):37–48 Li G, Yan H, Wang S, Xia Y (2005) Comparative analysis on value of information sharing in supply chains. Supply Chain Management: An International Journal 10(1):34–46 Lovejoy W (1998) Integrated operations: a proposal for operations management teaching and research. Production and Operations Management 7:106–124 Mapes J, Szwejczewski M, New C (2000) Process variability and its effect on plant performance. International Journal of Operations and Production Management 20(7):792–808 Mitroff I, Betz F, Pondy L, Sagasti F (1974) On managing science in the systems age: Two schemas for the study of science as a whole systems phenomenon. Interfaces 4(3):46–58 Reiner G, Fichtinger J (2009) Demand forecasting for supply processes in consideration of pricing and market information. International Journal of Production Economics 118(1):55–62 Rungtusanatham M, Choi T, Hollingworth D, Wu Z, Forza C (2003) Survey research in operations management: historical analyses. Journal of Operations management 21(4):475–488 Schmidt G (2005) The OM Triangle. Operations Management Education Review 1(1):87–104 Silver E (2004) Process management instead of operations management. Manufacturing & Service Operations Management 6(4):273–279 Suri R, Diehl G, de Treville S, Tomsicek M (1995) From CAN-Q to MPX: Evolution of queuing software for manufacturing. Interfaces 25(5):128–150 de Treville S, van Ackere A (2006) Equipping students to reduce lead times: The role of queuing-theory-based modeling. Interfaces 36(2):165

Improving Business Processes with Rapid Modeling: the Case of Digger Reinhold Schodl, Nathan Kunz, Gerald Reiner and Gil Gomes dos Santos

Abstract Rapid Modeling is a method for modeling operational processes as a network of queues and analyzing them by applying queuing theory. Despite its great potential for solving problems in operations management, Rapid Modeling receives rather limited attention among practitioners. Therefore, this paper presents a teaching case for students and managers to illustrate the possibilities of Rapid Modeling. Moreover, the teaching case aims to raise awareness of the importance of considering both financial aspects and process management principles when improving processes. For that purpose, theoretical foundations for an integrated analysis and evaluation are discussed and applied in a practical context. The system under study is a real-life non-profit organization that is challenged by its growth strategy. The paper demonstrates how software based on queuing theory can be applied to find a solution in a quick and structured way. Key words: teaching case, operations management, queuing theory, Rapid Modeling

Reinhold Schodl (B), Nathan Kunz, Gerald Reiner and Gil Gomes dos Santos Enterprise Institute, Faculty of Economics, University of Neuchˆatel, Avenue A.-L. Breguet 1, 2000 Neuchˆatel, Switzerland, e-mail: [email protected] Nathan Kunz e-mail: [email protected] Gerald Reiner e-mail: [email protected] Gil Gomes dos Santos e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 6, 

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1 Theoretical Foundations A fundamental question of operations management is which level of capacity is appropriate for production. In other words, how many employees and machines are required to meet expected demand? As an introduction for the teaching case, the theoretical aspects of these questions are discussed in this chapter. Capacity utilization is an indicator measuring to what degree provided capacity is actually used. Therefore, capacity utilization helps to identify whether there is too much or too little capacity. Capacity utilization can be expressed as the ratio between actual working time and available time of labor and machinery. Labor and machinery costs are important contributing factors to the cost of goods sold and, as a consequence, determine profit or loss. Therefore, determining the right capacity level has great financial implications. From a financial point of view, high productivity is a major goal. Productivity expresses the output of the production per one unit of input, where the measurement units can be defined as quantity or time. For instance, labor productivity can be expressed as the ratio between produced output in time equivalents and available time of labor. Generally, there is a positive relationship between utilization and productivity, as illustrated in Fig. 1 (see Al-Darrab, 2000). From a financial perspective, utilization should be maximized because a high utilization level implies a small proportion of unused capacity, which leads to high productivity. For a given utilization level, productivity can be further increased by improving efficiency (see Al-Darrab, 2000). Efficiency is an indicator for “doing things right” and can be expressed as the ratio between produced output in time equivalents and actual working time of labor or machinery. The impact of efficiency can be seen in Fig. 1. To conclude, managers must address utilization and efficiency to ultimately achieve good financial results.

Fig. 1 Productivity

However, obeying that rule will not necessarily result in a successful business. Apart from “doing things right,” an additional aim must be achieved. That aim

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is high effectiveness, or “doing the right things.” Regardless of the industry and company type, customer requirements must always be taken into account when production processes are designed and managed. Customer satisfaction is defined by not only the physical quality of the products but also the quality in terms of time. Delivery speed and reliability are important competitive factors and, to a great extent, a direct result of the production processes’ design. The initial question, namely which level of capacity is appropriate for production, must be answered from a process management point of view as well. For that purpose, the relationship between capacity utilization and lead time will be discussed. A production order’s lead time is the time span between entering and leaving the production process. The lead time consists of process times, that is, the time spent adding value by utilizing labor and machines, and waiting times, which occur when labor or machines are required but not available. The probability that a resource will be requested but unavailable, resulting in waiting time, increases with higher capacity utilization. Figure 2 illustrates the relationship between capacity utilization and lead time (see Anupindi et al, 1999). Lead time

High variability

Low variability Process time Utilization 100%

Fig. 2 Lead time

Accordingly, for a given maximum average lead time, which derives from customer requirements, the maximum capacity utilization can be determined. The relationship between capacity utilization and lead time is not linear because waiting time rises progressively with increasing utilization. The variability of time (e.g., process times, setup times, the time between the arrival of orders, and machine failure times) is the reason that lead time approaches infinity when capacity utilization approaches 100% (see Hopp and Spearman, 1996). Variability means that for a group of entities, the value of an attribute varies between the entities. For instance, consider a batch in production as an entity and the time needed to set up a machine to process that batch as its attribute. In a real production system, the setup time would usually vary for each batch. Variability can be the effect of random (e.g., quality problems and machine failures) or controllable (e.g., diverse product portfolio and planned maintenance) causes. The coefficient of variability is a common measure for variability

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and is defined as the ratio of the standard deviation to the mean. Figure 2 shows the impact of variability on lead time. If variability can be reduced (e.g., through employee training and standardized products), the lead time for a given utilization can be decreased (see Suri, 1998). To conclude, managers must monitor utilization and variability to achieve process outputs that are in line with customer requirements. It follows from the above discussion that the search for the optimal capacity must not only focus on financial aspects but also take into account process management principles. Such an integrated analysis will be applied to solve the problem described in the next chapter.

2 Problem Description 2.1 Company Background Numerous countries suffer the plague of anti-personnel mines. To restore decent living conditions for many people around the world, millions of mines must be removed. However, manual demining can only clear 5 to 50 square meters per day. The Digger Foundation (see Digger, 2010) is a Swiss humanitarian organization whose aim is to develop and manufacture machines for demining in order to increase working efficiency and reduce the danger for demining personnel. In a market dominated by commercial companies, the Digger Foundation is the only non-profit manufacturer of demining machines. Digger’s current product, the DIGGER D-3, is an armored machine that can withstand detonations, while being lightweight enough to be transported into areas with poor road infrastructure. Furthermore, it can be remote controlled in order to guarantee total safety for the operator, who stays a safe distance away. The DIGGER D-3 is not the only demining machine on the market, but several unique characteristics differentiate it from those of competitors. This machine was designed and manufactured to meet the specific needs of humanitarian mine clearing, such as affordable purchase price, low operating cost, simple maintenance, and high flexibility due to its multi-tool concept. Digger’s production facility is located in the canton of Bern, in Switzerland, in an old military compound. In order to benefit from high flexibility at low fixed costs, Digger works with several subcontractors for the supply of prefabricated components. In the past, only the final assembly has been carried out in Digger’s workshop. However, Digger has started to internalize almost all welding tasks by hiring welders. This configuration allows Digger to better adapt production lead time to customer requirements.

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2.2 Challenge Due to its success, Digger’s product demand is steadily increasing. This requires the company to adapt its strategy to fulfill its humanitarian mission as effectively as possible. Since being established, Digger has received financial support from thousands of donors from both the public and private sector. However, donations are neither easily scalable nor predictable in nature. Therefore, Digger’s long-term objective is to become financially independent and rely primarily on the sale of its products. However, this increases the pressure to raise revenues while keeping production cost low. In recent years, production capacity has been increased by the employment of more salaried staff and the acquisition of additional equipment. Forecasts suggest capacity still might not be sufficient in the future. An important success factor for Digger is its capacity for fast and reliable delivery. However, there are concerns that its labor and machinery cannot cope with increasing demand, resulting in too long delivery times. This could endanger Digger’s growth plans. To successfully supply as much demining equipment as possible, Digger has to find the right balance between acceptable cost and sufficient capacity. Consequently, the following questions must be answered: How much labor and equipment is adequate for the planned growth strategy? Can process performance be improved without increasing capacity?

2.3 Production Details1 Digger produces one main product, the DIGGER D-3, with an expected lead time of, at maximum, six months. Details of product demand and the bill of material can be seen in Tables 1 and 2, respectively. Table 1 Demand data Year 0

Year 1

Year 2

3 units

6 units

9 units

The strategy for manufacturing the DIGGER D-3 is make-to-order. Production operates 8 hours per day, 260 days per year. Data describing the current production’s resources is represented in Table 3, and the production steps are given in Table 4. For each activity, labor and machinery are simultaneously utilized. The production lot size of the product is 1, while the lot size of each component equals the quantity needed to manufacture 1 unit of the product. Batch sizes for the transport between 1

In order to respect Digger’s confidentiality requirements, the market, operational, and financial figures presented in this case were changed and do not represent the actual circumstances.

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Table 2 Bill of material

Product / Components DIGGER D-3 ├Translation │ ├Tensioner Support │ ├Rear Support │ └Oscillating Support ├Rear Hood ├Front Hood ├Internal Assembly 5 │ ├Internal Assembly 4 │ │ └Internal Assembly 3 │ │ ├Internal Assembly 2 │ │ │ ├Internal Assembly 1 │ │ │ │ └Chassis │ │ │ │ └Hull │ │ │ └Power Train │ │ │ └Engine │ │ └Arm Support │ └Diesel Tank ├Support Triangle ├Front Arm └Electronics ├Electronic Control Unit ├Remote Control Unit └Cable Strand

Units 1 1 2 2 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

stations are the same as the production lot sizes. The variability of the production system is generally moderate. So for the time between the arrival of orders, the operations’ setup and process times, and the machines’ repair times, the coefficient of variability is generally estimated to equal 1. Financial data related to revenue, as well as cost for labor, machinery, and raw materials and supplies, is summarized in Table 5.

3 Solution For different levels of demand, based on the pursued growth strategy, suitable capacity levels in terms of labor and machines have to be determined. Digger is a humanitarian organization, but profit must be generated and reinvested to achieve future growth. Therefore, the right amount of capacity must be found to maximize profit in the long run.

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Table 3 Resource data Resource

Type

Number in Pool

Unavailability

Station

Electric Lab

Machine

1

On average 10 h every 1000 h

ElectronicAssembling

Hoist 1

Machine

1

On average 10 h every 500 h

WeldingAssembling

Hoist 2

Machine

3

On average 10 h every 500 h

GeneralAssembling

Painting Cabin

Machine

1

On average 20 h every 500 h

Painting

Welding Machine

Machine

2

On average 20 h every 1000 h

Welding

Assembler

Labor

3

12% of total time

Painting, General-Assembling

Electronics Technician

Labor

1

12% of total time

ElectronicAssembling

Welder

Labor

2

12% of total time

Welding-Assembling, Welding

profit = (revenue per unit − variable cost per unit) × output − fixed cost. Moreover, lead time constraints, resulting from customer requirements, must be taken into account. average lead time ≤ accepted average lead time. In this way, the financial and process perspective are integrated when determining the optimal capacity. Dynamic cause and effect relationships and process variability prevent a simple deterministic calculation of the output and lead time. In general, simulation can be applied, but the building and application of simulation models is often time consuming and fairly complex. Rapid Modeling can help to overcome those limitations. Rapid Modeling is a method with which to model operational processes as a network of queues and analyze them based on analytical queuing theory (see Suri, 1989). Specialized software can be used to model and analyze production processes in a quick and intuitive way (see Suri et al, 1995; Rabta et al, 2009). For this case study, the software “Rapid Modeler” is used, which has been developed within the project “Keeping Jobs in EU,” funded by the European Community’s Seventh Framework Programme (see University of Neuchˆatel, 2010).

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Table 4 Routings Product / Component DIGGER D-3 Translation Tensioner Support

Rear Support

Oscillating Support

Rear Hood

Front Hood Internal Assembly 5 Internal Assembly 4 Internal Assembly 3 Internal Assembly 2 Internal Assembly 1 Chassis Hull Power Train Engine Arm Support

Diesel Tank

Support Triangle

Front Arm Electronics Electronic Control Unit Remote Control Unit Cable Strand

Step

Station

Setup Time per Batch

Process Time per Unit

1 1 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1 1 1 1 2 3 1 1 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1

General-Assembling General-Assembling Welding-Assembling Welding Painting Welding –Assembling Welding Painting Welding-Assembling Welding Painting Welding-Assembling Welding Painting Welding-Assembling Welding Painting General-Assembling General-Assembling General-Assembling General-Assembling General-Assembling General-Assembling Welding-Assembling Welding Painting General-Assembling General-Assembling Welding-Assembling Welding Painting Welding-Assembling Welding Painting Welding-Assembling Welding Painting Welding-Assembling Welding Painting Electronic-Assembling Electronic-Assembling Electronic-Assembling Electronic-Assembling

20 h 20 h 5h 5h 1h 2h 5h 1h 2h 5h 1h 5h 5h 1h 3h 5h 1h 10 h 4h 4h 4h 4h 10 h 5h 10 h 2h 5h 2h 3h 5h 1h 2h 5h 1h 2h 5h 1h 3h 10 h 2h 5h 5h 2h 10 h

320 h 120 h 10 h 25 h 3h 5h 20 h 1h 3h 15 h 1h 15 h 80 h 9h 10 h 50 h 6h 35 h 20 h 30 h 28 h 18 h 35 h 25 h 110 h 18 h 12 h 13 h 18 h 55 h 7h 8h 30 h 5h 4h 20 h 3h 18 h 100 h 6h 65 h 30 h 33 h 110 h

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Table 5 Financial data Financial Parameter

Value

Revenue Labor cost - all types Depreciation - Hoist Depreciation - Welding Machine Depreciation - Painting Cabin Depreciation - Electronic Lab Cost of raw materials and supplies

520 000 CHF/unit 80 000 CHF/year 10 000 CHF/year 35 000 CHF/year 30 000 CHF/year 20 000 CHF/year 300 000 CHF/unit

By analyzing the effects of different capacity configurations, solutions can be determined, which lead to high profits and comply with lead time restrictions. When comparing different scenarios, capacity utilization should be examined to find a satisfactory solution in a structured way. To avoid unbalanced production, capacity utilization should be below a chosen threshold level for all stations. Table 6 shows the number of resources in the current situation (i.e., Year 0) and that of a possible solution for the next 2 years. Operational and financial results based on these capacity levels are summarized in Table 7 for each year. With the chosen amount of labor and machinery for Year 1 and Year 2, output in accordance with the defined growth strategy can be achieved, lead time will fall below the accepted value, and profit can be generated. Note that the applied method allows for a comparison of the effects of different scenarios but does not guarantee that an optimal solution will be found. Therefore, other reasonable configurations might exist. Table 6 Capacity of solution Resource Electronic Lab Hoist 1 Hoist 2 Painting Cabin Welding Machine Assembler Electronics Technician Welder

Year 0

Year 1

Year 2

1 1 3 1 2 3 1 2

2 2 4 1 4 5 2 5

2 2 6 1 5 6 2 6

So far the focus has been on the adjustment of process capacity. But can process performance also be improved without changing capacity levels? Generally, that is possible by measures increasing efficiency and by measures reducing variability. Different cases demonstrate how organizational measures can improve performance before investing in additional capacity (see Bourland and Suri, 1992; Enns, 1996). Efficiency can be positively influenced, for instance, through the use of advanced technology. For example, Digger could introduce new welding techniques

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Table 7 Results of solution Measure

Year 0

Year 1

Year 2

Output Lead time Revenue Depreciation Labor cost Cost of raw material and supplies Profit

3 units 5.8 months 1 560 000 CHF 160 000 CHF 480 000 CHF 900 000 CHF 20 000 CHF

6 units 5.9 months 3 120 000 CHF 270 000 CHF 960 000 CHF 1 800 000 CHF 90 000 CHF

9 units 5.9 months 4 680 000 CHF 325 000 CHF 1 120 000 CHF 2 700 000 CHF 535 000 CHF

that shorten process times. Variability can be addressed, for example, through learning effects. If Digger scales up production, employees might have more routine, leading to less variable process times. Apart from these examples, other possibilities for influencing efficiency and variability should be considered. The effects on process performance can be analyzed with Rapid Modeling by adjusting the respective parameters in the model.

4 Conclusion Decision makers in the field of operations management are frequently challenged by complex problems, which have to be solved with limited time and resources. Therefore, managers and students should be aware of possibilities allowing for the improvement of processes with reasonable effort, while also avoiding oversimplification. This teaching case outlines the potential of Rapid Modeling and highlights the importance of an integrated analysis of processes by combining financial and operational performance measures. The solution of the real-life problem of this teaching case illustrates basic concepts in operations management as well as the possibilities provided by Rapid Modeling software. Acknowledgements This work is supported by the European Union’s Seventh Framework Programme (Marie Curie Industry-Academia Partnerships and Pathways, Keeping Jobs in EU, project number 217891) and the University of Neuchˆatel (Initiative de soutien a` des d´emarches d’enseignement innovantes 2009/2010).

References Al-Darrab I (2000) Relationships between productivity, efficiency, utilization, and quality. Work Study 49(3):97–103

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Anupindi R, Deshmukh S, Chopra S, van Mieghem J, Zemel E (1999) Managing business process flows. Prentice-Hall, Upper Saddle River Bourland K, Suri R (1992) Spartan industries. Tech. rep., Case Study, Tuck School of Business, Dartmouth College, Hanover, NH Digger (2010) Digger D.T.R. URL http://www.digger.ch Enns S (1996) Analysis of a process improvement path using Rapid Modeling. Total Quality Management & Business Excellence 7(3):283–291 Hopp W, Spearman M (1996) Factory Physics: foundations of manufacturing management. Irwin Inc., Chicago Rabta B, Alp A, Reiner G (2009) Queueing networks modelling software for manufacturing. In: Reiner G (ed) Rapid Modelling for Increasing Competitiveness: Tools and Mindset, Springer, London Suri R (1989) Lead time reduction through Rapid Modeling. Manufacturing Systems 7(7):66–68 Suri R (1998) Quick Response Manufacturing: a companywide approach to reducing lead times. Productivity Press Suri R, Diehl G, de Treville S, Tomsicek M (1995) From CAN-Q to MPX: Evolution of queuing software for manufacturing. Interfaces 25(5):128–150 University of Neuchˆatel (2010) Keeping Jobs in Europe. URL http://www2.unine. ch/iene-kje

Part III

Rapid Modelling in Services

Quick Response Service: The Case of a Non-Profit Humanitarian Service Organization Arda Alp, Gerald Reiner and Jeffrey S. Petty

Abstract The focus of this paper is to explore and discuss the applicability of traditional operations management principles within the context of humanitarian service operations (HSO), illustrated by a nonprofit humanitarian service organization (HNPSO). We want to make two major contributions related to performance improvement based on lead time reduction and performance measurement. First, we develop an improvement framework to analyze and reduce the service lead time in parallel with provision of an improved capacity management. The results of this study show that lead time reduction strategies in combination with queuing theory based modeling techniques (Suri, 1998, 2002; Reiner, 2009) help the HNPSO managers effectively manage their service providing processes. Such an integrated and profound capacity management enables organization to deal with short-term demand fluctuations and long-term growth. In this way managers can find the balance between the provision of daily operations as well as the maintenance of monetary income to secure the growth of the organization and continuous improvement. Furthermore, we highlight the benefits and challenges of an aggregated performance measurement approach in a HNPSO. Our approach links operational, customer oriented, and financial performance measures, gives management competitive advantage more relevant than that of a traditional performance system. Considering the relatively limited operations management applications in nonprofit performance measurement systems, this paper contributes to both research and practice.

Arda Alp (B) and Gerald Reiner Entreprise Institute, University of Neuchˆatel, Rue A.L. Breguet 1, 2000 Neuchˆatel, Switzerland, e-mail: [email protected] Gerald Reiner e-mail: [email protected] Jeffrey S. Petty Lancer Callon Ltd., Suite 298, 56 Gloucester Road, UK-SW7 4UB London, e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 7, 

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Key words: non-profit humanitarian service organization, performance measurement, lead time reduction, rapid modeling

1 Introduction During recent years humanitarian organizations have become the focus of more and more interest among researchers in the field of operations management as well as supply chain management. Since the 90ies, academic literature in these fields made remarkable developments (Chandes and Pach´e, 2010). Furthermore, the non-profit sector is increasingly perceived as ‘big business’ and is also becoming important in several countries (US: ∼10% of GDP, Ireland:∼8.8% of GDP, UK:∼2.25% of GDP Wainwright, 2003; Gunn, 2004; Taylor et al, 2009). Our study will focus on specific organizations in this broad field, i.e., humanitarian nonprofit service organizations (HNPSO) will be investigated in more detail. Our aim is to explore and discuss the applicability of traditional operations management principles within the context of humanitarian service operations (HSO). Furthermore, we want to make two major contributions related to performance measurement and performance improvement based on lead time reduction. As previously mentioned, traditional operations management (developed in a manufacturing context) may not be sufficient because, owing to the differences between manufacturing and services, specific concepts and approaches should be developed for service operations management. Craighead and Meredith (2008) and Voss (2005) provided some evidence for this requirement. Within service environments and the related processes can typically be characterized as ”make-to-order”. Therefore, specific strategies have to applied, e.g. demand or process variability cannot be hedged by safety inventory. For analyzing ”make-to-order” systems queuing theory plays an important role. A queuing system usually consists of an arrival process, a production or service process, and a queue (Hopp and Spearman, 2007). This model clearly shows that waiting times rise with process time, with average utilization and with variability. For managers it follows that in case of high variability it is not reasonable to maximize average utilization, as flow time will rise exponentially, and customers may have to wait too long. Hence, an important performance measure in service operations is the waiting time. Furthermore, (Bielen and Demoulin, 2007; Davis and Vollmann, 1990) provided evidence for the strong link between waiting time and customer satisfaction. It is well known in standard Operations Management (OM) literature that waiting times can be reduced by integrated demand and supply management (Heikkil¨a, 2002). We want to investigate how this approach can be transferred to HSO in general as well as to HNPSOs more specifically. The performance measurements of a HSO is an important issue as donors require accounts of activities, particularly from HNPSO, without forgetting that the clients (volunteers, affected populations, etc.) run the risk of being the silent victims of poor performance in terms of processes and output (Chandes and Pach´e, 2010).

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A fundamental idea in the field of OM for the evaluation of process improvement is to first analyze the alternatives based on performance measures (e.g. time, WIP, etc.). Afterwards these performance measures will provide the input for the financial evaluation. We will show that this approach (in particular the performance evaluation) is vital for nonprofit organizations (NPOs) and voluntary and charitable organizations (VCOs), as opposed to relying upon the classical financial performance measures. Current performance measures of nonprofit organizations are accountability focused with weak linkages to their continuous improvement strategy or without any link to operational, customer-oriented measures (Moxham, 2009a; Poister, 2003). A recent study of Oloruntoba and Gray (2009) indicates that only a limited number of studies focus on the link between customer satisfaction and non-profit service organizations/providers performance. Beamon and Balcik (2008) presented some performance metrics in the humanitarian organization context that addresses the above mentioned problem. The main aspect is the differentiation between efficiency (utilization, costs, etc.) and effectiveness (customer satisfaction) dimensions. What managers need to do is to find the cause and effect relationships between the different performance measures rather than choosing between them (Behn, 2003; Banker et al, 2000). Our overall objective is to develop and present an approach for a HNPSO that enhances their operations and, ultimately, increases their performance. This increased performance should improve their competitiveness and positioning among other HNPSOs. Since customer satisfaction is a step towards competitiveness (Johri, 2009), our minor objective is to improve their customer satisfaction (internal and external) by matching capacity with demand (efficient management of capacity). At this point we demonstrate how lead time reduction strategies and rapid modeling techniques (De Treville et al, 2004; Suri, 1998, 2002; Singh, 2009; Reiner, 2009) can help VCOs and NPOs to achieve these objectives. Recent studies in the VCO and NPO sector (Taylor et al, 2009; Moxham and Boaden, 2007; Moxham, 2009b) underscore the importance of the link between an organization’s improvement strategy, financial decisions, and its performance measurement system. NPOs and VCO’s gain more competitive advantage if the performance measurement is used as a part of the improvement strategy (Moxham, 2009a,b). Therefore, NPOs and VCO’s need more support to link their continuous improvement strategy and performance measurement systems. This study will be of interest to both managers and academics because our findings add to limited knowledge on OM, rapid modeling, and lead time reduction techniques in the non-profit sector. In Section 2, we explain the improvement framework, focusing on the analysis phase and the modeling. In Section 3, we provide an illustration of the improvement framework. We review the current status of the HNPSO and describe how we implement the analysis and modeling within the organization. We then discuss our initial findings and highlight improvement possibilities. Finally, in Section 4, we provide our conclusions, highlight limitations, and discuss further research possibilities.

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2 Improvement Framework: Analysis Phase and Modeling In this section we would like to introduce a generic improvement framework that can be used for HNPSOs. The starting point is the analysis phase, where we use several observations to analyze the current state of the HNPSO based on three main subject areas. (i) Communication: We will examine how the business processes are connected to each other and whether the communication is completed effectively or not. If not, what makes for effective communication? We also focus on how the HNPSO’s departments and employees transfer information internally and how this information is stored. (ii) Capacity management: We will analyze how the HNPSO can meet current and future business requirements in a cost-effective manner using their existing capacity. At this point, it is very crucial for the HNPSO’s management to know whether existing capacity is enough to achieve the organization’s goals. We also determine which activities should be focused on in order to improve the organization’s performance and efficiency. (iii) Performance management: Here we pose two fundamental questions. Do performance measures adequately give information about services and the processes? Do performance measures provide sufficient information necessary to make smart management decisions? As we have mentioned above, it is necessary to identify the cause and effect relationships between the different performance measures. Furthermore, we have to collect specific data that is necessary for the generation of the analysis model to support the generic improvement framework, namely: • • •

Process flow data Information flow data Performance measures

We use queuing networks software tools Rapid Modeler (http://www.unine.ch/ienekje) and MPX, to build the analysis model. These queuing networks software tools are designed for manufacturing as well as service processes. The model development process is automatic and embedded into the software (Rabta et al, 2009). The idea behind these tools is based on modeling processes as a network of queues considering the interaction of the nodes. Analysis is based on principles of queuing theory-node decomposition approaches. The analytical model uses several types of data (i.e. demand data, routing data, resource data, bill of material, production lot sizes) in order to yield critical process performance measures such as capacity utilizations, process lead times, WIP, etc. (Schodl, 2009). Before we are able to analyze different process improvements we have to ensure that the model behaves as intended (verification) and that the model provides the same output data as the real system (validation) (see in detail Kelton et al, 1998; Law et al, 1991). Accurate decision support is only possible with a ‘valid’ model which is an accurate representation of the system being modeled. Thus, verification and validation are crucial to ensure that the decisions made with the modeling tools are consistent with those that would be made by physically experimenting with the

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system (Fig. 1). For instance, validation can be done by comparing the mean production lead time with actual system’s average lead time (Schodl, 2009). In the next section we will highlight the validation of our model in greater detail. Validation Verification Establish credibility System

Conceptual model

Analysis and data

Modeling

Establish credibility

Validation

Rapid modeler

‘Correct’ results Running model

Results implemented

Sell results to management

Fig. 1 Relationship of verification, validation ans establishing credibility (adapted from Chapter 5, pp. 298, Law et al (1991))

3 Illustration of the Improvement Framework 3.1 Current Status of the Organization In this section we describe the HNPSO case in detail. In pursuit of its mission, this organization employs a holistic business model which integrates international, local, and online programs that are available to volunteers worldwide. Each program provides meaningful volunteer experiences (i.e., cross-cultural relationships) and support (i.e. development projects, teaching, medical aid, etc.) for communities in over 40 countries (developed as well as developing countries). Here we will describe the current status of HNPSO within the context of the aforementioned three subject areas: (i) Communication: Current communication and synchronization problems internally among program coordinators; externally with customers- lead to time and capacity related inefficiencies. Some of these inefficiencies come from waiting times: waiting for documents/information from colleagues (in progress) or for approval (i.e. signature). Lack of coherence among departments, including ill-defined, unstructured ways of communication, lack of common documentation and information, causes extra workloads, repeated activities and increased processing times. This naturally leads to unexpected delays, jobs waiting in queues, and over utilization of the labor force. (ii) Capacity management: In the context of the HNPSO’s programs, fluctuations in customer applications during peak periods create a capacity complication. This problem leads to increased workloads, unexpected waiting times, increased response times and finally there is an impact on internal and external customer satisfaction.

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Management needs better decision support for the allocation of the current workforce to solve the problem of potential customer loss. Currently it is not clear if the current workforce is sufficient for proper customer tracking and other activities. (iii) Performance management: Currently two major points are not very clear for the management team. Do the HNPSO’s performance measures adequately give information about the organization’s services and the processes? Do the HNPSO’s performance measures provide sufficient information necessary to make smart management decisions? More attention is needed on how their performance measures are defined and being used. Relatively, there is less importance placed on operational measures and inadequate focus on customer related measures. As such, much effort is needed on customer related evaluations and customer satisfaction related definitions. In this sense, tracking of customers needs to be integrated with the customer related measures. Also, the HNPSO management team should consider performance criteria important for their funders, which are primarily donors, since, donors may wish to measure the effective and efficient use of their donations (Wainwright, 2003).

3.2 Rapid Modeling In this subsection we describe how we implemented the analysis model within the context of the organization. The model uses only labor force (e.g. paid, volunteer, etc.) as its resources and there are not any machine type resources. Products are all in the form of services such as volunteer programs, development projects, etc. Outputs are the lead time values for each program, which include labor utilizations and the number of programs accomplished. Using the number of programs accomplished, we can also calculate the number of customers served. Resources, services, outputs and outcomes are illustrated in Fig. 2. Currently there is no activity to measure the effects of outputs on outcomes (instant, intermediate, long term). Verification and validation of the model is done using current customer order lead times as a benchmark value (Fig. 3). Data collection and evaluation were conducted during a three-month period. We collected data on current measurement practices using structured and ad-hoc interviews. In addition, we also used other text based records such as the annual board meeting reports, documentation associated with the monitoring and evaluation of customers (i.e. alumni surveys), performance (i.e. applications numbers and volumes, etc.) and website related metrics, figures, rankings, process definitions, etc. In addition, we also examined the existing process flow and collected the related task completion times in order to use them in our capacity related calculations.

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Outcomes (intermediate)

Outcomes (initial)

Outputs External Influences Services

Resources

NOT CURRENTLY MEASURED

Lead time values for each program Labour utilizations Number of customers served Number of programs/projects accomplished Volunteer Program–International Development Projects – Local Alumni Clubs – Local Awareness Project– Local

7 paid staff, full-time 6 volunteer, full-time 11 volunteer, part-time 1 building Funds, donations, program related incomes

Fig. 2 Generic program logic model for the case NPO; Adapted from Poister (2003) and Moxham and Boaden (2007)

Fig. 3 Validation of the model

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3.3 Initial Findings We observe significant variation among labor utilizations. Current utilization and idle time percentages are given in Table 1. Subsequently, Table 2 shows the calculated flow time efficiency of HNPSO using current lead-time and the real service time to indicate the amount of waiting time associated with the process. We observe that none of the HNPSO’s programs achieve a benchmark value representative for the industry (Chopra, 2010). Table 1 System utilization

Table 2 System flow time efficiency

We analyze the effect of possible process time variations on flow time and utilization of all programs: how much the service times of critical processes have to be reduced in order to achieve a level of overall process utilization that guarantees reliable system performance. For instance, current service time of some internal activities can take up to ∼1−2 days with a labor utilization of 81.3%–93.7%. However, it is possible to reduce the service time of these bottleneck activities from days to hours of 480–20 minutes with a level of ∼76.3%–77% of labor utilization.

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3.4 Improvement Possibilities In this section, we observe the possibility of several improvements in parallel – driven by communication related, capacity management related, and performance system related actions: (i) Communication related improvement and related effects: The main goals should be to make communications and collaboration secure, cheap, efficient and easier to manage. Efficiency can be increased with communications and collaboration platforms, as well as unified communications technologies. Integration of processes among several departments is inevitable. Better process and definitions (i.e. data hierarchy, proper documentation), simplified processes (i.e. decrease of unnecessary tasks), written procedures (i.e. less preparations, less mistakes) will also help. Communication improvement has a significant positive effect on the system performance. Better process definitions facilitate cross-functional communication, which lowers workloads and dramatically decreases the flow time. A lead time of a few days is achievable compared to the current situation with several months of lead time (Table 3). Table 3 Effects of communication improvements

(ii) Capacity related improvement and related effects: Improving the capacity management of the HNPSO (i.e. application of an integrated capacity management) can help the organization to compensate short-term demand fluctuations and longterm growth. Better training and an increase of a multi-skilled workforce can provide benefits such as a reduction of the total required workforce, labor cost savings and flexibility (i.e. less dependency of program managers and top management). We analyze capacity related effects under four scenarios: (1) the current situation, (2) a moderate demand increase, (3) demand increases realistic for long-term growth, and (4) demand increases for short-term fluctuations. The demand increase factors are based upon actual demand data. Table 4 shows how active capacity management can address a growth of demand while ensuring process performance. The results indicate that the current system is very sensitive to any small variation in demand. If improvement suggestions are not taken into consideration, the current system has very limited capacity to hedge any demand variation. Therefore,

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Table 4 Effect of capacity management

these capacity-based recommendations can help the organization to deal with shortterm demand fluctuations and long-term growth. In this sense, if the paid workforce is increased by one, the system will be able to hedge moderate demand increases (Scenario 2). Scenario 3 reflects a realistic demand increase for long-term growth and system can handle this with one more paid employee. For the given capacity adjustments, flow time decreases for both Program 1 and 2. Scenario 4 reflects a realistic demand increase for short-term growth and system can handle this with two additional paid employees, while flow time decreases for all programs. (iii) Performance system related improvement and related effects: A performance system, which links operational, customer oriented, and financial performance measures gives management information, which is more relevant than that of a traditional performance system. Management needs better decision support and for this, a comprehensive performance measurement system is essential. Our performance related calculations consider fundamental financials such as cost of goods sold, unit payroll per person, revenues, gross margin, net income, profit and loss statements. In addition, we calculate a theoretical value for the number of lost customers. We assume that customers can be lost due to unsatisfied demand and long lead times. In this sense: (i) lost customers(demand) = actual demand – satisfied demand; (ii) lost customers(leadtime) = actual demandt−1 * proportion of lost customers. The proportion of lost customers has its maximum and minimum values between the maximum acceptable customer delivery time and the average ideal lead time (30% shorter process times with process improvements). Each lost customer represents a potential indirect loss for the HNPSO in magnitude of unit revenue per program. We consider the same scenarios and analyzed two situations for each scenario; ‘Situation A’ where we assume the HNPSO’s management does not prefer to proceed with a workforce increase; ‘Situation B’ where the HNPSO management

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prefers to proceed with a workforce increase. In addition to results of the workforce increase on utilization and lead time values, this section illustrates the related effects selected financial measures. Current system has a negative income. Regarding a moderate demand increase, current capacity is not able to cope with this increase. This results also in lost sales. However, if capacity is increased, demand can be satisfied and this results in slightly decreased negative income. Regarding the long-term growth, current capacity is not able to cope with assumed growth and the outcome of this is lost sales which is almost 3.5 times higher than the initial lost sales value. The system can handle a demand increase with additional workforce, and this yields a positive income (relatively 78.70% of the absolute value of initial lost sales). Regarding the short-term fluctuations, on average 100% variation is likely to happen as short-term fluctuation. Currently the system is not able to handle this demand variation, which results in lost sales (almost 9 times larger than the initial value). With two additional employees, the system can handle this demand variation, resulting in a positive net income (relatively 694.55% of the absolute value of initial lost sale). The analysis of the four scenarios illustrates an approach, which integrates: (i) operational data (lead time, achievable output); (ii) customer-oriented data (lead time-dependent customers satisfaction); (iii) financial data (unit revenue, unit payroll).

4 Conclusion Current performance measurement practices of the nonprofit sector are still limited (Moxham and Boaden, 2007; Moxham, 2009a; Taylor et al, 2009). However, the findings of this study can encourage managers of VCOs and NPOs to consider rapid modeling techniques and lead time reduction strategies in their operational and strategic decisions. We observe that a performance system, which links operational, customer oriented, and financial performance measures, gives management information that is more relevant than that of a traditional performance management system. In addition, an increased emphasis on efficient capacity management should be given a very high priority. If capacity is not adjusted to cope with the increase in demand, this leads to increased lead times, decreased customer satisfaction, and, ultimately, lost sales (theoretically). In this sense, rapid modeling techniques can help the managers of NPOs and VCO’s to manage their capacity, which will gradually help them to reduce their service delivery lead times. In return, this will provide a benefit as improved client satisfaction and better economic situation (i.e. increased sales). Our approach of linking operational, customer oriented, and financial performance measures is necessary to enable NPOs and VCOs to maintain their continuous improvement (Tersine and Hummingbird, 1995; Bozzo and Hall, 1999) in pursuit of their social missions.

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Considering the relatively limited operations management applications in nonprofit performance measurement systems, this paper contributes to both research and practice. The study provides a detailed examination of performance measurement at strategic and operational levels, as well as impact of lead time reduction strategies and rapid modeling techniques on an HNPSO’s organizations strategy. However, we need to expand the scope of our study in order to validate the applicability of our findings to the diverse nonprofit sector. Acknowledgements This work is supported by the SEVENTH FRAMEWORK PROGRAMME – THE PEOPLE PROGRAMME – Marie Curie Industry-Academia Partnerships and Pathways Project (No. 217891) ”Keeping jobs in Europe”.

References Banker R, Potter G, Srinivasan D (2000) An empirical investigation of an incentive plan that includes nonfinancial performance measures. The Accounting Review 75(1):65–92 Beamon B, Balcik B (2008) Performance measurement in humanitarian relief chains. International Journal of Public Sector Management 21(1):4–25 Behn R (2003) Why measure performance? Different purposes require different measures. Public Administration Review 63(5):586–606 Bielen F, Demoulin N (2007) Waiting time influence on the satisfaction-loyalty relationship in services. Managing Service Quality 17(2):174–193 Bozzo SL, Hall MH (1999) A review of evaluation resources for nonprofit organizations. Tech. rep., Canadian Centre for Philanthropy Research Chandes J, Pach´e G (2010) Investigating humanitarian logistics issues: From operations management to strategic action. Journal of Manufacturing Technology Management 21(3):320–340 Chopra S (2010) Flow time efficiencies in white collar processes. URL Available at: http://www.csun.edu/ aa2035/SOM686/powerpoints/Part(Retrieved: April, 2010) Craighead CW, Meredith J (2008) Operations management research: Evolution and alternative future paths. International Journal of Operations & Production Management 28(8):710–726 Davis M, Vollmann T (1990) A framework for relating waiting time and customer satisfaction in a service operation. Journal of Services Marketing 4(1):61–69 De Treville S, Shapiro R, Hameri A (2004) From supply chain to demand chain: The role of lead time reduction in improving demand chain performance. Journal of Operations Management 21(6):613–627 Gunn C (2004) Third-sector development: Making up for the market. Cornell University Press Heikkil¨a J (2002) From supply to demand chain management: Efficiency and customer satisfaction. Journal of Operations Management 20(6):747–767

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Hopp W, Spearman M (2007) Factory physics: Foundations of manufacturing management. McGraw-Hill/Irwin Johri G (2009) Customer satisfaction in general insurance industry - A step towards competitiveness. Journal of Risk & Insurance Pravartak 4(3):1–9 Kelton W, Sadowski R, Sadowski D (1998) Simulation with ARENA. McGraw-Hill, USA Law A, Kelton W, Kelton W (1991) Simulation modeling and analysis. McGrawHill, New York, USA Moxham C (2009a) Performance measurement: Examining the applicability of the existing body of knowledge to nonprofit organisations. International Journal of Operations & Production Management 29(7):740–763 Moxham C (2009b) Quality or quantity? Examining the role of performance measurement in nonprofit organizations in the UK. In: Proceedings of 16th International Annual EurOMA Conference, G¨oteborg, Sweden Moxham C, Boaden R (2007) The impact of performance measurement in the voluntary sector: Identification of contextual and processual factors. International Journal of Operations & Production Management 27(8):826–845 Oloruntoba R, Gray R (2009) Customer service in emergency relief chains. International Journal of Physical Distribution & Logistics Management 39(6):486–505 Poister T (2003) Measuring performance in public and nonprofit organizations. Wiley, New York, NY Rabta B, Alp A, Reiner G (2009) Queueing networks modeling software for manufacturing. In: Reiner G (ed) Rapid Modelling for Increasing Competitiveness Tools and Mindset, Springer, London, chap 2, pp 15–23 Reiner G (2009) Preface. In: Reiner G (ed) Rapid modelling for increasing competitiveness: Tools and Mindset, Springer Publishing Company, Incorporated, London, Springer, London Schodl R (2009) The best of both worlds - Integrated application of analytic methods and simulation in supply chain management. In: Reiner G (ed) Rapid Modelling for Increasing Competitiveness - Tools and Mindset, Springer, London, chap 13, pp 155–162 Singh P (2009) Improving lead times through collaboration with supply chainpartners: Evidence From Australian Manufacturing Firms. In: Reiner G (ed) Rapid Modelling for Increasing Competitiveness - Tools and Mindset, Springer, London, chap 23, pp 293–305 Suri R (1998) Quick response manufacturing: A companywide approach to reducing lead times. Productivity Press Suri R (2002) Quick response manufacturing: A competitive strategy for the 21st century. In: Proceedings of the POLCA Implementation Workshop Taylor M, Heppinstall M, Liao M, Taylor A (2009) Performance management and funding in the Third Sector: A research agenda. In: Proceedings of 16th International Annual EurOMA Conference, G¨oteborg, Sweden Tersine R, Hummingbird E (1995) Lead-time reduction: The search for competitive advantage. International Journal of Operations and Production Management 15(2):8–18

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Applying Operations Management Principles on Optimisation of Scientific Computing Clusters Ari-Pekka Hameri and Tapio Niemi

Abstract We apply operations management principles on production scheduling and allocation to computing clusters and their storage resources to increase throughput and reduce lead time of scientific computing jobs. In addition, we study how this approach affects the amount of energy consumed by a computing job comprised of hundreds of calculation tasks. Methodologically we use the design science approach by applying domain knowledge of operations management and efficient resource allocation on the efficient management of the computing resources. Using a test cluster we collected data on CPU and memory utilisation along with energy consumption on different ways of allocating the jobs. We challenge the traditional one job per one processor core method of scheduling scientific clusters with parallel processing and bottleneck management. We observed that by increasing the utilisation rate of the cluster memory increases throughput and decreases energy consumption. We studied also scheduling methods running multiple tasks per CPU core and scheduling based on the amount of free memory available. The test results showed that, at best these methods both decreased energy consumption down to 45% and increased throughput up to 100% compared to the standard practices used in scientific computing. The results are being further tested to eventually support LHC computing of CERN.

1 Introduction Scientific computing clusters are widely used in many research disciplines, especially in experimental physics, astronomy and bio sciences. Computing intensive research deploys easily thousands, even hundreds of thousands of CPUs to analyse Ari-Pekka Hameri (B) HEC, University of Lausanne, CH-1015 Lausanne, Switzerland, e-mail: [email protected] Tapio Niemi Helsinki Institute of Physics, CERN, CH-1211 Geneva, Switzerland, e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 8, 

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various data sets and models. These clusters can be viewed as production resources processing jobs consisting of numerous tasks. The tasks can be processed by different resources and finally the jobs are assembled together to be delivered back to the customers of the cluster. Jobs are in the backlog of the cluster and they may have severity priorities, deadline constraints and each production resource has its capacity and utilisation levels. Operating such a cluster has its own cost structure related to capital invested, energy consumed, maintenance work, and facility related costs. In all, a computing cluster resembles closely an industrial production unit, thus the working hypothesis for this paper is to apply operations management principles to improve computing cluster productivity, overall efficiency, and customer satisfaction. We set to study this opportunity by using real experimental physics data, computing jobs and dedicated computing cluster. As CERN, the European Organization of Nuclear Research at Geneva, provides us with a unique possibility to experiment and test the hypothesis. Once fully operational, CERN and its Large Hadron Collider (LHC) experiment will produce about 15 petabytes of data in a year. One copy of this data is stored at CERN (so called Tier-0 site) and another copy of the data is distributed to 11 Tier-1 sites around the world. From Tier-1 sites the data is further forwarded to Tier-2 sites where simulations and user specific analyses based on the experimental data are performed. The overall computing infrastructure comprises numerous computing clusters of alternating size, yet the total amount of CPUs is well over 100 000. Efficient management of these computing resources is vital for the success of the project, which is foreseen to be active the next 20 years. Our approach is based on operations management and the used research approach follows that of design science by applying the domain knowledge from operations management to efficient management of the computing resources. Past research in the field of computing system efficiency has mostly focused on hardware and infrastructure aspects, e.g. the development of more efficient hardware or optimising cooling of computer centres. The theoretical background of our research comes from production optimisation research. Production engineers have always been searching for ways to increase throughput of a facility with limited capacity. The principles of production dictate several challenges to them, like the law of capacity dictating that in a steady state, all plants will release work at an average rate that is strictly less than average capacity. This means that the organization and allocation of work in production system affects the eventual performance of the system. Lead time, the time designated for a job to traverse a designated portion of the production process, is vital for customer satisfaction and competitiveness of the system. Throughput, lead time and cycle time, the average time between the release and completion of a job along a given routing, are all vital characteristics when assessing the efficiency of the system. Adding the recycling and energy consumption of the system, we have all critical parameters of any production and value adding system. In IT systems, around 50% of the energy consumption comes from cooling. Therefore reducing the electricity consumption of a computational task by n units actually, reduces the total energy consumption twice the amount. When applying new optimising methods to large scale computing resources, the results of the

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project can bring remarkable savings, especially in large computing installations like the one used by CERN where the total energy consumption is several megawatt hours a year. The results of our earlier work have shown that optimising the configuration of workload management system can both decrease energy consumption and increase the total efficiency. The measured improvements so far have been increasing throughput up to 100% and decreasing energy consumption down to 40-50% compared to the standard practice in the LHC computing. Generally, value adding production systems with high throughput and short lead times have been proven to generate other benefits than the pure output performance. Statistically, these systems are also producing better quality and less waste, thus having an overall better environmental efficiency. Also systems which are operationally better performing have more satisfied customers and tend to be more competitive in the market. Based on this, the optimized computing system will, in addition to reduced electricity consumption, also work more reliably and offer more computing power. In the following, we briefly describe earlier research in the field, and then we detail our methodological approach and the research hypotheses together with our test setting. The results on using different loading principles to run the jobs through the test cluster are discussed and compared before final conclusions are drawn.

2 Related Work On the operations management point of view the mathematical link between throughput or operational speed and inventory level was originally demonstrated by Little (1961) who showed that speed is directly proportional to average inventory. This means that inventory speed will increase as inventory reductions are obtained and the throughput rate remains constant. Scale and cost-centric manufacturing dominated operations management research until the 1970s when the quality movement turned the focus to continuous improvement and errorless operations. This meant that ways of reducing inventory would improve the lead time and punctuality of the system. Numerous scheduling techniques were developed to make production planning easier, as the fundamental sequencing problem leads to polynomial growth of alternative solutions. Tackling this challenge lead to the simplification of production facilities and to the birth of the Just-in-Time principles, which have been widely adopted by assembly industries ever since. Further, Goldratt and Cox (1984) came out with their optimised production technology that focused on bottlenecks and the throughput of the system. Hopp and Spearman (1996) compiled the key set of the mathematical principles determining lead time and its implication to the performance of the production facility. To improve computing centre efficiency has attained a fashionable label, the so called green computing. It is a wide topic incorporating issues like data centre location near cheap energy sources (Brown and Reams, 2010), minimising so called e-waste (Hanselman and Pegah, 2007), designing optimal cooling infrastructure

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and running the centre in an optimal way (Marwah et al, 2009). Generally we can say that energy and resource optimisation has mostly focused on hardware and infrastructure issues, not so much on operational methods such as workload management and even less on operating system or application software optimisation for energy-efficiency. We first present methods focusing on the whole data centre or at least the cluster level. For example, Lefurgy et al (2007) suggested a method to control peak power consumption of servers. The method is based on power measurement information on each computing server. Controlling peak power makes it possible to use smaller and more cost- and energy-effective power supplies. Poweraware schedulers can be seen belonging to the same category. For example, Rajan and Yu (2008) and Mukherjee et al (2009) have studied this topic. Scheduling is a widely studied topic and most of the work focuses on finding optimal schedules when jobs have preconditioned constraints and/or strict time limitations. Usually this work has been related to high-performance computing in which the aim is to optimise processing time of individual computing jobs. These jobs can have strict deadlines or require massive parallelism. Our focus area is closer to so called high-throughput computing in which individual tasks are not very time critical and the aim is to optimise the total throughput of the system over a longer period of time. This has received less research interest so far. Actually some studies suggest clearly opposite approaches, e.g. Koole and Righter (2008) suggest a scheduling model in which tasks are replicated to several computers. There are some studies on energy-aware scheduling, like Bunde (2006), who has studied power aware scheduling methods for minimising energy consumption and not reducing system performance by applying dynamic voltage scaling technologies. Goes et al (2005) have studied scheduling of irregular I/O intensive parallel jobs. They note that CPU load alone is not enough but all other system resources (memory, network, storage) must be taken into account in scheduling decisions. Santos-Neto et al (2004) studied scheduling in case of data-intensive data mining applications. Wang et al (2009) have studied optimal scheduling methods in a case of identical jobs and different computers. They aim to maximise the throughput and minimise the total load. They give an on-line algorithm to solve the problem. The second group of studies focus on the server level. Venkatachalam and Franz (2005) gave a detailed overview on techniques that can be used to reduce energy consumption of computer systems. Li et al (2005) studied performance guaranteed control algorithms for energy management of disk and main memory. Ge et al (2005) studied methods based on dynamic voltage scaling technology of microprocessors and created a software framework to implement and evaluate their method.

3 Methodology and Research Problem Our research approach follows that of design science, which according to Hevner et al (2004) must produce a viable artifact in the form of a construct, a model, a method, or an instantiation. We follow Simon (1973) definition of design science

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that emphasises the process of exploration through design: design science is research that seeks (i) to explore new solution alternatives to solve problems, (ii) to explain this explorative process, and (iii) to improve the problem-solving process. Following Holmstrom et al (2009) this research aims to complete all four steps of design science approach. 1. Solution incubation: development of an initial solution design. Our earlier research has shown that better allocation of computing tasks produces better throughput and energy-efficiency. By using the existing theory in different domain, namely production and operations management, the initial working solution was established and tested. This needs to be refined and tested in larger scale, with much larger load and computing resources. 2. Solution refinement: solving the problem. This step forms the main part of this research project. The intended consequences of the solution need to be confirmed with a real production environment. The obtained results need to be verified and tested in different configurations. 3. Explanation I: Development of substantive theory and establish theoretical relevance. The theoretical implications of the refined solution design are analysed and documented. Their impact on to exiting body-of-knowledge is to be discussed. 4. Explanation II: Development of formal theory, i.e. strengthen theoretical and statistical generalisability. This will be covered in our future work. We aim to complete theoretical and empirical examination of relevant contingencies and develop a formal representation of the solution design. If possible we aim also to refine the solution design in multiple contexts, e.g. for other computational problems. The research problem for our study questions how it is possible to improve throughput and energy-efficiency by scheduling the jobs based on their estimated properties and loads of all components (CPU, memory and network) of computing nodes. Further on we seek ways on how to schedule task and jobs in order to maximise throughput and minimise energy-consumption in computing clusters (see Figure 1). The main difference of this setting when compared to manufacturing systems is that in a computer it is possible to run several tasks in parallel, and that tasks are not dedicated to any specific resources. Reasonable overloading can improve throughput but at some point it causes task processing to slow down. The total amount of tasks is also limited by the amount of memory. Another difference with manufacturing is that it is difficult to see what is going on inside the computer, thus all measures are based on in- and output of the system, as actual work in progress takes place literally in a black-box. Our work concentrates on studying, developing, and finally testing different scheduling methods for high energy physics jobs in a computer cluster. Further development of the traditional methods, or configuration and tuning of existing ones will be based on results by the analysis of log data on workload management systems, existing literature, interviewing operators, and own test results. Statistical methods are used to analyse existing data from computing clusters to find out typical

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Fig. 1 Scheduling system

use scenarios and properties of computing jobs such as memory requirements and running times. We used the log files of a Finnish LHC computing facility forming part of the international grid computing infrastructure. The data contain all relevant information on submitted computing jobs during January 2010. This data was used both estimating properties of future jobs and developing realistic test scenarios. The data sets used were mainly related to CMS and Atlas experiments at CERN. In all the data set contained data from around 50 000 jobs. The cluster used for collecting data contained 128 nodes each having two dual-core processors, making the total amount of cores to 512. Before moving on to actual tests and results we briefly analyse 1) what are the properties of average high energy physics (HEP) analysis jobs, 2) what is the utilisation rate of the resources, and 3) whether previously ran jobs can be used to predict properties of future jobs. First we measured memory and I/O utilisation of an individual analysis HEP job. These data intensive analysis jobs read the data from a disk or from a file server through the network in order to perform the analysis for the data. The I/O traffic is relatively high all the time but the memory usage stays low until enough data have been retrieved (see Figures 2 and 3). For scheduling this means that the memory need of a computing node can dramatically increase although no new tasks have been started.

Fig. 2 I/O usage of a HEP analysis job

Fig. 3 Memory usage of a HEP analysis job

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When analysing the log data, we noticed that lead times of jobs are 15 to 50% longer than the actual CPU times. Since the cluster configuration was set to process one job per CPU core, this means that there is a bottleneck slowing down the processing. The most obvious bottleneck is I/O access to the disk and network. When estimating memory utilisation of jobs compared to their CPU utilisation, the memory utilisation rate was about a half from the CPU utilisation rate. One reason for this is an irregular memory utilisation curve of physics jobs. We assume that the other reason is related to I/O waiting times. Based on characteristics on CPU and memory utilisation, we analysed how many jobs were run in parallel in the cluster. The maximum number of jobs running was 481 during the test period (Figure 4). The y axis of the histogram indicates the number of jobs run with the number of simultaneous jobs indicated in x axis. Most of the jobs run simultaneously with 100-150 other running jobs. Since the utilisation rate of the memory is less than half of the utilisation of CPU, we could process more jobs in parallel. Figure 5 shows how many jobs based on our data could have been processed in parallel in a computing node having 8 gigabytes of memory.

Fig. 4 Simultaneously running jobs in the cluster during the time of our sample

Fig. 5 Hypothetical maximum number of possible jobs in a 8GB node

The processing times and memory utilisation of jobs are illustrated in Figures 6 and 7. Variation in these is significant. Since the data does not show any clear pattern, we tested running mean as an estimate for these values for the next job. We used average square error to evaluate different models. In our case small errors are not essential, but it is important to avoid large errors. The best estimate for both CPU time and total memory usage was given by running mean of past 10 values, while for the maximum memory the best was the average of 14 previous values. For comparison, we computed square errors using a random guess (white noise with the same mean and deviation as the real data), the mean of the data, and the previous value. Mean square errors for these were clearly larger than using the running mean.

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Fig. 6 Maximum memory usage

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Fig. 7 CPU usage

4 Tests and Test Results Our test method was to run similar sets of test applications by using different scheduling methods. We used three different test jobs: an I/O intensive data analysis job, a CPU intensive physics simulation job, and a mixed job that contained both I/O and CPU intensive tasks. Each test run contained 50 runs of the same job to reduce variance. We measured the run time and electricity consumption of each test run and calculated throughput and electricity consumption per job based on the measured values. Finally each of these 50-run groups were processed five times and the final numbers are an average of these results. The test computer was similar as the ones used in production clusters. It has two Intel Xeon E5520 quad core 2.27 GHz processors, 16 GB memory, and Linux operating system. The workload management system was Sun Grid Engine (SGE) (Sun Microsystems, 2008). In scientific computing clusters, so called ”single job (task) per core” -scheduling method is practically the default setting. It is also the default setting of the SGE workload management system. It simply means that each CPU core can run maximum one job. Usually the jobs are distributed equally to all computing nodes of the cluster. This inflexible scheduling method is at its best when jobs are pure CPU or memory jobs, but with I/O intensive applications, I/O traffic becomes a bottleneck that makes the processor waiting for I/O operations causing CPU utilisation dropping down from 100%. To improve efficiency we tested three different methods to remove the bottleneck of I/O traffic. The first of the methods is to run more than one task per CPU core, i.e. applying a multi-tasking approach. Improvements depend heavily on the application. With CPU intensive simulations or mixed workload, the 2 tasks per core setting improved throughput compared to the 1 task/CPU setting 40% and energyefficiency 18%. While in the case of data analysis jobs, the improvements were 9% in throughput and 8% in energy-efficiency. Running more tasks per core does not affect much the numbers of mixed or CPU intensive workload but improves remark-

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ably the efficiency of the I/O workload: with 4 tasks per core throughput increased 41% and energy consumption decreased 25%. This validates our assumption that I/O access is a bottleneck. A challenge with the fixed scheduling is to determine the number of tasks to be run per core, since we have no information how much resources, e.g. memory, jobs need. In high throughput computing CPU power is not usually the limiting factor and running (too) many simultaneous jobs just slows down the processing time of an individual job but it does not have a dramatic effect on throughput. Instead, memory or I/O access are usually bottlenecks and overloading them has a negative effect on performance. For example, allocating too much memory dramatically slows down the system because of the need to use swap memory, or even cause a system crash. Since flexible resource allocation has been shown to be more efficient and our earlier work shows increasing memory utilisation improving efficiency (Niemi et al, 2009), we developed a scheduling method based on the amount of the free memory in a computing node. The method is more complex than the fixed number of tasks per CPU core scheduling but it is also much more flexible, since the memory consumption of jobs does not need to be known in advance. In the memory-based scheduling, a new job is submitted to the computing node, if the memory utilisation level is lower than a given memory threshold. We determined the memory threshold as follows: the memory threshold = the total amount of memory - operating system memory requirements - the estimated memory usage of the next task. We used a moving average method to estimate the memory requirement of the next task. Implementing the memory-based scheduling method would be easy if each job allocated all its memory as the first task after it has started. Since this is not the case, workload management systems use the load adjustment method to deal with the problem: the scheduler reserves some fixed amount of memory for a fixed period of time to compensate the amount of memory that the just started job will allocate later. If no adjustments were used, the scheduler would schedule an almost infinite number of tasks to start immediately, causing the system to crash. Assuming that all jobs are similar and their memory usage is now in advance, this system works well. However, it can easily reserve too much memory in the beginning of processing and it cannot take into account complex memory profiles. In our tests, the fixed memory-based scheduling gave similar results than running 4 jobs per core in cases of mixed and CPU intensive workloads but with I/O intensive data analysis it was clearly better: throughput was 27% better and energy consumption 17% lower. As mentioned above, it is difficult to define the exactly right value for the load adjustment. If the load adjustment value is too big, some amount of memory remains unused all the time because of the cumulative effect of adjustments. In the opposite case, the system can become overloaded. The fixed load threshold can also make the CPU load and memory utilisation fluctuate, heavily reducing efficiency. To alleviate this problem, we developed an additional mechanism based on the fuzzy logic: a fuzzy controller tunes the load threshold value based on the actual memory utilisation and its change. The fuzzy control also stabilises the memory usage level as we can see in Figure 4. The fuzzy method is based on a set

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Fig. 8 Memory usage of fixed load threshold and fuzzy scheduling

of simple fuzzy rules that are given in a form IF x AND y THEN z. For example: IF Memory is Positive AND Memory change is Positive THEN Control is Negative. When comparing fuzzy scheduling to fixed memory-based scheduling, fuzzy scheduling gave similar results for both in throughput and energy-efficiency than fixed memory-based scheduling in a case of mixed workload and it was around 1% better in CPU intensive workload. In the case of I/O intensive data analysis fuzzy scheduling worked better. It outperformed the fixed memory-based scheduling method by 17% in throughput and 11% in energy-efficiency. A summary of test results is given in Figure 9 that shows percentagel improvements compared to the 1 tasks per CPU core case. The first group of columns comes from the mixed jobs containing both I/O intensive data analysis and CPU intensive simulation jobs. As it can be seen, all three scheduling methods gave similar improvements, around 40% in throughput and 20% in energy-efficiency, compared to the 1 task per core setting. We assume that adding CPU intensive tasks stabilises resource utilisation, since CPU power can be focused on them while I/O tasks must wait for new data from the disk. In the middle columns, the I/O intensive data analysis, memory-based scheduling methods, especially the fuzzy scheduling method, gave clearly better results than 2 or 4 tasks per core scheduling. One reason is that the fuzzy method processed even more than 4 parallel tasks per core but we also assume that more stable resource utilisation helped reducing the I/O bottleneck. The last columns represent the pure CPU intensive workload. The results are very similar to the mixed workload. This follows from two issues: 1) the processing time of an I/O intensive job is only around 25% of the processing time of a CPU job, and 2) in the CPU intensive workload there is not a clear bottleneck to remove as there is in the case of I/O intensive tasks.

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Fig. 9 Improvements of 2, 4, fixed memory, and fuzzy memory threshold scheduling methods compared to the 1 task / CPU setting

5 Conclusions Since the conventional one task per processor core scheduling practice seems to under utilise memory, we studied different scheduling methods and noticed that memory usage -based scheduling gives better results. However, it is difficult to find a fixed memory threshold giving the optimum results. Therefore, we developed a fuzzy logic -based algorithm measuring memory consumption and changes in it. By applying operation management principles on capacity utilisation and bottleneck planning we devised simple fuzzy rules than manage to keep the system load more stable and also achieved better efficiency than using a fix threshold or fix amount of tasks per CPU core. Our tests showed that optimising the configuration of the workload management system can both decrease energy consumption down to 45% and increase throughput up to 100% compared to the current standard practices in scientific computing. The results are also in line with Schmenner (2010), who proposed that companies having emphasised flow, implying a focus on speed and variability reduction, would outperform companies emphasizing other goals. The principle emphasizes focus on value adding tasks and removing of non-value adding tasks, while at the same time trying to eliminate bottlenecks in order to introduce even flow and short lead time. The main application area of these research results is scientific computing, especially grid computing related to the LHC experiment of CERN. Later, the results can also be applied to other areas such as computer clusters running web services. It has been estimated that the electricity cost can easily exceed the hardware cost during the lifetime of the computing system. In computing intensive science energyefficient solutions are important since these computing systems process millions of jobs each of which consist of thousands of individual computing tasks. The results of our research are implemented as software tools which can directly be applied to LHC computing clusters.

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References Brown D, Reams C (2010) Toward energy-efficient computing. Queue 8(2):30–43 Bunde D (2006) Power-aware scheduling for makespan and flow. In: SPAA’06: Proceedings of the 18th annual ACM symposium on Parallelism in algorithms and architectures, ACM, New York, NY, USA, pp 190–196 Ge R, Feng X, Cameron K (2005) Performance-constrained distributed dvs scheduling for scientific applications on power-aware clusters. In: SC ’05: Proceedings of the 2005 ACM/IEEE conference on Supercomputing, IEEE Computer Society, Washington, DC, USA, p 34 Goes L, Guerra P, Coutinho B, Rocha L, Meira W, Ferreira R, Guedes D, Cirne W (2005) AnthillSched: A scheduling strategy for irregular and iterative I/Ointensive parallel jobs. In: Job Scheduling Strategies for Parallel Processing, Springer, pp 108–122 Goldratt EM, Cox J (1984) The Goal. North River Press, Croton-on-Hudson Hanselman S, Pegah M (2007) The wild wild waste: e-waste. In: Proceedings of the 35th annual ACM SIGUCCS conference on user services, ACM, New York, NY, USA, pp 157–162 Hevner A, March S, Park J, Ram S (2004) Design science in information systems research. MIS Quarterly 28(1):75–105 Holmstrom J, Ketokivi M, Hameri A (2009) Operations management as a problemsolving discipline: A design science approach. Decision Sciences 40(1):65–87 Hopp W, Spearman M (1996) Factory physics. Irwin, Chicago Koole G, Righter R (2008) Resource allocation in grid computing. Journal of Scheduling 11(3):163–173 Lefurgy C, Wang X, Ware M (2007) Server-level power control. In: ICAC’07: Proceedings of the Fourth International Conference on Autonomic Computing, Washington, DC, USA, IEEE Computer Society, pp 4–4 Li X, Li Z, Zhou Y, Adve S (2005) Performance directed energy management for main memory and disks. Transactions on Storage 1(3):346–380 Little J (1961) A proof for the queuing formula: L= λ w. Operations Research 9(3):383–387 Marwah M, Sharma R, Shih R, Patel C, Bhatia V, Mekanapurath M, Velumani R, Velayudhan S (2009) Data analysis, visualization and knowledge discovery in sustainable data centers. In: Proceedings of the 2nd Bangalore Annual Compute Conference, ACM, New York, NY, USA, ACM, pp 1–8 Mukherjee T, Banerjee A, Varsamopoulos G, Gupta S, Rungta S (2009) Spatiotemporal thermal-aware job scheduling to minimize energy consumption in virtualized heterogeneous data centers. Computer Networks 53(17):2888–2904 Niemi T, Kommeri J, Ari-Pekka H (2009) Energy-efficient scheduling of grid computing clusters. In: Proceedings of the 17th Annual International Conference on Advanced Computing and Communications (ADCOM 2009), Bengaluru, India Rajan D, Yu P (2008) Temperature-aware scheduling: When is system-throttling good enough? In: WAIM ’08: Proceedings of the 2008 The Ninth Interna-

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Increasing Customer Satisfaction in Queuing Systems with Rapid Modelling No´emi Kall´o and Tam´as Koltai

Abstract Companies have to increase their customers’ satisfaction to keep their competitiveness. In services, waiting has great impact on service level and customer satisfaction. Consequently, in time-based competition, one of the main objectives of service companies is to minimize customer waiting. Waiting can be defined in several ways; however, the ultimate management objective should be the maximization of customer satisfaction. The paper shows how customer satisfaction can be approximated with utility functions and establishes a theoretical background for utility transformation of waiting time. The case study of the checkout system of a real do-it-yourself superstore is used to illustrate the application of the suggested method. The results show that utility related objective function may justify queuing system changes even if the average waiting time does not improve.

1 Introduction Queuing theory is frequently criticised for not being appropriate to solve practical problems because of its applied simplifications (Bhat, 1969). This is a widespread opinion, although the first results of queuing theory were all answers to practical problems (Bhat, 2008). These results were related to the operation of telephone switching systems. The work of Engset at the Telegrafverket and of Erlang at the

No´emi Kall´o (B) and Tam´as Koltai Department of Management and Corporate Economics, Budapest University of Technology and Economics, 1111 Budapest, Muegyetem rkp. 9. T. e´ p. IV. em., Hungary, e-mail: [email protected] Tam´as Koltai e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 9, 

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Copenhagen Telephone Company dealt only with practical, operational problems (Stordahl, 2007). Nowadays, managers generally use simulation to analyze the operation of production systems (Suri, 2009). In spite of this, the benefits of queuing theory cannot be denied. The application of the analytical models needs only little data and provides quick results. These results, unfortunately in most cases, are only approximations – as the analytical models cannot take into consideration several aspects of real operation. Moreover, only a few measures can be calculated by queuing theory. These measures are generally parameters of descriptive statistics: means and standard deviations. In the different fields of economics and management science, these values, however, are sufficient for the approximate models or heuristics which provide practically relevant information. One of these models is the mean-variance (or the twomoment) decision model which can be used as a simplification of the expected utility model. With the help of this model, utility maximization can be simplified by approximating expected utilities with functions of the means and variances of the decision variables. As time can be considered as a resource and it can be spent waiting, some (negative) utility can be assigned to waits. This decrease of utility caused by waiting corresponds to the effect of waits on satisfaction. Using the two-moment decision model, to approximate customer dissatisfaction caused by waiting, only the means and variances of waiting times are needed. Therefore, rapid modelling based on the formulae of queuing theory gives sufficient information to approximate customer satisfaction in different queuing systems. The structure of this paper is the following. Section 2 shows how customer satisfaction can be approximated with utility functions and establishes a theoretical framework for utility transformation of waiting time. In Section 3, a case study of the checkout system of a do-it-yourself superstore is presented to illustrate the application of the suggested method. Finally, Section 4 summarizes the main conclusions.

2 Customer Satisfaction as a Function of Waiting Time In time-based competition, one of the main objectives of service companies is to minimize customer waiting. Waiting, however, can be defined in several ways. Different customer groups can have different average waiting times. Besides the mean, the standard deviation or the distribution of waiting times are also important characteristics of the waiting process. Moreover, the actual and perceived length of waits can also be differentiated. Independently of how waiting is defined and measured, the main objective of operation optimization is to maximize customer satisfaction, that is, to minimize customer dissatisfaction related to waiting times. Time can be considered as a resource. As such, a resource, like money, time can be gained and can be lost, that is, can be saved and can be wasted. In this

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sense, waiting time is a kind of loss. The loss represented by waiting time (that is, the decrease of satisfaction) is a subjective value. A formal technique to quantify subjective factors is the application of utility functions (Keeney and Raiffa, 1993). A utility function describing the relationship between waiting and satisfaction has to possess the following characteristics: • Utility functions are generally exponential (Keeney and Raiffa, 1993). • People are generally risk averse in terms of time (Leclerc et al, 1995). Therefore, to describe satisfaction, negative exponential utility functions should be used. • Satisfaction is a function of the difference between the actual and expected performance (Gr¨onroos, 2001). Consequently, the negative exponential utility function should determine satisfaction as the difference of expected and actual waiting times. Based on these assumptions and the results in the related scientific literature (see for example Kumar et al, 1997), the following function can be used to determine the relationship between waiting time and customer satisfaction, γ

E [U (W, T0 )] = E[−AT0 e−r(T0 −W ) ],

(1)

where A, γ and r are positive values. A expresses the assumed worth of customers’ time. Parameter γ describes the direct effect of the expected waiting time (T0 ) on satisfaction. Parameter r denotes the extent of customers’ risk averseness in terms of time. W is a stochastic variable describing actual waiting time of a customer. Notations used in the paper are summarised in Table 1. Table 1 Notations Parameter A γ T0 r W λe t S S0 S1 C E R L s0 i pi s0E rE s0R rR

Explanation assumed worth of customers’ time direct effect of the expected waiting time on satisfaction expected waiting time extent of customers’ risk averseness in terms of time actual waiting time (stochastic variable) parameter of the exponential distribution average waiting time average satisfaction initial satisfaction variable satisfaction number of all checkouts number of express checkouts number of regular checkouts limit value functional satisfaction level the number of items bought the probability of buying i items functional satisfaction level in the express lines risk averseness in the express lines functional satisfaction level in the regular lines risk averseness in the regular lines

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Equation (1) can be used to transform a certain waiting time to the corresponding satisfaction level. Knowing these values, the average satisfaction can be calculated. That is, the waiting time of each customer has to be known (measured or simulated), which makes the calculation of customer satisfaction difficult. The calculation of the average utility (average satisfaction), however, can be simplified if the distribution function of the waiting time is known. Equation (1) contains only one stochastic variable, the waiting time (W ). Consequently, the calculation of the mean can be simplified as follows, γ

γ

E [U (W, T0 )] = −AT0 E[e−r(T0 −W ) ] = −AT0 E[e−rT0 erW ] γ

= −AT0 e−rT0 E[erW ].

(2)

To get the average satisfaction level for a stochastic variable (W ), an exponential function (erW ) has to be determined and the mean of this function has to be calculated. The calculation of this mean for the most frequently used distribution functions can be found in the literature (e.g. Keeney and Raiffa, 1993). In many queuing systems, the arrival process of the customers can be described with Poisson distribution – as, according to the Palm-Khintchine theorem, the superposition of many independent and properly normalized renewal processes forms a Poisson process (Kleinrock, 1975). If the arrival process can be described with Poisson distribution, and there is a high traffic, the distribution of waiting times is approximately exponential (Kimura, 1983). The mean of the exponential transformation (e−rX ) of a stochastic variable with exponential distribution (λe e−λe x ) can be calculated according to the following formula (Keeney and Raiffa, 1993),     −λe x E e−rX X=λe e−λe x = E e(−rλe e ) =

λe . λe + r

(3)

Equation (3) contains two parameters. Parameter r expresses the extent of customers’ risk averseness in terms of time. λe is the parameter of the exponential distribution of the stochastic variable (that is, the reciprocal of its mean). Consequently, (3) can be written as follows,     1 1 1 = 1/t = 1 + rt. E erW = E −rW = (4) −rW e E[e ] 1/t+r

Knowing the mean of the exponential transformation of the waiting time, a function to determine the average customer satisfaction (the mean utility) as a function of waiting time can be formulated, γ

γ

γ

S = E [U (W, T0 )] = −AT0 e−rT0 (1 + rtx ) = −AT0 e−rT0 − AT0 e−rT0 rt.

(5)

According to (5), the satisfaction level has two parts. The first part is independent of the waiting time. It can be interpreted as the initial satisfaction level (S0 ) and mainly depends on the expectations of customers. The second part depends on the waiting

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time (S1 ) – longer waits cause lower satisfactions. From (5), it can also be seen that the extent of risk averseness connects the initial and the variable satisfaction levels (S1 = rS0 ). With these two parts of satisfaction, the satisfaction function can be simplified in the following way, S = E [U (W, T0 )] = S0 + S1 t.

(6)

Equation (6) can be used to determine the average satisfaction of different customer groups. The mean satisfaction level of all customers can easily be calculated as a weighted average of the mean satisfaction level of customers in the express and in the regular lines. Equation (6) shows that to determine the average satisfaction of customers only the descriptive statistics of the waiting time are needed. These values can be calculated with analytical models. That is, to analyze customer satisfaction, the analytical models give sufficient information; there is no need for complex simulation models. The next section shows how customer satisfaction may change as a consequence of the application of express checkouts in a do-it-yourself superstore.

3 Case Study to Illustrate the Maximization of Customer Satisfaction Related to Waiting 3.1 Description of the System and Preliminary Results The application of express checkouts is a frequently used tool for waiting time reduction. When express lines are applied, two customer groups are created. Customers buying more items than a certain amount have to use the regular checkouts. Only people buying no more than this quantity can join the express lines. The number of items that controls line-type selection is called limit value. Our former analyses revealed that one of the main parameters which influences waiting time and which can be easily controlled by the management is the limit value (Koltai et al, 2008). In this section, the real data of a do-it-yourself superstore has been used (Koltai et al, 2008). In this store, generally five checkouts operate. Using the data provided by the checkout information system, the arrival rates for the different days and for the different parts of the days has been estimated. For all periods the Poisson arrival process has been acceptable according to Kolmogorov-Smirnov tests. Based on R´enyi’s limiting distribution theorem and its generalizations, the distribution function of the time interval between two consecutive events remains the same after a rarefaction and coordinate transformation (R´enyi, 1956; Sz´antai, 1971a,b). That is, the arrival processes of the two customer groups can also be approximated with Poisson processes, the distribution of the interarrival times can be considered exponential. The density function of the number of items bought by customers has also been provided by the checkout information system, and for describing it a truncated

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geometric distribution with a mean of 3.089 has been found acceptable by a chisquare test. The service time of customers, as it could not be obtained from any information systems, was measured manually. The linear relationship between the number of items bought and the service time was tested with regression analysis. The assumption of linearity was accepted based on a 0.777 correlation coefficient. According to the results of linear regression, service time has two parts. The first part (constant) is independent of the number of items bought and is equal to 0.5463 minute. The second part (slope) is proportional with the number of items bought and its unit value is equal to 0.1622 minute. Linear regression can be applied to determine the standard deviation of these parameters and the service times of customers buying different amounts as well (Koltai et al, 2008). Results presented in this paper are valid for a midday traffic intensity with an arrival rate most characteristic for the store (λ = 95 customers/hour). According to the geometric distribution, customers buy generally only few items. Therefore, two of the 5 working checkouts was considered to be express servers (C = 5, E = 2). For analyzing the waiting times when different limit values are applied, a numerical Excel model has been developed (Koltai et al, 2008). This model calculates the special characteristics of the express line system (of the express and regular checkouts) with different limit values based on the main parameter of the original queuing system (system without express checkouts). With these parameters, using the formulae of analytical models, an upper and a lower estimation can be given for the average waiting times. Based on the estimations of the waiting times of customers in express and in regular waiting lines, the average waiting time of all customers can be calculated. In Table 2, the average waiting times in the different queue-types can be seen. These results are in accordance with the basic assumptions: small limit values reduce waiting in the express lines and increase waiting in the regular lines; large limit values have inverse effects. Based on these different effects, the average waiting times in all lines as a function of the limit value can be described with a U-shaped curve. Table 2 Average waiting times in queue Average waiting time in queue Express lines Regular lines All lines

Limit value L=1 0.0121 0.0663 0.0488

L=2 0.0487 0.0330 0.0415

L=3 0.1102 0.0160 0.0811

L=4 0.1957 0.0075 0.1563

With different limit values, different waiting times can be achieved. An important management objective related to express line systems is to determine the limit value which optimizes the operation, minimizes waiting times, and maximizes customer satisfaction. The minimal average waiting time in all lines, that is, the highest service

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level can be achieved by using 2 items as limit value (L = 2) – as it can be seen in Table 2. Service level, even if it is denoted as a function of waiting, can be defined in several ways. However, independently of the way how waiting is defined and measured, the main objective of operation optimization is to maximize customer satisfaction, that is, to minimize customer dissatisfaction related to waiting.

3.2 Analysis of Customer Satisfaction Related to Waiting Time Based on the method presented in Section 2, the average satisfaction level of the different customer groups (using express or regular checkouts) can be quantified. If the average waiting times of the different customer groups is known, the customer satisfaction can be calculated according to (6) for both customer groups. Equation (6) uses only common customer characteristics which do not make distinctions between customers buying small and large amounts. Customers using the express and the regular checkouts, however, have different attitudes toward waiting. People’s tolerance for waiting highly depends on the perceived value of the service for which they wait (Maister, 1985). These differences can be taken into consideration in the parameter values used for the different customer groups. Customer satisfaction has a strong relationship with quality. In the case of services, two forms of quality can be distinguished: technical quality (the quality of what is provided) and functional quality (the quality of how it is provided) (Gr¨onroos, 2001). Technical quality or the value of service can be defined as a function of the number of items bought. In this way, the initial satisfaction level (S0 ) can be expressed as the function of the average number of items bought,   S0 = s0 × f

∑ i × pi

,

(7)

i

where s0 is the functional satisfaction level (independent of the amount bought), i is the number of items bought and pi is the probability of buying i items. The average satisfaction level of a customer group is determined by three factors. Functional satisfaction is based on the service process (s0 ). Technical satisfaction is determined by the number of items bought (i). These two values define initial satisfaction (S0 ). From this value, the variable service level (S1 ) can be determined by using the average waiting time (t) and the parameter denoting the extent of the customers’ risk averseness (r). The values of these parameters, however, are not the same in the different customer groups: • In the express lines, customers buying only a few items can be found. The application of express checkouts favours them; however, they are not getting a really valuable service. Their functional satisfaction is high, while their technical satisfaction is low. Customers buying only few items and waiting during shorter

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periods are more responsive to the variation in waiting times. Consequently, they are more risk averse than people in regular lines. If the limit value increases, the services become more valuable at these checkouts. In the same time, as the average waiting time increases, the benefit from using express checkouts becomes lower. Technical satisfaction increases, while functional satisfaction and risk averseness decrease. • In the regular lines, customers are waiting to buy large amounts of goods. The service claimed is valuable, but their waiting will be long. Their technical satisfaction is high, while their functional satisfaction is low. Customers buying large amounts and waiting to longer periods are less risk averse than people in express lines. If the limit value increases, the services become more valuable at these checkouts, and the average waiting time decreases. Technical and functional satisfaction, and risk averseness increase. • In a queuing system without express checkouts, customers buying small and large amounts are waiting in common lines. As the application of express checkouts expresses the managers’ commitment to waiting reduction, the functional satisfaction level in a queuing system without express lines is lower than in a queuing system with express lines. The average number of items bought is higher in a system without special servers than the average number of items bought in the express lines and lower than the average number of items bought in the regular lines. Accordingly, the technical satisfaction and risk averseness is between the values valid for the two customer groups. Based on these findings, different parameter values (s0 , r) have been assigned to each customer group. Using the results of the analytical formulae (with the help of the numerical model), the average satisfaction level with different parameter values has been calculated. The results presented in this paper were obtained using functional satisfaction value 10 for a queuing system without express checkouts (s0 ). As the application of express checkouts is used for reducing waiting times, the functional satisfaction of customers in express lines is assumed to be 8 (s0E ) and the functional satisfaction of customers in regular lines is assumed to be 9 (s0R ). The risk averseness of all customers (r) is indicated by 1. People in express lines are less tolerant toward waiting; therefore their risk averseness (rE ) is assumed to be 1.2. The risk averseness of the more tolerant customers in regular lines (rR ) is assumed to be 0.8. Although there are several parameter values which are in accordance with the consideration above, similar results have been obtained independently of the actual parameter values. That is, the main conclusion is not sensitive to the change of parameter values. The average satisfaction level as a function of the limit value can be seen in Fig. 1. Satisfaction in a queuing system with express lines is influenced by the limit parameter, while in a queuing system without special servers the value of average satisfaction is constant. Figure 1 shows that that average satisfaction level in an express line system has a distinct maximum. The limit value inducing maximal satisfaction can be used to

Average satisfaction

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0 -0.2 Queuing system with express lines Queuing system without express lines

-0.4 -0.6 -0.8 -1 -1.2 1

4

7

10

13

16

19

22

25

Limit value Fig. 1 Average waiting time and average satisfaction as functions of the limit parameter (s0 = 10, r = 1; s0E = 8, rE = 1.2; s0R = 9, rR = 0.8)

optimize operation. Based on Fig. 1, it can also be concluded that using a limit value which is near to the optimal value causes no significant decrease in satisfaction. Comparing the satisfaction levels in queuing systems with and without express lines, it can be concluded that, with an appropriate limit value, the application of express lines may increase average satisfaction. Moreover, if not the optimal but a near-optimal limit value is used, the satisfaction will be higher than in the original system. That is, if the optimal limit value cannot be applied (e.g. for some operational reasons), it will not have serious consequences. It can also be seen in Fig. 1, that if a limit value significantly different from the optimal one is applied, the average satisfaction will be lower than in the original system. That is, managers have to know at least roughly the optimal limit value if they do not want to decrease customer satisfaction to a great extent. The analyses of the average service levels has proved that, in terms of satisfaction, higher service level is offered in queuing systems with express checkouts than in systems without express checkouts. In our former analyses, we have found that, based on different waiting time measures (the mean and standard deviation of the actual and perceived waiting times), the application of express checkouts cannot decrease customer waiting significantly (Kall´o and Koltai, 2009). These waiting measures are lower in queuing systems without express lines than in express line systems – even if they are operating with optimal limit values. Therefore, the application of express checkouts cannot be justified by the decrease of average waiting time. Despite this finding, the analysis of satisfaction levels has proved that customer satisfaction can be improved with express checkouts. These results are illustrated with Fig. 2. Figure 2 describes the difference between the performance of queuing systems with and without express checkouts. In the terms of waiting times (gray solid curve and line), a queuing system without express checkouts performs better than an express line system – using any limit value. In terms of satisfaction (black dashed curve and line), an express line system – using optimal or near-optimal limit value – performs better than a queuing system without special servers.

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Average waiting time with express checkouts Average waiting time without express checkouts Average satisfaction with express checkouts Average satisfaction without express checkouts 1

3

5

7

9 11 13 15 17 19 21 23 25 Limit value

Fig. 2 Average waiting time and average satisfaction as functions of the limit parameter (s0 = 10, r = 1; s0E = 8, rE = 1.2; s0R = 9, rR = 0.8)

Based on the results expressed by Fig. 2, it also can be concluded that using an objective function maximizing customer satisfaction does not refine the optimum obtained by average waiting time minimization. Independently of the applied objective functions (minimization of average waiting time or maximization of average satisfaction), the optimal limit value is equal to two (L = 2). Consequently, there is no need for applying difficult objective functions and complex tools to determine the optimal operation of an express line system. The minimization of average waiting time determines the optimal operation. Optimal operation, however, does not necessarily lead to the decrease of the average waiting time, but definitely leads to the improvement of customer satisfaction.

4 Conclusions To keep their competitiveness, companies have to increase customers’ satisfaction. In the era of time-based competition, customer satisfaction should be defined as a function of waiting. Waiting times can be analyzed with the help of queuing theory. This paper shows that the results of rapid modelling can be used to analyze customer satisfaction. Our analysis has proved that customer satisfaction can be approximated with utility functions and established a theoretical framework for utility transformation of waiting times. Our results underline that express line systems using appropriate limit value can increase customers’ satisfaction. That is, companies can gain competitive advantage by applying express checkouts. Comparing the effects of the application of express checkouts on average waiting time and on average satisfaction it can be concluded that the main benefit of express checkouts is not the decrease of average waiting time but the better allocation of short and long waiting times among the different customer groups. That is, even in time-based competition, overall waiting time minimization is not necessarily the main objective. In some cases, the better allocation

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of short and long waiting times among the different customer groups can have more favourable effects. Theoretically the objective is to increase customer satisfaction. Technically, however, the optimization (minimization) of the average waiting time leads to the maximization of customer satisfaction. Therefore, (6) is necessary only for justifying the usefulness of express checkouts. During the daily operation, however, queuing formulae developed for the calculation of the average waiting time can be used. The results of the analysis of express line systems presented in this paper can be used in other areas as well. Several other line structuring rules can be analyzed in service systems (lines for business class passengers in airports, privileged customers in special services, etc.), and the application of line structuring rules can be extended to production systems as well. These are topics of further researches.

References Bhat U (1969) Sixty years of queueing theory. Management Science 15(6):280–294 Bhat U (2008) An introduction to queueing theory. Birkhauser Gr¨onroos C (2001) Service management and marketing. John Wiley & Sons, Inc. Kall´o N, Koltai T (2009) Rapid modeling of express line systems for improving waiting processes. In: Reiner G (ed) Rapid Modelling for increasing competitiveness, Springer Keeney R, Raiffa H (1993) Decisions with multiple objectives. Cambridge University Press Kimura T (1983) Diffusion approximation for an M/G/m queue. Operations Research 31(2):304–321 Kleinrock L (1975) Queueing systems, volume I. John Wiley & Sons, Inc. Koltai T, Kall´o N, Lakatos L (2008) Optimization of express line performance: numerical examination and management considerations. Optimization and Engineering 10(3):377–396 Kumar P, Kalwani M, Dada M (1997) The impact of waiting time guarantees on customers’ waiting experiences. Marketing Science 16(4):295–314 Leclerc F, Schmitt B, Dube L (1995) Waiting time and decision making: is time like money? Journal of Consumer Research 22(1):110 Maister D (1985) The psychology of waiting lines. In: Cziepel J, Solomon M, Surprenant C (eds) The service encounter, Lexington Books R´enyi A (1956) A Poisson folyamat egy jellemz´ese (A possible characterization of the Poisson process). MTA Mat Kut Int K¨ozl 1:519–527 Stordahl K (2007) The history behind the probability theory and the queuing theory. Telektronikk 2:123–140 Suri R (2009) A perspective on two decades of Rapid Modeling (foreword). In: Reiner G (ed) Rapid Modelling for increasing competitiveness, Springer Sz´antai T (1971a) On an invariance problem related to different rarefactions of recurrent processes. Studia Sci Math Hungarica 6:453–456

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Sz´antai T (1971b) On limiting distributions for the sums of random number of random variables concerning the rarefaction of recurrent processes. Studia Sci Math Hungarica 6:443–452

Rapid Modelling of Patient Flow in a Health Care Setting: Integrating Simulation with Lean Claire Worthington, Stewart Robinson, Nicola Burgess and Zoe Radnor

Abstract This paper provides an evaluation of an experiment in using discrete event simulation modelling at a Rapid Improvement Event in a large hospital trust. It presents empirical findings about the challenges of building a model rapidly within a time constrained event. The aim was also to help introduce Lean principles so the learning and understanding acquired by participants, their interaction with the model, and the experimentation promoted and facilitated by the model are all considered. Our learning from this action research about rapid modelling and the modifications made to our approach as a result of the experiment are described.

1 Purpose This paper explores and reports the use of discrete event simulation (‘simulation’) modelling as a part of a Rapid Improvement Event (RIE) in a hospital trust. The work is part of a bigger project that seeks to investigate how simulation modelling can support and sustain Lean improvements in health care. Our research posits three

Claire Worthington (B), Stewart Robinson, Nicola Burgess and Zoe Radnor Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK, e-mail: [email protected] Stewart Robinson e-mail: [email protected] Nicola Burgess e-mail: [email protected] Zoe Radnor e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 10, 

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potential applications of simulation modelling in relation to Lean implementation as illustrated in Fig. 1: • Educate: simulation provides a basis for training in Lean ideas e.g. process flow • Engage/facilitate: simulation is used as part of a Lean improvement initiative to better understand the issues and to help identify Lean improvements. • Experiment/evaluate: simulation is used to determine the effectiveness of Lean improvements that emerge from a Lean initiative. The nature of these activities suggests an ordering of, respectively, before, during and after the Lean initiative.

During Educate

Lean Initiative

Experiment/ Evaluate

Engage/ Facilitate

Before

After

Fig. 1 Simulation modelling and lean initiatives

There is much published work on discrete event simulation in health applications (as reviewed by Jacobson et al 2006) but this mainly concentrates on the evaluation/experimental stage, with the simulation usually being built for the practitioners by external consultants. Although a lot of models have been built, barriers exist to the implementation of their findings or their sustained use (Mahachek 2002 and Lowery 1996). Our approach is to use discrete event simulation modelling during a Lean initiative (Engage/Facilitate) where the model is built with the practitioners. We present the findings of our action research case study on the importance of this approach in relation to the learning and understanding acquired by participants, their interaction with the model, and the experimentation promoted and facilitated by the model. We also present our findings in relation to the challenges of building a model rapidly within a time constrained event.

2 Approach The first phase of our research has sought to characterise the nature of complexities within healthcare in relation to Lean methodologies and explore if and how simulation modelling can support Lean implementation. This has been done by means of interviews (approximately 50) with a cross section of staff in two case study hospital trusts. Our empirical findings suggest that simulation modelling can play a

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potentially key role in supporting and sustaining Lean implementation in a hospital trust (Burgess and Radnor, 2010). The process mapping activity, which forms the basis of a majority of Lean led RIEs (and other Lean interventions), is a pivotal experience in which participants can see the whole process and appreciate their role within it, sometimes for the first time. This activity is often conducted using brown paper and post-it notes in a workshop setting; typically over 2 days. We use simulation modelling at this point to elevate (but not replace) the brown paper exercise to a new level in which key metrics such as queuing or utilisation are recorded together with cycle and throughput times (McDonald et al, 2002). The simulation needs to be developed rapidly as the RIE is time constrained. The fidelity of the simulation must be sufficient for the practitioners’ credibility and for evaluation of suggested changes to the existing process, but low enough for speedy model development. This situation lent itself to an action research study (Coughlan and Coghlan, 2002) as we were actively involved in contributing to improving the process. We had the opportunity to try this approach by attending an RIE at a 700 bed hospital in NW England serving a population of 265 000. This opportunity occurred at very short notice. The RIE lasted 4.5 days (from 08.30 to 16.30) with the final morning being used to report back to senior management at an outbrief session. Fifteen members of hospital staff and one Governor attended the RIE. Consultants, ward clerks, nurses (community and ward) and managers (nursing, operational and clinical) were all represented. The RIE was facilitated by two of the hospital’s Lean facilitators and two researchers attended as observers, participants and modeller. The issue of concern was their paediatric observation and assessment unit (OAU) where it was felt that too often, children (and carers) had to wait for unacceptably long periods of time for tests, treatments and decisions. The problem was exacerbated because there was no separate room for waiting so the children’s playroom had to be used. The layout is displayed in Fig. 2.

Play Room + Wait Area

Triage Room Ward Clerk

Treatment Room

Fig. 2 OAU Layout

During the first morning of the RIE participants were divided into groups and visits were made to the paediatric OAU. One group spoke to patients and their carers to gather the opinions of the patients. We were asked to identify the paths followed by patients through the unit as a sequence of tasks and for each task to record esti-

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mates of the ‘touch’ times, resource requirements and their availabilities and waiting times. This information was then used to build up a post-it process map of OAU as displayed in Fig. 3.

Fig. 3 OAU post-it process map

Children arriving at the Unit checked in with the Ward Clerk, waited in the play room until they could be triaged in the triage room and then they waited again in the playroom until one of the 6 beds was free for them to be assessed. Following this they returned to the playroom to await the results of tests, or the availability of a doctor and a bed for a decision on admission, discharge or further tests. As in Ntungo (2007) building the process map revealed features of the process that had not been known or appreciated by all, it was a useful focus for discussion, clarification and agreement on the ‘as is’ process for those who worked within it and for us, a useful accepted current state process map of OAU. While the RIE participants considered where there was waste in the current process, the post-it diagram was typed into Microsoft Visio as a flowchart; the display was chosen to resemble the post-its. This diagram was then imported into Simul8 (the package used for the discrete event simulation: www.simul8.com). Some Visio shapes, such as processes, were recognized by Simul8 and treated as ‘Work Centres’ while others, such as storage icons, had to be manually altered within Simul8. The information collected for the process map included estimates of touch times (service times) for each task so these provided parameters for the service time distributions in the simulation. Triangular distributions were used for bounded processes and Erlang3 used for other process times. The number of children arriving each day was also available and in this initial model an exponential distribution with a constant daily mean was used to represent arrivals to OAU. This model was presented to the RIE at the start of day 2. It served as a useful reminder of what they had agreed the previous day. An explanation was given of

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how the model corresponded to the post-its before it was run, and then examples of modifications that could be easily made (e.g. altering the number of beds) were demonstrated. Assumptions that had been necessary were explained and agreed or modified. Data, essential for the simulation that had not been provided, were discussed and values agreed. These missing data were mainly the proportions going along the different routes in the process map, as indicated by question marks in Fig. 4. Radiologist + nurse + porter

Nurse 0 0

17 0

0

0 ?

1 0

Radiology 40-60

0

34 Referred GP M/W AE

Snr Doc 1

0

Treatment started drugs oxygen

Triage5 mins

Arrive on G5

16

?

0

? To bed or cubicle 1 min

? 0

?

Seen by J Dr or ANP 15-30 mins Ward clerk nurses bed bureau

Treatment room blood terst 20-60

? Qualified Nurse

J DR ANP

0

Snr Dr Review 15 mins

0

?

? ?

Nurse ANP + dr Playworker

0

Fig. 4 Simulation model of OAU

Face validation was then undertaken and it included consideration of the routes that children followed through the model, the percentage of patients taking at least 4 h or 6 h to get through OAU, inspection of the charts of utilization of beds and numbers in the playroom. Agreement was reached on modifications that were to be incorporated into the simulation for consideration the next day (day 3). These fell in the following four areas and included: input values – refined to reflect the varying hourly arrival rates result collection – included the no. in the playroom and times in OAU by hour alternative scenario – triage performed on a bed: – senior decision maker absent; – one bed reserved for admissions from ED; simulation display – queue to display people It is worth noting that our approach followed the usual protocol of an action research study with day 1 providing the pre-set up where the aim was to understand the context and purpose of the study. Information was gathered, analysed and fed back on day 2 with an iterative process following, in which options for improvement and implementation were evaluated.

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3 Findings Our findings from this action research are considered from the perspectives of the hospital, simulation modelling and our contribution to the Rapid Improvement Event.

3.1 Hospital Findings When the simulation model was displayed for the first time all participants at the RIE engaged with it, to the extent that all were impressed by its complexity i.e. the complexity of the process they worked with every day. It was the managers including the consultants, however, who quickly appreciated the simulation’s usefulness in evaluating alternatives and also its dependence on the data provided as input. This was also apparent in the engagement with the results presented on the 3rd day, when understandably those who had suggested alterations were more engaged than those who had not. It is worth pointing out that the main aspects that the RIE had identified for consideration were applying 5S to the room where triage took place (i.e. sorting it out so that work could be done, according to agreed procedures in well ordered surroundings), providing visual management boards to monitor the progress of patients through the OAU and thinking about whether and what treatment ‘tollgates’ could be established. Thus flow improvement was not the explicit focus of this event. Ward level staff were therefore engrossed in the practicalities of improving their every day circumstances so, it was primarily the managers who showed pleasing understanding, interaction and experimentation with the simulation model. As a result of this two rapid experiments were run based on the simulation. In the first, the patient goes straight to a bed for triage and stays there until the decision is taken that leads to departure from OAU. In the second experiment one bed is reserved for the exclusive use of patients coming from ED, as these are usually the more ill children in OAU. The simulation was described as a ‘myth buster’ as it showed that these alterations had much less impact on performance than expected. Additionally a more detailed model of part of the process is to be built and is an example of post initiative evaluation/experimentation.

3.2 Simulation Modelling Findings From the perspective of the simulation model our findings mainly refer to the need for speedy (and accurate) development of the post-it diagram and clear communication of results. Although each situation will differ, there are some standard features that will be required. The development of templates for input data

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(arrivals, tasks, resources etc.) and results calculation (e.g. time in system) is seen as necessary. We found that the feature linking Visio to Simul8 did not provide any particular advantage. Indeed, it slowed the process of model creation down. As a result, in future events we will directly convert the post-it process map to a Simul8 model. The speed of model development adopted in this RIE allowed little time for checking, or thinking, thus the approach required an experienced, careful modeller so that the simulation was ‘good enough’ but denied the modeller (and the model) chance to contribute all that they might. The way that results are presented can affect their availability and usefulness to Lean intervention participants. The choice of information for feedback and how best to display these results therefore needs careful consideration, and by doing it quickly opportunities may have been lost.

3.3 Event Findings Our initial ideas for simulation modelling contributing to Lean initiatives are illustrated in Fig. 1. It has separate pre, during and post roles for simulation. In our experiments we have found that these roles are not discrete, for instance it is possible to educate during a Lean Initiative. So, for instance, education about flow behaviour that is pertinent for Lean improvement could be undertaken with simulation models and presented during a Lean initiative. Consequently we have removed the time dimension from our initial diagram (Fig. 5). The arrows now denote the flow of information between the simulation and the Lean initiative (inputs).

Educate

LEAN INITIATIVE

Experiment/ Evaluate

Training or interlude SimLean - basic

Engage/ Facilitate SimLean - full

Legend Input

Fig. 5 Simulation and Lean initiatives – revised

In terms of the educational role of simulation, we still consider that this could entail training events prior to a Lean initiative, but it could also involve brief ‘educational interludes’ during a Lean initiative. The idea would be to use simulation to convey particularly challenging concepts during a Lean intervention using predesigned working models. For instance, a general model of a hospital ward could be

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used to demonstrate the need to balance the output rate with the input rate, something that hospital staff do not always grasp as being essential. These educational interludes could be adopted at any point during a Lean intervention when the need to understand an issue demonstrated by a model arose. As a result of this, we have identified two styles of simulation use during a Lean intervention (referred to as SimLean): (a)

(b)

SimLean (basic): for a basic event where the concentration is on visual management, 5 or 6S and working towards standardized working practices. In these events the post-it map and discussion will be used as triggers for educational episodes to show for instance, the effect of not matching resources to input or the effect on performance of variability in demand. The post-it map may not be modelled – when it is, it would be for illustrative purposes and not for detailed consideration of generated results. Fidelity of model and data have less importance here than in style (b). SimLean (full): for an event where the focus is on the flow of patients, the postits will be transformed into a simulation model with the close involvement of participants and their commitment to provide necessary data for input and validation. The building of the model will be expedited by the use of preprepared input templates (set up in Excel as this package is readily available); pre-prepared charts clearly displaying default performance measures; and predesigned components of commonly used Simul8 objects. The participants in the Lean intervention will be involved in the iterative process of developing and evaluating current and alternative scenario.

The structure of these proposed SimLean events is given in appendices 1 and 2 respectively. We see these as two extremes on a continuum of simulation use during Lean interventions. We are planning to implement SimLean later this year.

3.4 Summary Where the simulation model only adds animation to the post-it process map it is doubtful that the effort involved is warranted in terms of the benefits that can reasonably be expected. Despite this we have learnt from our rapid experiment in that we now have much greater knowledge about the 3Es (Educate, Engage, and Experiment). In particular we have better insight into: • what to build for Educate i.e. what flow concepts are found challenging to convey through Lean training; • what kind of event is appropriate for simulation model Engagement; • how to allow the participants to choose the next steps on their terms i.e. how to refine the model for Experiment. Developing these ideas and opportunities could be instrumental in making significant and sustainable improvements that would not have taken place if simulation modelling were not employed (Robinson, 2001).

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4 Relevance/Contribution This paper provides an evaluation of our rapid experiment of using simulation modelling at a Rapid Improvement Event in a large hospital trust. By adopting an action research approach we have been able to modify our approach to after an evaluation of our initial methodology. From a Lean perspective we have positive indications that it is useful to incorporate simulation modelling into the Lean improvement process. The insights that it gives into the behaviour of interacting patient flows through the care process by evaluating comparative output measures are seen to be valuable. Our work in developing the SimLean approach will contribute to this opportunity. From a modelling perspective we have experienced working in an environment where the participants decide on the next steps to be taken, even when they are not the “experts”’ choice. The resulting ownership of outcomes and results engenders interest in the continuing success of the shared process. This approach is worthy of consideration by modellers who have developed many models of health applications which have resulted in many interventions but relatively few have resulted in sustained improvement (Brailsford et al, 2009). In terms of rapid modelling, we have identified two styles of simulation use during a Lean initiative: SimLean (basic) and SimLean (full). Given the need to implement these models during a Lean improvement event, we need to be able to facilitate rapid model development and use. This can be achieved through the use of pre-designed models and model and data templates related to the healthcare and Lean improvement environment. Our work to date has provided evidence of the potential importance of this approach in leading to implementation of improvements in process flows which follow Lean principles by developing simulation models with the practitioners. Acknowledgements The financial support of the Strategic Lean Implementation Methodology Project (SLIM) (www2.warwick.ac.uk/fac/soc/wbs/projects/slim) is acknowledged. SLIM is funded by the Warwick Innovative Manufacturing Research Centre. The helpful comments from referees are also acknowledged. This paper is adapted from Worthington, C. and Robinson, S.: Integrating simulation with Lean: Rapid low fidelity modelling in healthcare: Submitted to EUROMA 2010.

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Appendix 1: SimLean (basic): Educational Lean implementation at a basic level; the focus of the RIE is likely to be in creating the foundational conditions for sustainable Lean implementation: mapping the process, identifying non-value adding steps, introduction of visual management tools and 5S. The contribution of simulation at this stage is ‘seed corn’ i.e. to introduce a simple simulation of in-out flows. Emphasis necessary on visual impact to engage (based on early experiments). The objective is to nurture a sense of what simulation could be used for and also to help embed the key principles of Lean such as pull and flow.

SimLean – an introduction 30 mins. eg. Illustration of a busy Assessment Unit

Introduction and welcome to RIE

Organisation’s process

Day 1 Data collection by participants (Guidance)

Process Map (post-its)

Educational Interlude –30 Probably Day 2

Legend

mins. Generic illustrations designed to illustrate Lean principles

Total intervention: 1 hour

Lean Thinking Value, Flow, Pull, Perfection

SimLean intervention

SimLean intervention (behind scenes)

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Appendix 2: SimLean (full): Interactive/Participatory This level of SImLean is suitable for Lean implementation where expected outputs are likely to be: changes to process, resources and facility configuration. Participants should be decision makers eg. A departmental manager, general manager, executive manager, consultant, nurse etc.

SimLean – an

Day 1

Data collection by participants (Guidance)

introduction 30 mins. A generic illustration eg. Increase number of beds

Process Map (post-its)

Data critical, if not robust or more is needed, go back to data collection stage

Simulate (Behind scenes)

Validate

SimLean – scenario testing approx 1 hour. Some generic examples if necessary eg. If we took out this process step, this is what happens to flow

Lean Thinking Value, Flow, Pull, Perfection

Implement Total intervention: 3 hours plus

Scenario testing to be facilitated either during the SimLean interlude or when the model is ready the next day

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References Brailsford S, Harper P, Patel B, Pitt M (2009) An analysis of the academic literature on simulation and modelling in health care. Journal of Simulation 3(3):130–140 Burgess N, Radnor Z (2010) Lean implementation in health care: Complexities and tensions. In: Euroma, Porto Coughlan P, Coghlan D (2002) Action reseach for operations management. International Journal of Operations and Production Management 22(2):220–240 Jacobson S, Hall S, Swisher J (2006) Discrete-event simulation of health care systems. Patient Flow: Reducing Delay in Healthcare Delivery pp 211–252 Lowery J (1996) Introduction to simulation in health care. In: Proceedings of the 28th conference on Winter simulation, IEEE Computer Society, p 84 Mahachek A (2002) An introduction to patient flow simulation for health-care managers. journal of the Society for Health Systems 3(3):73–81 McDonald T, Van Aken E, Rentes A (2002) Utilising simulation to enhance value stream mapping: A manufacturing case application. International Journal of Logistics Research and Applications 5(2):213–232 Ntungo C (2007) Quality culture in government: the pursuit of a quality management model. Total Quality Management & Business Excellence 18(1):135–145 Robinson S (2001) Soft with a hard centre: discrete-event simulation in facilitation. Journal of the Operational Research Society 52(8):905–915

Part IV

Rapid Modelling and Financial Performance Measurement

Evaluation of the Dynamic Impacts of Lead Time Reduction on Finance Based on Open Queueing Networks Dominik Gl¨aßer, Boualem Rabta, Gerald Reiner and Arda Alp

Abstract The basic principles of rapid modelling based on queueing theory, that provide the theoretical foundations for lead time reduction, are well known in research. We are globally observing an underinvestment in lead time reduction at top management levels. In particular, the maximization of resource utilization is still a wide-spread aim for managers in many companies around the world. This is due to inappropriate performance measurement systems as well as compensation systems for managers which neglect the monetary effects of lead time reduction. Therefore, we developed a model based on open queueing networks to evaluate the financial impacts of lead time reduction. Illustrated by an empirical case from the polymer industry, we will demonstrate the impact of performance measures on financial measures. That is why we will take into consideration efficiency performance measures (work in process, lead time, etc.) as well as effectiveness performance measure (e.g., customer satisfaction, retention rate). Based on our evaluation model, we will be able to investigate different scenarios to reduce lead time for the given case and evaluate these, based on the developed overall performance measurement model, i.e., optimization of the batch size, resource pooling, de/increase in the number of resources. In particular, we achieved a 75% lead time reduction and a 11% overall

Dominik Gl¨aßer (B), Boualem Rabta, Gerald Reiner and Arda Alp Institut de l’entreprise, Universit´e de Neuchˆatel - Rue A.-L. Breguet 1, CH-2000 Neuchˆatel, Switzerland, e-mail: [email protected] Boualem Rabta e-mail: [email protected] Gerald Reiner e-mail: [email protected] Arda Alp e-mail: [email protected] G. Reiner (ed.), Rapid Modelling and Quick Response, c Springer-Verlag London Limited 2010 DOI 10.1007/978-1-84996-525-5 11, 

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cost reduction (resource costs, setup costs, WIP costs, penalty costs, inventory costs) without changing the whole production layout or making high investments. Key words: Queueing Networks, Rapid Modelling, Manufacturing Systems, Lead Time Reduction, Financial Evaluation

1 Introduction It is well known that lead time reduction will increase the competitiveness of companies as well as supply chains in many industries (Suri et al, 1993; Suri, 1998; de Treville et al, 2004). Long lead times that decrease customer satisfaction are causing companies to lose market shares and are missing opportunities to tailor operations to company needs. Time-based dimensions are critical determining factors for companies in assessing their strategic position and maintaining their competitive move especially if they have to operate in an agile supply chain environment (Naylor et al, 1999; Christopher and Towill, 2000; Mason-Jones et al, 2000; Lee, 2002). Regarding time-based competition (Stalk and Hout, 1990; Askenazy et al, 2006), lead time reduction is one of the leading and effective mechanisms. Furthermore, supply chains are filled with useless inventory when lead times are long. Unfortunately, despite this demonstrated importance of lead time reduction, we are globally observing an underinvestment in lead time reduction at top management levels. Nevertheless, the basic principles of rapid modelling based on queueing theory that provide the theoretical foundations for lead time reduction, are well known in research (see actual developments in Reiner, 2009) but the transfer of knowledge has to be facilitated since e.g., the maximization of resource utilization is still a wide-spread aim for managers in many companies around the world (de Treville and van Ackere, 2006). Cutting lead time is understood by experienced managers but is rarely given sufficient importance. For many key managers, lead time reduction still means working faster, harder and longer in order to complete the job in less time instead of fully understanding the functional dependency between lead times, capacity utilization and variability (Suri, 1998). Many managers believe that machines and employees have to be kept busy to accelerate production which is wrong. This leads to the widely spread aim of maximizing capacity utilization (de Treville and van Ackere, 2006). This behaviour is contrary to the real operations management requirements in many of circumstances, e.g., the importance of cutting lead time becomes even more important if a manufacturer has to act in an agile supply chain environment since the responsiveness to customer needs, plays a decisive role (Christopher and Towill, 2000). Based on the classic formula for calculating safety stock that is used as part of a reorder point replenishment policy, it is evident that delivery time mean and variance directly affect the safety stock level (Silver et al, 1998; Nahmias, 2005). For this reason, long lead times will fill the supply chain with endless inventory

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that increases the cash-to-cash cycle time (Hammel et al, 2002). On the other hand, reducing the variance as well as average lead time, result in reduced safety stock that is reflected by lower stock keeping costs, without worsening the service level, i.e., the number of stock outs, etc. Another problem in this context is that some managers consider huge batch sizes to be a possibility to increase capacity and to reduce setup costs. Very often, the sophisticated extensions of the economic order quantity (EOQ) (Silver et al, 1998; Nahmias, 2005) are taken to calculate the optimal batch size. However, this fails to cover the “real” costs of huge batch sizes and neglects the importance of responsiveness (Suri, 1998). In general, based on Little’s Law (Little, 1961) it can be demonstrated that cutting cycle times will reduce work in process as well as working capital and finally increases capital turnover and therefore also the ROI (Hopp and Spearman, 2000). These dependencies have to be integrated also into accounting processes (Maskell and Kennedy, 2007). Unfortunately, traditional accounting and reward systems are based on managing scale and costs and disregard the importance of lead time reduction and the interrelations with the overall process performance and financial impacts (Suri, 1998). Traditional cost accounting systems motivate mass-production measurements (e.g., increased labor efficiency, maximized machine utilization). Contrarily, this leads to higher inventories, longer lead times and finally, waste of the overcapacity and inventory which is opposed to lean thinking, etc. However, companies applying modern operations management approaches should use value stream costing rather than traditional-based (e.g., ABC) based costing (Maskell and Kennedy, 2007; Van der Merwe, 2008). This also provides better-decision making, i.e., wrong simple short-term, cost-focused decision support with traditional systems should be avoided. For instance, as we have mentioned above, classical machine utilization maximization motivates people to reduce excess capacity and in parallel, keeps machines busy while resulting in excess inventories. Contrarily, modern key operations management principles place emphasis on creating and keeping capacity and use this capacity for growth or hedging uncertainties. Product costs should be highly related to process flow and thus one of the key factors of success (controlling speed and improving efficiency) is to control the flow (Johnson, 2006; Maskell and Kennedy, 2007; Maynard, 2008; Van der Merwe, 2008). As we have discussed above, underlying system dynamics and relations between utilization, lot size, variability as well as layout to lead time and finally revenue as well as costs are not straightforward and difficult to comprehend (de Treville et al, 2009). In our paper we will provide an evaluation framework which takes into consideration these dependencies and the financial implications. The investigations will be carried out on the basis of quantitative models illustrated by empirical data. With this open queueing model it is possible to reproduce causal correlations between the production and the financial performance measures. In order to give a precise evaluation of the financial effect of lead time reduction, we determine the side effects of changes in lead time. Compared to previous studies, where the benefits of lead time reduction is only shown through few cost measures, our approach provides a more general framework by considering all cost components that are directly affected

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by lead time reduction. Furthermore, our framework considers feedback of system changes, i.e., the reciprocal effect of the input (e.g., capacity) and the output (lead time, cost, service level. . . ) on each other. We describe the importance and the interactions between performance measures in Section 2. Section 3 explains the evaluation frame as well as the underlying open queueing model. In particular, the links between financial and production performance measures are described. Section 4 shows a numerical illustration from the polymer industry. Additionally, a guideline of how it is possible to reduce the lead time within a production facility is highlighted. We will provide also some techniques to reduce lead time in a fast and efficient way. We will optimize the overall system performance by reducing the overall lead time without changing the complete production design. Afterwards, the results are presented. In Section 5 a conclusion and outlook for further research is given.

2 Performance Measurement Traditionally, businesses have used financially orientated measures of business performance. Dissatisfaction with these cost-accounting-based performance measures is increasing, because they are somewhat obsolete as their focus is too narrow, and they fail to provide insight into