Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, H. Dawid, K. Inderfurth, W. Kürsten
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Martin Albrecht
Supply Chain Coordination Mechanisms New Approaches for Collaborative Planning
ABC
Dr. Martin Albrecht PAUL HARTMANN AG
[email protected]
ISSN 0075-8442 ISBN 978-3-642-02832-8 e-ISBN 978-3-642-02833-5 DOI 10.1007/978-3-642-02833-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009931327 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
For Rita and Amalia Isabel
Foreword
Inter-organizational supply chains have to coordinate their material, information, and financial flows efficiently to be competitive. However, legally independent supply chain (SC) partners are often reluctant to share critical data such as costs or capacity utilization, which is a prerequisite for central planning or hierarchical planning – the planning paradigm of today’s Advanced Planning Systems (APS). Consequently, concepts for collaborative planning are needed, considering a joint decision making process for aligning plans of individual SC members with the aim of achieving coordination in the light of information asymmetry. This is the starting point and challenge of the PhD thesis of Martin Albrecht because little is known about how to design a solution for this difficult decision problem. Starting from an initial solution – that may be generated by upstream planning – improved solutions are looked for. This is achieved by computer-supported negotiations, i.e., an exchange of different order proposals within the planning interval among the SC partners involved, where partners are free to accept or reject proposals. One challenge in this negotiation process is to find new proposals and counterproposals which have a good chance of acceptance while improving the competitive position of a SC as a whole. Here, Albrecht devised new generic coordination schemes for planning tasks which can be modeled either by Linear Programming (LP) or Mixed Integer Linear Programming. For the LP case finite convergence to the optimum has been proved. While previous research on collaborative planning stopped with a clever coordination scheme Albrecht also considered a further, very important aspect of negotiations: How to get the partners to tell the truth when exchanging information and to accept a very promising solution for the supply chain as a whole. Formally speaking, coordination mechanisms are needed where the coordination schemes can be embedded. One of the coordination mechanisms advocated by Albrecht is the surplus sharing by an initially agreed upon lump sum payment to one party. He has been able to show that the corresponding mechanism results in truth-telling as a weakly dominant strategy. The reader can expect both analytical results as well as computational tests of collaborative planning schemes for various lot-sizing problems including some from industrial practice – and there is a lot more to be gained
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Foreword
from reading this thesis but I will not reveal more details here. I wish this excellent thesis a wide audience of interested and very satisfied readers and a large impact on collaborative planning. Hamburg, April 2009
Hartmut Stadtler
Preface
When I started my research, most known collaborative planning approaches dealing with mathematical programming models were based on a serious oversimplification of reality: They presumed a team setting, where parties honestly disclose information and sometimes even accept deteriorations if this benefits the supply chain as a whole. One of the major contributions of this work has been to relax this assumption. I have developed mechanisms, which achieve coordination despite self-interested behavior of parties. Without wanting to relativize the importance of this contribution, I would like to point out the existence of a particular real-world team: The people supporting me when I was writing this thesis. First of all, I am indebted to Prof. Dr. Hartmut Stadtler. He not only set the example for my research, but also provided (sometimes incredibly) generous advice and professional and personal support. Among many other things, he has patiently read my papers many times and supplied several insightful suggestions at all stages of this work. I am also grateful to Prof. Dr. Karl-Werner Hansmann for his willingness to serve as the co-referee for this thesis. Apart from my academic advisers, I am indebted to my colleagues and collaborating researchers. Particularly, I want to thank Carolin P¨uttmann for her great teamwork in the EU-project InCoCo-S, for listening to many of my (not always fully worked out) ideas, and for carefully proofreading the whole dissertation. Dr. Bernd Wagner and Volker Windeck also read parts of the thesis and provided many valuable suggestions. Last, but not least, I am thankful to Prof. Dr. Heinrich Braun and Benedikt Scheckenbach from the SAP AG for challenging discussions and for making available the real-world test data used in this work. I also thank the Gesellschaft f¨ur Logistik und Verkehr for subsidizing the printing of this work. Certainly most important for this dissertation has been my family, although not interested in supply chain management at all. My parents supported my education, without expecting anything in return. My wife Rita not only renounced to much shared time, but encouraged me with all her love to keep on researching until I have (finally) been satisfied with this work. Thank you, everybody. Heidenheim, May 2009
Martin Albrecht ix
Contents
Abbreviations .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xiii Nomenclature .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xv 1
Introduction .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.1 Motivation and Goals of This Work . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.2 Methodology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.3 Outline . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
1 1 2 2
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Supply Chain Planning and Coordination .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1 Supply Chain Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1.1 Definitions and Overview .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1.2 Master Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2 Model Formulations for Master Planning . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2.1 Generic Master Planning Model .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2.2 Extension to Lot-Sizing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3 Decentralized Planning and Coordination .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3.2 Decentralized Supply Chain Planning .. . . . . . . . . . . . . . .. . . . . . . . . . . 2.3.3 Upstream Vs. Collaborative Planning . . . . . . . . . . . . . . . .. . . . . . . . . . .
5 5 5 8 9 10 12 20 20 24 30
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Coordination Mechanisms for Supply Chain Planning . . . . . . . .. . . . . . . . . . . 3.1 Symmetric Information .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.1.1 Non-cooperative Game Theory .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.1.2 Cooperative Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2 One-Sided Information Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.1 Signaling .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.2 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3 Multilateral Information Asymmetry .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.1 Auctions and Their Application to Supply Chain Coordination 3.3.2 Mechanisms with Focus on Proposal Generation .. . .. . . . . . . . . . .
35 35 36 41 43 43 45 48 48 51
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New Coordination Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 63 4.1 Generic Scheme for Linear Programming and Analytical Results . . . . . 64 4.1.1 Version with Iterative, Unilateral Exchange of Cost Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 64 4.1.2 Version with One-Shot Exchange of Cost Information .. . . . . . . . 80 4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 82 4.3 Application to Master Planning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 99 4.3.1 Linearization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 99 4.3.2 Adaptation to Master Planning . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .103 4.3.3 Generic Modifications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .111 4.3.4 Modifications for Master Planning . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .114 4.4 Customizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .120 4.4.1 Master Planning with Lot-Sizing . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .120 4.4.2 Voluntary Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .122 4.4.3 Lost Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .123 4.4.4 Multiple Suppliers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .126
5
New Coordination Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .129 5.1 Surplus Sharing Determined by the Informed Party . . . . . . . . .. . . . . . . . . . .130 5.2 Surplus Sharing Determined by Lump-Sum Payments . . . . . .. . . . . . . . . . .133 5.3 Surplus Sharing by a Double Auction . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .141 5.4 Comparison of Mechanisms and Discussion . . . . . . . . . . . . . . . . .. . . . . . . . . . .149 5.5 Application with Rolling Schedules . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .151
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Computational Tests of Coordination Schemes . . . . . . . . . . . . . . . . .. . . . . . . . . . .155 6.1 General Master Planning Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .155 6.1.1 Generation of Test Instances and Performance Indicators.. . . . .155 6.1.2 Analysis of Solutions for the Generic Scheme .. . . . . .. . . . . . . . . . .162 6.1.3 Analysis of Solutions for the Modified Scheme . . . . .. . . . . . . . . . .164 6.2 Uncapacitated Lot-Sizing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169 6.2.1 Generation of Test Instances .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169 6.2.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .171 6.3 Multi-level Capacitated Lot-Sizing Problem . . . . . . . . . . . . . . . . .. . . . . . . . . . .174 6.4 Models for Campaign Planning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .179 6.4.1 Generation of Test Instances .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .179 6.4.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .180 6.5 Real-World Supply Chain Planning Problems . . . . . . . . . . . . . . .. . . . . . . . . . .184 6.5.1 Planning Problems and Model Formulation . . . . . . . . .. . . . . . . . . . .185 6.5.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .192
7
Summary and Outlook .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .197
References .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .201
Abbreviations
AGC Average gap closure achieved by the scheme AGS Average gap after the application of the scheme AGU Average gap of the uncoordinated solution APO Advanced planner and optimizer APS Advanced planning system arb. Arbitrary B Buyer BOM Bill of material B2B Business-to-business CLSP Capacitated lot-sizing problem CLSPL Capacitated lot-sizing problem with linked lot sizes cos. Cosinus CSLP Continuous setup lot-sizing problem CU Capacity unit DLSP Discrete lot-sizing and scheduling problem EOQ Economic order quantity GC Gap closure achieved by the scheme GLSP General lot-sizing and scheduling problem GM Generic master planning model GS Gap after the application of the scheme H High IGFR Increasing generalized failure rate IP Informed party L Lot size (driver) L Low (type of demand forecast) LB Lower bound LP Linear programming MLCLSP Multi-level capacitated lot-sizing problem MLCLSPL Multi-level capacitated lot-sizing problem with linked lot sizes MLPLSP Multi-level proportional lot-sizing and scheduling problem MLULSP Multi-level uncapacitated lot-sizing problem MINLP Mixed-integer nonlinear programming MIP Mixed-integer programming
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MU NLP OEM Q RHS RP PPM S Sched. SMP SNP SOS2 spl. T TBO TC TS U UB UT
Abbreviations
Monetary unit Nonlinear programming Original equipment manufacturer Quantity Right-hand side Reporting party Production process model Supplier Scheduling Single machine processor Supply network planning Special ordered set of type 2 Split Time Time between orders Time for solving the centralized model Time for running the scheme Unit Upper bound Unit time
Nomenclature
Indices, Sets, and Index Sets ˘E Set of proposals already found ˘iE Set of proposals previously generated by the scheme ˘B Set of proposals identified by the buyer ˘S Set of proposals identified by the supplier ˘ up Set of proposals identified by GMupS E i Set of cost changes associated with proposals for central resource use a Arc linking two locations ABa Location at the beginning of arc a AEa Location at the end of arc a Br .x/ r-neighborhood of x CS Set of solutions identified by the scheme for the MLULSP DS Set of proposals with delayed supply compared to the starting proposal ES Set of proposals with early supply compared to the starting proposal f Superindex denoting the first proposal generated i Decentralized parties i Suppliers init Superindex denoting the initial solution J Set of items j Items or operations JB Set of items produced by the buyer JD Set of items supplied JE Subset of items sold to external customers JlE Items sold at location l S Set of items produced by the supplier J Jm Set of items produced on resource m L Set of locations l Location, l 2 L M Set of resources m Resources (e.g., personnel, machines, production lines) MB Set of resources of the buyer MS Set of resources of the supplier NDS Set of proposals without delayed supply compared to the starting proposal
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Nomenclature
NES Set of proposals without early supply compared to the starting proposal new Superindex indicating a new proposal P Set of PPM P Set of parties p PPM, p 2 P Pc Set of parties for which coordinated proposals have been chosen PS Set of suppliers PB Set of supply proposals optimal for the buyer subject to any N B Rj Set of predecessor items of item j S Set of customer classes s Customer class, s 2 S Sj Set of immediate successors of item j in the BOM st Superindex indicating the starting solution T Set of periods t Periods Xis Subset of feasible solutions to CS1i XiE Set of proposals found so far Subset of vertex solutions to DPi Xivd Xiv Set of vertex solutions identified so far Parameters and Random Variables tiB Periods between subsequent setups of the buyer tiS Periods between subsequent setups of the supplier Unit penalty costs for arbitrary deviations O ; A ; B ; C Weights i e Scalars i e Scalars aj Cumulated capacity requirements of an item j ei Number of different solutions found so far for party i ej Average secondary demand for item j Proposal PB Proposal out of PB with the same N B and N S as in the systemwide optimal solution ie Proposal e for the central resource use by party i ij Resource use for the j th proposal generated by party i p Production lead time of PPM p aj Transportation lead time for item j along arc a j Lead time for item j Type of demand forecast hB Lower bound for the probability distribution of hB
b be lc
blcj j
ie a; b
Maximum costs for backorders that are caused by a shortage in the supply or the production of item j Potential cost impact of backorders in the supply of item j Cost effects of the proposal e by party i Lower and upper bounds for S
Nomenclature
Ai amj amp b Bk1 ,Bk2
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Use of the central resources by decisions xi Capacity needed on resource m for one unit of item j Capacity needed on resource m for one unit of PPM p Purchase price Prior knowledge of parties #1, #2 about the other party’s bids for proposals
k (upper bounded by bj1 ,bj2 ) Bids by parties #1, #2 for proposal k Total amount of the central resources Bi Use of the decentralized resources by decisions xi bi Total amount of the decentralized resources bjt Large number, not limiting feasible lot size of item j in period t bpt Big number indicating the maximum production quantity of PPM p in period t blj0 Amount of backorders for item j at the beginning of the planning interval bljT Amount of backorders for item j at the end of the planning interval blcj Backorder costs for one unit of item j in a period blljs Backorder costs for one unit of item j of customer class s in a period at location l bsj Batch size for item j bsp Batch size for PPM p cB Buyer’s costs of the systemwide optimal solution csys Overall costs in the systemwide optimum PB cB Buyer’s costs for the proposal out of PB with the same N B and N S as in the systemwide optimal solution c0 Production costs at the supplier’s site ci Costs associated with decisions xi cce n;n Costs of the solution to the centralized model for test instance n ccor;n;i Costs after i iterations of the scheme csys ./ Systemwide costs resulting from an implementation of CS csys Costs for the best solution out of CS PB csys Costs of an implementation of the best proposal out of PB cunc;n Costs of the uncoordinated solution (usually determined by upstream planning) cam0j Initial campaign quantity for item j cb e Buyer’s costs change of the previous proposal e compared to the initial solution conoc Average ratio between backorder and overtime costs cp Unit penalty costs i cpB Penalty costs for supplier i cs e Supplier’s costs change of the previous proposal e csie Costs of proposal e for supplier i csel Costs for one unit of storage capacity increase at location l csslj Penalty costs for one unit of stock below the required safety stock of item j at location l bk1 ,bk2 b0
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Nomenclature
ctaj cvp D d dj0 djt dljst
Transportation costs for one unit of item j along arc a Variable production costs of PPM p Demand (random variable) Demand per unit time Value of the j -th dimension of Ai xist xi Primary, gross demand for item j in period t Primary, gross demand for item j of customer class s in period t at location l dlbejt Deviation of proposal e that is due to lost sales and relevant for the buyer dlsejt Deviation of proposal e due to lost sales for the supplier cum ejt Cumulated secondary demand for item j in period t F ./ Cumulated density function f ./ Probability density function fmt Randomly generated factor determining capacity profiles g max Expected surplus of the best solution identified by the scheme g mech Expected surplus that can be realized by the mechanism gRP .l/ Gains of the RP from coordination subject to l h Unit holding cost per unit time hB Limit for acceptable hB hj Holding cost for one unit of item j in a period hS Holding cost of the supplier hlj Holding cost for one unit of item j at location l in a period hbj Buyer’s unit costs for inventory holding of the supplied item j ij0 Inventory of item j at the beginning of the planning interval P rev ijt Inflow of item j in period t originating from earlier production periods i clmax Maximum storage capacity at location l i clj Consumption of storage capacity at location l by one unit of item j i celmax Maximum extension of storage capacity at location l i nilj Inventory of item j at location l at the beginning of the planning interval K Constant of an arbitrary value (e.g., 1) used for the correct transformation of the unit of aj k Parameter for surplus sharing in the sealed bid double auction kj0 Value of the j -th dimension of kiT kmt Available capacity of resource m in period t L; LO Lump sum payment l; lO Markup (above the lump sum) L Optimal lump sum L1 ,L2 Prior knowledge of parties #1, #2 about the leeway in general, (upper bounded by lj1 ) 1 2 Lk ,Lk Prior knowledge of parties #1, #2 about the leeway for proposals k (upper i
L lk lb
bounded by lj1 ) Lump sum required by party i Markup for proposal k Lower bound
Nomenclature 0
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lbjd Lower bound for dj0 0 lbjk Lower bound for kj0 lmaxljs Maximum lateness for demand fulfillment of item j of customer class s at location l lscj Costs for lost sales of one unit of item j in a period lsljs Costs for lost sales of item j of customer class s at location l m Number of approximation intervals m1 ,m2 (General) markdowns chosen by parties #1, #2 m1k ,m2k Markdowns chosen by parties #1, #2 for proposal k Mi Vector made up of big numbers that exceed marginal cost savings resulting from increases in central resource use mjt Big number, denoting the maximum cost change per unit deviation in the supply quantities mfjp Material flow of item j from PPM p minlotp Minimum lot size for item p N B;up Number of setups in the upstream planning solution NkB Number of buyer’s setups in the planning interval for items k 2 SJ D \ J B NjS Number of supplier’s setups in the planning interval for items j 2 J D pre nk Number of items preceding item k Number of the buyer’s orders within tiS ; tiSC1 oi Overtime costs for one unit of resource m ocm P Set of decentralized parties p Selling price P .Q/ Purchase price dependent on the purchase quantity Q r Parameter denoting the ratio between N B and N S (rounded down) r .l/ Function that maps the expected reduction of S with l cum rjk Number of units of item j required to produce one unit of the (direct or indirect) successor item k rjk Number of units of item j required to produce one unit of the immediate successor item k S Subset of decentralized parties S Systemwide surplus from coordination (random variable) s Selling price s Share of the revenue generated sk1 ,sk2 Savings by parties #1, #2 for proposal k Si Marginal i i surplus from coordination for party i defined within the interval a ; b (random variable) S sys Expected surplus for the whole system (random variable) sji Savings of party i with proposal j Sk Systemwide surplus for proposal k (random variable) sc Setup cost scB Setup cost of the buyer scj Setup cost for a lot of item j scS Setup cost of the supplier
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Nomenclature
stj tiB TL TU tiS C tcB tcSC u u ub 0 ubjd 0 ubjk ut v v .S / w w w0j X f;C xnjt
Setup time for item j Periods in which setups of the buyer occur Time horizon in setting L Time horizon in setting U Periods in which setups of the supplier occur Reservation value of the buyer Reservation value of the supplier Prices for central resource use Utility vector Upper bound Upper bound for dj0 Upper bound for kj0 Average capacity utilization Salvage value Surplus from forming set S Target for the reduction of the number of setups for the items supplied Wholesale price Initial setup state of item j Random variable denoting perturbation C st Node n (x-coordinate) for the linearization of f f Kjt ; XBjt xbjt f; st xnjt Node n (x-coordinate) for the linearization of f f Kjt ; XBjt xbjt s;C C st xnjt Node n (x-coordinate) for the linearization of f s Kjt ; XBjt xbjt s; st xnjt Node n (x-coordinate) for the linearization of f s Kjt ; XBjt xbjt Solution previously found (in step e of the scheme) xie xiv Value taken by variables xi in the vertex v identified so far xO ; xA ; xB ; xC Breakpoints 0 xtjt Modified target supply quantity of item j in period t e xtjt Amount of item j supplied in period t in the previous proposal e e;i xtjt
Supply quantity of item j in period t delivered by supplier i and specified by proposal e max xtaj Maximum transportation quantity of item j along arc a in a period mi n xtaj Minimum transportation quantity of item j along arc a in a period xtjt Target for the supply quantity of item j in period t ysjup Number of orders for item j in the proposal from upstream planning z Parameter indicating the ratio between T and N B (rounded down) Variables C f st f;C K Weight for node n for the linearization of f ; XB xb jt njt jt jt f; st f njt Kjt ; XBjt xbjt Weight for node n for the linearization of f C st s;C Weight for node n for the linearization of f s Kjt ; XBjt xbjt njt s st s; K Weight for node n for the linearization of f ; XB xb jt njt jt jt
Nomenclature
xxi
i e
Decision variables defining a linear combination of previous proposals about the central resource use of party i BLjt Amount of backorders for item j in period t BLljst Amount of backorders of item j of customer class s at location l in period t p;C Cjt Penalties or bonuses for greater supply of item j in period t p; Cjt Penalties or bonuses for less supply of item j in period t CBd Costs for the decisions of the buyer’s planning domain Costs for the decisions of the supplier’s planning domain CSd CMLCLSP Value of the objective function of the MLCLSP CAMjt Campaign variable for item j in period t (quantity of the current campaign up to period t) CAMpt Campaign variable for PPM p in period t (quantity of the current campaign up to period t) ls Djt Difference in the supply quantity of item j in period t due to lost sales B g Profit of the buyer I .Q/ (Leftover) inventory Ijt Inventory of item j at the end of period t Iljt Amount of inventory of item j at location l at the end of period t IBjt Inventory of the (supplied) item j at the buyer’s site in period t ICElt Increase of storage capacity at location l in period t ISjt Inventory of item j at the supplier’s site in period t ki Prices for changes in central resource use C Kjt Endogenously determined unit prices for positive deviations from the startst ing proposal xjt of item j in period t Kjt Unit prices for negative deviations of item j in period t ls Kjt Penalty costs for lost sales of item j in period t Kjagg;C Endogenously determined unit penalty costs for shifts of the supply of item j to later periods compared to the starting supply pattern Kjagg; Endogenously determined unit penalty costs for shifts of the supply of item j to earlier periods compared to the starting supply pattern LSjt Amount of lost sales of item j in period t LSljst Amount of lost sales of item j of customer class s at location l in period t M .Q/ Quantity sold to the market Omt Amount of overtime on resource m in period t Q Order quantity QB Optimal order quantity for the buyer QSC Optimal order quantity for the supply chain Rjt Integer number of full batches produced in the current campaign of item j up to period t Rpt Integer number of full batches produced in the current campaign of PPM p up to period t Sjt Quantity of the last batch of item j in period t which is not finished in t Spt Quantity of the last batch of PPM p in period t which is not finished in t S Sljt Undershot of safety stock of item j at location l in period t
xxii
Nomenclature
Wjt
Setup state indicator variable (=1 if item j is set up at the end of period t, 0, otherwise) Setup state indicator variable (=1 if PPM p is set up at the end of period t, =0 otherwise) Production quantity of item j at the beginning of period t (i.e., of the first campaign in t) Production quantity of PPM p at the beginning of period t Production quantity of item j that is not produced at the beginning of period t (i.e., not part of the first campaign in t) Production quantity of PPM p that is not produced at the beginning of period t Decision variables in the generic LP model Production amount of item j in period t Amount of item j delivered to the buyer in period t Amount of item j delivered by the supplier in period t st Increase in the supply of item j in period t compared to xjt Decrease in the supply of item j in period t Transportation quantity of item j along arc a in period t Binary setup variable (=1 if item j is produced in period t, =0 otherwise) Binary setup variable (=1, if PPM p is produced in period t, =0 otherwise) Setup operation indicator for resource m in period t (=1 if a setup occurs on resource m in period t, =0 otherwise) Indicator variable, =1 if item j is ordered in period t, =0 otherwise Objective function value of CS1i Binary variable (=1 if proposal i of party j is implemented, =0 otherwise).
Wpt b Xjt b Xpt e Xjt e Xpt
xi Xjt XBjt XSjt X TjtC X Tjt X Tajt Yjt Ypt Y Imt Y Sjt ZCS1i Zij
Chapter 1
Introduction
1.1 Motivation and Goals of This Work Supply chain planning is concerned with the determination of integrated operational plans for all functional areas and members within a supply chain. Depending on the organizational structure of the supply chain, this task can either be considered as the state-of-the-art or as a challenge for future supply chain excellence. State-of-the-art is the planning in intra-organizational supply chains. This task is supported by a broad range of procedures elaborated in the literature during the last decades as well as modeling tools, APS (Advanced Planning Systems), which are widely used by practitioners.1 This, however, is not the case for inter-organizational supply chains consisting of multiple, legally independent parties. Current APS only provide interfaces for data exchange between parties, but do not support inter-organizational collaborative planning. In APS, an integrated planning requires a (central) entity equipped with all relevant data and the decision authority to implement the systemwide optimal plan. However, this approach comes with a number of downsides: The need for disclosing potentially confidential information by the decentralized parties, the conflict of central targets with the incentive structure in decentralized organizations, and the missing guarantee for truthful information disclosure; indeed, very few applications of this approach have been reported so far.2 This result stands in sharp contrast to the literature, where coordination has been widely recognized as one of the key drivers of supply chain performance in the last 10–15 years. A large number of papers evaluating the benefits from coordination and proposing new coordination mechanisms have been produced. Unfortunately, these mechanisms have severe limitations making it impossible to apply them to inter-organizational supply chain planning. Among these limitations are a complete knowledge about the others’ model data and team behavior by the participating parties as well as the restriction on economic order quantity or newsvendor models. 1 2
See, e.g., the case studies reported by Stadtler and Kilger (2007, p. 367). E.g., Shirodkar and Kempf (2006, p. 420).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 1,
1
2
1 Introduction
In this study, we augment the existing literature by new coordination mechanisms, which lift the major limitations as needed for a potential practical application. These mechanisms are the first to simultaneously include several generic features like: The assumption of multilateral information asymmetry about other parties’
detailed data (before, during, and after the application of the mechanism) No need for involving a third party Self-interest of parties Complex mathematical programming models including discrete decisions run by
the decentralized parties In addition to a theoretical development and foundation, computational tests for randomly generated data as well as real-world data indicate that substantial savings can be obtained by the mechanisms proposed.
1.2 Methodology The coordination mechanisms have to identify an improvement compared to an initial, uncoordinated solution and to include incentives to implement the improved solution. For that purpose, this work combines methodologies from two different areas of economic research: Operations research and game theory. The improvements are identified by innovative mathematical programming models. We assume that such models are used by the parties for their decentralized planning and develop extensions, which can be applied in an iterative manner for the generation and identification of potentially coordinating supply proposals; the single steps undertaken for this purpose are called a coordination scheme. The effectiveness of the schemes proposed is demonstrated by analytical and computational results. We analytically prove the convergence of the schemes for specific model classes, and show by computational tests that the schemes are able to substantially mitigate the suboptimality from decentralized planning. To determine the incentives for the decentralized parties to follow the rules of the schemes, the mechanisms rely on concepts from the area of game theory. Strategic (and potentially untruthful) behavior of decentralized parties is explicitly taken into account. We build on insights and ideas from bilateral bargaining and behavioral research to design several mechanisms that can be applied in different organizational structures. For two of these mechanisms, upper bounds for the losses due to information asymmetry will be derived.
1.3 Outline The thesis is organized as follows. Chapter 2 provides the basis for the mechanisms developed in this work. First, we describe the task of Master (i.e., mid-term)
1.3 Outline
3
Planning and corresponding mathematical model formulations. Second, we define basic terms used in this work and discuss the consequences of decentralized planning. We show how the resulting planning processes can be modeled mathematically and identify drivers for the systemwide suboptimality of unilateral targets, which are established without coordination. Finally, we outline upstream planning, which we assume as the standard planning procedure without coordination, as well as our concept for collaborative planning. Chapter 3 surveys the literature on coordination mechanisms. We have structured this review according to the assumptions on the knowledge about the other parties’ data, i.e., we distinguish between symmetric, one-sided asymmetric, and multilateral asymmetric information. Moreover, we elaborate the basic ideas regarding the design of the related mechanisms and provide classifications for the literature of this area. Chapters 4 and 5 comprise the core contributions of this work. In Chap. 4, the different coordination schemes are outlined. We begin with two versions of a generic scheme for coordinating decentralized parties running arbitrary linear programming models (in one version, even one of the parties may run a mixed-integer programming model). Moreover, we present a scheme for coordinating uncapacitated lot-sizing models in supply chains of one buyer and one or multiple suppliers. For both schemes, analytical results about their convergence behavior can be derived. Apart from that, for a two-party supply chain, we present modified versions of these schemes with improved convergence rate and improved applicability for Master Planning problems that include discrete decisions. Amongst others, we cover extensions of these modified versions to voluntary compliance by the supplier, the modeling of lot-sizing and lost sales, and settings with multiple suppliers. As a second component of the coordination mechanisms, several contractual frameworks are outlined and analyzed in Chap. 5. Resulting are mechanisms applicable for different organizational structures and different distributions of the bargaining power between the decentralized parties. The strategies adopted by the decentralized parties using these mechanisms are discussed in light of behavioral theories and analytical reasoning. As a third issue in the chapter, we outline how the mechanisms can be adapted for rolling schedules, which are frequently used in real-world production planning. Computational tests are provided in Chap. 6. We examine the performance of the schemes based on randomly generated test instances for different Master Planning models as well as for real-world Master Planning data provided by the SAP AG. For all problems investigated, significant improvements compared to upstream planning can be identified after a modest number of iterations. Finally, Chap. 7 summarizes the contributions of this work and outlines opportunities for further research.
Chapter 2
Supply Chain Planning and Coordination
The aim of this chapter is to familiarize the reader with the topic of this work, how to coordinate mid-term planning in decentralized supply chains, i.e., supply chains that comprise several independent, legally separated parties with their own decision authorities. We start with a description of planning in centralized supply chains (Sect. 2.1), where the decision authority and the knowledge of all relevant planning data is hold by a single party. In Sect. 2.2 we provide centralized mathematical model formulations for mid-term supply chain planning (Master Planning). Section 2.3 deals with supply chain planning in decentralized environments. We describe the differences in the planning processes compared to centralized planning, provide reasons for the potential suboptimality of decentralized planning and introduce coordination and, more specifically, collaborative planning as approaches to mitigate this suboptimality.
2.1 Supply Chain Planning 2.1.1 Definitions and Overview We begin with an abstract and often cited definition of a supply chain: A supply chain is a “. . . network of organizations that are involved, through upstream and downstream linkages, in the different processes and activities that produce value in the form of products and services in the hands of the ultimate consumer.”1 As an illustration, we depict in Fig. 2.1 a supply chain consisting of a set of vendors, plants, distribution centers, and customers that are linked by material flows.2 From a business economics perspective, supply chains require supply chain management, that can be defined as “the task of integrating organizational units along a supply chain and coordinating material, information and financial flows in order to fulfill (ultimate) customer demands with the aim of improving the 1 2
Christopher (2005, p. 17). For similar representations, see, e.g., Shapiro (2001, p. 6) and Stadtler (2007b, p. 10).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 2,
5
6
2 Supply Chain Planning and Coordination
Fig. 2.1 Sketch of a supply chain (example)
competitiveness of a supply chain as a whole.”3 One of the building blocks of supply chain management is (advanced) supply chain planning.4 The aim of supply chain planning is to determine an integrated plan for the whole supply chain; referring to Fig. 2.1, such a plan comprises appropriate quantities of the raw materials procured, of the products manufactured in the plants, of the products distributed, and of the products sold to the customers. Of course, this is a very complex task especially for real-world organizations, which may comprise a large number of facilities, customers, and products. Therefore, it has been proposed in the literature5 to organize (production) planning in a hierarchical way.6 The basic idea of hierarchical planning is the separation of decisions according to their impact, e.g., on the profitability of the supply chain. The decisions at the upper levels, i.e., those with greater impact, are determined first and implemented as targets for the planning of the lower levels. Further important characteristics are the aggregation of data and decisions at the upper levels and the provision of feedback by the lower levels.7 A common representation for the individual tasks of supply chain planning is the Supply Chain Planning Matrix (see Fig. 2.2).8 Frequently, these planning tasks are supported by software tools in practice. As a standard software, Advanced Planning Systems (APS) have been developed by different companies (e.g., SAP, Oracle9 ). Each APS contains several modules that cover in part the planning tasks stated in Fig. 2.2. The Advanced Planning Matrix provides an overview of these modules (see Fig. 2.3).10 Long-term planning is the object of Strategic 3
Stadtler (2007b, p. 11). See Stadtler (2007b, p. 27). 5 See, Hax and Meal (1975, p. 53) and the work of Stadtler (1988) as a comprehensive introduction into hierarchical planning. 6 Although originally proposed for production planning, this concept equally applies to planning in supply chains. 7 See, e.g., Stadtler (2007b, p. 32). 8 See Fleischmann et al. (2007, p. 102). 9 See SAP (2008) and Oracle (2008). The market share of APS is substantial and steadily growing; e.g., in 2005, revenues increased by $40 million up to $741 million, see White et al. (2006, p. 2). 10 Figure 2.3 has been adapted from Meyr et al. (2007b, p. 109) with slight modifications (see below). 4
2.1 Supply Chain Planning
7
procurement
production
long-term
• materials programme • supplier selection • cooperations
• plant location • production system
mid-term
• personnel planning • material requirements planning • contracts
• master production scheduling • capacity planning
short-term
• personnel planning • ordering materials
• lot-sizing • machine scheduling • shop floor control
distribution
sales
• physical distribution • product programme structure • strategic sales planning
• distribution planning
• mid-term sales planning
• warehouse replenishment • transport planning
• short-term sales planning information flows
flow of goods
Fig. 2.2 Supply Chain Planning Matrix
procurement
long-term
distribution
sales
Strategic Network Design
mid-term
short-term
production
Master Planning Master Planning
Purchasing & Material Requirements Planning
Production Planning
Distribution Planning
Scheduling
Transport Planning
Demand Planning
Demand Fulfillment & ATP
Fig. 2.3 Advanced Planning Matrix
Network Design, whereas the mid-term planning tasks are covered by Purchasing & Material Requirements Planning, Master Planning, Production Planning, Distribution Planning, and Demand Planning. Analogously, the short-term planning tasks are tackled by Purchasing & Material Requirements Planning, Production Planning, Scheduling, Transport Planning, Demand Fulfillment & Available-To-Promise (ATP). Note that Fig. 2.3 differs from the original Advanced Planning Matrix by the
8
2 Supply Chain Planning and Coordination
overlap of Master Planning into sales (see the shadowed area in Fig. 2.3). In our opinion, this is more concise than the original representation since Master Planning frequently involves sales-related decisions like backorders and lost sales.11 In the following, we will provide a more detailed description of Master Planning, which is the main focus of this work. For an in-depth explanation of the other modules, we refer to the textbook of Stadtler and Kilger (2007).12
2.1.2 Master Planning Master Planning means mid-term operational decision-making carried out simultaneously for all functional areas participating in the order fulfillment process: Procurement, production, distribution, and sales. In this work, we focus on this planning level since the potential financial impact for collaborating enterprises is largest here. Master Planning is based on monthly or weekly time buckets; hence, planning horizons of, e.g., 12, 24, or 52 periods are used. It is important that the planning horizon is chosen such large that effects due to seasonal demand are considered. This requires that the planning interval comprises one seasonal cycle at least. Table 2.1 provides an overview about the basic decisions made within Master Planning. Most of these decisions are likewise mentioned in other descriptions of Master Planning.13 A somewhat ambiguous role in this context plays lot-sizing,
Table 2.1 Basic decisions of Master Planning Functional area Decision Procurement Quantities of raw materials to purchase Production Production quantities Resource utilization Inventory levels Utilization of overtime Lot-sizing Distribution Quantities of products to transport Inventory levels Sales Quantities of products to deliver to the customers (including backorders and lost sales)
11
See also Sect. 2.1.2. The importance of (lost) sales for integrated mid-term planning is further supported by the practitioner-oriented literature, where mid-term planning models with the aim of profit maximization have been proposed, see, e.g., Timpe and Kallrath (2001, p. 423) and Kallrath (2002, p. 315) for a model that additionally includes strategic aspects. 12 See Stadtler and Kilger (2007, p. 117). 13 See Rohde and Wagner (2007, p. 160) and G¨unther (2006, p. 20).
2.2 Model Formulations for Master Planning
9
which is often regarded as a short-term planning task in the literature.14 Indeed, lot-sizing is not an issue for mid-term planning in many industries.15 A noteworthy exception, however, are process industries, where lot-sizing decisions have a substantial impact on the quality of the resulting plans. Due to large setup times and expensive setup activities, practical mid-term planning model formulations originating from this area usually include lot-sizing.16 The results of these decisions (e.g., the planned amount of stock or the planned use of overtime17) constitute targets for the lower-level modules Purchasing & Material Requirements Planning, Production Planning, and Distribution Planning. Note that, according to the principles of hierarchical planning, the corresponding data has to be disaggregated for this purpose. The input data for Master Planning is deterministic. Real-world data, especially demand forecasts, however, always comprise some uncertainty. Because of that, mid-term planning models incorporating stochastic data have been elaborated in the literature.18 In practice, however, stochastic decision models are rarely used for mid-term planning. Instead, planning is done based on rolling schedules.19 This means that only the first periods of plans (i.e., the periods before the frozen horizon) are implemented; the rest is determined later by re-planning, which is undertaken periodically (e.g., once a month). This approach provides the flexibility to react with plan changes if uncertainty is revealed in future periods.
2.2 Model Formulations for Master Planning In this section, we present mathematical model formulations for Master Planning that are used throughout this work (among others, for the computational verification of the coordination schemes proposed). First of all, we provide a generic linear programming (LP) model in Sect. 2.2.1. Compared to mixed-integer programming (MIP) or nonlinear programming (NLP), the mathematical structure of LP is much simpler, which makes approaches based on LP particularly suited for structural analyses. However, results valid for LP do not necessarily extend to other model classes.20 These classes include MIP models, which are required for modeling
14
As an example, see the Supply Chain Planning Matrix on p. 7 of this work. This holds, e.g., for the (German) automotive industry, see Meyr (2004, p. 447). 16 E.g., Timpe and Kallrath (2001, p. 42) and Grunow et al. (2003, p. 109). 17 See, e.g., Rohde and Wagner (2007, p. 161). 18 E.g., Leung et al. (2007, p. 2282), Escudero et al. (1993, p. 311), and Scholl (2001, p. 295). 19 See, e.g., Fleischmann et al. (2007, p. 84). 20 As an example, consider the simplex algorithm, a common solution procedure for LP models, which yields the optimal solution to, e.g., MIP models only in case of the total unimodularity of the underlying matrix of coefficients, see Domschke and Drexl (2006, p. 91); for a description of the simplex algorithm and of solution procedures for MIP models such as branch and bound, see, e.g., Domschke and Drexl (2006, p. 21 and p. 126). 15
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2 Supply Chain Planning and Coordination
lot-sizing decisions at the Master Planning level. Therefore, we additionally provide model formulations accounting for lot-sizing in Sect. 2.2.2.21
2.2.1 Generic Master Planning Model Before presenting the mathematical formulation of the generic Master Planning model (GM), we state its underlying assumptions: Several items are arranged in a general bill of material (BOM) and produced on
one or more specific resources. Production results in variable capacity loads. The capacities of the resources are finite and can be extended by costly, infinitely
available overtime. Demand is dynamic and deterministic for all items. Unfulfilled demand can either be backlogged or lost; both actions incur additional
costs. Inventory holding of items is possible and results in holding costs.
min
XX
hj Ijt C
j 2J t 2T
X X
ocm Omt C
m2M t 2T
X X
blcj BLjt C lscj LSjt
j 2J E t 2T
s.t. BLjt C LSjt C Ijt 1 C Xjt D djt C BLjt1 C Ijt 8j 2 J E ; t 2 T (2.1) X (GM) Ijt 1 C Xjt D rjk Xkt C Ijt 8j 2 J nJ E ; t 2 T (2.2) X
k2Sj
amj Xjt kmt C Omt
8m 2 M; t 2 T
(2.3)
j 2J
BLj 0 D blj0
8j 2 J E
BLj jT j D
bljT
Ij 0 D
8j 2 J
ij0
8j 2 J
(2.4) E
(2.6) E
BLjt 0 8j 2 J ; t 2 T [ f0g Ijt 0 8j 2 J; t 2 T [ f0g
(2.7) (2.8)
LSjt 0
(2.9)
8j 2 J E ; t 2 T
Omt 0 8m 2 M; t 2 T Xjt 0 8j 2 J; t 2 T :
21
(2.5)
(2.10) (2.11)
Note that there are further decisions and restrictions in Master Planning that rely on MIP for their modeling. An example is the restriction that overtime can only be taken in an integer number of shifts, which is used in our computational tests of Sect. 6.1 for analyzing the sensitivity of the scheme regarding the presence of integer variables.
2.2 Model Formulations for Master Planning
11
Indices and Index Sets j Items or operations (e.g., end products, intermediate products, raw materials), j 2 J ; J E is the subset of (end) items sold to external customers m Resources (e.g., personnel, machines, production lines), m 2 M t Periods, t 2 T , with T D 1; : : : ; jT j Sj Set of immediate successors of item j in the BOM Data amj Capacity needed on resource m for one unit of item j blcj Backorder costs for one unit of item j in a period bl0j Amount of backorders for item j at the beginning of the planning interval bljT Amount of backorders for item j at the end of the planning interval djt Primary, gross demand for item j in period t ij0 Inventory of item j at the beginning of the planning interval hj Holding costs for one unit of item j in a period kmt Available capacity of resource m in period t lscj Costs for lost sales of one unit of item j in a period ocm Overtime costs for one unit of resource m rjk Number of units of item j required to produce one unit of the immediate successor item k Variables BLjt Amount of backorders for item j in period t Ijt Inventory of item j at the end of period t LSjt Amount of lost sales of item j in period t Omt Amount of overtime on resource m in period t Xjt Production amount of item j in period t The objective function minimizes the costs for inventory holding, overtime, backorders, and lost sales. Constraints (2.1) determine the quantities of the external demand that are backlogged and lost. Constraints (2.2) ensure the fulfillment of the secondary demand.22 Constraints (2.3) limit the capacity used for production to the sum of normal capacity and overtime. Constraints (2.4)–(2.6) fix the amounts of backorders and inventories at the borders of the planning interval.23 Finally, (2.7)–(2.11) determine the nonnegativity of the decision variables. Note that due to its generic character, this model formulation does not cover all decisions potentially relevant for Master Planning.24 For lot-sizing, we refer to 22
For ease of exposition, we have separated items into two groups: End items with external demand (J E ) and intermediate items used for production (J nJ E ). In a setting where an item is used for production and has external demand, this model formulation would have to be adapted accordingly. 23 This fixation seems the most straightforward possibility for the modeling of backorders. Note that when planning is based on rolling schedules, the modeling of maximum latenesses may be more adequate. 24 I.e., those listed in Table 2.1 (see p. 8).
12
2 Supply Chain Planning and Coordination
the following subsection. Moreover, an exemplary modeling of storage capacities and transportation is included in the model for the real-world planning problems presented in Sect. 6.5.
2.2.2 Extension to Lot-Sizing A broad variety of models incorporating lot-sizing decisions has been proposed in the literature. We begin with some basic formulations and discuss how to integrate them into Master Planning. Built on this, we present a Master Planning model that is extended to production campaigns. Campaign planning is a variant of lot-sizing with practical relevance for process industries and additional difficulties for supply chain coordination,25 which makes this extension particularly suited for examining the performance of the coordination schemes proposed in this work. 2.2.2.1 Basic Models One of the first production planning problems analyzed in the literature is the determination of the economic order quantity (EOQ),26 i.e., the optimal order quantity of an item based on a number of restrictive assumptions. The most important assumptions are:27
One single item is considered. Demand is deterministic and constant. The replenishment lead time is zero. Inventory holding is possible and results in holding costs. Each replenishment requires fixed ordering costs.
Then the total relevant costs per unit time can be expressed by c .Q/ D
hQ sc d C : Q 2
Data d Demand per unit time h Unit holding cost per unit time sc Cost for one replenishment order (D setup cost) Variables Q Order quantity
25
See Example 2.6 as an illustration of this issue. See Harris (1913, p. 135). 27 For alternative listings of assumptions underlying the EOQ model, see, e.g., Neumann (1996, p. 28) and Silver et al. (1998, p. 150). 26
2.2 Model Formulations for Master Planning
13
p c .Q/ is a convex function. It takes its minimum with the EOQ Q D 2sc d= h. Due to the restrictive assumptions, direct applications of this model are rather rare in practice.28 In spite of that, this model has proved useful as a basis for analyzing lot-sizing decisions in broader contexts, which include the potential cost impact of deviations of lot sizes from the EOQ29 and supply chain coordination mechanisms.30 The term lot-sizing also means the determination of optimal order quantities, but – in contrast to the EOQ – without the limitation to constant demand.31 The most basic lot-sizing model is the uncapacitated dynamic single-item lot-sizing model developed by Wagner and Within (1958).32 Since the applicability of this model is again rather limited, we state a more relevant extension to several items and several levels of the BOM, the MLULSP (D Multi-Level Uncapacitated Lot-Sizing Problem).33 XX XX min hj Ijt C scj Yjt (2.12) j 2J t 2T
j 2J t 2T
s.t. (2.2), (2.6), (2.8), (2.11) (MLULSP) Ijt 1 C Xjt D djt C Ijt
8j 2 J E ; t 2 T
Xjt bjt Yjt 8j 2 J; t 2 T Yjt 2 f0; 1g 8j 2 J; t 2 T :
(2.13) (2.14) (2.15)
Data bjt Large number, not limiting feasible lot size of item j in period t, PjT j e.g., bjt D Dt dj for j 2 J E ; for j 2 J nJ E , bjt can be calculated P recursively by bjt D k2Sj rjk bkt hj Holding cost for one unit of item j in a period scj Setup cost for a lot of item j Variables Yjt Binary setup variable (D 1, if item j is produced in period t, D 0 otherwise) The objective function (2.12) minimizes the sum of inventory holding and setup costs. Constraints (2.2), (2.6), (2.8), and (2.11) are taken from GM. Constraints (2.13) ensure together with (2.2) the fulfillment of external and secondary demand, respectively. Setup constraints (2.14) enforce variables Yjt to 1 if a lot of item j is produced in period t. Constraints (2.15) define variables Yjt as binary. 28
Note that this only holds for this model in its pure form presented above. For some extensions (e.g., to a multi-level BOM), real-world applications have been reported, see, e.g., Muckstadt and Roundy (1993, p. 61) for an automotive manufacturer and Stadtler (1992, p. 217) for a light alloy foundry. 29 See, e.g., Zangwill (1987, p. 1209) and Stadtler (2007a, p. 407). 30 See Chap. 3 for examples. 31 See, e.g., Silver et al. (1998, p. 198). Note that the optimal quantities for single orders usually differ from each other in case of dynamic demand. 32 See Wagner and Within (1958, p. 89). 33 See, e.g., Domschke et al. (1997, p. 154).
14
2 Supply Chain Planning and Coordination
Combining the MLULSP and GM, we obtain a Multi-Level Capacitated Lot-Sizing Problem (MLCLSP) with backorders and lost sales as a generic Master Planning model that includes decisions related to lot-sizing. This model differs from the original MLCLSP developed by Billington et al. (1983)34 by the negligence of penalties for undertime and by the inclusion of backorders and lost sales. min CMLCLSP D (MLCLSP)
X X m2M t 2T
XX
hj Ijt C
j 2J t 2T
ocm Omt C
X X
j 2J E
XX
scj Yjt C
j 2J t 2T
blcj BLjt C
t 2T
X X j 2J E
lscj LSjt
t 2T
s.t. (2.1)–(2.11), (2.14), (2.15): Variables CMLCLSP Value of the objective function of the MLCLSP 2.2.2.2 Extension to Campaign Planning Campaign planning is a variant of lot-sizing, which raises additional challenges for an efficient mathematical modeling and is of great importance in process industries.35 Analogously to a production lot, a campaign means the production of several units of items without performing any additional setup operation.36 The sizes of these units usually cannot be chosen continuously; due to technical restrictions, e.g., fixed tank or reaction volumes, whole batches, i.e., prespecified amounts of items, have to be produced. Hence, a campaign length corresponds to an integer number of batch sizes.37 For the modeling of campaign planning, the MLCLSP has to be altered in two respects. First, of course, we have to assure that only complete batches are produced. Second, the MLCLSP contains a representation defect, which affects the applicability of this model for campaign planning. The MLCLSP comprises the restrictive assumption that, whenever an item is produced in a period, a setup has to be performed for this item. The setup is required irrespectively whether the resource has already been set up for this item at the end of the preceding period, i.e., the setup state could have been preserved. This representation defect can affect the optimality of the resulting production plans in general.38 For campaign planning, this effect is significantly aggravated, particularly if the production of a single batch requires a considerable share of the available capacity. 34
See Billington et al. (1983, p. 6). For real-world applications and case studies for campaign planning in process industries, see, e.g., Brandenburg and T¨olle (2008), Grunow et al. (2002, p. 281), and Rajaram and Karmarkar (2004, p. 253). 36 See, e.g., Suerie (2005c, p. 2). 37 See Kallrath (2005, p. 341). 38 For an extensive discussion of this issue, see Suerie (2005c, p. 34). 35
2.2 Model Formulations for Master Planning
15
Table 2.2 Comparison of lot-sizing models with the preservation of setup states DLSP CSLP PLSP CLSPL GLSP Maximum number of items per period 1 1 2 Arbitrary Arbitrary Sequence-dependent setups allowed? No No No No Yes
In the literature, several model formulations have been developed that overcome the above mentioned representation defect by allowing lot sizes that overlap period boundaries.39 These models differ by their scope and their computational complexity. Table 2.2 provides a comparison of five basic models with this property, the DLSP (Discrete Lot-sizing and Scheduling Problem),40 the CSLP (Continuous Setup Lot-sizing Problem),41 the PLSP (Proportional Lot-sizing and Scheduling Problem),42 the GLSP (General Lot-sizing and Scheduling Problem),43 and the CLSPL (Capacitated Lot-Sizing Problem with Linked lot sizes).44 The scopes of these models differ by the maximum number of items that can be produced per time period45 and by the question whether sequence-dependent setups can be modeled.46 We choose the CLSPL as the basis for evaluating the coordination schemes proposed. The main reason for this is that the CLSPL allows the production of an arbitrary number of items per period. In anticipation to Sect. 2.3, we want to point out that coordination becomes most relevant if several items are ordered and if decentralized parties have little leeway for adapting their production plans (e.g., due to tight capacities and elevated costs for shortages). In such situation, parties would hardly confine themselves to models which artificially restrict the production to one or two items per period. Potential modest increases in inventory holding and setup costs with the CLSPL are of secondary relevance then. Such increases may be caused by the greater computational complexity of the CLSPL, which usually results in larger optimality gaps compared to the DLSP, CSLP, and PLSP, provided that a limit on the solution time is applied.47 Sequence-dependent setups, in turn, 39
For comprehensive surveys of these models see, e.g., Drexl and Kimms (1997, p. 221) and Jans and Degraeve (2008, p. 1619). 40 See Fleischmann (1990, p. 338). 41 See Karmarkar and Schrage (1985, p. 328). 42 See Drexl and Haase (1995, p. 75). 43 See Fleischmann and Meyr (1997, p. 12). 44 See Dillenberger et al. (1993, p. 112). More recently, this model has been investigated by Gopalakrishnan et al. (2001, p. 851) and Suerie and Stadtler (2003, p. 1039). 45 Note that this classification – although sufficient for our subsequent argumentation – is too rough to capture the difference between the DLSP and the CSLP, which is the all-or-nothing condition required by the DLSP and relaxed in the CSLP. 46 Note that we state here whether the basic formulations of these models (which have been reported in Drexl and Kimms (1997, p. 221) and (Suerie, 2005a, p. 16)) include sequencedependent setups. This does not always apply to the model type in general; e.g., Fleischmann (1994, p. 397) proposes a variant of the DLSP which allows sequence-dependent setups. 47 See Suerie (2005c, p. 164) for computational results for campaign planning models based on the PLSP and the CLSPL. Regarding the best solutions found, however, the CLSPL can outperform
16
2 Supply Chain Planning and Coordination
are usually not included at the Master Planning level;48 hence, there is no need for a computationally more demanding model like the GLSP that additionally covers this issue. Concerning the restrictions on feasible campaigns, we limit here to single-item campaigns with fixed batch sizes.49 For sake of simplicity, we neither include minimum campaign lengths50 nor batch availability.51 Further discussion deserves the modeling of lead times in the multi-level version of the CLSPL considered here. Lead times of zero, which have implicitly been assumed for the MLCLSP, may cause infeasibility of solutions obtained by multi-level lot-sizing models with the preservation of the setup states. Such infeasibility may arise if a successor item is not produced at the end of a period (i.e., its setup state is not carried over into the next period) and one of its predecessor items is produced at the end of the same period (i.e., its setup state is carried over into the next period). With insufficient inventories of the predecessor item at the beginning of the period, the secondary demand of the successor item might not be fulfilled in time.52 In order to exclude the generation of infeasible solutions, we choose in analogy to Kimms (1996)53 lead times equal to or greater than one period length for intermediate items.54 Below we present the adaptation of the MLCLSP to the preservation of setup states and campaign restrictions. This model extends the single-level formulation proposed by Suerie (2005b)55 to a multi-level BOM structure. min CMLCLSP (MLCLSPL-C) s.t. (2.4)–(2.11), (2.15) LSjt C BLjt C Ijt1 C Xjtb C Xjte D djt C BLjt1 C Ijt 8j 2 J E ; t 2 T
(2.16)
the PLSP (without campaign restrictions) in spite of the larger optimality gaps, see Suerie (2005c, p. 157). 48 This holds, e.g., for Timpe and Kallrath (2001) and Grunow et al. (2003), the papers cited in Sect. 2.1.2 as examples for lot-sizing decisions in Master Planning. 49 The modeling of these campaigns in lot-sizing has been introduced by Kallrath (1999, p. 330) and further been improved by Suerie (2005a, p. 49). 50 For their modeling, see, e.g., Suerie (2005c, p. 95). 51 For the modeling of batch availability, see, e.g., Br¨uggemann and Jahnke (1994, p. 755) for the DLSP and Suerie (2005c, p. 98) for the CLSPL. 52 Note that this problem is only relevant for models with the preservation of setup states and, hence, does not apply to the MLCLSP. 53 See the formulation of the Multi-Level Proportional Lot-sizing and Scheduling Problem (MLPLSP) in Kimms (1996, p. 87). 54 This modeling is only exact if exogenous lead times of this duration occur. For single-machine problems, Stadtler (2008) has proposed an extension of the MLPLSP, which allows the modeling of zero lead times. For multi-machine problems (like the problem considered here), however, we are not aware of any corresponding practicable formulation. 55 Suerie (2005b, p. 102).
2.2 Model Formulations for Master Planning prev Ijt 1 C ijt
17
X b e C Ijt D rjk Xkt C Xkt k2Sj
8j 2 J nJ ; t D 1; : : : ; j E
X
b e Ijt 1 C Xjt j C Xjt j D
b e C Ijt rjk Xkt C Xkt
(2.17)
k2Sj
8j 2 J nJ ; t D j C 1; : : : ; jT j X X b e C amj Xjt C Xjt stj Yjt kmt C Omt E
j 2J
(2.18) 8m 2 M; t 2 T (2.19)
j 2J amj >0
e Xjt bjt Yjt
8j 2 J; t 2 T
(2.20)
b Xjt
bjt Wjt 1 8j 2 J; t 2 T Wjt Yjt C Wjt 1 8j 2 J; t 2 T n fjT jg X Wjt 1 8m 2 M; t 2 T n fjT jg
(2.21) (2.22) (2.23)
j 2J amj >0
Wjt 1 C Wjt Yjt C Ykt 2 8m 2 M; j; k 2 J; k ¤ j; amj ; amk > 0; t 2 T n fjT jg CAM jt CAM jt 1 C
(2.24)
C bjt Yjt
8j 2 J; t 2 T
(2.25)
b CAM jt CAM jt 1 C Xjt bjt Yjt
8j 2 J; t 2 T
(2.26)
b Xjt
CAM j 0 cam0j 8j 2 J e CAM jt Xjt C bjt 1 Yjt e 8j 2 J; t 2 T CAM jt Xjt
(2.27) 8j 2 J; t 2 T
(2.28) (2.29)
b CAM jt 1 C Xjt bsj Y Imt 8m 2 M; j 2 J; amj > 0; t 2 T
(2.30)
D bsj Rjt C Sjt 8j 2 J; t 2 T n f1g CAM jt 1 C Sjt bsj .1 Y Imt / 8m 2 M; j 2 J; amj > 0; t 2 T n f1g
(2.31) (2.32)
Y Imt Yjt 8m 2 M; j 2 J; amj > 0; t 2 T X Y Imt Yjt 8m 2 M; t 2 T
(2.33)
b Xjt
(2.34)
j 2J amj >0
Wj 0 D w0j
8j 2 J
(2.35)
CAM jt 0 8j 2 J; t 2 T [ f0g Rjt 2 N0 8j 2 J; t 2 T
(2.36) (2.37)
Wjt 2 f0; 1g 8j 2 J; t 2 T n fT g Sjt 0 8j 2 J; t 2 T n f1g
(2.38) (2.39)
YI mt 0 8m 2 M; t 2 T:
(2.40)
18
Data bsj cam0j prev ijt stj j w0j
2 Supply Chain Planning and Coordination
Batch size for item j Initial campaign quantity for item j Inflow of item j in period t originating from production in earlier periods Setup time for item j Lead time for item j Initial setup state of item j
Variables CAM jt Campaign variable for item j in period t (quantity of the current campaign up to period t) Integer number of full batches produced in the current campaign of Rjt item j up to period t Quantity of the last batch of item j in period t which is not Sjt finished in t Wjt Setup state indicator variable (D 1 if item j is set up at the end of period t, D 0 otherwise) Xjtb Production quantity of item j at the beginning of period t (i.e., of the first campaign in t) e Xjt Production quantity of item j that is not produced at the beginning of period t (i.e., not part of the first campaign in t) YI mt Setup operation indicator for resource m in period t (D 1 if a setup occurs on resource m in period t, D 0 otherwise) The objective function and some constraints are taken from the MLCLSP. The inventory balance constraints (2.16)–(2.18) now account for nonzero production lead prev times.56 In the left-hand side of constraints (2.16), inflows ijt from earlier production periods have been considered, which are relevant if the MLCLSPL-C is applied with rolling schedules. As a prerequisite for modeling production campaigns, we b have replaced Xjt , the standard variables denoting the production quantities, by Xjt e and Xjt in (2.16)–(2.18). The same change has been applied to constraints (2.19), where setup times have additionally been included.57 The aim of constraints (2.20)–(2.24) is to ensure that in case of production of item j in period t the corresponding resource has been set up for j . Constraints (2.20) force variables Yjt to 1 if item j is produced in period t, but not at the beginning of t.58 The alternative to a setup operation is a setup carry-over from the preceding period. With a setup carry-over for item j , i.e., Wjt 1 D 1, the production of j can 56
Note that we have omitted inventory holding of items during their production lead times. The only relevant effect of these inventories is on the relative profitability of items. This, however, can be easily considered by altering the unit penalty costs for lost sales accordingly. 57 Note that for the validness of these constraints, we implicitly assume that setup times for an item j on a machine m only incur with nonzero variable capacity load for the production of j on m. 58 Note that in order to avoid infeasibilities, the maximum lot size bj has to exceed the batch size bsjt .
2.2 Model Formulations for Master Planning
19
take place at the beginning of period t [see constraints (2.21)]. Constraints (2.22)– (2.24) ensure correct values for Wjt . According to constraints (2.22), Wjt can only take 1 if a setup for item j has been performed in period t or if the corresponding resource has already been set up for j at the end of t 1. Moreover, each resource can be set up at most for one item at period boundaries [see constraints (2.23)].59 Finally, we have to exclude a positive setup state for item j both at the beginning and the end of a period if another item k is produced in this period and no setup has been performed for j [see (2.24)]. The properties of feasible campaigns are defined by constraints (2.25)–(2.32). For that purpose, we have introduced variables CAMjt indicating the quantity of the campaign of item j that is finished in the period where, relative to t, the last setup has been performed for j (i.e., in one of the periods t, t 1, . . . ).60 Constraints (2.25) and (2.26) fix the campaign quantities of item j in period t to those of t 1 augmented by the production amount of j at the beginning of t, given that there has been no setup for j in t. Otherwise, these constraints are inactive. The campaign quantities are initialized by the batch size in (2.27).61 Further bounds on the campaign quantities are provided by (2.28)–(2.30). Constraints (2.28) ensure that the campaign quantity of item j at the end of t must not exceed the quantity of j produced after a setup of j in t. A lower bound for the campaign quantities is set by (2.29). Constraints (2.30) ensure that at least the batch size is produced in each campaign. Moreover, constraints (2.31) and (2.32) restrict the campaign length to an integer number of batches. Constraints (2.31) identify the quantity of item j in period t within an unfinished batch (i.e., the production quantity within an unfinished campaign that exceeds an integer multiple of bsj ) and assign this value to variable Sjt . Constraints (2.32), in turn, imply the end of the current campaign (i.e., Sjt D 0) if there has been a setup on the resource on which j is produced. The next sets of constraints, (2.33) and (2.34), fix the values for Y Imt . Due to (2.33), variable Y Imt has to take a value greater than or equal to 1 if resource m has been set up in period t.62 Furthermore, Y Imt is set to zero if no setup occurs on m in t (2.34).
59 Motivated by the real-world test data of Sect. 6.5, we have formulated this constraint differently to Suerie (2005c, p. 102), where a setup state of 1 is strictly required for all periods T n fjT jg. A setup state of 0 in periods T n fjT jg can be superior to a setup state of 1 if setups in these periods lead to suboptimal (e.g., due to zero demand in early periods) or even infeasible solutions (e.g., if the available capacity in period 1 is smaller than the corresponding setup time, e.g., due to prescheduled maintenance). 60 Note that variables CAM jt therefore do not necessarily reflect the actual campaign quantity of item j in the current period t ; i.e., CAM jt may take positive values although j is not produced in t . This information, however, is sufficient for modeling purposes here. 61 Again we have chosen a formulation that differs somewhat from that of Suerie (2005c, p. 95). Suerie (2005c) has initialized CAM j 0 by 0 if Yjt D 1, which, however, leads to wrong results if the minimum batch size exceeds one period length. Then, constraints (2.30) would require that the complete batch quantity is produced within a single period if there has not been any setup carryover into this period [which holds, e.g., in period 1 due to constraints (2.35)]. This may result in infeasible plans or excessive overtime costs. 62 Note that due to constraints (2.32), variables YI mt cannot exceed 1.
20
2 Supply Chain Planning and Coordination
Variables Wjt are initialized by (2.35). The remaining constraints (2.36)–(2.40) define decision variables as nonnegative, binary, and integer, respectively.
2.3 Decentralized Planning and Coordination In this section, we shift our focus from centralized to decentralized supply chain planning. We analyze the inherent lack of coordination in decentralized planning, point out the resulting drawbacks, and motivate the need for coordination, specifically collaborative planning. We start with some basic definitions which are related to coordination and used throughout this work.
2.3.1 Basic Definitions Supply Chain Coordination According to Horv´ath (2001), the term coordination is among the most “dazzling” ones of business economics.63 First, we consider the meaning of coordination within organizational theory, a classical field of economic research, where coordination is of crucial importance.64 There, the need for coordination is a direct consequence of the division of labor, which creates single activities with interdependencies among them.65 Thus, coordination can be regarded as complementary to the division of labor: The re-adjustment of these single activities in order to reach superordinate aims.66 The essence of this definition is commonly accepted in the literature,67 although some authors prefer application-oriented definitions, such as: “Coordination is the meshing and balancing of all factors of production or service and of all the departments and business functions so that the company can meet its objectives.”68 With respect to supply chains, the definitions of coordination in the literature are somewhat more concrete than those mentioned above, but differ substantially from each other. A first stream of literature defines a supply chain as coordinated if and only if actions leading to the supply chain optimum are implemented.69 A weaker requirement for the actions implemented is an improvement for the whole supply chain compared to the default solution, i.e., the solution that would have 63
Horv´ath (2001, p. 113). See, e.g., Horv´ath (2001, p. 113). 65 See, e.g., Laux and Liermann (1993, p. 7). 66 See, e.g., Frese (1975, column 2263). 67 See, e.g., Hansmann (2001, p. 255), Horv´ath (2001, p. 114), and Kieser and Walgenbach (2003, p. 101). 68 Horngren et al. (2006, p. 180) and similar Bhatnagar et al. (1993, p. 142). 69 See, e.g., Cachon (2003, p. 230), who limits the set of contracts that coordinate supply chains to those where “the set of supply chain optimal actions is a Nash equilibrium, i.e., no firm has a profitable unilateral deviation from the set of supply chain optimal actions.” 64
2.3 Decentralized Planning and Coordination
21
been implemented without coordination. This definition is implicitly supported by Corbett and de Groote (2000), who devise a menu of contracts which results in systemwide costs that are equal to or lower than those of the default solution, but not necessarily equal to those of the systemwide optimum, and call the output “coordinated.”70 Note that the usual default solution in supply chains is characterized by double marginalization, where the more powerful parties decide on actions with impact for the whole supply chain and thereby consider their own margins only.71 A third alternative is to regard any systemwide feasible solution as coordinated, which seems to be favored by Schneeweiß.72 We adopt a variant of the alternative secondly mentioned and define that: actions coordinate a supply chain if the resulting systemwide profit is greater than that in the default solution.
Our reasoning for this is as follows: The identification of plans which assuredly lead to the exact systemwide optimum is rather an exception in the context of supply chain planning, especially if information asymmetries are present. Information asymmetry means that (at least) one party has information that is relevant for decisions concerning all parties but not known by at least one other party.73 Hence, it would seem misleading to classify the vast majority of approaches which only lead to improvements as non-coordinating. Therefore, we regard the implementation of the optimal actions as a special case of coordination (“optimal coordination”). Note that also in organizational theory,74 usually no emphasis is placed on the question whether the readjustment of the single actions does lead to the global optimum. The third alternative, i.e., only to require feasibility for coordination, seems least appropriate since it completely ignores aspects of solution quality. Coordination Scheme, Coordination Mechanism Next, we introduce two further terms that are closely related to the establishment of coordination and of central importance in this work.75 We begin with the definition of a coordination scheme: A coordination scheme is a set of rules specifying actions whose implementation by decentralized parties potentially coordinates a system.
70
See Corbett and de Groote (2000, p. 449); for a detailed description of this approach, see Sect. 3.2.2. 71 This phenomenon has been recognized first by Spengler (1950, p. 347), much earlier than corresponding advances in the field of supply chain management. 72 “Worst-case coordination,” see Schneeweiss (2003, p. 278). 73 Explicit definitions of information asymmetry are rare in the literature. See Schneeweiss (2003, p. 219) for a notion of this term similar to ours. 74 See the literature cited above. 75 Here, we limit to the definitions of these terms. For examples refer to Chap. 3. Note that in the following, we will abbreviate “coordination mechanism” by “mechanism” and “coordination scheme” by “scheme” if the meaning of these terms is clear from the context.
22
2 Supply Chain Planning and Coordination
Note that we do not require here that the application of a coordination scheme always leads to the coordination of a system, which would reduce the scope of this term considerably. Furthermore, a coordination scheme does not necessarily incentives for decentralized parties to follow its rules, i.e., to implement the actions proposed. The definition of a coordination mechanism goes a step further in this respect. It is built on the definition of a mechanism in the field of mechanism design,76 where a mechanism constitutes a framework that specifies the outcomes (e.g., surplus sharing) of decentralized parties depending on the actions undertaken by them (e.g., their information disclosed):77 A coordination mechanism is a mechanism for which the implementation of the optimal strategies by decentralized, self-interested parties may lead to a coordinated outcome and neither violates the individual rationality of the participating parties nor the budget balance of the system.
Hence, we regard a coordination mechanism as a specific mechanism requiring a potential coordinated outcome and two basic properties in mechanism design, individual rationality and budget balance.78 (Ex post) individual rationality requires that no party is worse off by participating in the mechanism, i.e., that the profits of all participating parties are at least equal to their profits achieved in the default solution. Budget balance means that the payments of parties in the mechanism sum up to zero, i.e., that the mechanism does not require an outside subsidy.79 Moreover, note that a coordination mechanism extends the tasks of a coordination scheme; hence, as in the work, a scheme is often embedded in a mechanism.
Collaborative Supply Chain Planning The term “collaborative planning” is generally known as a part (and sometimes even regarded as a synonym80) of the business practice “Collaborative Planning, Forecasting and Replenishment.” As a formalized process, CPFR has been worked out by the standardization committee VICS (Voluntary Interindustry Commerce Standards) and implemented within over 300 companies.81 The CPFR process model consists of eight planning tasks, which can be subsumed under four main activities: Strategy and planning, demand and supply management, execution, and analysis.82 76
Mechanism design is an area of game theory concerned with the aggregation of unobservable individual preferences into a collective decision, see, e.g., Mas-Colell et al. (1995, p. 857). 77 Somewhat more technically, in Jackson (2003, p. 2), a mechanism is defined as a “specification of a message space for each individual and an outcome function that maps vectors of messages into social decisions and transfers.” 78 See, e.g., Chu and Shen (2006, p. 1215), who argue that these properties are necessary for mechanisms to be practicable. 79 See, e.g., Chu and Shen (2006, p. 1215). 80 See, e.g., Li (2007, p. 159). 81 See VICS (2008). 82 See VICS (2008).
2.3 Decentralized Planning and Coordination
23
“Planning” in this context does not refer to the alignment of operational plans, but to the identification and communication of events which may affect demand, such as promotional activities or product introductions. Whereas collaboration is restricted to mere information exchange in the original process model, some authors broaden the scope of this term to decision-making. Raghunathan (1999) explicitly includes production scheduling within CPFR.83 Danese (2005) mentions a concept called “limited CPFR collaboration,” where plans are jointly synchronized by the partners (e.g., the replenishment between a central company and a distribution center).84 Akkermans et al. (2004) describe a business process called “collaborative planning,” where companies “jointly take decisions regarding production and shipments for a large part of their collective supply chains.”85 This notion of collaboration is also included in the definition proposed here: Collaborative supply chain planning is a joint decision making process for the alignment of plans of independent, legally separated supply chain parties.
This definition is closely related to that of Stadtler (2009), except for two minor differences. First, we have provided a definition for the term “collaborative supply chain planning” instead of “collaborative planning” in order to avoid potential contradictions to other fields of science dealing with planning for objects different to supply chains. However, since we are only concerned with supply chains in this work, we will use the terms “collaborative supply chain planning” and “collaborative planning” as synonyms in the following. Second, we do not require information asymmetry. In fact, information asymmetry is one of the main reasons for the application of collaborative planning. However, it seems reasonable not to exclude important developments in the field of supply chain coordination that cover uncertain demand or issues of coalition forming, but no asymmetric information.86 Finally, it is important to note that a coordination of supply chains made up by legally separated parties cannot be achieved without collaboration. Such supply chains do not comprise a central entity which is entitled to determine the actions of the decentralized parties directly or by means of incentive schemes.87 As a
83
See Raghunathan (1999, p. 1054); note that the terminology CFAR instead of CPFR is used there. 84 See Danese (2005, p. 458). 85 Akkermans et al. (2004, p. 445). 86 For a survey of these approaches, see Sect. 3.1. 87 Mechanisms relying on such entities have been proposed by, e.g., Lee and Whang (1999, p. 633), Pfeiffer (1999, p. 319), Bukchin and Hanany (2007, p. 273), and Kutanoglu and Wu (2006, p. 421). However, these approaches do not ensure the individual rationality of parties. This is feasible if a central entity can entirely determine the costs allocated to the decentralized parties, but seems problematic for general independent parties, who want to improve their gains compared to a default solution. Note in this context that we do not categorically exclude the participation of a central entity in collaborative planning. We only exclude central entities that influence the default solution and thus the individual rationality of parties, but not those that act as mediators and only determine the allocation of the systemwide surplus from coordination (see Sect. 3.3 for examples).
24
2 Supply Chain Planning and Coordination
consequence, we will also use the terms “collaborative (supply chain) planning” and “coordination mechanism for supply chain planning” as synonyms in this work.
2.3.2 Decentralized Supply Chain Planning In this section, we analyze the impact of decentralization on supply chain planning. We argue that decentralized planning may result in suboptimality for a supply chain as a whole and identify the major drivers for that. For ease of exposition, we focus on supply chains consisting of one buyer and one supplier that run the Master Planning models presented in Sect. 2.2. Thereby, we adopt the convention to consider the buyer as female (“she”) and the supplier as male (“he”).88 Note that most of our analysis is equally relevant for more general settings, i.e., supply chains with more than two parties and other planning levels.
2.3.2.1 Models for Decentralized Planning Due to the increasing concentration on core competencies and subsequent outsourcing activities, many supply chains are not run by a single enterprise, but by several, legally separated business units. Under these conditions, an integrated planning for a supply chain as a whole might fail. A major reason for this is information asymmetry: Usually, none of the decentralized parties has the knowledge about all data required for integrated planning. Sharing local production data among decentralized parties, however, is problematic.89 Private data may be sensitive (especially capacity data90 ) and constitute a strategic advantage for bargaining, which is lost after revelation. Although this problem is fundamental and matters in most real-world supply chains, current APS do not offer any convincing solution for it. In fact, APS often include tools for collaboration in their planning suites;91 however, these tools only facilitate the exchange of information, but do not account for the reluctance of decentralized parties to interchange detailed (production) data. Hence, instead of solving the centralized model, a decentralized party can only determine plans that are valid for its own planning domain, i.e., the “part of the supply chain and the related planning processes that is under the control and in the responsibility of one planning organization.”92 For developing mathematical models 88 This convention is also used within the supply chain contracting literature, e.g., Cachon (2003, p. 230). 89 This has also been pointed out by , e.g., Arikapuram and Veeramani (2004, p. 111). 90 As an indicator for the reluctance of practitioners for exchanging production capacities, see the empirical study of Kersten (2003, p. 332). 91 E.g., SAP (2008) and Oracle (2008). 92 Kilger et al. (2007, p. 263).
2.3 Decentralized Planning and Coordination
25
for decentralized planning, consider the interdependencies between the decentralized models that are made up by the inventory balance constraints for the items supplied:93 Ijt 1 C Xjt D
X
rjk Xkt C Ijt
8j 2 J D ; t 2 T:
(2.41)
8j 2 J D ; t 2 T;
(2.42)
ISjt1 C Xjt D XSjt C ISjt
8j 2 J D ; t 2 T;
(2.43)
Ijt D ISjt C IBjt
D
8j 2 J ; t 2 T ;
(2.44)
XBjt D XSjt
D
8j 2 J ; t 2 T ;
(2.45)
IBjt 0
D
8j 2 J ; t 2 T;
(2.46)
ISjt 0
D
8j 2 J ; t 2 T;
(2.47)
XBjt 0
8j 2 J ; t 2 T;
(2.48)
XSjt 0 8j 2 J ; t 2 T:
(2.49)
k2Sj \J B
Sets J B Set of items produced by the buyer J D Set of items supplied These constraints can be reformulated as X IBjt1 C XBjt D rjk Xkt C IBjt k2Sj
\J B
D
D
Variables IBjt Inventory of the (supplied) item j at the buyer’s site in period t; IBj 0 D 0 ISjt Inventory of item j at the supplier’s site in period t; ISj 0 D 0 XBjt Amount of item j delivered to the buyer in period t XSjt Amount of item j delivered by the supplier in period t (2.42) and (2.43) are inventory balance constraints for the decentralized models. Constraints (2.44) are necessary for the correct computation of the systemwide costs for inventory holding. Constraints (2.45) exclusively comprise the interdependent decisions and link the decentralized models. Constraints (2.46)–(2.49) ensure the nonnegativity of the new variables IBjt ; ISjt ; XBjt , and XSjt . The formulations for the decentralized models of buyer and supplier can be derived from a given centralized model by limiting the items and resources modeled to those owned by buyer and supplier and augmenting these models by (2.42) and (2.43), respectively. We denote the decentralized versions of the models of buyer and supplier by adding “S” and “B,” respectively, to the model names. Throughout this dissertation, we abbreviate “buyer” by “B” and “supplier” by “S.”
93
With “items supplied,” we mean the subset of items that may be ordered by the buyer (and subsequently supplied by the supplier)
26
2 Supply Chain Planning and Coordination
We take GM as the base model in the following. Since (2.45) comprise decisions of both planning domains, a direct inclusion of these constraints is neither possible in GM-S nor in GM-B. Instead, for modeling decentralized planning, targets for the supply quantities have to be incorporated into the decentralized models. I.e., GM-B has to be augmented by xtjt D XBjt
8j 2 J D ; t 2 T
(2.50)
xtjt D XSjt
8j 2 J D ; t 2 T:
(2.51)
and GM-S by Data xtjt Target for the supply quantity of item j in period t Without the knowledge of the solution to the centralized model, the target quantities leading to the supply chain optimum can only be determined by chance. In practice, myopic procedures for (unilaterally) determining these targets are employed. Before dealing with these procedures in Sect. 2.3.3, we will point out the major reasons why such myopic procedures often fall short.
2.3.2.2 Drivers for Suboptimality of Decentralized Planning We have identified a couple of major drivers for systemwide suboptimality due to inappropriate targets for the supply quantities, thereby focusing on the Master Planning models presented in Sect. 2.2. These drivers correspond to the characteristics of these targets that cause elevated costs for one decentralized party and, thus, potentially for the whole supply chain. The knowledge of these drivers is useful for both characterizing settings with a need for coordination and classifying collaborative planning approaches; in fact, all approaches mentioned in the literature review of Chap. 3 can be classified according to the drivers identified here. In Table 2.3, we provide an overview of the drivers and state their effects on the parties’ costs. Three main drivers can be distinguished. The category “time” refers to a temporal over- or undersupply, “lot cycles” to a misalignment of the lot cycles in the supply target, and “quantity” to inappropriate choices of the absolute quantities ordered within the planning interval. In the following, we will explain in more detail how these drivers can lead to increases in the supplier’s costs. Regarding the driver T, two main effects can be distinguished. First, the target supply quantities in early periods of the planning horizon may be too large to be
Table 2.3 Drivers for increases in the supplier’s costs Driver Impact on parties’ costs Time (T) Lot cycles (L) Quantity (Q)
Increase of overtime, setup, holding, and backorder costs Increase of holding, setup costs Increase of costs for overtime, lost sales
2.3 Decentralized Planning and Coordination
27
covered by the supplier’s stock and production with normal capacity. In order to fulfill this target, the supplier has to employ costly overtime (see Example 2.1). If the supplier’s costs for early supply exceed the costs of the buyer for a potential delay, suboptimality for the whole supply chain results. Example 2.1 Consider the supply target for an item in the left of Fig. 2.4. Assume zero initial inventories. Since the possible build-up of stock of this item in period 1 is smaller than the excess supply in period 2, the use of (expensive) overtime is necessary for the supplier. Second, the contrary may happen, i.e., the supply may take place too late from the supplier’s point of view. Then, elevated holding costs for the supplier may incur if the supplier has to prepone part of his production due to, e.g., restricted production capacity in a later period or scale effects of lot-sizing. The systemwide costs will increase, too, provided that these additional holding costs exceed the cost increase for the buyer due for an earlier order (see Example 2.2). Example 2.2 In the right of Fig. 2.4, the supply target for item 3 is depicted. Since the supplier’s capacity is partially reserved in period 2, a part of the production of item 3 has to take place in period 1. Let there be end items 1 and 2 with equal holding costs and demand. Let each of these end items require one period length and item 1 one unit (U) of the predecessor item 3 for production. Then the buyer is indifferent which item to produce first, and his choice, i.e., producing item 1 in period 2 (instead of period 1), increases systemwide costs. The driver L has two manifestations, too. On the one hand, considering single items, their TBO (D time between orders) may be inappropriate. If an item is ordered too often, the additional costs for supplier’s setups or inventory holding may cause systemwide suboptimality (see the following example). Example 2.3 Consider a serial supply chain with one end item and one item supplied. For the production of 1[U] of the end item, 1[U] of the item supplied is required. Assume six periods with level end item demand of 1[U] per unit time (UT=1 period), holding costs of 1 monetary unit (MU) per unit and unit time, and setup costs of 2[MU] for both parties. Let the TBO for the supply target be 2[UT] (see Fig. 2.5), as preferred in the local solution of the buyer. This target leads to
Fig. 2.4 Example for early and late supply
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2 Supply Chain Planning and Coordination
Fig. 2.5 Example for inappropriate TBO
Fig. 2.6 Example for misalignment of lot sizes
systemwide costs of 15[MU] (composed by setup costs of 6[MU] for buyer and supplier as well as holding costs of 3[MU] for the buyer), which exceed 14[MU], the systemwide optimal costs for a TBO of 3[UT]. On the other hand, a misalignment of lot cycles among successor items may further increase the supplier’s costs (see Example 2.4). Example 2.4 Consider Fig. 2.6. Assume equal demand for all end items (1,2,3) at the end of period 2 and zero initial inventories. Let the production capacity in period 2 be half the capacity in period 1 and setup and inventory holding costs be equal for all end items. Assume that the supply target is one of the buyer’s locally optimal plans where items 1 and 3 are produced in period 1 and item 2 in period 2. This plan involves a positive secondary demand for items 4 and 5 in period 1 and for item 4 in period 2. Due to restricted production capacities, the supplier has to set up item 4 twice during the planning horizon to fulfill this demand. The systemwide optimal order, in contrast, goes along with the production of items 1 and 2 in period 1 and the production of item 3 in period 2. This plan involves the same costs for the buyer, but lower costs for the supplier. Since item 4 is exclusively demanded in period 1, the supplier can limit to one setup of item 4 and thus reduce his setup costs. The last driver for elevated costs of the supplier is the quantity, i.e., inappropriate relations among the absolute quantities ordered. First, consider one single item supplied. Here, too large target quantities are suboptimal if the resulting overtime of the
2.3 Decentralized Planning and Coordination
29
Fig. 2.7 Example for production program as driver
supplier is more expensive than the buyer’s lost sales that would occur for smaller supply quantities. Second, for several interdependent items, the relation of their target quantities may be inappropriate as illustrated by the subsequent example. Example 2.5 Consider one period and two items supplied. Assume that the buyer produces two items (1,2) with contribution margins of 4[MU] and 3[MU], respectively. Let the maximum demand for these items be 5[U]. The capacity of the buyer’s resource is 5 capacity units (CU). Both items 1 and 2 require 1[CU/U] of this resource for their production as well as 1[U] of their predecessor items 3,4 (see Fig. 2.7). Items 3 and 4 are produced on a resource with a capacity of 10[CU]. The production of item 3 requires 4[CU/U] on this resource, and the production of item 4 1[CU/U]. The capacity of this resource can be extended by overtime at costs of 1[MU/U]. Consider a supply target of 5[U] of item 3 and 0[U] of item 4. Although maximizing the buyer’s profits, this target is not optimal for the whole supply chain (systemwide profits: 4 5 10 D 10 [MU]). Due to the elevated overtime costs for the supplier, the production of 1.25[U] of item 3 and 5[U] of item 4 is more profitable instead (4 1:25 C 3 3 D 14[MU]). For the buyer, analogous drivers can be identified, with the difference that early and late supply have contrary consequences here and that backorders and lost sales may become necessary apart from overtime. Note that often combinations of these drivers occur, e.g., for the MLCLSP, which comprises capacity restrictions and multi-level lot-sizing. In the presence of specific model characteristics, the impact of the drivers can be considerably aggravated (see Example 2.6). Example 2.6 Assume a BOM with two end items which require each 1[U] of an intermediate item for production (see Fig. 2.8a). Let the target supply quantities be equal to the secondary demand for the items supplied (Fig. 2.8b). An implementation of this target avoids overtime at a buyer planning based on the MLCLSP94 (Fig. 2.8c), an insight that proved useful for the design of coordination schemes.95 With the presence of large setup times, however, overtime at the buyer may be required if the capacity consumption of a setup for item 2 is greater than the available
94 95
See Sect. 2.2.2 for this model. See the scheme of Dudek and Stadtler (2005, p. 673) and that proposed in Chap. 4.
30
2 Supply Chain Planning and Coordination A
C
B
D
E
Fig. 2.8 Aggravation of the suboptimality of decentralized planning with setup times and campaign restrictions
slack capacity (Fig. 2.8d). Even worse may be the consequences of minimum campaign quantities (Fig. 2.8e). Here, the campaign for item 1 cannot be started since the supply of item 3 is smaller than the minimum campaign quantity.
2.3.3 Upstream Vs. Collaborative Planning A common procedure for determining targets for supply quantities is upstream planning, i.e., sequential planning starting with the locally optimal plan for the downstream party (here: the buyer). Then this party derives an order for the raw materials required to fulfill her plan and communicates this order to the upstream party (here: the supplier).96 Depending on the supplier’s leeway for order fulfillment, two cases can be distinguished, full and voluntary compliance by the supplier. Forced compliance means that the supplier fulfills the buyer’s order without any changes. In the literature, this assumption underlies simulation experiments for evaluating the systemwide suboptimality due to decentralization97 as well as coordination schemes.98 As a motivation for the forced compliance case, refer to the use of
96
In supply chains with more than two tiers, the upstream party would act as an upstream party again and communicate his order to the party at the next tier upstream, and so on. 97 See Simpson and Erenguc (2001, p. 119) and Simpson (2007, p. 127). Note that in these papers, other terminologies (“pull style planning,” “local planning”) have been employed instead of “upstream planning.” 98 See, e.g., Dudek and Stadtler (2005, p. 668). There, the term “upstream planning” has been introduced.
2.3 Decentralized Planning and Coordination
31
Fig. 2.9 Upstream planning with forced compliance by the supplier
quantity flexibility contracts in practice. There, the supplier has to fulfill the buyer’s order completely if the order quantities are kept within a prespecified corridor.99 We illustrate upstream planning with forced compliance by an example (see also Fig. 2.9). Example 2.7 In order to fulfill her end item demand (1), the buyer determines the production plan that is optimal for her planning domain (2). She prefers to produce the demand of period 3 already in period 2 since the associated setup cost reduction exceeds the additional inventory holding costs. Next, the buyer derives via BOMexplosion the amounts of the intermediate items which are necessary to fulfill her production plan and places orders for these intermediate items (3). Finally, the supplier determines a production plan for his planning domain with the restriction of fulfilling the buyer’s orders (4). Note that the production of the supplier may differ from the order schedule. Here, the supplier has to prepone a part of production because of tight capacities. The mathematical modeling of this planning process is as follows: First, the buyer solves GM-B, thereby assuming an unrestricted delivery of the items supplied. Then she communicates to the supplier the supply quantities xtjt , which are determined by X xtjt D rjk Xkt : k2Sj \J B
Next, the supplier solves GM-S augmented by (2.51). An obvious drawback of this procedure is that the supply target is unilaterally determined by the buyer. This renders the resulting solution vulnerable for the drivers for suboptimality affecting the supplier’s costs.100 For uncapacitated lot-sizing problems, the resulting suboptimality has been evaluated by simulation studies. For different parameter settings, average costs increases of 11.5% and 4% compared to upstream planning have been reported.101 If capacity restrictions come into play, greater suboptimalities can be expected since the overtime costs caused by
99
See Tsay (1999, p. 1340) for examples for quantity flexibility contracts in industry. See Table 2.3 on p. 26. 101 See Simpson and Erenguc (2001, p. 123) and Simpson (2007, p. 133). 100
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2 Supply Chain Planning and Coordination
Fig. 2.10 Upstream planning with voluntary compliance by the supplier
inappropriate orders are usually of a higher magnitude than the costs for inventory holding and setups.102 Usually, this suboptimality increases in case small safety stocks are held by the supplier. Although forced compliance allows a straightforward determination of the target supply quantities, this assumption cannot always be sustained. Apart from different contractual agreements between buyer and supplier that concede the supplier some (or even infinite) leeway for shortages,103 the supplier simply might not be able to fulfill the buyer’s order – even if he is willing to do so. This occurs, for example, if the supplier has scarce production capacities and is not able to extend them.104 Therefore, we further consider voluntary compliance, which means that the supplier is not obliged to exactly fulfill the buyer’s order, but can freely choose the extent of his order fulfillment. Below, we provide an example for this (Fig. 2.10). Example 2.8 Analogously to Example 2.7, the buyer combines the demands of periods 1 and 2 (1) in her production plan (2) in order to save setup costs and places a corresponding order (3). Here the supplier cannot fulfill this order completely. His optimal production plan (4) results in a reduced supply in period 1 (5). The order fulfillment by the supplier determines the initial, uncoordinated solution. Here, this solution is feasible for the whole supply chain, provided that the resulting lost sales are feasible for the buyer. Again, we provide the mathematical modeling for this procedure. First, the buyer solves GM-B and communicates the supply quantities xtjt to the supplier. Next, the supplier determines the extent he is willing to fulfill the buyer’s order, thereby taking into account penalties for shortages and lost sales. For this purpose, he solves GM-S with the modified objective function 102 Some indications on the resulting suboptimality in capacitated settings provide our computational tests in Chap. 6. 103 As an example for such agreements, consider the contractual practice in the video rental industry, see, e.g., Cachon and Lariviere (2001b, p. 20). 104 This may be due, e.g., to technical constraints on the production process such as batch production (see Sect. 2.2.2.2). There, the batch sizes (e.g., volumes of tanks) might not be extendable in medium term.
2.3 Decentralized Planning and Coordination
X X j 2J S t 2T
hj Ijt C
X X
ocm Omt C
m2M S t 2T
33
X X
X X
blcj BLjt C
j 2J D t 2T
lscj LSjt ;
j 2J D t 2T
(2.52) S
with M as the set of resources of the supplier and modified inventory balances for the items supplied: Ijt 1 C Xjt C BLjt C LSjt D BLjt 1 C Ijt C xtjt
8j 2 J D ; t 2 T: (2.53)
Finally, the buyer solves GM-B augmented by 0 D xtjt
X k2Sj
rjk Xkt
8j 2 J D ; t 2 T;
\J B
0 derived from the outcome of the supplier’s with modified supply quantities xtjt model: 0 xtjt D xtjt BLjt LSjt C BLjt 1 :
Like in the forced compliance case, systemwide suboptimality may result from inappropriate orders of the buyer. Moreover, inappropriate shortages by the supplier may trigger elevated costs for the buyer. A further potential drawback of this procedure is the missing guarantee for feasibility in the presence of particular restrictions such as shelf-life or storage capacity restrictions (see Example 2.9). Example 2.9 Assume that the supplier delivers two intermediate items that are assembled by the buyer to an end item. Given zero initial inventories, the buyer will order these items in an equal relation, of course. Assume that the supplier can only fulfill the order of one of these items due to restrictions on his production capacities. Since the buyer cannot perform the assembly then, she would have to keep the supplier’s delivery in stock. If her storage capacities are too scarce, then the supplier’s delivery might be infeasible for her. Simply choosing a lower supply than offered by the supplier, however, does not necessarily establish systemwide feasibility if the supplier’s storage capacity is restricted, too. Summarizing, upstream planning may lead to suboptimal plans and even bears the risk of generating infeasible solutions. In this work, we present a remedy for this: Collaborative planning. We propose that decentralized parties do not settle for the solution from upstream planning, but collaborate in order to identify a feasible supply target with costs coming near to the systemwide optimum. While upstream planning is based on unilateral targets, collaborative planning takes the preferences of all decentralized parties into account. It aims to combine the advantages of centralized and decentralized planning, i.e., to reach an appropriate solution quality without exchanging detailed information about the decentralized parties’ production processes. The collaborative planning mechanisms proposed in this work operate as add-ons to improve a default solution, which, e.g., may result from upstream planning. They try to identify an improvement by an iterative
34
2 Supply Chain Planning and Coordination
Fig. 2.11 Collaborative planning
exchange of several supply proposals, with the chance that one of these proposals is superior to the target determined by upstream planning. We illustrate this process by an example (see also Fig. 2.11). Example 2.10 Consider upstream planning with forced compliance. The buyer’s demand (1) and purchase order (2) are taken from Example 2.7. Now, apart from upstream planning (2), parties iteratively interchange supply proposals in order to identify an improved solution for the whole supply chain. Here the supplier proposes a more equal temporal distribution of the supply quantities to increase his capacity utilization (3), and the buyer adapts this proposal by aggregating the supply of periods 3 and 4 (4). Steps (5),(6), : : : stand for further proposals generated, which are not depicted in Fig. 2.11 for ease of exposition. The main challenges for carrying out such a collaborative planning process are the identification of potentially coordinating proposals and the assurance that the coordination process is supported by the incentives of self-interested decentralized parties. Both issues are tackled by the new mechanisms proposed in Chaps. 4 and 5.
Chapter 3
Coordination Mechanisms for Supply Chain Planning
In recent years, a large number of papers have been produced that propose and analyze mechanisms for supply chain coordination. The number of existing surveys about the literature of this area is considerable, too.1 These surveys, however, are not exhaustive; surprisingly few emphasis has been placed on a central topic for the design of coordination mechanisms, the determination of appropriate incentives for decentralized parties in light of information asymmetry.2 In this chapter, we provide a literature review, which comprises new classifications as well as explanations of central ideas behind different types of mechanisms. We focus on mechanisms that are directly applicable or transferable to the coordination of Master Planning. In addition, we include approaches that address operational planning (e.g., scheduling) and provide interesting, novel ideas for the design of coordination mechanisms, but exclude those dealing with strategic planning tasks.3
3.1 Symmetric Information In this section, we review coordination mechanisms which require that at least one party has access to the information that is necessary for solving the centralized planning models. In principle, coordination in such settings could be achieved easily: The party disposing of all relevant data could determine the optimal plan for the whole system and offer the resulting supply target to the other parties together with some share of the surplus from coordination. This approach, however, is not regarded satisfactory in the literature; the largest body on supply chain coordination mechanisms has been developed for settings with complete information. 1 See, e.g., Whang (1995, p. 413), Fugate et al. (2006, p. 129), Cachon and Netessine (2004, p. 1), and the references provided below. 2 One exception is the paper by Cachon and Netessine (2004, p. 13), which, however, completely omits the literature on bilateral information asymmetry provided in Sect. 3.3. 3 See, e.g., Van Mieghem (1999, p. 954), B¨ockem and Schiller (2004, p. 219), Plambeck and Taylor (2007, p. 1872), and Ha and Tong (2008, p. 701).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 3,
35
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3 Coordination Mechanisms for Supply Chain Planning
First, this approach may fall short if multiple parties are to be coordinated. Then, parties may not be willing to follow the advice of a central planner if some of them can form subcoalitions which yield greater benefits than those to be allocated by the central planner. For the design of coordination mechanisms in this setting, the application of cooperative game theory is required; we review corresponding approaches in Sect. 3.1.2. Second, some research accounts for the inexpediency of central targets in decentralized organizations and advocate the establishment of coordinating contracts instead of these targets. One advantage of contracts is that parties can reduce the administrative burden by handling over some effort for decision-making and data collection (e.g., the generation of demand forecasts) to the responsibility of the other parties. This may lead to substantial simplifications in the planning, e.g., a buyer can usually collect demand information easier than a supplier acting as a central coordinator. Moreover, the establishment of contracts has the huge benefit that parties’ incentives are kept untouched, e.g., a buyer has potentially a better motivation for optimally determining sales quantities and prices on her own since she is directly affected by the results of these decisions.4 The effects of these contracts are investigated using concepts from non-cooperative game theory in the literature.5 We address them in the following subsection.
3.1.1 Non-cooperative Game Theory Mechanisms based on non-cooperative game theory usually propose the establishment of coordinating contracts. We review these approaches separately according to the drivers for suboptimality identified in Sect. 2.3. In Sect. 3.1.1.1, we survey approaches related with the drivers Q (quantity) and T (time), and in Sect. 3.1.1.2 those related with L (lot sizes).6
4 As a further obstacle in this context, note that if one party determines the optimal solution for the whole supply chain and asks the other parties for an implementation of this solution, the other parties might not consent due to their missing involvement into the decision-making process. 5 Non-cooperative and cooperative game theory are branches of game theory that basically differ by their focus: Whereas non-cooperative game theory yields a prediction about the outcome of the game (in monetary terms) without exactly specifying the actions taken, cooperative game theory focuses on the specific actions taken by players assuming that parties can negotiate effectively, see, e.g., Cachon and Netessine (2004, p. 26). 6 Note that in the section at hand, we will also include simple contracting schemes, where no sophisticated game theoretical analysis is needed since neither uncertainty nor coalition building is considered. This is a convention taken here since these approaches could be subsumed under both cooperative and non-cooperative game theory in principle. However, we do not include approaches that, instead of relying on contracts, advocate a unilateral determination of the systemwide optimum by one party and disregard the design of incentives for its implementation (e.g., Shirodkar and Kempf, 2006, p. 420; and Barbarosoˆglu, 2000, p. 732).
3.1 Symmetric Information
37
3.1.1.1 Coordination on Quantities and Time Except for those focusing on the coordination of order cycles (see the next subsection), most coordination mechanisms requiring symmetric information are built on the newsvendor model and extensions. To illustrate the basic ideas of these mechanisms, we exemplarily discuss the application of revenue-sharing contracts in this setting. Consider a buyer–supplier supply chain with a single item supplied. This item is sold by the buyer at a price p in an external market (see Fig. 3.1). Thereby, the buyer faces a newsvendor problem:7 The market demand is stochastic and the quantities ordered by the buyer have to be specified before the start of the selling season. Denote the market demand as a random variable D with the probability density function f .D/ and the cumulated density function F .D/, the quantity ordered by the buyer by Q, and the quantity sold to the market by M .Q/. Neither the order of the buyer nor the market demand have to be fulfilled completely, i.e., voluntary compliance is assumed for both buyer and supplier. Furthermore, the buyer has the possibility to sell leftover inventory I.Q/ D .Q M.Q//C at a salvage value v. Production costs at the supplier’s site are denoted by c0 . In the default planning process without coordination, the supplier specifies a wholesale price w for the traded good. Unfortunately, such a wholesale price (contract) does not coordinate the supply chain under standard assumptions (amongst them, risk-neutrality and self-interest of parties8 ). This can be recognized by the following analysis.9 Consider gB .Q/, the expected profit of the buyer dependent on Q. Z Q gB .Q/ D .p w/ Q .p v/ f .D/ dD: (3.1) 0
gB .Q/ is made up by the profit for selling all units ordered in the external market less the difference of selling price and salvage value for the salvaged units. Since
Fig. 3.1 Coordination by revenue sharing
7
For an introduction to the newsvendor model, see, e.g., Silver et al. (1998, p. 385). See, e.g., Cui et al. (2007, p. 1303), for a paper that does not rely on these standard assumptions. There, it is shown that wholesale price contracts may lead to supply chain coordination, given fair (and not self-interested) behavior of parties. 9 Similar analyses can be found, e.g., in Cachon (2003, p. 233), Lariviere (1999, p. 233), and Thonemann (2005, p. 215). 8
38
3 Coordination Mechanisms for Supply Chain Planning
gB .Q/ is strictly concave, QB , the optimal order quantity of the buyer, can be obtained by differentiating (3.1). Setting
@gB .Q/ D p w .p v/F .Q/ @Q
equal to zero yields QB
DF
1
pw : pv
(3.2)
To find out whether the choice of a profit-maximizing wholesale price is a Nash equilibrium,10 compare (3.2) to QSC , the order quantity optimal for the whole sup ply chain, which can be derived analogously to QB : QSC
DF
1
p c0 pv
:
It is easy to see that c0 is the only wholesale price which induces the buyer to place a systemwide optimal order. Such a choice, however, would lead to a profit of zero for the supplier. Hence, for a rational supplier, the Nash equilibrium is to choose a wholesale price w > c0 . This results in double marginalization since the buyer will order inadequate (and too low) quantities then. In light of this problem, several contracts have been designed which actually provide incentives for the buyer to place a systemwide optimal order. Among those are revenue-sharing contracts which are used within the video retail industry.11 There, the supplier subsidizes larger order quantities by offering a lower wholesale price and requires a share s of the revenue generated by the buyer for compensation (see also Fig. 3.1). Like w, the share s is fixed before the start of the selling season. It can be shown that properly designed revenue-sharing contracts lead to an optimal coordination of the supply chain, independently from all other parameters like the market demand, and so on. With such a contract, the buyer’s gains become: Z
Q
gB .Q/ D .sp w/ Q .sp v/
F .D/ dD:
(3.3)
0 can be obtained by the partial derivative of (3.3) to Q: Again, QB QB
DF
1
sp w : sp v
(3.4)
10 A Nash equilibrium is a central solution concept in non-cooperative game theory, where “each player’s strategy choice is a best response [. . . ] to the strategies actually played by his rivals,” Mas-Colell et al. (1995, p. 246). 11 See, e.g., Cachon and Lariviere (2001a, p. 20).
3.1 Symmetric Information
39
Setting the RHS (right-hand side) of (3.4) equal to the RHS of (3.2) leads to a conditional equation for s and w with real-valued solutions for these parameters. As a consequence, always a contract can be designed where the choice of the systemwide optimal order quantity is a Nash equilibrium and both parties obtain greater profits than in the default solution (i.e., the mere application of a wholesale price contract). The distribution of the surplus depends on the concrete choices of s and w and has to be determined by additional negotiations between the parties. Table 3.2 provides a review of seminal and recent papers proposing and analyzing coordination by contracts. Note, however, that this review is not exhaustive; we refer to the extensive surveys by Tsay et al. (1998),12 Lariviere (1999),13 and Cachon (2003),14 for additional research done in this area. Moreover, Table 3.1 provides some nomenclature used for the classification. Most papers are based on the newsvendor model and, hence, address the driver Q (quantity) characterizing the suboptimality of the default setting. The buyer is usually assumed as the leader deciding about the order quantities. An exception is the paper of Corbett and DeCroix (2001),15 where the supplier decides about his supply of hazardous solvents causing additional disposal costs for the buyer. A further nonstandard setting is that considered by Ferguson et al. (2006).16 There, false failure returns by the retailer cause systemwide suboptimality. To incorporate these returns into our classification, they can be interpreted as inappropriate quantities required, which cause elevated production costs for the supplier. Papers that focus on the driver T (time) mostly consider multi-period inventory models with base stock policies adopted by the decentralized parties. Both discrete17 and continuous time structures with finite production capacities18 have been addressed. Concerning the supply chain structures, extensions on assembly supply chains and multiple retailers have been investigated apart from the basic buyer–supplier setting. The increased number of players at one tier results in competition effects,
Table 3.1 Nomenclature for the classification (structure)
12
See Tsay et al. (1998, p. 299). See Lariviere (1999, p. 233). 14 See Cachon (2003, p. 227). 15 See Corbett and DeCroix (2001, p. 881). 16 See Ferguson et al. (2006, p. 376). 17 E.g., Cachon and Zipkin (1999, p. 936). 18 E.g., Caldentey and Wein (2003, p. 1). 13
Group
Parameter
Comment
Structure
1-1 n-1 1-n Arb.
Buyer–supplier setting Multiple suppliers, one buyer One supplier, multiple buyers Arbitrary structure
40
3 Coordination Mechanisms for Supply Chain Planning Table 3.2 Coordination on quantities and time with symmetric information Author(s) Driver Structure Type of contract Bernstein and Federgruen (2005) Q 1-1 (Nonlinear) price discount Bernstein and Federgruen (2007) Q 1-n (Nonlinear) price discount Bernstein et al. (2006) Q 1-n Wholesale price Burer et al. (2008) Q 1-n Transfer payment Cachon (2004) Q 1-1 Wholesale, advanced purchase Cachon and Lariviere (2005) Q 1-n Revenue sharing Corbett and DeCroix (2001) Q 1-1 Shared-savings Cui et al. (2007) Q 1-1 Wholesale, quantity discount Dana Jr. and Spier (2001) Q 1-n Revenue sharing Dong and Zhu (2007) Q 1-1 Wholesale, advanced purchase Ferguson et al. (2006) Q 1-1 Target rebate Gerchak and Wang (2004) Q n-1 Revenue sharing Lariviere and Porteus (2001) Q 1-1 Wholesale Pasternack (1985) Q 1-1 Buyback Sethi et al. (2004) Q 1-1 Quantity flexibility Taylor (2001) Q 1-1 Return policies Taylor (2002) Q 1-1 Target rebate Tsay (1999) Q 1-1 Quantity flexibility Bassok and Anupindi (2008) T 1-1 Quantity flexibility Barnes-Schuster et al. (2002) T 1-1 Option Bernstein and DeCroix (2006) T n-1 Transfer payment Cachon and Zipkin (1999) T 1-1 Transfer payment Caldentey and Wein (2003) T 1-1 Transfer payment Gupta and Weerawat (2006) T 1-1 Two-part revenue sharing Zhang (2006) T n-1 Transfer payment
which – depending on the characteristics of the setting considered – may mitigate19 or exacerbate20 the suboptimality of (pure) wholesale price contracts. Although some contract types have been shown to dominate others or to be equivalent in some specific settings,21 no general preference for particular contract types can be recognized, a result, which underlines the activeness of this field of research.
3.1.1.2 Coordination of Lot Cycles A further stream of coordination mechanisms that assume symmetric information deals with the alignment of production and order lot cycles. The basic insight, that an independent or unilateral determination of lot cycles may cause systemwide
19
See, e.g., the analysis of Cachon (2003, p. 271). See, e.g., Gerchak and Wang (2004, p. 29). 21 See, e.g., Cachon and Lariviere (2005, p. 32), for the equivalence of revenue-sharing and buyback contracts in a buyer–supplier newsvendor setting. 20
3.1 Symmetric Information
41
suboptimality, has already been recognized by Goyal (1976).22 For illustrating the manifestation of double marginalization in this setting, consider a supply chain with one buyer and one supplier planning based on the EOQ model.23 Denote the buyer’s and supplier’s setup costs by scB ; scS > 0 and the buyer’s holdings costs by hB . Moreover, assume that the supplier follows a lot-for-lot policy, i.e., he directly produces each order of the buyer without any inventory holding. Then the order lot size chosen by the buyer under upstream planning (i.e., without coordination), s QB
D
2scB d ; hB
is smaller than the systemwide optimal order lot size s D QSC
2 .scB C scS / d : hB
Based on this insight, several authors have addressed the determination of coordinated production policies for more complex settings24 and the design of quantity discounts for their implementation.25 Compared to the mechanisms presented in the previous subsection, these approaches seem less appropriate for decentralized coordination since they usually require the computation of the centralized optimum by the party offering the rebate and – in contrast to those analyzed in the previous subsection – do not provide the advantage of handling over some effort for data collection and decision-making to the other party. Due to this fact and to the existence of several recent and comprehensive surveys on this topic,26 we omit a detailed listing of related papers.
3.1.2 Cooperative Game Theory Two basic objects in cooperative game theory are bargaining under complete information and coalition building. While few applications for supply chain coordination exist regarding the former,27 a considerable number of coordination
22
See Goyal (1976, p. 107). For the EOQ model, see also Sect. 2.2.2.1. 24 E.g., the relaxation of the lot-for-lot assumption (Lee and Rosenblatt 1986, p. 1177) and the coordination of production and shipment policies (e.g., Hill 1999, p. 2463). 25 E.g., Monahan (1984, p. 720) and Joglekar and Tharthare (1990, p. 492). 26 See, e.g., Sucky (2004b, p. 110), Sarmah et al. (2006, p. 1), Li and Wang (2007, p. 1), and Ben-Daya et al. (2008, p. 726). 27 Among those are Kohli and Park (1989, p. 693), Gjerdrum et al. (2002, p. 586), and the analysis in the paper by Nagarajan and So˘sic (2008, p. 725). 23
42
3 Coordination Mechanisms for Supply Chain Planning
mechanisms based on alliance formation have been proposed in the recent years. There, the question is whether the surplus from coordination can be shared such that all parties have incentives to participate in the grand coalition, i.e., the coalition comprising all parties, which maximizes systemwide profit. This is not necessarily the case, as shown in the following example. Example 3.1 Consider three parties that can achieve a joint surplus of 300[MU] from cooperating in the grand coalition. If two parties form a subcoalition, the surpluses would be 250[MU] for this coalition and 0[MU] for the remaining party. Although the grand coalition maximizes the systemwide surplus, a binding agreement for the formation of this coalition cannot be formed. For any rule for surplus sharing in the grand coalition (e.g., 100,100,100), two parties would benefit from building a subcoalition (e.g., with a surplus sharing of (125,125) among them). A formal answer to this question, i.e., whether an allocation of the surplus exists that provides incentives for forming the grand coalition, is given by the concept of the core. The core is defined as the set of utility vectors u D .u1 ; : : : ; uI / of parties i 2 P D f1; : : : ; I g that satisfy X i 2S
ui v .S / 8S P and
X
ui v .P / ;
i 2P
with v .S / ; v .P / as the surpluses (in terms of utility)28 from forming the subcoalition S and the grand coalition P , respectively, that can be allocated among parties (e.g., by a central planner). Apart from a unique utility vector, the core may comprise multiple elements or be empty (as in Example 3.1). Some approaches applying this concept to supply chain coordination indeed prove emptiness or non-emptiness of the core,29 whereas others calculate core cost allocations explicitly.30 The potential non-emptiness is an important disadvantage of the core. Therefore, other concepts such as the Shapley value and the nucleolus31 have been elaborated, which – in contrast to the core – can always be determined, but are not necessarily chosen by self-interested parties. In the last years, many application of cooperative game theory to supply chain coordination have been proposed. Note, however, that a transferability of these approaches to general settings with asymmetric information does not seem possible since cooperative game theory requires the specification of the parties’ concrete actions, which is only possible under complete information. Moreover, several current
28
Coalition analysis in cooperative game theory usually relies on games with transferable utility (TU games), where utility (most often money) is freely transferable among players, see, e.g., Myerson (1991, p. 422). 29 E.g., Dror and Hartman (2007, p. 78). 30 See, e.g., Drechsel and Kimms (2008) for capacitated lot-sizing and Houghtalen et al. (2007) for freight alliances. 31 See, e.g., Myerson (1991, p. 452) for more on these concepts.
3.2 One-Sided Information Asymmetry
43
surveys of this area are available. Therefore, instead of providing a detailed listing of corresponding papers, we refer to Meca and Timmer (2008)32 for a review of cooperative lot-sizing models and to Nagarajan and So˘sic (2008)33 for a general review about supply chain coordination based on cooperative game theory.
3.2 One-Sided Information Asymmetry Since symmetric information often does not reflect business reality properly, research on supply chain coordination mechanisms has been extended to information asymmetry. As a first step towards this extension, one-sided information asymmetry has been addressed by a series of papers. This assumption is particularly attractive since it allows for the transfer of well-known concepts that originally have been developed in economic theory to overcome the phenomenon of adverse selection. Adverse selection arises with private information held at least by one party and distorts the optimal strategies of the uninformed parties from the systemwide optimum.34 This phenomenon has been recognized by Akerlof (1970)35 in the context of markets for used cars, where qualitatively good cars can only be sold at low prices since buyers cannot distinguish the actual conditions of cars in advance. Basic mechanisms to mitigate the resulting inefficiencies are signaling and screening. In the following subsections, we present papers that apply these mechanisms to supply chain coordination.
3.2.1 Signaling Signaling has first been proposed by Spence (1973)36 in the context of labor markets. There, employers cannot directly infer the types of applicants (i.e., their skills), which leads to inefficiencies in these markets. A common remedy for that is signaling: Skillful applicants signal by investing into their education (which unskillful workers are not willing to do) in order to credibly communicate their types to potential employers. Cachon and Lariviere (2001a) have transferred this idea to supply chain coordination.37 They consider a setting reversed to that outlined in Sect. 3.1.1.1. The buyer determines the wholesale price for the item supplied and the supplier
32
See Meca and Timmer (2008, p. 1). See Nagarajan and So˘sic (2008, p. 719). 34 For a similar definition, see Mas-Colell et al. (1995, p. 436). 35 See Akerlof (1970, p. 488). 36 See Spence (1973, p. 355). 37 See Cachon and Lariviere (2001a, p. 629). 33
44
3 Coordination Mechanisms for Supply Chain Planning
Fig. 3.2 Setting considered by Cachon and Lariviere (2001a)
bears the demand risk: His capacity limits the maximum supply quantity and, hence, the fulfillment of market demand; a build-up of capacity is possible, but costly and has to take place before the start of the selling season. The buyer, in turn, can place her order after observing the market demand (see also Fig. 3.2). In this setting, parties have conflicting goals with respect to the build-up of capacity by the supplier. The buyer prefers a high build-up; she has no extra costs and, if demand turns out to be high, she can fulfill a larger share of it. The supplier, however, favors this build-up less since he might have spent the corresponding effort unnecessarily in case of low demand. Moreover, assume that the forecast of market demand is private information of the buyer and that the buyer communicates this forecast to the supplier. Then, in light of the above conflict, the forecast information provided by the buyer may be not credible for the supplier since the buyer benefits from an inflated forecast without any repercussions on her profit. Cachon and Lariviere (2001a) analyze the parties’ best strategies in this setting with the simplification that only two possible types 2 fH; Lg (H D high, L D low) for the buyer’s demand forecast exist: The forecast is D D X , with X as a publicly known random variable with the distribution function F .X /. Denote the level of the capacity built up by the supplier by k. Then the buyer’s profit dependent on and k becomes gB .k/ D .p w/ M.k/;
with M.k/ as the expected sales volume defined by M.k/ D E Œmin fk; D g : The buyer’s profit crucially depends on the supplier’s capacity choice, which in turn depends on the supplier’s believe about D and, hence, about . The buyer can follow two main strategies here. First, she can design a contract (e.g., choose an appropriate level of the wholesale price) without regard to the question whether the supplier believes the demand forecast. This strategy results in a pooling equilibrium, where the supplier has to rely on his prior knowledge about in order to determine his optimal capacity build-up. If the (buyer’s) forecast is of type H, this equilibrium may be unfavorable since it may imply a low build-up of capacity and result in systemwide forgone profits. The second strategy for the buyer is signaling, i.e., to provide a credible forecast to the supplier. To achieve this, the simplest possibility is to alter the wholesale
3.2 One-Sided Information Asymmetry
45
Fig. 3.3 Signaling
price (see Fig. 3.3). Assume that the forecast is of type H. In order to convince the supplier that the type is H, the buyer has to increase w to such a value that would be suboptimal for a buyer of the type L – even if the supplier believed the signal and built up capacity accordingly. That means, the optimal wholesale price w has to be chosen such that H @gB .w ; H / D 0; @w L .w ; H / @gB < 0; @w H L .w ; H / ; gB .w ; H / as the gains for a buyer of types H and L, with gB respectively, provided that the supplier believes that the buyer’s type is H .38 Apart from the increased wholesale price, Cachon and Lariviere (2001a) point out further signaling strategies for the supplier like burning money, minimum quantity commitments, and option contracts, that can be used with forced compliance of the supplier. Cachon and Lariviere (2001a) also show the preference of some strategies over the others, but did not determine the optimal signaling strategy under ¨ voluntary compliance. Further insights into this setting provides the paper of Ozer and Wei (2006),39 who take the perspective of the supplier and propose contracts that induce signaling of the buyer.
3.2.2 Screening Screening is a mechanism for the uninformed party to improve market efficiency.40 There, the uninformed party proposes a menu of choices and the party holding the private information self-selects, i.e., it selects the choice with the highest benefits and thereby reveals the private information. Often, an infinite number of potential (continuous or discrete) menus exist. Fortunately, the well-known revelation principle41 allows reducing the number of these menus considerably: It states that the optimal menu is incentive-compatible, i.e., the selecting party has incentives to reveal its private information truthfully. In the above example for labor markets,
38
See Cachon (2003, p. 325). ¨ See Ozer and Wei (2006, p. 1238). 40 Among the first to study screening have been Rothschild and Stiglitz (1976, p. 629), who applied this mechanism for coordination in insurance markets. 41 See Myerson (1979, p. 61). 39
46
3 Coordination Mechanisms for Supply Chain Planning
Fig. 3.4 Screening
a possibility for screening is to offer contracts specifically designed for different types of applicants (e.g., regarding their stay in the organization).42 Relying on the paper of Corbett and de Groote (2000),43 we will exemplarily describe how screening can be applied for supply chain coordination. We consider the setting outlined in Sect. 3.1.1.2, with the difference that the buyer’s unit holding cost hB is incomplete information for the supplier. Incomplete information means that “some players do not know the payoffs of the others.”44 Here we assume that the supplier has some estimates about this cost in the form of a cumulated probability distribution F .hB / lower bounded by hB . Further assume that the supplier has sufficient bargaining power to perform screening, i.e., to propose a menu of quantity discount contracts as a take-it-orleave-it offer to the buyer. Each of these contracts specifies a pair .Q; P .Q// made up by the order quantity Q and the corresponding price P .Q/. Choosing one of these contracts, the buyer reveals her private information hB (see Fig. 3.4). C Further assume given reservation values of parties tcB , tcSC , i.e., their costs (which can also be negative with profitable outside opportunities, e.g., trading with other parties) incurring if the buyer accepts neither of the contracts offered. Moreover, assume that the supplier opts to refuse trading with the buyer if hB exceeds a limit hB . Then the optimization problem for determining the optimal menu of contracts to be offered by the supplier can be stated as follows:
scS d C P .hB / C EhB >hB tcSC (3.5) Q .hB / hB scB d C Q .hB / P .hB / 8hB 2 hB ; hB (3.6) s.t. tcB Q .hB / 2 scB d hB @P .hB / D : (3.7) @hB 2 Q .hB /2 min EhB hB
The objective function (3.5) minimizes the sum of the expected costs for the supplier in case of acceptance and refusal of trade. We write P .hB / in short for P .Q .hB // here. Constraints (3.6) are individual rationality constraints. They ensure that the buyer’s costs from accepting one of the contracts offered (the RHS of 3.6) do not 42
See, e.g., Salop and Salop (1976, p. 619). See Corbett and de Groote (2000, p. 444). 44 See, e.g., Fudenberg and Tirole (1991, p. 209). The pioneering work for the modeling of incomplete information in game theory is the three-part essay of Harsanyi (1967, p. 159), Harsanyi (1968b, p. 486), and Harsanyi (1968a, p. 320), where it is shown how to model a game with incomplete information by a game with imperfect information. For a recent review on this topic, see Myerson (2004, p. 1818). 43
3.2 One-Sided Information Asymmetry
47
exceed the buyer’s reservation value, provided that hB 2 hB ; hB . The second constraint (3.7) determines the incentive compatibility of the menu. This constraint ensures that the buyer selects the contract designed for her private information (here: hB ). Here, this is achieved by setting the partial derivative of the quantity discount function P .hB / equal to the partial derivative of the buyer’s costs (the RHS of 3.7). Based on this model, Corbett and de Groote (2000) derive the optimal set of contracts that can be offered by the supplier. This set is continuous, i.e., for any holding cost rate reported by the buyer, a different contract is offered.45 The structure of the above model with the individual rationality and the incentive compatibility constraints is shared by all supply chain coordination mechanisms based on screening. In Table 3.3, we provide a classification of these mechanisms. Note that the criteria chosen for this classification differ from those for symmetric information.46 Since all approaches deal with buyer–supplier settings and rely on screening as the contract type, we omit these parameters and incorporate the type of the unknown data instead. As with symmetric information, most of these papers focus on the driver Q and base their analyses on newsvendor models47 or bilateral monopoly settings.48 However, also the other drivers mentioned in Sect. 2.3.2.2 have been covered.49 The variety of the unknown data considered has been substantial, too. Note that the Table 3.3 Coordination mechanisms based on screening Author(s) Driver Private data Arya and Mittendorf (2004) Burnetas et al. (2007) Corbett et al. (2005) Corbett et al. (2004) Ha (2001) ¨ Ozer and Wei (2006) Li et al. (2007) Schenk-Mathes (1995) Taylor (2006) Zhang et al. (2008) Cachon and Zhang (2006) ¨ Lutze and Ozer (2008) Corbett (2001) Corbett and de Groote (2000) Sucky (2004a)
45
Q Q Q Q Q Q Q Q Q Q T T T, L L L
Demand (consumer’s valuations) Market demand (distribution) No private, but non-contractible data Variable costs of the buyer Variable costs of the buyer Market demand (distribution) Market demand (distribution) Supplier’s type (cost function) Market demand (distribution) Buyer’s inventory level Supplier’s costs for lead time reduction Shortage costs of the buyer Buyer’s setup, backorder costs Buyer’s holding costs Buyer’s type (setup and holding costs)
For an approach that assumes discrete choices, see, e.g., Schenk-Mathes (1995, p. 176). See Table 3.2 on p. 40. 47 E.g., Burnetas et al. (2007, p. 465). 48 E.g., Corbett et al. (2004, p. 550). 49 The driver T has been considered by Corbett (2001, p. 487), Cachon and Zhang (2006, p. 881), ¨ and Lutze and Ozer (2008, p. 898), and L by Corbett and de Groote (2000, p. 444), and Sucky (2004a, p. 493). Note in this context that there are further publications of Sucky (e.g., Sucky 2006) with almost identical scopes. We did not include them separately into our classification. 46
48
3 Coordination Mechanisms for Supply Chain Planning
approach by Corbett et al. (2005)50 plays a particular role in this context. There, all information is commonly known, but some relevant parameters, the parties’ efforts, are non-contractible ex ante. Corbett et al. (2005) show that screening can coordinate decisions in this setting, too. They propose that the supplier offers to the buyer an optimal set of contracts with the buyer’s non-contractible effort as the parameter to be selected by her.
3.3 Multilateral Information Asymmetry In this section, we review mechanisms designed for multilateral information asymmetry. We begin with an introduction to the concept of auctions, that plays a central role for the determination of the decentralized parties’ incentives and the allocation of the surplus, and provide an overview of related papers focusing on supply chain coordination (Sect. 3.3.1). In Sect. 3.3.2, we address papers that additionally deal with the generation of coordinating proposals, a crucial issue that is not considered by conventional auctions, and describe the Dantzig–Wolfe decomposition as a basic approach, which underlies many of these papers.
3.3.1 Auctions and Their Application to Supply Chain Coordination Standard Auctions Consider a single good traded, an auctioneer, and a set of potential buyers with private information about their valuations. To determine the allocation of the good among the buyers, four basic auction procedures can be applied.51 In the ascendingbid auction, the auctioneer successively raises prices until a single bidder remains, who has to buy the good at the final price. The descending-bid auction works the other way round. There, the auctioneer decreases prices, and the first buyer that accepts a price obtains the good. Further auction types are first-price sealed bid and second-price sealed bid (Vickrey) auctions, where bidders simultaneously submit sealed bids and the object is sold to the buyer with the highest bid at the highest and second-highest price, respectively. Two important topics in the design of (optimal) auctions are revenue maximization and incentive compatibility. A famous insight is the revenue equivalence theorem52 here: If the buyers are risk-neutral and if their private valuations of the good are independently drawn from some random distributions, then any auction mechanism where the highest bid wins the good and bidders with valuations at the 50
See Corbett et al. (2005, p. 653). E.g., Klemperer (1999, p. 229). 52 See Myerson (1981, p. 65). 51
3.3 Multilateral Information Asymmetry
49
lowest possible level obtain zero benefits yields the same expected revenue. Truthtelling is ensured if the good is sold at the second-highest price, which holds both for the ascending-bid auction and the second-price sealed bid auction.53 Double Auctions A double auction is a variant of auctions that lifts the requirement of a central auctioneer and treats buyers and sellers of the traded good(s) symmetrically. Its simplest form is bilateral trade. There, a potential buyer and a potential seller of an indivisible good simultaneously submit sealed bids consisting of purchasing and selling prices (b and s), respectively. After breaking the seals, the conditions of trade are determined by the following rule: If the purchasing bid is lower than the selling bid, i.e., if b < s, nothing happens. Otherwise, the good is sold at the price kb C.1k/s. The factor k determines the allocation of the surplus among parties. A natural choice for k is 1=2, leading to a price equal to the average of seller’s and buyer’s bids. In the literature, this mechanism has been investigated extensively.54 A basic insight is that buyer and seller have incentives to underbid and overbid, respectively. The fact, that they will not exactly bid their reservation values, their valuations of the traded good (which they receive if trade does not take place), can also be interpreted that parties do not truthfully reveal their private information. Analyses of this mechanism assume that the probability distributions of parties’ reservation values are common knowledge. This common knowledge does not only refer to the first-order believes, i.e., the knowledge of parties about the probability distributions of the other parties’ reservation values, but also to higher-order believes, i.e., the additional knowledge that the own distribution is known by the other party, that the latter knowledge is known by the other party, and so on.55 Then, parties’ best bidding strategies are interdependent since the bids of the seller depend on his expectations about the buyer’s bids and vice versa. It has been shown that the resulting game has multiple Nash equilibria.56 Fortunately, players prefer the most efficient equilibrium in empirical tests,57 they adopt the linear equilibrium strategies first characterized by Chatterjee and Samuelson (1983).58 For uniformly distributed prior knowledge with equal probability density functions of parties, the efficiency of this equilibrium is 27=32 D 84:375% on average.59 53
See, e.g., Klemperer (1999, p. 230). The proof of truth-telling in the second-price sealed bid auction is due to Vickrey (1961, p. 8) and is among the most famous results in auction theory. 54 See Chatterjee and Samuelson (1983, p. 835) and Myerson and Satterthwaite (1983, p. 265) for seminal papers on this topic. 55 For a formal description of this type of knowledge, see Aumann (1976, p. 1236). 56 See Satterthwaite and Williams (1989, p. 107) for characterizations of equilibria when parties play differential strategies and Leininger et al. (1989, p. 63) when parties’ strategies are stepfunctions. 57 See Radner and Schotter (1989, p. 179) and Rapoport and Fuller (1995, p. 179). 58 See Chatterjee and Samuelson (1983, p. 838). 59 See Chatterjee and Samuelson (1983, p. 844).
50
3 Coordination Mechanisms for Supply Chain Planning
This mechanism is regarded to capture main features of bargaining in both small and large markets.60 In fact, its generalization to multiple sellers and buyers can be carried out straightforwardly. We refer to Williams (1993),61 Rustichini et al. (1994),62 and Cripps and Swinkels (2006)63 for the analysis of efficiency and parties’ equilibrium strategies in these settings, and Chu and Shen (2006)64 and to Chu and Shen (2008)65 for recently developed variants inducing truth-telling of participants.
Multiunit and Combinatorial Auctions A recent focus of auction research is on multiunit auctions. Even in the simplest case, where each bidder wants to purchase one of several nonidentical items, additional problems compared to standard auctions have to be addressed, like the question, which item is allocated to which bidder.66 Combinatorial auctions form a more complex variant of multiunit auctions, which is of considerable relevance in practice.67 They can be defined as “simultaneous multiple-item auctions that allow submission of all or nothing bids for combinations of the items being sold.”68 Compared to other auction types, combinatorial auctions involve new challenges like the non-existence of market prices and the computational difficulty of the winner determination problem. In principle, extensions of standard auction mechanisms can be applied (e.g., ascending-bid or Vickrey auctions),69 but their analysis and optimal design become considerably more difficult here. Table 3.4 provides an overview about auctions applied for supply chain coordination. All approaches address inappropriate quantities as the driver for systemwide suboptimality. Here, we again take the supply chain structure as a criterion for the classification. Thereby, we write an “(e)” behind the identifier of the structure if a central entity, i.e., a third-party auctioneer, is required. Moreover, we distinguish between the types of the auctions and state their additional properties.
60
See, e.g., Satterthwaite and Williams (1989, p. 108). See Williams (1993, p. 1101). 62 See Rustichini et al. (1994, p. 1041). 63 See Cripps and Swinkels (2006, p. 47). 64 See Chu and Shen (2006, p. 1215). 65 See Chu and Shen (2008, p. 102). This paper, in fact, synthesizes double auctions with multiunit auctions (see the next paragraph) since the buyers bid on bundles of goods there. 66 See, e.g., Milgrom (2004, p. 251). 67 See, e.g., the application of combinatorial auctions for the US spectrums for telephone services reported by Milgrom (2004, p. 297). 68 See Pekec and Rothkopf (2003, p. 1485). For recent surveys on combinatorial auctions, refer further to Abrache et al. (2007, p. 131), de Vries and Vohra (2004, p. 247), de Vries and Vohra (2003, p. 284), and Milgrom (2004, Chap. 8). 69 See, e.g., Milgrom (2007, p. 935). 61
3.3 Multilateral Information Asymmetry Table 3.4 Auctions for supply chain coordination Author(s) Type Beil and Wein (2003) Ascending Chen (2007) All standard formats Chen et al. (2005) Second-price Gallien and Wein (2005) Ascending Mishra and Veeramani (2006) Ascending Mishra and Veeramani (2007) Descending Parkes and Kalagnanam (2005) Iterative
51
Structure 1-n 1-n 1-n(e) 1-n(e) 1-n(e) 1-n(e) 1-n(e)
Additional properties Multiattribute Object: contracts Multiunit Multiunit Multiunit Multiunit Multiattribute
All papers address procurement auctions, and most of them rely on standard auctions formats. The aim of the auctions and, hence, the structures of the underlying supply chains, differ: Some papers aim at maximizing the buyer’s profit, and others at maximizing the profit of the whole system and, hence, are designed for use with a third-party auctioneer, e.g., an e-marketplace.70 Regarding the objects of the auctions, most often multiple units of items or items with multiple attributes are traded. An exception is Chen (2007),71 who applies auctions for determining supply contracts in a newsvendor-type setting. Interestingly, neither combinatorial auctions nor double auctions have been proposed in this context yet. Although they do not consider the auctioning of bundles, most approaches rely on multiunit auctions. Since the cost functions assumed are rather simple, the generation of potentially coordinating purchase quantities does not involve major difficulties. In fact, the approach of Gallien and Wein (2005)72 is the only to include one bottleneck in form of a capacity restriction at the supplier. The applicability of this approach, however, is limited by the additional assumption that the auctioneer can estimate the suppliers’ capacity constraints, i.e., the capacity needed for the production of items and the total available capacity at the suppliers’ bottlenecks. In Master Planning, however, where a plenty of decisions and interdependencies exist, a direct application of these approaches seems less appropriate.
3.3.2 Mechanisms with Focus on Proposal Generation If optimization problems with multiple bottlenecks (e.g., multiple periods) are used for modeling the decisions of the decentralized parties, a multidimensional set of potentially coordinating supply quantities has to be considered. Here, apart from designing incentives for the decentralized parties to reveal their costs and from determining the allocation of the surplus, the generation of potentially coordinating supply proposals becomes a further task to resolve.
70
See Elmaghraby (2004, p. 213) for a survey about the application of auctions in e-marketplaces. See Chen (2007, p. 1562). 72 See Gallien and Wein (2005, p. 76). 71
52
3 Coordination Mechanisms for Supply Chain Planning
We begin with a description of dual decomposition as a generic approach for generating such proposals. Subsequently, we survey the corresponding literature on coordination mechanisms. Since these mechanisms are most closely related to the topic of this work, we describe in detail the characteristics of each of them. Finally, we discuss to what extent generic requirements for the coordination of Master Planning are fulfilled by these approaches and the other mechanisms reviewed in this chapter. 3.3.2.1 Outline of Dual Decomposition Consider a generic centralized decision model. (C) min ZC D s.t.
X
X
ciT xi
i 2P
Ai xi b0
(3.8)
i 2P
Bi xi bi
8i 2 P
(3.9)
xi 0 8i 2 P :
(3.10)
Indices and Sets i Decentralized parties, i 2 P D f1; : : : ; I g. Data Ai Use of the central resources by decisions xi , Ai 2 Qnoi Bi Use of the decentralized resources by decisions xi , Bi 2 Qmi oi b0 Total amount of the central resources, b0 2 Rn bi Total amount of the decentralized resources, bi 2 Rmi ci Costs associated with decisions xi , ci 2 Roi Variables xi Decision variables, xi 2 Roi The objective function of C minimizes the costs resulting from the decisions xi . The first constraint (3.8) links the interdependent decisions of the decentralized parties i by establishing an upper bound on the use of the central resources. Note here that the rationality of the constraint matrix Ai is necessary and sufficient for ensuring the consistency of (3.8).73 Constraints (3.9) and (3.10) capture the decentralized restrictions for variables xi . By skipping the interdependent constraint (3.8), C decomposes into the decentralized problems of parties i : (DPi ) min ciT xi s.t. Bi xi bi xi 2 Xi :
73
See Meyer (1974, p. 223).
8 (3.11) < 8i 2 P (3.12) :
3.3 Multilateral Information Asymmetry
53
Fig. 3.5 Dantzig–Wolfe decomposition
To illustrate the main ideas of dual decomposition, we will present Dantzig– Wolfe decomposition as a basic procedure and outline subgradient optimization, a variant which is frequently used here. The idea of Dantzig–Wolfe decomposition is that the decentralized parties solve DSi , i.e., DPi augmented by (dual) prices for central resource use and communicate the costs (ciT xi ) and the central resource use (Ai xi ) of the resulting solution (xi ) to a central entity. Using a master model (MP), the central entity recombines the proposals for central resource use obtained so far and derives new prices to be sent to the decentralized parties. All parties carry out these steps iteratively until neither DPi nor MP yield any new solutions and, hence, the optimal solution to C has been identified (see also Fig. 3.5).74 In the following, we will provide a formal motivation for this procedure:75 Consider C with a Lagrangian relaxation of constraints (3.8).
.LRu / min
X
ciT xi
i 2P
Cu
X
! Ai xi b0
i 2P
(3.9), (3.10): Variables u Prices for central resource use, u 2 Rn For an arbitrary choice of u, the optimal solution to LRu is a lower bound on the costs of C. For a given u, LRu can be decomposed into the subproblems solved by the decentralized parties:76 (DSi ) min ZDSi D ciT xi C uAi xi s.t. (3.11), (3.12):
74
8 < :
8i 2 P
This procedure can be initialized, e.g., be solving DPi with prices of zero for central resource use, see Ho and Loute (1981, p. 306). 75 With our exposition of this approach, we follow Holmberg (1995, p. 61) and Klose (2001, p. 74). Alternative descriptions provide, e.g., Dantzig and Wolfe (1960, p. 101) and Wolsey (1998, p. 185). 76 Note that the constant ub0 is omitted in the objective functions of these problems for ease of exposition.
54
3 Coordination Mechanisms for Supply Chain Planning
Next, consider the maximum of LRu for u 0: (LD)
max min u0
xi
X
ciT xi
Cu
i 2P
X
! Ai xi b0
i 2P
s.t. (3.9), (3.10): Due to Lagrangian duality,77 both the objective function values and the optimal solutions to C and LD are equal. Obviously, for each party i , the optimal solution to LD will include one of the solutions to DSi that are optimal for a given u. These solutions are vertices of the polyhedron defined by (3.11) and (3.12), and are successively generated in the course of the decomposition scheme; hence, before the termination of the scheme, only a subset of the vertices has been identified. Accounting for that, LD can be rewritten as ! X X 0 T v v (LD ) max vminvs (3.13) ci xi C u Ai xi b0 : u0 x 2X i i
i 2P
i 2P
Sets Xivs Set of vertex solutions identified so far Data xiv Value taken by xi in the vertex v identified so far, xiv 2 Xivs Taking the dual of LD0 , we obtain the master problem that is solved by the central entity. v X X min ZMP D ciT i v xiv i 2P
(MP) s.t.
X
ATi
i 2P v X
v X
vD1
i v xiv b0
(3.14)
vD1
i v xiv D 1
8i 2 P
(3.15)
vD1
i v 0 8i 2 P; v D 1; : : : ; v: Index v Vertex identified so far, v D 1; : : : ; v Variables i v Variables indicating the share of the vertex solution v of party i that is included in the new allocation determined by MP
77
See, e.g., Minoux (1986, p. 212).
(3.16)
3.3 Multilateral Information Asymmetry
55
In MP, the central entity identifies a systemwide feasible convex combination of the proposals previously submitted by the decentralized parties with minimum systemwide costs.78 Constraint (3.14) ensures that the central resource use by this convex combination does not exceed the totally available capacity. Constraints (3.15) and (3.16) determine that a convex combination must be chosen. During the course of the decomposition scheme, neither the optimal (dual) prices u nor the set of all relevant vertex Psolutions (and the proposals derived from them) are known. The P former implies i 2P ZDSi ZC , whereas the latter ZMP ZC . We obtain i 2P ZDSi D ZMP after the identification of all vertices that are not dominated in terms of central resource use. This is also the reason for finite convergence of Dantzig–Wolfe decomposition: Since the number of vertices is finite and at least one new vertex solution is found in each iteration, a finite number of iterations is sufficient for identifying the optimal solution to C. Apart from solving MP, the prices for the use of the central resources can be determined via subgradient optimization.79 There, these prices are updated by a step into the direction of the subgradient of the optimal objective function value of LD in each iteration. This procedure has asymptotic convergence towards the optimal prices and the optimal objective function value of MP, but does not provide the primal solution, which has to be determined by an additional heuristic then.80 Since both Dantzig–Wolfe decomposition and subgradient optimization are based on the solution of LD0 , they are subsumed under the term dual decomposition. Apart from dual decomposition, a couple of further mathematical decomposition techniques have been developed.81 Among those is primal decomposition,82 where parties follow a reversed methodology, i.e., the central entity determines the allocation of the central resources relying on dual prices supplied by the decentralized parties. We will not deal in-depth with these approaches here since they – with one exception83 – have not been used for the design of supply chain coordination mechanisms yet and show drawbacks similar to Dantzig–Wolfe decomposition due to the exchange of dual information.84
3.3.2.2 Literature For classifying the literature of this subsection, we again distinguish between the structures of the supply chains considered (“Arb.” stands for arbitrary) and the Note that for this purpose, it is sufficient for the central entity to know proposals Ai xiv and the cost changes ciT xiv , which are actually supplied by the decentralized parties (instead of xiv ). 79 See Held et al. (1974, p. 62). 80 See Holmberg (1995, p. 76). 81 For an overview of these techniques, see, e.g., Holmberg (1995, p. 61). 82 Within primal decomposition, Benders decomposition (see Benders 1962, p. 238) is the counterpart to Dantzig–Wolfe decomposition. 83 See the paper by Arikapuram and Veeramani (2004) discussed below. 84 See also our discussion of these drawbacks at the end of this section. 78
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Table 3.5 Nomenclature for the classification (self-interest) Parameter Comment SI Disc. The applicability of the mechanism is discussed in light of self-interested behavior by parties, e.g., by means of simulation studies SI Proof Analytically proven results for self-interest Team Assumption of team behavior
drivers for suboptimality. Thereby, we include “GLP” as a generic driver which characterizes mechanisms that can be applied for general LP problems and comprises the drivers T and Q. Further note that we have included two approaches that have been developed for scheduling (Sched.), but whose methodologies seem relevant for the coordination of Master Planning in principle. In addition, we state (if applicable) the decomposition procedures the mechanisms are based on and distinguish between the underlying assumptions about parties’ behavior. With the exception of Cui et al. (2007), all approaches analyzed in the previous sections assume self-interest of parties. Here, in contrast, most authors postulate team behavior, which allows significant simplifications for the design of the mechanisms.85 In summary, three representations of self-interest can be distinguished (see Table 3.5). The classification is provided in Table 3.6. Note that there are further surveys available in the literature like Stadtler (2009) for collaborative planning and Heydenreich et al. (2007)86 for scheduling, which employ somewhat different criteria for their classifications and only cover a part of the literature listed here. Two properties are shared by most papers: The assumption of team behavior and the application of mathematical decomposition procedures based on the exchange of dual information. We describe the approaches grouped according to the parties’ behavior, starting with team behavior. The paper of Arikapuram and Veeramani (2004)87 proposes a coordination scheme based on the L-shaped method. The L-shaped method is a decomposition procedure, which builds on ideas from primal decomposition and is usually applied within stochastic programming.88 There, the central entity iteratively generates primal solutions, and the subproblems supply constraints, which are added to the master problem. Arikapuram and Veeramani (2004) propose customized model formulations for the application of the L-shaped method to supply chain planning and present the results of corresponding computational tests. Dudek and Stadtler (2005, 2007)89 present coordination schemes relying on an iterative exchange of supply proposals and their associated cost effects between 85
Team behavior trivially satisfies the important requirement that parties should have incentives to implement the actions specified by the mechanism. 86 See Heydenreich et al. (2007, p. 437). 87 See Arikapuram and Veeramani (2004, p. 111). 88 See, e.g., Birge and Louveaux (1997, p. 155). 89 See Dudek and Stadtler (2005, p. 668) and Dudek and Stadtler (2007, p. 465).
3.3 Multilateral Information Asymmetry Table 3.6 Mechanisms focusing on proposal generation Author(s) Driver Behavior Arikapuram and Veeramani (2004) GLP Team Dudek and Stadtler (2005) T Team Dudek and Stadtler (2007) T Team Ertogral and Wu (2000) T Team Jeong and Leon (2002) GLP Team Jung and Jeong (2005) T Team Karabuk and Wu (2002) T Team Schneeweiss and Zimmer (2004) T Team Walther et al. (2008) Q Team Chu and Leon (2008) L SI Disc. Fink (2006) T SI Disc. Kutanoglu and Wu (1999) Sched. SI Disc. Lee and Kumara (2007) T SI Disc. Guo et al. (2007) GLP SI Disc. Fan et al. (2003) GLP SI Proof Nisan and Ronen (2001) Sched. SI Proof
57
Structure n-1(e) 1-1 1-n 1-1-1(e) Arb. 1-1 1-1 1-1 n-1 1-n 2-1(e) N.a. Arb. Arb.(e) Arb.(e) N.a.
Procedure Primal decomposition Other Other Dual decomposition Dual decomposition Other Dual decomposition Other Dual decomposition Other Other Dual decomposition Other Dual decomposition Dual decomposition Other
decentralized parties. The aim of this exchange is to improve upstream planning in supply chains, where parties plan based on the MLCLSP. The proposals are generated by optimization models that penalize deviations from the last proposal communicated by the other party. Dudek and Stadtler (2005, 2007) evaluate the effectiveness of their schemes by computational tests. These papers differ from each other by the assumptions on the supply chain structures and on the cost exchange (bilaterally, unilaterally). The papers of Ertogral and Wu (2000),90 Jeong and Leon (2002),91 and Walther et al. (2008)92 are based on dual decomposition with subgradient optimization for the solution of LD and mainly differ by their focus. Whereas Ertogral and Wu (2000) address the coordination of decentralized MLCLSP with a fairness objective and an artificially introduced central agent, Walther et al. (2008) coordinate a recycling network with multiple bottlenecks, where the focal company can naturally serve as the party updating the prices sent to the recycling companies. Jeong and Leon (2002), in turn, address a general decentralized environment consisting of different types of agents. Jung and Jeong (2005) propose a simplistic scheme for the coordination of plans of a manufacturer and a distributor.93 They address a supply chain where orders determined by the distributor cause infeasibilities or high penalty costs at
90
See Ertogral and Wu (2000, p. 931). See Jeong and Leon (2002, p. 789). 92 See Walther et al. (2008, p. 334). 93 See Jung and Jeong (2005, p. 167). Note that these authors have published further papers (e.g., Jung et al. 2008) relying on essentially the same setting and the same approach. Hence, we did not include these papers separately in our classification. 91
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3 Coordination Mechanisms for Supply Chain Planning
the manufacturer. The scheme consists of the communication of feasible production plans by the manufacturer, which are incorporated in the form of additional constraints into the planning of the distributor thereafter. Karabuk and Wu (2002) present a further approach using dual decomposition.94 They address the capacity planning between marketing and manufacturing departments and model this problem with formulations similar to the Master Planning models presented in Sect. 2.1.2. Their coordination scheme relies on the Augmented Lagrangian procedure, a variant of dual decomposition, with a dual problem that differs from LD by an additional quadratic perturbation term in the objective function. Karabuk and Wu (2002) propose to solve this dual problem by an updating procedure similar to subgradient optimization and show the effectiveness of their scheme by a case study for a real-world semiconductor capacity allocation problem. Finally, Schneeweiss and Zimmer (2004) present a coordination scheme based on hierarchical anticipation.95 There, the buyer estimates the supplier’s model parameters in order to identify a supply plan with low systemwide costs. This paper relies on an extreme requirement regarding team behavior: It not only assumes truthful information exchange, but also the willingness of parties to accept solutions inferior to the initial solution, provided that a systemwide improvement is obtained. Chu and Leon (2008) address the coordination of uncapacitated, static single item lot-sizing problems of one supplier and multiple buyers.96 They provide a heuristic for the solution of the centralized problem and a coordination scheme that yields the same solution, given truth-telling by parties. In addition, they show by means of (deterministic) simulations that the outcome of their mechanism is not affected by strategic (and potential untruthful) behavior of parties. Fink (2006)97 proposes a scheme for the coordination of supply chains of two suppliers and one buyer. This scheme relies on a mediator, that is informed about the restrictions, but not about the objective functions of the decentralized problems. This mediator generates systemwide feasible solutions, which are either accepted or rejected by the decentralized parties. Fink (2006) shows in computational tests that a good performance of this mechanism can be achieved if parties are willing to accept a substantial share of the proposals generated. As a potential incentive for this, he suggests minimum acceptance ratios, however without further exploring their concrete effects. Guo et al. (2007) propose a coordination scheme along the lines of Dantzig– Wolfe decomposition.98 They assume a central auctioneer with initial inventories of the goods traded. This auctioneer iteratively recombines proposals of the decentralized parties and determines market-clearing prices. Moreover, Guo et al. (2007) investigate the effects of different bidding strategies where parties do not follow the
94
See Karabuk and Wu (2002, p. 743). See Schneeweiss and Zimmer (2004, p. 687). 96 See Chu and Leon (2008, p. 484). 97 See Fink (2006, p. 351). 98 See Guo et al. (2007, p. 1345). 95
3.3 Multilateral Information Asymmetry
59
rules of the scheme, but try to usurp greater shares of the surplus by distorted cost reporting. Since these strategies only showed a minor impact on the performance of the scheme in their computational study, Guo et al. (2007) infer that their mechanism is robust regarding strategic behavior of parties. Similarly, Kutanoglu and Wu (1999) also rely on dual decomposition, but propose subgradient optimization to determine the prices communicated to the decentralized parties.99 Apart from a general introduction into the concept of combining auctions with dual decomposition, Kutanoglu and Wu (1999) provide customized model formulations for scheduling. Finite convergence of their scheme, however, cannot be guaranteed due to the presence of binary variables in the scheduling problems considered there. Lee and Kumara (2007)100 propose a coordination mechanism based on a “double auction market.” They formulate a Master Planning model with inventory decisions and call it a lot-sizing problem, which seems incorrect since this model neither includes any lot-sizing constraints of the type of (2.14) nor any binary variables. Their scheme requires that, apart from orders, the suppliers communicate their inventory cost functions and inventory capacities to the buyers. Lee and Kumara (2007) claim that parties adopt a truth-telling behavior, but they did not provide any analytical proof for that.101 Finally, there are two papers that include analytical proofs for the incentive compatibility of the mechanisms presented there. Fan et al. (2003) propose a mechanism based on ideas from dual decomposition.102 Similarly to Guo et al. (2007), decentralized parties iteratively generate and communicate proposals and their associated costs to a central entity, which determines the market-clearing prices. Fan et al. (2003) show the incentive compatibility of this mechanism, but do not provide any (analytical or empirical) insights about the convergence behavior, which does not allow further conclusions about the potential practical applicability of this mechanism.103 The paper of Nisan and Ronen (2001) provides a theoretical framework and analytical results regarding the design of incentive-compatible mechanisms in the area of task scheduling. Moreover, Nisan and Ronen (2001) propose a new mechanism based on randomization and derive a lower bound for its efficiency.
99
See Kutanoglu and Wu (1999, p. 813). See Lee and Kumara (2007, p. 4715). 101 Lee and Kumara (2007, p. 4724), argue that truth-telling is ensured since the final allocation is determined by a Vickrey-type auction. However, amongst others, they did not address the suppliers’ incentives to reveal their inventory cost functions and capacities truthfully. 102 See Fan et al. (2003, p. 1). 103 The latter drawback has also been pointed out by Guo et al. (2007, p. 1346), who in addition provide a simple numerical example, for which the scheme developed by Fan et al. (2003) does not converge. 100
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3.3.2.3 Discussion and Conclusions Summarizing the findings from our literature review, a series of generic requirements for the decentralized coordination of Master Planning can be recognized. In the following, we mention four of them, which seem mandatory for coordination in many practical settings and have been advocated in part by the literature, but have never been simultaneously covered within a single approach:
Bilateral information asymmetry Self-interested behavior by parties No involvement of a third party Applicability for complex LP or MIP
Bilateral information asymmetry, which is the most natural information status of independent parties in supply chains,104 has only been assumed by a small share of coordination mechanisms. Most of these mechanisms overcome the resulting difficulties by postulating team behavior of parties. Such behavior has the obvious advantage in case of information asymmetry that reliable data can be exchanged additionally, which helps the decentralized parties to identify coordinated solutions. Unfortunately, team behavior does not seem justifiable if a supply chain consists of independent, profit-maximizing (and potentially competing) parties. The party adopting team behavior by, e.g., freely disclosing the cost changes of proposals, runs the risk that the other party shows unfair behavior and usurps the lion’s share of the surplus generated. Moreover, many approaches rely on the involvement of a third party. Important advantages of such a third party are that its existence allows a direct application of Dantzig–Wolfe decomposition and that the third party can participate in bidding and surplus sharing.105 However, a third party is a strong assumption and not naturally given in supply chains. An artificial establishment of a (trusted) third party and its acceptance by all participating parties seems very problematic.106 A final and trivial requirement for coordinating Master Planning is the applicability of the coordination mechanism to complex LP or MIP. Complex LP can be tackled by mathematical decomposition procedures, whereas their transfer to MIP raises additional problems. For the application of primal decomposition, the identification of dual functions would become necessary. Their identifiability for MIP has be shown;107 however, both the associated computational effort for their generation108 and the amount of data involved and, hence, exchanged among parties may cause difficulties, like the risk that parties’ private data can be reconstructed by other
104
See, e.g., Arikapuram and Veeramani (2004, p. 111). See, e.g., Wu (2003, p. 67), for a generic description of the possibilities of third parties to mediate bargaining processes. 106 See, e.g., Chu and Leon (2008, p. 484), for a paper supporting this argument. 107 E.g., Wolsey (1981, p. 173). 108 E.g., Guzelsoy and Ralphs (2008, p. 118). 105
3.3 Multilateral Information Asymmetry
61
parties.109 In dual decomposition, in turn, dual prices are not an effective guidance towards a systemwide improvement if duality gaps are large. A final argument against the application of approaches based on classical decomposition schemes is that not all managers have an understanding about the meaning of dual information and, hence, may be reluctant towards a disclosure of information of that kind. The lack of existing mechanisms that cover all of these requirements for the coordination of Master Planning constitutes the starting point for this thesis: In the following, we present novel mechanisms incorporating these requirements and evaluate the properties of these mechanisms by analytical reasoning and computational tests.
109
E.g., using inverse optimization, see Troutt et al. (2006, p. 422).
Chapter 4
New Coordination Schemes
This chapter and Chap. 5 contain the core of this thesis, new mechanisms for collaborative supply chain planning. Recapitulating the definitions of Sect. 2.3.1, a coordination mechanism is a contractual framework for coordinating the outcomes of (self-interested) actions of the decentralized parties. Each of the mechanisms presented here comprises a scheme specifying the proposal generation and the information exchange. For all schemes developed in this thesis, two variants will be provided that cover the different requirements for organizing the information exchange raised by the contractual frameworks presented in Chap. 5: An iterative, unilateral exchange of cost changes and a one-shot exchange by both parties. This allows us to describe and analyze the different schemes and frameworks separately. In this chapter, different schemes are presented and customized for the Master Planning models described in Sect. 2.2. We begin with schemes for the coordination of general decentralized LP problems (Sect. 4.1) and of one buyer and several suppliers planning based on uncapacitated dynamic lot-sizing models (Sect. 4.2), and derive analytical results about their convergence behavior. For a practical application of these schemes, however, two major limitations have to be considered: First, the scopes of the schemes are restricted to specific problem classes, i.e., LP and the MLULSP, which do not include some Master Planning problems such as the MLCLSP. Second, the solution quality of the generic scheme for LP decreases considerably with an increase of the underlying optimization models.1 Therefore, apart from adapting the schemes to a generic Master Planning setting in a two-party supply chain (Sect. 4.3), we additionally propose (heuristic) modifications, which yield a favorable performance even for complex MIP. Further extensions to capacitated lot-sizing, voluntary compliance by the supplier, lost sales, and Master Planning with several suppliers are provided in Sect. 4.4. Figure 4.1 illustrates the relationships between the different schemes proposed. Note that the single extensions presented in Sect. 4.4 can again be regarded as building blocks, e.g., the scheme for the MLCLSP is not restricted to full compliance, but can be applied in a voluntary compliance setting, too, as done in our computational study of Sect. 6.4. 1
See also the results of our computational tests in Sect. 6.1.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 4,
63
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4 New Coordination Schemes
Fig. 4.1 Relationships between Master Planning models
4.1 Generic Scheme for Linear Programming and Analytical Results In this section, we outline two versions of the generic scheme for LP that differ by their requirements on the information exchange. In the first, cost changes are reported by all but one party in an iterative fashion, whereas the second relies on a one-shot disclosure of cost changes by all parties.
4.1.1 Version with Iterative, Unilateral Exchange of Cost Information Consider a decentralized organization with the centralized and decentralized problems outlined in Sect. 3.3.2.1. In this version of the scheme, decentralized parties iteratively exchange proposals for the use of the central resources by parties i D 1; : : : ; I 1 compared to an initial solution.2 In addition to these proposals, the associated cost changes are reported unilaterally, i.e., parties i D 1; : : : ; I 1 report to party I both the cost changes due to their own proposals and the cost changes resulting from a (potential) feasible implementation of party I ’s proposals. Exemplarily for a three-party setting, the resulting information exchange is depicted in Fig. 4.2. Here, party I D #3 determines the first proposal (action 1), and so on.3 One iteration of the scheme comprises the submission of one proposal by each party and of the associated cost changes by the cost-reporting parties. For ease of exposition, we have assumed in Fig. 4.2 that the information is processed in a synchronous fashion. If proposals are exchanged asynchronously (e.g., the proposal by #1 and the associated cost changes are known to #3 earlier than the proposal by #2), the scheme can be applied without further changes. 2 For the scheme, the initial solution might be chosen arbitrarily; when applying the mechanisms proposed in Chap. 5, the initial solution will correspond to the default solution, the solution established without coordination. 3 This sequence is chosen for ease of exposition; other variants are possible, too (see Example 4.2).
4.1 Generic Scheme for Linear Programming and Analytical Results
65
Fig. 4.2 Information exchange within the generic scheme with unilateral exchange of cost information
Remarkably, the information exchange is restricted to proposals and their associated cost changes compared to the initial solution. Thereby, sensitive information like capacities and absolute (unit) costs are kept private. Both supply proposals and cost changes can be subsumed under primal information. The limitation of the exchange to primal information is one of the major advantages of the scheme proposed here since difficulties arising from the generation and the exchange of dual information can be avoided that way.4 The crucial feature that determines the performance of this scheme as well as the performance of any coordination scheme in general is the proposal generation. An obvious requirement for new proposals is that they potentially result in systemwide improvements. Moreover, the number of proposals generated in the course of the scheme should be modest since each proposal generated and each cost change disclosed involve transaction costs for the participating parties and some rough knowledge about the others’ modeling data. In all the schemes presented in this work, the proposals are generated by appropriately designed optimization models. In the scheme at hand, we use two different models for that purpose. The first of them, CS1i , is run by parties i D 1; : : : ; I 1. Its aim is twofold: Proposals are generated that show a potential for systemwide savings and address regions of the solution space that have not been investigated in previous iterations of the scheme.5 (4.1) .CS1i / max ZCS1i D ciT xist xi C kiT Ai xist xi s.t. ATi .xist xie /ki ciT .xist xie /
for e D 1; : : : ; e i
ki M ki 0 (3.11); (3.12): Sets XiE Set of proposals found so far 4 5
See our corresponding discussion at the end of Sect. 3.3. See Sect. 3.3.2.1 for basic models and input data used for the exposition of this scheme.
(4.2) (4.3) (4.4)
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4 New Coordination Schemes
Data Mi Vector consisting of big numbers that exceed marginal cost savings resulting in the central from increases resource use, i.e., MiT Ai xi1 xi2 > ciT xi1 xi2 with Mi 2 Rn ˇ ˇ e i Number of different solutions found so far for party i (e i D ˇXiE ˇ) xie Solution previously found, xie 2 XiE ; xist denotes the starting solution Variables ki Prices for changes in central resource use, ki 2 Rn The objective function of CS1i maximizes the potential savings of a new solution xi compared to a fixed starting solution xist .6 Potential starting solutions for party i comprise the solutions obtained by CS1i in previous iterations of the scheme and those resulting from an evaluation of proposals generated by the other parties.7 The second term of the objective function comprises penalties or bonuses for changes in the use of the central resources. Variables ki are prices for these changes (i.e., unit penalty costs and bonuses, respectively). They are determined by constraints (4.2) such that ZCS1i 0 with the repetition of a previous solution xie 2 XiE .8 In order to avoid unboundedness, constraint (4.3) establishes an upper bound for variables ki . Finally, (4.4) is a nonnegativity constraint for ki . After solving CS1i , the new supply proposal i D Ai xi together with i D ciT xi xii ni t , the associated cost change compared to xii ni t , are communicated to party I . xii ni t is the initial solution and is determined based on the initial resource allocation ii ni t . This data is retained by parties in form of the ordered sets ˘iE , the proposals previously identified for party i , and iE , the associated cost changes. Note that with fixed values for ki , CS1i corresponds to the subproblem of dual decomposition DSi .9 The fact that ki is determined endogenously in CS1i and not supplied by a master problem (or a central updating procedure like subgradient optimization) is the crucial point that allows us to avoid an exchange of dual information. Like in the subproblem of dual decomposition, the optimal solutions to CS1i are elements of Xiv , the set of vertices of DPi . To prove this, we start with a preliminary result characterizing the solutions to CS1i . f
f
Lemma 4.1 Consider CS1i with xi fixed to an arbitrary value xi . For ki , T the optimal solution to this problem, there are scalars i e 0 with kif
6 Note here the different meaning of “starting solution,” which is the solution that is used as the starting point for proposal generation, compared to that of “initial solution,” which is the first solution used in the scheme. 7 See also p. 71 for the statement of the model used for this evaluation. 8 Note that for ease of exposition, the redundant constraint with e D st has been included here. 9 See p. 53 for this model.
4.1 Generic Scheme for Linear Programming and Analytical Results
67
T T P ei Pei st xi xie Ai xist xif D M T i 0 eD1 ci xist xie i e and Ti0 C eD1 T ATi i e xist xif ATi . Proof. Write CS1fi for CS1i with xi D xif . CS1fi can be reduced to n max kiT Ai xist xif j kiT Ai xist xie ciT xist xie ki
8e D 1; : : : ; ei ; ki Mi ; ki 0g : CS1fi -D, the problem dual to CS1fi , is 8 ei < X T T ci xist xte i e j min Mi i 0 i : eD1
X ei
Ti0 C
e T
xist xt
T i e ATi xist xif ATi ; i e 0
8e D 0; : : : ; ei
eD1
9 = ;
:
According to duality theory, the objective function values of the optimal solutions to CS1fi and CS1fi -D are equal, which proves this lemma. In Lemma 4.2, we show that the outcome of CS1i is a feasible vertex solution, provided that the starting solution is not dominated by a convex combination of already existing solutions and that the objective function takes a positive value. Lemma 4.2 Let DPi only comprise continuous variables. Then: P ei 1. CS1i has a feasible solution if ciT xist ciT eD1 i e xie for all i e 0 with P P ei ei Ai xist Ai eD1 i e xie and eD1 i e D 1. 2. For xi , the optimal solution to CS1i , xi 2 Xiv holds. Proof. 1. Since C and, hence, DPi are feasible, it suffices to show that constraints (4.2)–(4.4) do not render CS1i infeasible. Since these constraints do not depend on xi , CS1i is feasible, provided the feasibility of CS1fi for at least one xif . To demonstrate the latter, we prove the feasibility and the boundedness of the dual problem CS1fi -D, which has been stated in the proof of Lemma 4.1. CS1fi -D has a feasible solution since i 0 can take any value greater than or equal to zero. The optimal solution to CS1fi -D is bounded, e.g., if ei X
ei T X st f ciT xist xie ie 0 8i 0; Ti0 C ATi : xi xie ATi ; ie xist xi
eD1
eD1
(4.5)
Due to the assumptions (1) of this lemma, ei X eD1
ciT
i i X X st xi xie i e 0 8i 0; Ai i e xist xie 0; ie D 1: (4.6)
e
e
eD1
eD1
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4 New Coordination Schemes
P ei P ei The condition eD1 i e D 1 can be omitted in (4.6) since (4.6) implies eD1 P ei ciT xist xie i e ˛ 0 and Ai eD1 i e xist xie ˛ 0 for any scalar ˛ 0 P ei with eD1 ˛i e D 1. Hence, together with (4.5) we get that CS1fi -D is bounded if T ATi 8Ti0 0. This inequality is fulfilled for, e.g., xif D xist . Ti0 xist xif Hence, CS1fi -D has a feasible, bounded solution and, according to duality theory, also CS1fi , and thus CS1i . 2. Since we only consider the properties of xi , we can omit (4.2)–(4.4). Then, CS1i reduces to max ciT xist xi C kiT Ai xist xi s.t. (3.11); (3.12): Each solution xi to this problem is a vertex of the polytope defined by inequalities (3.11), (3.12), and hence, a vertex solution to DPi .10 We further illustrate the effect of CS1i by an example. Example 4.1 Assume that there are two dimensions of central resource use, i.e., two dimensions of vector Ai xi contain non-zero elements.11 Denote the value of these dimensions by the scalars d1 and d2 . Moreover, set d1st D 0 and d2st D 0, i.e., the starting proposal is situated at the origin of the coordinate system. corresponding Further assume two proposals C D d C D d1C ; d2C and D D d D obtained in previous iterations of the scheme (see the left of Fig. 4.3). Note that this example comprises a special case, where the coordinates of new proposals always exceed those of the starting proposal. Hence, only penalties ki and no bonuses incur here (i.e., ki 0). We consider two new proposals, differing by the question whether they can be expressed by a scalar multiple of a convex combination of the proposals identified so far.12 First, we consider a proposal by such a PNew1, whiche can be Peexpressed New1 scalar multiple ˛ > 0, i.e., d New1 D eeD1 New1 d with D ˛. The e eD1 e penalty costs for New1 are determined endogenously in CS1i to ˛ times the savings resulting from a convex combination of these proposals compared to the starting proposal. The determination whether New1 has a potential for being found as a new proposal (which is the case if the associated penalty costs are smaller than the savings compared to the starting proposal) is illustrated in the right of Fig. 4.3. Compare the costs of New1 with the costs of the starting solution and the costs of Lk,
10
Note that DPi and CS1i may have several optimal solutions including non-vertex solutions. Here we assume that CS1i is solved using an algorithm that limits its search to vertex solutions (e.g., a primal simplex). 11 For ease of exposition, we skip the index i within this example. 12 A proposal Ai xin can be expressed by this multiple if and only if Ai xin lies within the cone spanned by the starting proposal and the proposals found so far (here: C, D), see the left of Fig. 4.3.
4.1 Generic Scheme for Linear Programming and Analytical Results
69
-> Fig. 4.3 Proposal generation by CS1i
P Pe e Lk a scalar multiple of New1, for which d Lk D eeD1 Lk e d with eD1 e D 1. If the cost change at New1 relative to its distance to the starting proposal is smaller than the relative cost change at Lk (like in the right of Fig. 4.3), then New1 has a potential for being found as a new proposal. Next, consider the proposal New2, for which no New2 exists such that d New2 D e Pe New2 e d . This is because in one dimension (d2 ), the coordinates of New2 eD1 e are not sufficiently large to be covered by a linear combination of proposals already found, unless incurring excess resource use in the other dimension (d1 ). When determining the penalty costs, CS1i will internally attribute all cost changes between the starting proposal and the other proposals identified to the other dimension (d1 ). For New2, we have an extreme case. Since the d2 -coordinate is zero, the penalty costs attributed to New2 will be zero, too. What is finally done in model CS1i , is an endogenous evaluation of the (real) costs and the penalty costs of all points of the solution space (among those New1 and New2). The solution with the smallest sum of the real costs and penalty costs will be chosen as the new proposal. The second model for proposal generation, CS2I , is only run by party I . Its mathematical formulation is as follows: min ZCS2I D cIT xI C
ei XX i 2P; i ¤I
.CS2I / s.t. AI xI C
ei XX
eD1
ie i e b0
(4.7)
i e D 1 8i 2 P; i ¤ I
(4.8)
i 2P; i ¤I
ei X
ie i e
eD1
eD1
i e 0 8i 2 P; i ¤ I; e D 1; : : : ; e i (3.11); (3.12):
(4.9)
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4 New Coordination Schemes
Variables i e Decision variables defining a convex combination of previous proposals about the central resource use of party i The objective function of CS2I minimizes the costs of party I that are due to xI and to a convex combination (defined by variables i e ) of the cost changes of other parties’ previous proposals. Constraint (4.7) takes care that the total capacity of the central resource exceeds the systemwide resource use, i.e., the sum of the new proposal of party I and the convex combination of the other parties’ proposals determined by i e . Constraints (4.8) ensure that i e actually define a convex combination of previous proposals, and constraints (4.9) enforce the nonnegativity of i e . Pei Note that in (3.11) and (3.12), i is set to I here. CS2I yields i D eD1 i e ie , the new proposals for the use of the central resources by parties i D 1; : : : ; I 1. These proposals are communicated to parties i D 1; : : : ; I 1 in each iteration of the scheme. CS2I is closely related to the master problem of Dantzig–Wolfe decomposition. The basic difference is that, instead of using two separate optimization models, the proposal of party I and the allocation of the central resources are determined simultaneously in CS2I (see also Fig. 4.4). The importance of this technical trick is stressed by its general applicability to other (classical) decomposition techniques. In principle, common primal and dual decomposition procedures can be transformed analogously and subsequently used for decentralized coordination without a central entity. Note, however, that not all disadvantages of classical decomposition techniques for supply chain coordination can be removed that way. When coordinating MIP models, the difficulties due to the exchange of dual information13 persist. Moreover, the mechanisms presented in Chap. 5 cannot be applied in combination with dual decomposition procedures since these procedures require bilateral or multilateral exchange of cost information (either directly or in form of dual prices), which is not feasible in these mechanisms.
Fig. 4.4 Comparison of classical decomposition techniques with the scheme with unilateral cost exchange
13
See p. 60 for a discussion of these difficulties.
4.1 Generic Scheme for Linear Programming and Analytical Results
71
Apart from CS1i and CS2I , a model for the evaluation of the other parties’ proposals is run by parties i D 1; : : : ; I 1. (CS-EVALi ) min ciT xi s.t. Ai xi il
(4.10)
(3.11); (3.12):
Data il Last proposal communicated by party I to i This model extends DPi by the additional constraint (4.10), which ensures that the resource use by party i does not exceed il . Relevant outcomes of CS-EVALi are
il D ciT xi xii ni t , the cost changes of a potential implementation of il , and the solution xi , that is used by party i as the starting solution for the next application of CS1i . Based on these models, a coordination scheme can be devised which identifies the systemwide optimum in a finite number of steps. We first present a basic version of this scheme in Algorithm 1, which requires all parties to run LP models. After that, we show how this scheme can be easily adapted such that convergence is achieved even if one party (party I) runs an MIP model. The function solve( ), that is used Algorithm 1, runs the corresponding models and extracts their outputs as specified above. Note that for reasons of computational effectiveness, CS1i should only be run if ZCS1i > 0 for party i in the last application of CS1i ; we did not include this in Algorithm 1 for ease of exposure. In Table 4.1, we summarize the input and output data for the models run within the scheme. The superscript new denotes the new proposals generated by CS1i . The solutions XiE remain private knowledge of parties i during the scheme. The set ˘iE consisting of proposals inew generated by party i and proposals il generated by party I , as well as the set of cost changes iE become known bilaterally among parties i and I .
Algorithm 1: GenericSchemeLPUnilateralCostExchange for i
1 to I 1 do solve(CS-EVAL i ) with i D ii nit
/* initialization */
repeat /* iteration */ for i 1 to I 1 do solve(CS1i ) with xist as the last outcome of CS-EVALi and communicate i and i to I solve(CS2I ) and communicate i separately to i D 1; : : : ; I 1 for i 1 to I 1 do solve(CS EVALi ) with i as the last proposal of I and communicate i to I until ZCS1i 0 8i and ZCS2I has not been improved compared to the last run of CS2I .
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4 New Coordination Schemes
Table 4.1 Input and output for the models solved within the generic scheme
CS1i
CS-EVALi
CS2I
Input
xil XiE
il
˘iE iE
Output
inew
inew xinew
il xil
il
Finite convergence of the scheme is shown by Theorem 4.1. Theorem 4.1 Following the scheme specified above, the optimal solution to C can be identified within a finite number of iterations if all decentralized decision problems can be formulated as LP models. Proof. The proof of this theorem is structured into five parts. In (a), we prove that CS1i always yields a feasible solution. This allows us to derive a lower bound on the costs of the solutions xi 2 Xi that have not been identified after the termination of the scheme (b). Based on that, we can characterize a subset of the solutions to C for systemwide costs are greater than or equal to the costs of which the x l D x1l ; : : : ; xIl , the systemwide solution resulting from the last run of CS2I (c). Since this subset comprises an r-neighborhood of x l , the optimality of x l follows (d). Last (e), we show that the convergence is finite. P ei (a) By Lemma 4.2, CS1i yields a feasible solution if ciT xist ciT eD1 i e xie Pei P ei st e for all i e 0 with Ai xi Ai eD1 i e xi and eD1 i e D 1. As a contraPei diction, assume the existence of scalars i e 0 with ciT xist > ciT eD1 i e xie , P P P 0 e e e i i i e Ai xist Ai eD1 i e xie and eD1 i e D 1. Set xi D eD1 i e xi . Since starting solutions xist are the optimal solutions for the last run of CS-EVALi , 0 ciT xist ciT xi holds for all feasible xi with Ai xist Ai xi . Hence, ciT xist ciT xi D P ei ciT eD1 i e xie and a contradiction results. (b) Hence, with the termination of the scheme, ZCS1i 0 holds. Then we get f ciT xist xi C kiT Ai xist xi 0 for any (fixed) solution xi . Setting xi D xi we get by Lemma 4.1 that there are i e 0 with ciT xi ciT xist C i 0 MiT ci
ei X st T xi xie i e ; eD1
Ti0 C
ei X st T T xi xie ATi i e xist xi ATi :
(4.11)
eD1
Next, we show by contradiction that there are i e 0 .i D 1; : : : ; e i / with ciT xi ciT xist ci
ei ei X X st T st T T xi xie ie ; xi xie ATi ie xist xi ATi : (4.12) eD1
eD1
4.1 Generic Scheme for Linear Programming and Analytical Results
73
T 0 Pei st 0 xi xie i e 8i e 0 Assume the contrary, i.e., that ciT xi < ciT xist ci eD1 T T Pei st 0 xi xie ATi i e xist xi ATi . Then we get with (4.12) that with eD1 there are i e 0 with i 0 MiT < ci
ei X
xist xie
T 0 i e i e
eD1
0 X T xist xie ATi i e i e : ei
0
8i e 0; Ti0
(4.13)
eD1
Due to the definition of Mi , MiT Ai xi1 xi2 > ciT xi1 xi2 holds 8xi1 ; xi2 2 T 0 T Pei st Pei st xi xie i e and xi2 D eD1 xi xie i e , we Xi . Setting xi1 D eD1 T Pei 1 P ei xi xi2 . i 0 MiT MiT Ai .xi1 xi2 / > ciT eD1 obtain i 0 Ai eD1 st 0 xi xie i e i e follows, which contradicts (4.13). (c) Consider Xis , the subset of feasible solutions to CS1i characterized by (4.12) P ei and eD1 i e 1. Obviously, 8xi 2 Xis , there are i e 0 with ciT xi P ei P 0 0 T ei xist xie i e C xist xist i e 8i e 0, eD1 i e 1, ciT xist ci eD1 Pei st 0 T T T and eD1 xi xie ATi i e C xist xist ATi i e xist xi ATi . Set P ei 0 0 0 i e for g D st, g D 0 otherwise, and define i e D i e C i e g D 1 eD1 P T ei xist xie i e , 8e D 1; : : : ; ei . Since xist 2 XiE , ciT xi ciT xist ci eD1 P Pei st T ei e T ATi i e xist xi ATi , and eD1 i e D 1 8xi 2 Xis . eD1 xi xi s Hence, 8xi 2 Xi , there are i e > 0 with ciT xi ci
ei ei ei X X e T T T T X e T T xi i e ; xi Ai xi Ai i e ; i e D 1: eD1
eD1
(4.14)
eD1
With the termination of the scheme, party I is not able to identify an improvement l l compared to xI ; 11 ; : : : ; li 1;ei , the last solution determined by CS2I . Then, cIT xIl C
ei XX i 2P
i ¤I
eD1
li e ie cIT xI C
ei XX i 2P
i ¤I
b0 AI xI C
i e ie ; 8i e 0; xI 2 XI ; i ¤ I;
eD1
ei XX i 2P
i ¤I
eD1
i e ie :
(4.15)
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4 New Coordination Schemes
P ei Let xil be the optimal solution to CS-EVALi subject to i D eD1 l A x e . Re Pi e i Ti l e T e i ni t e e and i D Ai xi in (4.15), we get that i 2P ci xi placing i D ci xi xi P P ei T T cI xI C i 2P ci eD1 i e xi 8i e 0 for i ¤ I and xI 2 XI with b0 P i ¤I Pei P P AI xI C i 2P Ai eD1 i e xie . With (4.14), we get that i 2P ciT xil i 2P ciT xi i ¤I P 8xi 2 Xis with b0 i 2P Ai xi . P ei i e D > 1, (d) For any xi … XIs characterized by (4.12) with xist D xil and eD1 k s k l l there is a xi 2 Xi with xi D xi C xi xi k and k D 1=. Consider Br xist , an r-neighborhood of xil , with an Euclidean metric and r > 0 chosen such small 2 that r < xik xil 8xik . This neighborhood comprises solely xi 2 Xis . Then, P P T l T B 8xiB 2 Xi . Hence, on each edge outgoing i 2P ci xi i 2P ci xi followsP P l from xi , there is a solution xi with i 2P ciT xil i 2P ciT xi . Since no further improvement is possible when moving from xist in the direction of any of its outgoing edges for any i 2 P , x l is the optimal solution to C. (e) By Lemma 4.2, only vertices and, hence, a finite number of different solutions are identified by CS1i . Since each application of CS1i yields a new solution, provided that ZCS1i > 0, CS1i and, hence, CS2I are run a finite number of times. Theorem 4.1 provides the mathematical foundation of the scheme: For a large class of optimization problems, the systemwide optimal solution can be identified within a finite number of steps. This indicates that the scheme proposed is built on an analytically solid basis and suggests a wide applicability of the scheme. Of course, the finiteness does not guarantee that the optimum is always found within a practicable number of iterations. To speed up the convergence rates for huge problems, some (heuristic) modifications are provided in Sects. 4.3 and 4.4. To further illustrate the single steps undertaken by the scheme, we provide a numerical example. Example 4.2 Consider the following centralized optimization problem with a block-angular structure. max 5x1 C 6x2 C 8x3 C 5x4 C 5x5 C 6x6 s.t. x1 C 4x2 C 2x3 C 4x4 C 6x5 C 4x6 24 2x1 C 3x2 C 6x3 C 4x4 C 3x5 C 4x6 25 x1 C x2 x1 C 2x2
5 8 4x3 C 3x4
12 x5 C 2x6 6
x1 ; x2 ; x3 ; x4 ; x5 ; x6 0:
(4.16) (4.17) (4.18) (4.19) (4.20)
4.1 Generic Scheme for Linear Programming and Analytical Results
75
Fig. 4.5 Graphical representations of DPi
Assume that there are three decentralized parties #1, #2, and #3 that respond for decisions .x1 ; x2 /, .x3 ; x4 /, and .x5 ; x6 /, respectively. Then the decentralized problems DP#1 , DP#2 , and DP#3 can be stated as follows: max 5x1 C 6x2 s.t. (4.17); (4.18) x1 ; x2 0 (4.21)
max 8x3 C 5x4 s.t. (4.19) x3 ; x4 0
max 5x5 C 6x6 s.t. (4.20) x5 ; x6 0:
In Fig. 4.5, we display the solution spaces and objective functions of DPi graphically. Further assume that the initial solution is determined by upstream planning and that parties #1 and #2 report their cost changes to party #3 during the scheme. Let #3 be the leader determining his central resource use first and #2 and #1 successively determine their production plans using the remaining central resources. The optimal solution to DP#3 is to produce x5 D 6 and x6 D 0. Although this proposal violates (4.16) and, hence, is infeasible for the whole system, it can be used as the initial solution for running the scheme. As the responses of #2 and #1 to this proposal, we assume x1 D x2 D x3 D x4 D 0 [other responses would lead to a greater violation of (4.16)]. The resulting proposals 11 D .0; 0/ and 21 D .0; 0/ are communicated to #3. Next, models CS1#1 and CS1#2 are run. Since x1 D x2 D x3 D x4 D 0, these models correspond to DP#1 and DP#2 in this step. Their optimal solution is to produce x1 D 2, x2 D 3, x3 D 3, and x4 D 0, leading to objective function values of 28 and 24, respectively. Resulting are proposals about central resource use 12 D .2 C 12 D 14; 4 C 9 D 13/ and 22 D .6; 18/ and cost changes 12 D 28,
22 D 24 to the initial solution that are communicated to #3 by #1 and #2 ( 11 and
21 are 0).
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4 New Coordination Schemes
Next, #3 solves CS2#3 . With c5 D 5 and c6 D 6, this model can be formulated as follows: max 011 C 021 C 2812 C 2422 C 5x5 C 6x6 s.t.
011 C 1412 C 021 C 622 C 6x5 C 4x6 24 011 C 1312 C 021 C 1822 C 3x5 C 4x6 25 11 C 12 D 1 21 C 22 D 1 11 ; 21 ; 12 ; 22 0:
The optimal solution to this model is x5 D 0, x6 D 2:25, 11 D 0, 12 D 1, 21 D 0:833, 22 D 0:167, leading to a new lower bound of 45.5 on the systemwide optimum. #3 communicates the resulting proposals 13 D .14; 13/ and 23 D .1; 3/ to #1 and #2, respectively. Then, the turn is again of parties #1 and #2. They evaluate these proposals by models CS-EVALi . Exemplarily, we provide the formulation of CS-EVAL#1 . max 5x1 C 6x2 s.t. x1 C 4x2 14 2x1 C 3x2 13 (4.17); (4.18); (4.21): Models CS-EVALi yield x1 D 2, x2 D 3, x3 D 0:5, x4 D 0, with associated cost changes of 13 D 28 and 23 D 4, which are communicated to #3 in turn. The next step of the scheme is that #1 and #2 run models CS1i again. Noting that 14 AD , we can exemplarily state CS1#1 below. 23 max 4 .x1 2/ C 6 .x2 3/ C k1 ..2 x1 / C 4 .3 x2 // C k2 .2 .2 x1 / C 3 .3 x2 // s.t. ..2 0/ k1 C 4 .3 0// C .2 .2 0/ k2 C 3 .3 0// 28 0 (4.22) ..2 2/ k1 C 4 .3 3// C .2 .2 2/ C 3 .3 3// k2 28 0 (4.23) (4.24) k1 4:00001 k2 2:00001
(4.25)
k1 ; k2 0 (4.17), (4.18), (4.21): Since the starting solution is equal to one of the solutions previously found, (4.23) is redundant. The upper bounds for k1 and k2 in (4.24) and (4.25) have been set to 4.00001 and 2.00001, respectively. We exemplarily provide the reasoning for determining the upper bound for k1 : An increase in the use of the first central resource
4.1 Generic Scheme for Linear Programming and Analytical Results
77
by 1 can be caused by increases of 1x1 or 1=4x2 , which correspond to increases in the objective function of DP#1 of 4 and 3=2, respectively. Hence, the maximum marginal cost savings are 4. Noting that the upper bound for k1 has to exceed this value, we have set this bound to 4:00001.14 The optimal solution to this model is the point of the solution space where #1 expects the maximum increase in the systemwide savings compared to the initial solution. Since only one proposal different from the starting proposal has been found so far,15 the complete cost difference between the proposals will be (endogenously) attributed to one single resource in (4.22). That way, the penalties imputed in the objective function can be minimized and the bonuses can be maximized. Here it is optimal to attribute this cost difference to the first resource, i.e., to set k1 to its maximum (4) and k2 to 0. The optimal solution is x1 D 5, x2 D 0 with an objective function value of 61 consisting of gains of 25 from the decentralized decisions and a bonus of 36. Analogously, the optimal solution to CS1#2 can be determined to x3 D 0, x4 D 4. As before, the resulting proposals 14 D .5; 10/, 24 D .16; 16/ and the cost changes 14 D 25, 24 D 20 are communicated to #3. The subsequent solution of CS2I by #3 yields the systemwide optimum: x1 D 5, x2 D 0, x3 0:23, x4 D 0, x5 1:64, x6 2:18, with an objective function value of 48:09. This solution comprises a vertex of #1, and solution vectors of the interiors of the solution spaces of DP#2 and DP#3 . However, the scheme is not finished here. In the last iteration, the vertex solution x1 D 0, x2 D 4 is identified by CS1#1 . After that, the scheme terminates since neither CS2i nor CS1i yield further new solutions. Table 4.2 summarizes the proposals generated together with the associated cost changes and solutions in each step of the scheme. Next, we show how to extend this scheme such that it also yields finite convergence if party I runs an MIP. The mere change is that in case CS1i yields no new solution, CS1i is run based on other starting solutions among XiE . For proving convergence, no specific rule for determining the starting solutions is
Table 4.2 Single steps in the numerical example Iteration Models x1 x2 x3 x4 x5
x6
0 2 2
0 3 3
0 3 0.5
0 0 0
6 – 0
0 – 2.25
2 .0; 0/ .6; 18/ .1; 3/
2
Initialization CS1i CS2I , CS-EVALi
1 .0; 0/ .14; 13/ .14; 13/
1
0 1 1
0 28 28
0 24 4
2 2
CS1i CS2I , CS-EVALi
5 5
0 0
0 0.2
4 0
– 1.6
– 2.2
.5; 10/ .5; 10/
.16; 16/ .0:4; 1:2/
25 25
20 1.6
3 3
CS1i CS2I , CS-EVALi
0 5
4 0
0.2 0.2
0 0
– 1.6
– 2.2
.16; 12/ .5; 10/
.0:4; 1:2/ .0:4; 1:2/
24 25
1.6 1.6
14 15
Other possible choices are, e.g., 4.00000001, 5, or even 1,000. This is because the proposals determined by CS-EVAL#1 and CS1#1 have been equal here.
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4 New Coordination Schemes
Algorithm 2: GenericSchemeUnilateralCostExchangeOneMIP for i
1 to I 1 do i nit xii nit ˚solve(CS-EVAL i ) with i D i
XiE xii nit
/* initialization */
repeat /* iteration */ for i 1 to I 1 do XiST XiE repeat xist random(XiST ) XiST XiST nxist xinew solve(CS1i ,xist ) until ZCS1i > 0 or XiST D fg XiE XiE [ xinew communicate inew and inew to I solve(CS2I ) and communicate i separately to i D 1; : : : ; I 1 for i 1 to I 1 do solve(CS EVALi ) with i as the last proposal of I and communicate i to I until ZCS1i 0 8i D 1; : : : ; I 1 and ZCS2I has not been improved compared to the last run of CS2I .
necessary; they, e.g., can be chosen randomly as done in our computational study of Sect. 6.1.16 The scheme terminates if no new solution can be found for any choice of xist 2 XiE . Algorithm 2 describes the extended scheme formally. There, the function random(XiST ) randomly chooses the starting solution among those previously found, except for such solutions which have already yielded ZCS1i 0 in earlier steps.17 Theorem 4.2 shows that the extended scheme yields finite convergence even in case of one decentralized MIP. Define Xivd as a set of vertex solutions to DPi that weakly dominates Xiv , i.e., for all xi 2 Xiv , there are scalars i e with ciT xi P ei Pei Pei vd T e e e eD1 i e ci xi ; Ai xi eD1 i e Ai xi , xi 2 Xi , and eD1 i e D 1. (Note that several dominating sets Xivd may exist for each Xiv ; the choice among them is irrelevant for our subsequent demonstration.) Theorem 4.2 Following the scheme specified above, the optimal solution to C can be identified within a finite number of iterations if the decision problems of the cost-reporting parties can be formulated as LP models. The decision problem of the remaining party may be of an LP or MIP type.
16
A further potentially favorable rule is to choose the initial solution as the starting solution, and only if no new proposal can be found, to randomly select another solution. 17 If ZCS1i 0 has held in earlier steps, this obviously extends to the further steps of the scheme, where the only difference in CS1i is that XiE is augmented by additional solutions.
4.1 Generic Scheme for Linear Programming and Analytical Results
79
Proof. The proof is structured into three steps. We begin with demonstrating that all elements xi of a set Xivd are identified (a). Based on that, we show that CS2i identifies the optimal solution to C (b). Finiteness is proven in (c). (a) We prove by contradiction that all xi 2 Xivd for all i D 1; : : : ; I 1 have been identified if ZCS1i 0 8i D 1; : : : ; I 1. As the contradiction, assume that there P ei is a xif , for which no scalars i e 0 exist with ciT xif ciT eD1 i e xie ; Ai xif P P ei ei Ai eD1 i e xi e , xie 2 Xivd , and eD1 i e D 1. Recall the following insight from Theorem 4.1 [see (4.11)]: For each solution xi , and hence, for each xif , there are scalars i e 0 with ciT xif ciT xist ci
ei ei T X X st T st T xi xie ie ; xi xie ATi i e xist xif ATi : eD1
eD1
(4.26) xif
As the next step of the proof, we show that for each , there is at least one xist P ei with i e and eD1 i e 1 complying with (4.26). For that purpose, we consider a set XiP XiE for whose elements xip scalars ip > 0 exist with ciT
X
xip ip
ciT
p
ei X
xie 0i e ; Ai
80i e ; Ai
ei X
xip ip Ai xif ;
p
eD1
p2Xi
X
ip D 1
p
p2Xi
p2Xi
0i e xie Ai xif ;
eD1
X
ei X
0i e D 1:
(4.27)
eD1
Define scalars i e with i e D i e for e ¤ st and xie 2 XiP , i e D 0 otherwise. Then (4.27) can be transferred to ciT xist
ci
ei X
xist
T xie
eD1
i e
ciT xist
ci
ei X
xist
T xie
eD1
0i e ;
ei X
i e < 1
eD1
ei T X st T f xi xie ATi i e xist xi ATi ; eD1
80i e ;
ei ei T X X st T f xi xie ATi 0i e xist xi ATi ; 0i e D 1: eD1
(4.28)
eD1
T Pei st xi xie i e in (4.28) corresponds to the obThe expression ciT xist ci eD1 jective function of CS1-fi 18 with i e D i e for e D 1; : : : ; ei and i 0 D 0. Due to
18
See Lemma 4.1 for this model.
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4 New Coordination Schemes
Pei the convexity of CS1-fi in i e , (4.28) is also valid for all 0i e with eD1 0i e > 1. Pei st P 0 Hence, for any xi 2 Xi , if (4.26) holds for a scalar i e 0 with eD1 0i e > 1, P ei then there are scalars i e 0 with eD1 i e 1 for which (4.26) holds. As a conP ei sequence, there is at least one xist 2 XiE , for which there are i e with eD1 i e 1 that comply with (4.26). T Pei Pei st Since ciT xist ci eD1 xi xie i e remains constant if eD1 i e 1 and only i e with e D st is varied, there is at least one xist with i e 0 such P ei Pei Pei f that eD1 i e D 1, ciT xi ciT xist ci eD1 .xist xie /T i e , and eD1 .xist P ei f T T e T T st e E T f T xi / Ai i e .xi xi / Ai . Hence, there are xi 2 Xi with ci xi ci eD1 P ei P ei f i e xie , Ai xi Ai eD1 i e xie , and eD1 i e D 1, which contradicts the definif tion of xi . (b) Next, we prove by contradiction that solving CS2i yields the optimal solution to C, given that all elements of Xivd have been identified. As the contradiction, assume that the decisions xi of (at least) one decentralized party i , though being optimal in C, cannot be expressed by a linear combination of elements among Xivd . For each xi … Xivd , there is a convex combination of xie 2 Xivd leading to the same or P ei P ei lower costs as xi , i.e., ciT eD1 i e xie ciT xi with eD1 i e Ai xie Ai xi and P ei vd eD1 i e D 1 hold due to the definition of Xi . Substituting xi in the optimal Pei solution to CS2i by eD1 i e xie yields an objective function value smaller than or equal to that for xi and does not violate the constraints of CS2i , (4.7), (4.8), (4.9), (3.11), and (3.12). Hence, a contradiction results. ˇ vˇ ˇ (c) Since CS1i yields vertex solutions only, CS1i is run at most ˇX ˇ i ˇtimes for each proposal generated. Since the number of vertices and, hence, ˇXiv ˇ is finite, the validness of this theorem follows directly. Note that in contrast to the base version of the scheme, a considerably larger share of the solution space has to be investigated in order to prove optimality. All nondominated vertices adjacent to all solutions previously found have to be generated here, whereas the base version only requires the identification of the non-dominated vertices within the neighborhood of the best solutions found so far.
4.1.2 Version with One-Shot Exchange of Cost Information The scheme outlined above can only be applied in combination with coordination mechanisms that allow for an iterative disclosure of cost information. In the following, we provide a modification that is applicable for a different form of information exchange, a one-shot disclosure of cost changes by all parties. This version shows somewhat increased computational complexity, but finite convergence, provided that all decentralized problems can be modeled as LP problems.
4.1 Generic Scheme for Linear Programming and Analytical Results
81
Here we also assume an (iterative) exchange of supply proposals among parties, but without the disclosure of any cost effects throughout the proposal generation. Model CS1i is used for proposal generation, too. However, this model is applied by all parties and the proposal generation terminates if models CS1i yield no further improvement for any party i D 1; : : : ; I and any starting solution, which is chosen among the solutions previously generated – analogously to the scheme for one decentralized MIP (see Sect. 4.1.1). After proposal generation, parties simultaneously communicate the cost effects of their proposals to each other (see Fig. 4.6). The proposal finally implemented can be determined by the master model MP.19 This model can be solved by any decentralized party since all parties hold the information needed. We summarize the single steps of this scheme in Algorithm 3. Theorem 4.3 shows that for this version finite convergence holds, too, provided that all decentralized problems can be modeled as LP.
Fig. 4.6 Generic scheme for LP with one-shot exchange of cost information
Algorithm 3: GenericSchemeLPWithOneShotExchangeOfCostChanges for i
1 to I do xii nit ˚solve(CS EVALi ) with i D ii nit
XiE xii nit
repeat for i 1 to I do XiST XiE repeat xist random(XiST ) XiST XiST nxist xinew solve(CS1i ,xist ) until ZCS1i > 0 or XiST D fg communicate inew to the other parties i XjE XiE C xinew until XiST D fg 8i communicate all cost changes among parties solve(MP)
19
See p. 54.
/* initialization */
/* iteration */
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4 New Coordination Schemes
Theorem 4.3 Following the scheme outlined above, the optimal solution to C can be identified within a finite number of iterations if the decision problems of all parties can be formulated as LP models. Proof. This proof is closely related to that of Theorem 4.2. From the proof of Theorem 4.2 we know that all xi 2 Xivd are identified by iteratively running CS1i and that this only requires a finite number of steps. What remains to show here is that MP identifies the optimal solution to C after the generation of all xi 2 Xivd for all parties i . We proof this by contradiction. As the contradiction, assume the existence of a xi … Xivd for which MP yields a lower objective function value. For each xi … Xivd , there is a convex combination of xie 2 Xivd leading to the P ei P ei same or lower costs, i.e., ciT eD1 i e xie ciT xi with eD1 i e Ai xie Ai xi and P ei vd eD1 i e D 1 hold due to the definition of Xi . Substituting xi in the optimal P ei solution to MP by eD1 i e xie yields an objective function value smaller than or equal to that for xi and does not violate the central resource constraints (4.7). Hence, a contradiction results. Compared to the scheme for LP with an iterative exchange of cost changes, the different form of information exchange has been gained at the expense of the same disadvantage valid for the extended scheme for one decentralized MIP: All non-dominated vertices have to be generated. Anyway, like in Dantzig–Wolfe decomposition, the identification of all (relevant) vertices is too demanding in general since their number increases exponentially with the problem size; computational results for interrupting the schemes after a given number of iterations are provided in Sect. 6.1.2. Moreover, for tackling larger problem instances and multiple MIP models, heuristic versions are proposed in Sects. 4.3 and 4.4.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results In contrast to the generic scheme for LP, the scheme for uncapacitated lot-sizing only requires an exchange of proposals, which are directly evaluated in terms of a systemwide improvement. Moreover, the proposals are only generated by the buyer. Thereby, no knowledge about the others’ costs changes is needed, which makes this scheme directly applicable for all forms of information exchange specified by the mechanisms presented in Chap. 5. The scheme can be applied by one buyer performing single sourcing with one or multiple suppliers. We depict the information flow for a setting of one buyer and two suppliers in Fig. 4.7. The optimization model for the buyer’s proposal generation is: min .2.120 / (MLULSP-CS) s.t. .2.20 /; .2.60 /; .2.80 /; .2.110 /; .2.130 / .2.150 / Y Sjt Ykt
8j 2 J D ; k 2 Sj \ J B ; t 2 T (4.29)
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
83
Fig. 4.7 Scheme for uncapacitated lot-sizing
X X j 2J D t 2T
YSjt 0
Y Sjt
X
ysjup w
(4.30)
j 2J D
8j 2 J D ; t 2 T:
(4.31)
Data w Target for the reduction of the number of orders for the items supplied ysjup Number of orders for item j in the proposal from upstream planning Variables Y Sjt Indicator variable (=1 if item j is ordered in period t, =0 otherwise) The objective function of MLULSP-CS and constraints (2.20 ), (2.60), (2.80), (2.110), (2.130)–(2.150) have been taken from the MLULSP20 and modified such that they only comprise items of the buyer’s domain, i.e., j 2 J B . Constraints (4.29) imply that variable YSjt takes a value of 1 if one of the successor items of j has been set up in period t.21 Constraint (4.30) establishes an upper bound for the number of orders for the supplied items. (4.31) are nonnegativity constraints. The proposals generated by MLULSP-CS show two characteristics favorable for the coordination of lot-sizing problems: First, they comprise a reduction of the number of orders YSjt and, hence, of the number of the setups necessary for the supplier to produce the items supplied without any inventory holding. Second, this reduction affects the buyer’s costs as little as possible since the buyer can choose freely the items for which she performs the required reduction of her order frequency. The rationale behind the reduction of Y Sjt is that the supplier’s setup and holding costs may be reduced that way considerably (see also Example 4.3). In each iteration of the scheme, MLULSP-CS is successively run for a given target w for the reduction of the number of the buyer’s orders until a new proposal is found.22 Thereby, we initialize w by 1 (or by 0 if the default solution is different
20
See Sect. 2.2.2.1. Note that values greater than 1 are precluded by the internal logic of the optimization model. 22 Note here that MLULSP-CS does not always yield a proposal different to all previously found. If there is more than one predecessor item, a reduction of the number of setups for a buyer’s item might go along with a reduction of the number of setups for several items of the supplier and, hence, cause an overfulfillment of the setup target [i.e., some slack in (4.30)]. A proposal that has been generated under such an overfulfillment will be repeated at least once since further increases of the setup target by 1 will not enforce an additional reduction of the number of orders. 21
84
4 New Coordination Schemes
Algorithm 4: ProposalGenerationUncapacitatedLotsizing ys up solve(MLULSP-B) w 1 ˘E fg repeat new solve(MLULSP-CS, x st ,w) w wC1 ˘E ˘ E [ new ˇ ˇ P up ˇ Dˇ until new … ˘ E or j 2J D ysj w < J
from upstream planning) and increase this parameter by 1 with each application of MLULSP-CS. A description of the single steps for proposal generation is provided by Algorithm 4. Analogously to GM-B, MLULSP-B denotes the decentralized model of the buyer. After the termination of the proposal generation, the proposals are split according to the origin of the items supplied (i.e., which supplier delivers the items) and separately communicated to the suppliers, who evaluate the costs of these proposals thereafter. According to the requirements of the coordination mechanisms the scheme is embedded in, the cost effects can either be exchanged unilaterally by the suppliers or in form of a one-shot disclosure by all parties. Below, we illustrate the scheme by a numerical example for a buyer–supplier setting. Example 4.3 Consider a serial BOM structure with one item supplied. Assume a planning horizon of jT j=6[UT], setup costs of 8[MU], and holding costs of 1 [U/ (UT MU)] for both parties. Table 4.3 provides the demand data, the solution resulting from upstream planning, and the proposals generated by the scheme. In upstream planning, it is optimal for the buyer to place an order in each period. The same holds for the supplier when answering the buyer’s proposal, which in sum results in systemwide costs of 96[MU]. In her first proposal, the buyer is enforced to reduce her order frequency, and she aggregates the orders of the periods with the lowest demand, where the additional costs for the aggregation are lowest. This decreases the systemwide costs to 89[MU]. In the second proposal, the buyer is indifferent whether to aggregate the orders of periods 1 and 2 or 3 and 4; here she chooses periods 1 and 2. The third proposal constitutes the systemwide optimum. There, the buyer performs a further aggregation resulting in systemwide costs of 77[MU]. Interestingly, the decrease in the systemwide costs of 19[MU] compared to upstream planning went along with a modest cost increase of only 5[MU] for the buyer. The fourth and fifth proposals, in turn, show again greater systemwide costs due to the large increases in the holding costs incurred by the buyer. The scheme terminates after the generation of the fifth proposal.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results Table 4.3 Example for the scheme lot-sizing t D1 2 Proposal Party d D 10 10 0 (default) B 10 10 0 S 10 10
85
for the coordination of dynamic 3 10 10 10
4 10 10 10
5 9 9 9
6 9 9 9
costs[MU] 48 48
1 1
B S
10 10
10 10
10 10
10 10
18 18
0 0
49 40
2 2
B S
20 20
0 0
10 10
10 10
18 18
0 0
51 32
3 3
B S
20 20
0 0
20 20
0 0
18 18
0 0
53 24
4 4
B S
30 30
0 0
0 0
28 28
0 0
0 0
73 16
5 5
B S
58 58
0 0
0 0
0 0
0 0
0 0
149 8
For the special case of a two-stage serial or assembly supply chain with level end item demand,23 we are able to derive analytical results regarding the convergence behavior of the scheme. Note in this context that the condition of level end item demand is of special relevance for coordination. In simulation experiments for uncapacitated lot-sizing, the greatest peaks of the suboptimality with upstream planning occurred for test instances with level demand.24 First of all, we introduce a common assumption: Assumption 4.1 Marginal holding costs are greater than or equal to zero. Marginal holding costs are defined as the differences between the unit holding costs of items and the sum of the unit holding costs of their predecessor items. If holding costs are only made up by the costs for the capital bound,25 then the statement of Assumption 4.1 holds directly. Further necessary is the introduction of some basic notation
˚ (Fig. 4.8). We denote a single supply proposal for an item j 2 J D by j D xtj1 ; xtj 2 ; : : : ; xtj jT j . Furthermore, let be the proposal (matrix) comprising the single proposals j for all j 2 J D . Hence, an (optimal) implementation of a proposal is equivalent to the implementation of a given supply target.26 Note that an optimal implementation of a 23
Note that we consider end item demand only, i.e., we assume zero demand for intermediate items. A further (implicit) assumption is the time independence of holding costs hj . 24 See, e.g., Simpson (2007, p. 136) and our computational results for the MLULSP (p. 172). 25 In fact, other costs like those for shelf space use are often minor compared to the costs of capital bound in inventory, e.g., Schneeweiß (1981, p. 69). 26 See p. 26 for the modeling of the supplier’s implementation of a supply target.
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4 New Coordination Schemes
SJD ∩ J B
NB NS
JD
items
set of items
J
B
buyer
J S supplier „refers to this set“
Fig. 4.8 Illustration of some basic notation
proposal always leads to a nested solution here, where the supplier performs setups in the order periods of the buyer only. This characteristic is shared by all optimal solutions to the MLULSP, provided a serial or an assembly BOM.27 Denote the systemwide costs resulting from an implementation of by csys ./. Obviously, there is at least one proposal for which csys . / D csys holds, with csys as the costs of the optimal solution to the MLULSP. Moreover, define NjS as the number of the supplier’s setups in the planning interval for items j 2 J D and NkB as the number of setups for their successor items k 2 SJ D \ J B . Moreover, define PB as the set of supply proposals optimal for the buyer subject to any N B . I.e., PB is composed by proposals that are obtained by solving MLULSP-B augmented by X
Ykt NkB
8k 2 SJ D \ J B :
(4.32)
t 2T
PB contains all proposals resulting from any combination of NkB for all k 2 ˇ ˇ Sj \ J B . Therefore, jPBj increases exponentially with ˇJ D ˇ, which makes a direct enumeration of all elements of PB computationally expensive. Moreover, define PB csys D min2PB csys ./ as the systemwide costs for the implementation of the best proposal out of PB, CS as the set of solutions identified by the scheme, and CS csys as the costs for the best solution out of CS . In the following, we derive our analytical results about the convergence behaviour of this scheme. We begin with considering the quality of solutions resulting from the implementation of the best proposal among PB. In Lemma 4.3, we focus on the special case that NjB =NjS 2 N and jT j =NjS 2 N for each j 2 J D in the systemwide optimal solution. Lemma 4.3 For a two-stage serial or assembly supply chain with zero initial inventories and level end item demand, the following implication holds: If NkB =NjS 2 N
27
See Love (1972, p. 329) for the proof of this property for a serial BOM and Crowston and Wagner (1973, p. 16) for an assembly BOM.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
87
and jT j =NjS 2 N for each j 2 J D and each k 2 Sj \ J B in the systemwide optimal solution, 2 PB holds. Proof. We begin with considering a serial supply chain with one item supplied. Write N S and N B for the number of the supplier’s and buyer’s setups in the systemwide optimal solution. Denote the starts of the periods in which setups of the supplier ocS cur by tiS , with i D 1; : : : ; N S . Set tN S C1 D jT j and define the number of periods between subsequent setups of the supplier by tiS D tiSC1 tiS . Further define oi as the number of the buyer’s orders within tiS ; tiSC1 . Analogously, denote the buyer’s order periods of the systemwide optimal solution by tijB with i D 1; : : : ; N S and B B j D 1; : : : ; oi . Define tijB D tijBC1 tijB and set ti;e i C1 D ti C1;1 (see also Fig. 4.9). Then the buyer’s holding costs can be written as: t B 1
S
dhB
oi N X ij X X
t;
t D1
i D1 j D1
with hB as the unit holding cost of the item produced by the buyer. Obviously, the buyer prefers equal tijB for all i D 1; : : : ; N S and j D 1; : : : ; oi , which also holds for each 2 PB then. For the supplier, this is not necessarily true. In the following, we will show that the potential influence of the supplier’s costs is not sufficient to affect the systemwide optimality of equal tijB . Subsequently, we show that equal lengths for tijB ; : : : ; tiBoi are systemwide optimal for all i D 1; : : : ; N S and j D 1; : : : ; oi . The systemwide costs for an interval tiS ; tiSC1 are i csys
B tijB ; : : : ; tio i
t B 1
D oi scB C dhB
ij oi X X
t C scS C dhS
j D1 tD1
oX i 1
tijB
tiS
j D1
j X
! tikB
:
kD1
(4.33) hS denotes the holding cost of the item produced by the supplier and scS , scB the corresponding unit setup costs. The last term of (4.33) comprises the supplier’s holding costs. It is based on the rationale that the supplier’s inventory in periods Δt1S
N S=2
Δt2S
x N B=4 e.g., e1=2
x: setups supplier, e.g., t2S = 6
x
Δt11B
Δt12B
B Δt21
... x
x
x
x
B B x: setups buyer, e.g., t12 = 3, t21 =6
t (T=8) 1
Fig. 4.9 Illustration of some notation
5
88
4 New Coordination Schemes
between two adjacent orders of the buyer depends both on the duration of this order interval and on the inventory required for the fulfillment of subsequent orders within S S PtijB 1 ti ; ti C1 . Dissolving some of the sums, e.g., by t D1 t D tijB tijB 1 =2, Poi 1 B and setting tiBoi D tiS kD1 ti k , (4.33) can be written as 1 tijB tijB 1 i B @ AC csys D oi scB C scS C dhB (4.34) tijB ; : : : ; tio i 1 2 j D1 oP oP i 1 i 1 B B ! tiS tiS tik tik 1 j oX i 1 X kD1 kD1 B tijB tiS tik : C dhS 2 j D1 kD1
oX i 1
0
Calculate the partial derivative of (4.34) with respect to a tijB with j ¤ oi : i tijB ; : : : ; tiBoi 1 @csys @tijB
dhB D 2
!
oi 1
2tijB 1 2tiS C 2
X
kD1 oi 1
CdhS
tiS
tijB
X
tiBk C 1 !
tiBk
:
kD1
We obtain: i @csys tijB ; : : : ; tiBoi 1 @tijB
!
oi 1
D d .hB hS /
tiS
C
tijB
C
X
tiBk
:
kD1
(4.35) The associated Hessian matrix contains constants of 2d .hB hS / at its diagonal and of d .hB hS / at all other entries. Using Gaussian elimination, a matrix with entries of 0 except at its diagonal can be obtained. The entries at the diagonal are constant and take the sign of hB hS . Since hB hS due to Assumption 4.1, the Hessian matrix is positive definite and a minimum is obtained by equating (4.35) to zero. It follows tijB D tiBoi for all j ¤ oi . Since tiS and tijB are integer due to the assumptions of this lemma, equal values for all tijB are feasible and, hence, systemwide optimal. Next, we show that also the setups of the supplier are equidistant in the systemwide optimal solution. We assume an optimal choice for tijB , i.e., tijB D tiBk for all i D 1; : : : ; N S and j; k D 1; : : : ; oi , and consider the systemwide costs for the whole planning horizon: 2 tijB 1
tiS B tij
1
3
7 S X6 X 7 6 ti hB X csys tiS D N S scS C N B scB C d t C hS tijB tiS jtijB 7 : 6 B 5 4 tij iD1 tD1 j D1 NS
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
89
At first, we limit to the extreme case hB D hS . There, we obtain after some transformations (amongst others, dissolving some of the sums as above): NS
dhS X S 2 S S S B csys t1 ; : : : ; tN S D N scS C N scB C ti tiS : 2 i D1 (4.36) PN S 1 S S ti , we obtain for the partial derivatives: With tN S D jT j i D1 S @csys t1S ; : : : ; tN S @tiS
S D dhS tiS tN S
(4.37)
for all i ¤ N S . Setting (4.37) to zero yields the minimum of (4.36) since the associS ated Hessian matrix is positive definite. We obtain tiS D tN S as the condition for S B this minimum. Since ti and tij are integer due to the assumptions of this theorem, the optimality of equal order intervals, the buyer’s preferred solution, follows. The same holds for the case hB > hS a fortiori. Then the buyer’s preferences have an even greater impact such that the optimality of the buyer’s preferred solution is not affected. This result directly extends to general two-level BOM structures. In structures with more than one predecessor item (i.e., assembly), the sum of unit holding costs for the supplier’s predecessor items is equal to or smaller than the unit holding cost for the successor item due to Assumption 4.1. Since the supplier’s costs are not sufficient for any N B and N S to cause a deviation from the buyer’s preferred solution, this extends also for the above case, where the supplier’s holding costs are partitioned among different items. With more than one successor item, this result obviously extends due to the even greater impact of the buyer’s preferences. PB Based on this result, we derive a general upper bound for csys in Lemma 4.4.
Lemma 4.4 In serial and assembly supply chains where buyer and supplier comprise one production stage each, inventories are zero, and demand is time initial p p p PB invariant, csys =csys 18=17 11 4 6 25 6= 24 C 11 6 1:53. Proof. We have structured the proof of this lemma into five parts. First of all, we describe the basic methodology and provide an lower bound for csys (a). In (b)–(d), we derive an upper bound for a serial structure, thereby distinguishing between three subcases depending on the relationships between N B , N S , and jT j. In (e), finally, we show that this bound extends to assembly structures. PB PB (a) Obviously, cB cB with cB as the buyer’s costs for PB , the proposal B S out of PB with the same N and N as in the systemwide optimal solution, and cB as the buyer’s costs of the systemwide optimal solution. cSPB , the costs for an implementation of PB by the supplier, however, may exceed cS , the supplier’s
90
4 New Coordination Schemes
costs in the systemwide optimum. Hence, the extreme case for the cost excess of a proposal out of PB is reached for hB D hS , a setting, which we assume for the rest of this proof. In the following, we investigate an upper bound for the excess over the systemwide optimum, i.e., we determine ub
PB csys csys
:
(4.38)
A lower bound for csys can be determined via (4.36), replacing tiS by jT j =N S B B and tij by jT j =N there: scB N B C scS N S C csys
d jT j2 hB 2
1 1 : NS jT j
(4.39)
Next, we prove the result of this lemma for a serial supply chain with one item PB supplied. To determine an upper bound for csys , a variety of cases has to be distin B ˘ ˘ guished. Introduce the parameters r D N =N S and z D jT j =N B . In the rest of the proof for the serial structure, we distinguish the following cases and subcases: (b) r D 1 and z D 1, (c) r > 1 or z > 1 and T =N B 2 N, and (d) r > 1 or z > 1 and T =N B … N. PB Our basic methodology for determining the upper bound for csys is to consider PB the worst-case order pattern that may result from running MLULSP-CS for a given N B .28 For the supplier, a possible, not necessarily implementation of PB is considered. (b) Assume r D 1 and z D 1. Since the buyer prefers setup intervals with lengths differing at most by 1, each PB only comprises order intervals of the lengths 1 and 2. W.l.o.g., assume that in his implementation of PB , the supplier will only choose setup intervals that include 1 or 2 orders of the buyer. If N B =N S is integer (i.e., 1), the supplier will place setups in all order periods. The resulting costs directly correspond to those of the systemwide optimum then. Hence, we focus on the case that N B =N S is not integer. Consider the implementation of PB , where setup intervals contain either one or two orders of the buyer. For the intervals that comprise only one order, the supplier’s holding costs are zero. Hence, cSPB can only deviate from cS for setup intervals comprising two orders. cSPB exceeds cS if larger order intervals are subsumed into these setup intervals in the implementation of PB . This is due to the disproportional increase of the supplier’s costs with increased length of the setup intervals, an effect, which can be recognized by considering cSPB :
28
Especially with level demand, there may be several equally optimal outcomes of MLULSP-CS; which of them are determined by the solver running this model, depends on external factors like the model structure and the solver characteristics and cannot be determined a priori.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results S
cSPB N S scS C hS d
NB
NB
1
S N NX X
91
tijB
i D1 j D1
NS X
tiBk :
(4.40)
kDj C1
The value of the RHS of (4.40) depends on the temporal distribution of larger order intervals B ij within the supplier’s setup pattern (note that some notation introduced in the proof of Lemma 4.3 is used here). Consider NB
NB
S 1 NX NS X S S B ch ti D tij tiBk ;
j D1
kDj C1
the supplier’s holding costs between two subsequent setups dependent on tiS , the length of the setup interval. These costs increase disproportionally with an increase S S of tiS , which implies an increase of any B ij within ti ; ti C1 , i.e., chS tiS 1 C 1=tiS chS tiS 1 C 1=tiS :
(4.41)
As a consequence, the choice of setup intervals with equal lengths minimizes the supplier’s costs. However, there are two potential obstacles for such a choice by the supplier. First, order intervals of the buyer with same lengths may be adjacent, while a short order follows a large one and vice versa in the systemwide optimum. With equal-sized adjacent orders, the supplier may be enforced to replicate such a sequence in his order fulfillment and, hence, incur larger costs due to (4.41). The extreme case for that is reached for jT j ! 1 since this diminishes the effects of the fact that at least one setup interval contains only one order in both the implementation of PB and the systemwide optimum. The latter adds constant factors to nominator and denominator of (4.38) and, hence, increases this ratio. ub for jT j ! 1 is upper bounded by ub for the constellation r D 2 and z D 1 with integer N B =N S , with proposals comprising two adjacent orders of the length 1 and two further adjacent order intervals of the length 2 (see the left of Fig. 4.10) (Here each setup interval of the supplier contains an equal number of orders; the absolute value of jT j does not matter since this equally affects nominator and denominator of the RHS of (4.38)). In the systemwide optimum, instead, the buyer’s and supplier’s setups of period 3 would be postponed to period 4 there. Hence, the supplier’s holding costs are augmented by 5hB d 4hB d D hB d . Here as well as in further parts of this proof, the buyer’s and supplier’s setup costs can be omitted since they constitute constant factors in nominator and denominator of the RHS of (4.38) and, hence, do not increase this ratio. With the buyer’s holding costs of 2hB d , we obtain as an upper bound 5hB d C 2hB d 7 D : 4hB d C 2hB d 6
92
4 New Coordination Schemes N B/ N S integer
N B / N S not integer
z =1, r = 2, |T | = 6
z =1, r =1, |T | =4
Setups B
Setups B
Setups S
Setups S t |T |
t |T|
PB Fig. 4.10 Cases with maximum csys =csys in 1(a), (b)
Second, the buyer’s pattern may comprise two short order intervals and a large one in between them. Then this large interval and one of the shorter have to be combined by the supplier within one single setup. The case with an extreme cost increase over the systemwide optimum is reached for jT j D 4, N B D 3, and N S D 2 (see the right of Fig. 4.10). There, the supplier’s costs exceed by 2hB d hB d D hB d the systemwide optimum, where the buyer’s and supplier’s setups would take place in period 2 instead of 3. Again omitting the buyer’s and supplier’s setup costs, we obtain as an upper bound ub1 D
3 hB d C 2hB d D : hB d C hB d 2
Since 3=2 > 7=6, we get for ubb , the upper bound for the case (b), ubb D 3=2. (c) Assume that z > 1 or r > 1 and that jT j =N B is integer, but not necessarily N B =N S . For the proof of this case, we express the costs of the base setting (with N B and N S ) by a weighted sum of the costs of settings 10 and 20 , where N B =N S is integer. Consider a feasible, but not necessarily optimal implementation of the buyer’s proposal that comprises o D N B rN S setup intervals of the supplier with r C 1 orders of the buyer and N S o setup intervals with r orders of the buyer. Define 10 and 20 as variations of the base setting differing by the number of buyer’s 0 0 orders, N B1 D N B .r C 1/ and N B2 D rN B as well as by the number of time 0 0 horizons, T 1 D jT j .r C 1/ and T 2 D jT j r, but with equal numbers of supplier’s 0 0 setups, N S1 D N S2 D N B (as an illustration, see Fig. 4.11). In these settings, each setup interval comprises an equal number of buyer’s orders (r C 1 and r, respectively). Since jT j =N B is integer, the production schedule of the base setting and its associated costs can be directly expressed by the combination 10 20 of appropriate fractions of the schedules of 10 , 20 (and their costs csys , csys ), which comprises the same number of orders as the base setting. Hence, we obtain PB csys
o 10 N S o 20 c C c : sys NB N B sys
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
93
Base setting (| T | =6, N B = 3) N B =3 N S =2
Setups B Setups S
t
|T|
Setting 2‘ (T 2‘ = 6, NB2‘= 3, NS2‘= 3)
Setting 1‘ (T 1‘=12, N B1‘ = 6, N S1‘ = 3) Setups B Setups S t
t
T 2‘
T 1‘
Fig. 4.11 Illustration of settings 10 and 20
This expression can be written as PB csys
y .1 y/ 20 0 c1 C csys ; r C 1 sys r
(4.42)
rel;10 rel;20 and csys as the lower bounds with y D .r C 1/ 1 rN S =N B . Define csys for the costs with the relaxation of the integrality of tiS and tijB for settings 10 and 20 , respectively. Combining (4.42) with (4.38) and (4.39), we obtain as an upper bound: 1 r 1 2scB N B C 2scS N S C T 2 dhB y rC1 C .1 y/ B B jT j jT j N N : (4.43) 2 1 1 B S 2scB N C 2scS N C jT j dhB N S jT j Omitting the setup costs as above and introducing a new parameter ˛ with 0 ˛ 1 and N B D .r C ˛/N S , we obtain the upper bound: z r 2 C ˛ C 2r˛ r ˛ : .r C ˛/ .r C ˛z 1/
(4.44)
The partial derivative of (4.44) with respect to r is .˛ 1/ ˛z .1 C 2z .r C ˛// .r C ˛/2 .z .r C ˛/ 1/
:
(4.45)
Since 0 < ˛ < 1, z 1, r 1, and either z > 1 or r > 1, the value of (4.45) is always smaller than zero. We further calculate the partial derivative of (4.44) with respect to z: ˛ .˛ 1/ 0: .r C ˛/ z .r C ˛/2 1 Hence, ubc takes its maximum with either z D 1 and r D 2 or z D 2 and r D 1. In the former case, we obtain the bound
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4 New Coordination Schemes
2 C 4˛ : 2 C 3˛ C ˛ 2
(4.46)
p Equation (4.46) takes its maximum with ˛ D 1=2 3 1 , leading to an upper p p bound of 4 3= 3 C 2 3 1:07. For z D 2 and r D 1, we obtain the bound 1 C 5˛ : 1 C 3˛ C 2˛ 2
(4.47)
p p Equation (4.47) takes its maximum with ˛ D 1=5 6 1 . Since 25 6= p p p p p 24 C 11 6 > 4 3= 3 C 2 3 , ubc D 25 6= 24 C 11 6 1:2. (d) Assume that z > 1 or r > 1 and that jT j =N B is not integer. The strategy for PB this part of the proof is to express an upper bound on csys by the costs of settings B with time horizons of integer multiples of N . Upper bounds for the costs in these settings are ubc ubsys;U;L , with ubsys;U and ubsys;L as upper bounds on the systemwide costs in U and L, which can be calculated analytically ˙ ˘ by (4.39). Define T U D N B jT j =N B and T L D T U N B D N B jT j =N B . Then the number of larger order intervals, i.e., those with tijB D T U =N B , is equal to jT j T L . Hence, the buyer’s costs for PB with the number of buyer’s setups N B are 0
PB cB
1 TU TL B 1 B 1 NX NX B C D N B scB C hB d @ jT j T L t C T U jT j tA : t D1
t D1
Define U and L as variations of the base setting with the same parameterizations except for jT j; the time horizons of U and L are T U and T L , respectively (see Fig. 4.12 for an example). Denote the buyer’s costs for PB subject to N B in these settings by cB;U and cB;L , respectively. Introduce a new parameter ˇ D U L L with 0 < ˇ < 1 indicating the weights of the characjT j T = T T teristics of the setting U within the base setting solution. Then we obtain 0
TU NB
1
TL NB
1
1
X C B B X PB cB D N B scB ChB d B t C .1 ˇ/ N B tC @ˇN A D ˇcB;U C.1 ˇ/ cB;L : t D1
t D1
Next, consider an implementation of PB by the supplier, where the number of buyer’s orders contained in the setup intervals of the supplier differ by one at most. Due to (4.41), the supplier will try to accumulate the larger order intervals of the buyer into setup intervals comprising a smaller number of orders.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
95
Base setting (|T|=5) Setups B Setups S t |T |
Setting L (TL =4) Setups B Setups S
N B=4 N S=3
U Setting U (T =8)
t T
TU
L
t
Fig. 4.12 Example for settings U and L
Preliminarily, we will assume that the supplier can accomplish that there is no setup interval that comprises both more orders and a greater share of the larger setup intervals than other intervals (this assumption is relaxed below). Recall that for cSPB holds S
cSPB N S scS C hS d
NB
NB
1
S N NX X
i D1 j D1
tijB
NS X
tiBk :
(4.48)
kDj C1
Due to (4.41), for the supplier’s holding costs of a setup interval tiS with n1 orders of the buyer with lengths of tijB and n2 orders with lengths of tijB C 1 holds: chS tiS
n1 n2 chS n1 tijB C chS n2 tijB C 1 : n1 C n2 n1 C n2
(4.49)
Due to the above assumption that there is no setup interval that comprises both more orders and a greater share of the larger setup intervals than the others, we can set tijB C 1 D T U =N B , tijB D T L =N B , n1 D N S jT j T L , and n2 D N S T U jT j . (If T U =N B is not integer, this setting does not violate the validness of (4.48) since the supplier could then include shorter order intervals of the buyer into the setup intervals containing more orders of the buyer, which reduces his costs for the implementation of PB .) Then, combining (4.48) with (4.49), we obtain:
cSPB
0 B N BS 1 U N NS NS S L NX X X T N jT j T TU B N S scS C hS d C @ NB NB NB i D1
j D1
1
NB N BS 1 NS N S T U jT j NX T L X TL C A: NB NB NB
j D1
kDj C1
kDj C1
96
4 New Coordination Schemes
This expression can be written as cSPB ˇcS;U C .1 ˇ/ cS;L ;
(4.50)
; cS;L as the supplier’s costs in settings U and L, respectively. with cS;U Next, we consider the (more general) case where a setup interval of the supplier exists that contains a greater share of larger orders. This is only enforced if a large setup interval is located between two shorter ones within the buyer’s order pattern (see the right of Fig. 4.10). Again the extreme case is reached here for N S D 2 and N B D 3. (For other values of N S , N B , additional constant cost factors would be included for both buyer and supplier, which would lower the resulting relative suboptimality of PB .) In the systemwide optimal solution, the supplier will gather the two smaller order intervals into a single setup interval, resulting in costs of hB d z .z C 1/. The supplier’s costs of setting U, in turn, will amount up to hB d.z C 1/2 and of setting L, up to hB d z2 . Since ˇ D 1=3 holds here, the weighted sum of the costs of settings U and L (see 4.50) will amount up to hB d 1=3.z C 1/2 C 2=3z2 . Hence, the maximum excess of the supplier’s costs in the systemwide optimum over this weighted sum becomes
ex D
z .z C 1/ 2 2 z 3
C
1 3
.z C 1/2
:
(4.51)
p Equation (4.51) takes its maximum with z D 1 C 2. Since z 2 N and z > 1, the maximum cost excess is reached for z D 2. Then, D 18=17 follows. ex PB PB PB Hence, we obtain an upper bound of 18=17 xcU csys C .1 x/ cL , with PB PB PB cU and cL as the systemwide costs for the implementation of in settings U PB PB PB PB and L, respectively. As in (c), for cU and cL , cU ubc cU and cL ubc cL holds, where cU and cL can be calculated by analytically via (4.36). Hence, we obtain as a general upper bound: scB N B C scS N S 18ubc C 17 scB N B C scS N S C jT j2 hB d2 N1S jT1 j L 2 1 2 d 1 1 1 C .1 ˇ/ T h ˇ TU S U S L 2 B N T N T : 2 1 1 d B S scB N C scS N C jT j hB 2 N S jT j
(4.52)
Since the supplier’s and the buyer’s setup costs are constants in numerator and denominator, their omittance does not affect the validity of (4.52). With z D T L =N B and r D N B =N S , we obtain the upper bound rz2 z ˇ C rˇ C 2rˇz 18 ubc : 17 .ˇ C z/ .r .ˇ C z/ 1/
(4.53)
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
97
The partial derivative of (4.53) with respect to z is 18 r .ˇ 1/ ˇ .2r .ˇ C z/ 1/ ubc : 17 .ˇ C z/2 .r .ˇ C z/ 1/2
(4.54)
Since 0 < ˇ < 1, z 1, and r 1, (4.54) takes only values smaller than zero. The partial derivative of (4.53) with respect to r is 18 .ˇ 1/ ˇz .2rz C z 1/ ubc 0: 17 .ˇ 1 C r .z C rz 1//2 As above, we first consider the case z D 1 and r D 2. Then we obtain the bound 18 .3 ˇ/ .1 C ˇ/ ubc 1: 17 3Cˇ p This expressiontakes its maximum with ˇ D 2 3 3, leading to an upper bound p of 18=17 ubc 7 4 3 . With r D 1 and z D 2, we obtain the upper bound 18 2 C 3ˇ 2ˇ 2 ubc : 17 2Cˇ p This expression takesits maximum with ˇ D 6 2. Since the resulting bound p p 18ubc =17 11 4 6 > 18=17ubc 7 4 3 , p ubd D 18ubc =17 11 4 6 . Since ubd > ubb and ubd > ubc , p ubd D 18ub2a =17 11 4 6 1:53 is a general upper bound for a serial BOM. (e) For assembly structures, assume the following transformation of the BOM into a set of serial substructures (see Fig. 4.13): Each of the buyer’s items j is replaced by a set of items jk , with k 2 Rj and Rj as the set of the predecessor items of j , such that each item supplied can be combined with one specific successor P item. Moreover, determine the unit holding costs of items jk to hkj D hj hk = i 2Rj hi , with P hk as the holding costs of the respective predecessor item k. Since hj i 2Rj hi
due to Assumption 4.1, hkj hk holds. We assume any repartition of the buyer’s setup costs among the buyer’s items in the transformed BOM. If the same N B is chosen, then the costs for PB (for the worst possible setup pattern) of the transformed BOM are greater than or equal to those of the original BOM. For the transformed BOM, the costs of the systemwide optimal solution are smaller than or equal to those for the original BOM. Since the buyer’s setup costs are not needed for the proof for the serial structure presented above, ubd extends for each of these serial substructures and, hence, for the original BOM.
98
4 New Coordination Schemes
j= 1 Rj = {2,3,4}
2
1
Transformation
12
13
14
Buyer
3
4
2
3
4
Supplier
items
Fig. 4.13 Example for the transformation of the BOM
The main result of this section is presented in Theorem 4.4.29 Theorem 4.4 For a serial or assembly two-stage supply chain with level end item demand, zero initial inventories, and the production portfolio of the buyers consists of one item, the scheme identifies a solution costs exceed those ofthe sys whose p p p temwide optimum at maximum by 18=17 11 4 6 35 6 11 C 24 6 1 1:53. The number of iterations required increases linearly with the number of periods and the number of items supplied. Proof. At first, we show that (in the worst case) proposals of CS dominate the corresponding proposals out of PB, provided that the buyer produces one item only. Define N B;up as the number of setups in the upstream planning solution. Successively decreasing the number of orders for items j 2 J D leads to the coverage of all setup frequencies N B N B;up which are not dominated by other setup frequencies.30 To prove that the upper bound derived in Lemma 4.4 extends to N B > N B;up , consider 1 1 1 ; cSrel N B D scS N S C d jT j2 hS 2 NS NB the supplier’s costs subject to N B and relaxed integrality for tiS and tijB .31 Since cSrel N B increases with N B and the buyer’s costs are greater for N B > N B;up than for N B;up , the denominator of (4.38) is greater for N B than for N B;up . Hence, the upper bound derived in Lemma 4.4 for a proposal out of PB with N B;up is also valid with an order frequency of N B > N B;up in the systemwide optimal solution.
29
Note that a corollary to this theorem is stated in Albrecht (2008) (called Theorem 2 there), where convergence is shown under the additional requirements on N B =N S and jT j =N B specified in Lemma 4.3. 30 Since the buyer’s costs for a dominating frequency are lower, proposals with dominated frequencies – which are not identified by our procedure – are obviously inferior. 31 See also equation (4.39).
4.3 Application to Master Planning
99
What remains ˇ ˇ to show is that the number of proposals generated increases linearly with ˇJ D ˇ and jT j. In our procedure for proposal generation, w is initialized ˇ ˇ by P 1 (or 0) and increased in each iteration by 1 at most until w j 2J D ysjup D ˇJ D ˇ. ˇ ˇ ˇ ˇ P Since j 2J D ysjup ˇJ D ˇ jT j, the maximum number of steps is ˇJ D ˇ jT j.
4.3 Application to Master Planning In this and the next section, we present the necessary steps for customizing the generic scheme for LP to the Master Planning models presented in Sect. 2.2. We begin with a description of the linearization of the objective function of CS1i , a basic requirement for tackling this model with a standard solver. Next, we show how to adapt the generic scheme for LP to GM as the underlying model and provide modifications that improve the applicability of this scheme for MIP models and accelerate the convergence rate.
4.3.1 Linearization Model CS1i is nonlinear since two vectors of variables are multiplied in its objective function. Therefore, the optimal solutions to CS1i cannot be identified by standard MIP software like CPLEX or Xpress-MP, that can efficiently tackle large-scale practical optimization problems,32 but falls short in case of general nonlinearities like those in CS1i . Thus, the identification of an appropriate solution procedure for CS1i is an additional issue, that needs to be covered here. For nonlinear problems comprising continuous variables only (NLP), a broad range of solution procedures has been elaborated in the literature. A procedure frequently used is successive linear programming (SLP). SLP has been successfully applied for the solution of optimization models with few nonlinear terms, which are employed, e.g., in the petrochemical industry, where quality and volume of products have to be determined simultaneously.33 In SLP, a series of linear approximations of the nonlinear terms (by their first-order Taylor series expansions) is solved that often converges to a local optimum of the original problem. Many practical supply chain planning problems, however, comprise integer or binary variables. For the resulting nonlinear mixed-integer programming problems (MINLP), computational difficulties are considerably greater than for NLP.34 Most solution procedures for MINLP are based on Branch & Bound, Generalized Benders
32
E.g., ILOG (2008) and Dash (2008). See Baker and Lasdon (1985, p. 264). 34 See, e.g., Kallrath and Wilson (1997, p. 376). 33
100
4 New Coordination Schemes
Decomposition, and Outer Approximation.35 Like in the standard case (pure MIP), Branch & Bound uses the solutions of the subproblems with relaxed integrality constraints as lower bounds for the search tree.36 The computational effort of Branch & Bound for MINLP, however, is considerably greater than for MIP because of the limited possibilities for exploiting the information generated at nodes previously investigated. The two latter procedures, Generalized Benders Decomposition and Outer Approximation, iterate between solving a master problem and nonlinear subproblems with fixed integer variables. As the name indicates, Generalized Benders Decomposition37 directly extends Benders Decomposition to the solution of MINLP. In Outer Approximation,38 the master problem uses primal information from the nonlinear subproblems and constructs a linearization of the nonlinear terms in the objective function and the constraints around the primal solution. Global convergence of these procedures can only be shown for some additional conditions about the problem structure. One of them is concavity of the objective function for fixed values of the integer variables.39 Unfortunately, this condition is not fulfilled in CS1i , as shown by the following example. Example 4.4 Assume that the integer variables in CS1i have been fixed to arbitrary values and that the vector k comprises one dimension only. With ci D 0 and scalars k 0 D kiT and d 0 D Ai xist xi , the objective function of CS1i becomes k 0 d 0 . A sufficient and necessary condition for the concavity of k 0 d 0 is the positive definiteness of the associated Hessian matrix 0
0
H.k d / D
0 1 : 1 0
The eigenvalues of this Hessian matrix are the roots of the following equation: ˇ ˇ ˇ 1 ˇ ˇ ˇ ˇ 1 ˇ D 0: The roots are 1 D 1 and 2 D 1. Since both positive and negative eigenvalues exist, H .k 0 d 0 / is indefinite, from which non-concavity of the objective function of CS1i follows. The principal aim of the computational tests of this work40 is to evaluate whether the schemes are able to identify practicable solutions. For this purpose, a piecewise linear approximation of the nonlinear term of the objective function of CS1i turned
35
See, e.g., Floudas (1995, p. 112) and Kallrath and Wilson (1997, p. 379). See, e.g., Gupta and Ravindran (1985, p. 1534). 37 See Geoffrion (1972, p. 237). 38 See Duran and Grossmann (1986, p. 307). 39 See, e.g., Floudas (1995, p. 114 and p. 144). 40 See Chap. 6. 36
4.3 Application to Master Planning
101
out to be appropriate.41 In contrast to the procedures for MINLP mentioned above, this approximation can be carried out using standard mathematical programming software and showed sufficient accuracy for obtaining meaningful results in our computational tests. For ease of exposition, we apply the following transformation for the nonlinear term in the objective function of CS1i : Write kj0 for the value of the j -th dimension of vector kiT and dj0 for the value of the j -th dimension of vector Ai xist xi . P 0 0 Then, kiT Ai xist xi D N j D1 kj dj holds. Following a well-known programming trick,42 kj0 dj0 can be transformed into two quadratic expressions: 2 2 1 0 1 0 kj C dj0 kj dj0 kj0 dj0 D : (4.55) 2 2 0
0
0
0
ubjk , ubjd , lbjk , and lbjd , the upper and lower bounds for kj0 and dj0 , are determined 0 0 as follows: ubjd are lbjd are the maximum (one-sided) deviations of proposals about 0 central resource use from the starting proposal in CS1i . ubjk is set equal to M , a 0 large number, and lbjk equal to zero.43 2 Next, we address the linear approximations of f f kj0 ; dj0 D 1=2 kj0 C dj0 2 and f s kj0 ; dj0 D 1=2 kj0 dj0 , the quadratic terms of (4.55). Instead of focusing specifically on f f kj0 ; dj0 and f s kj0 ; dj0 ,44 we explain our methodology for a general quadratic function f .x/ D x 2 . We divide the curve of f .x/ into several intervals and approximate each of them by a linear function. In the example of Fig. 4.14, we have chosen three intervals for the approximation. The line-segments OA, AB, and BC approximate f .x/ for x 2 ŒxO ; xC . The nodes O and C are the lower and upper bounds of f .x/, and the nodes A and B are b .x/, breakpoints. xO , xA , xB , and xC are the values of these nodes on the x-axis. f the function defined by these line-segments, can be written as a weighted sum of the function values of the nodes: b .x/ D O x 2 C A x 2 C B x 2 C C x 2 : f O A B C O ,A ,B , and C are additional variables and can be interpreted as the weights of the nodes. If, e.g., x is the average of xB and xC , the weights for xB and xC will
41
A thorough investigation of the effectiveness of different MINLP solution procedures for CS1i , in turn, does not seem necessary for that purpose. 42 See, e.g., Williams (1993, p. 152). 0 0 43 For the specific determination of ubjd , lbjd , and M for GM used here, see p. 110 and p. 114. 44 In Sect. 4.3.2, we present a model where these functions are explicitly linearized for GM.
102
4 New Coordination Schemes
Fig. 4.14 Piecewise linear approximation of f .x/
C
f(x)
f (x) fˆ(x)
B
A O xO
x xA
xB
xc
be 1=2 each. The correct determination of these weights is assured by the following constraints: x D O xO C A xA C B xB C C xC ;
(4.56)
O C A C B C C D 1; O ; A ; B ; C 0;
(4.57) (4.58)
O ; A ; B ; C SOS 2:
(4.59)
Constraint (4.56) sets x equal to the weighted sum of the arguments of the functions used for the approximation, whereas (4.57) normalizes the sum of the weights to 1. Constraints (4.58) ensure the nonnegativity of the weights, and constraints (4.59) restrict variables O ,A ,B , and C to a Special Ordered Set of type 2 (SOS2), which implies that at most two adjacent variables within the SOS2 (e.g., B and C ) can be nonzero.45 Alternatively to the use of SOS2, constraints (4.59) could be reformulated using additional (standard) binary variables. Regardless of which formulation is chosen, the computational difficulty of the underlying model increases considerably due to the additional discrete decisions. Thereby, the SOS2 characteristic can be exploited by the optimizer solving the problem, which usually reduces solution time. Further note that constraints (4.59) can be omitted for the linearization of f s kj0 ; dj0 . Due to the negative sign of this term in (4.55), both the objective function value of CS1i and the slopes of the corresponding line-segments are decreasing with greater values for x. Hence, greater weights for breakpoints with smaller x are preferred in the optimal solution to CS1i , which ensures that the correct line-segment is chosen. The opposite holds for f f kj0 ; dj0 due to its concavity. Without (4.59) and small x, the optimizer solving the problem might choose, e.g., a value greater than zero for O and C , and one of zero for A and B .
45
See Beale and Tomlin (1970, p. 447).
4.3 Application to Master Planning
103
The lower and upper bounds for f f kj0 ; dj0 and f s kj0 ; dj0 are ubd =4 and 1=4 .ubk C ubd /. Further consideration deserves the choice of the breakpoints and the expected approximation error. Both issues are dealt with in Sharpe (1971).46 There it is shown that, given equal probabilities of each x 2 ŒxO ; xC , the (absolute) expected error for the approximation of a quadratic function can be minimized by choosing equal interval lengths. In this work, we follow this reasoning and also rely on equal interval lengths. Define the average approximation error by 1 ub lb
Z
ub
b .x/ f .x/ dx; f
lb
with ub and lb as the upper and lower bounds of the range that is subject to the approximation.47 With equal interval lengths, the approximation error becomes .ub lb/2 ; 6m2 with m as the number of approximation intervals. With, e.g., m D 5, we get an average approximation error of .ub lb/2 =150. Comparing this with the average of f .x/ within these bounds and assuming that each value for x is equally probable within the range considered, the average approximation error becomes 1=75 1:33% of the average of the approximated function.
4.3.2 Adaptation to Master Planning Next, we describe how to adapt the generic scheme for LP to a two-party supply chain planning based on GM.48 Let each party dispose of a subset of resources and produce a subset of items. The buyer uses items produced by the supplier as input for her production process. The interdependence between the parties’ submodels are (2.45), the inventory balance constraints for the items supplied. To make the central resources explicit, we reformulate these constraints as XBjt XSjt
8j 2 J D ; t 2 T;
(4.60)
XBjt XSjt
8j 2 J ; t 2 T:
(4.61)
D
Based on this decomposition, the optimization models used within the scheme can be derived directly.
46
See Sharpe (1971, p. 1269). In the preceding example, lb D xO and ub D xC hold. 48 See Sect. 2.2.1 for the mathematical formulation of this model. 47
104
4 New Coordination Schemes
First, we address the version of the scheme with unilateral iterative exchange of cost information. We assume for ease of exposure that the supplier communicates his cost changes to the buyer during the execution of the scheme. Then, models GM-CS2B , GM-CS1B , and GM-CS1S are run within the scheme.49 (GM-CS2B ) min CBd C
eN X
e cs e
eD1
s.t. .DC-B/ eN X
XBjt D
e e xbjt
8j 2 J D ; t 2 T
eD1 eN X
e D 1
(4.62)
eD1
e 0
for e D 1; : : : ; e: N X X C st (GM-CS1B ) min CBd C Kjt XBjt xtjt C Kjt
(4.63)
j 2J D t 2T
X X X Tjt C X TjtC j 2J D t 2T
s.t. .DC-B/ X X j 2J D
t 2T
X X
e C st xtjt xtjt cb st cb e 8e D 1; : : : ; eN Kjt
(4.64)
st e xtjt xtjt cb st cb e 8e D 1; : : : ; eN Kjt
(4.65)
j 2J D t 2T
st X TjtC X Tjt XBjt xtjt
8j 2 J D ; t 2 T
CBd C csys C Kjt mjt 8j 2 J D ; t 2 T mjt 8j 2 J D ; t 2 T Kjt C Kjt ; Kjt 0 8j 2 J D ; t 2 T X TjtC ; X Tjt 0 8j 2 J D ; t 2
49
(4.66) (4.67) (4.68) (4.69) (4.70)
T:
(4.71)
For a reversed information flow, GM-CS2S is applied instead of GM-CS2B . Further note that for adapting the models stated below to the MLCLSP and MLCLSP-C presented in Sect. 2.1.2, these models simply have to be augmented by the additional features for lot-sizing and campaign planning.
4.3 Application to Master Planning
105
X X C st Kjt XSjt xtjt C (GM-CS1S ) min CSd C Kjt j 2J D t 2T
X X X Tjt C X TjtC j 2J D t 2T
s.t. .DC-S/; (4.68)–(4.71) X X e C st xtjt xtjt cs st cs e 8e D 1; : : : ; eN Kjt j 2J D t 2T
X X
j 2J D
st e xtjt xtjt cs st cs e 8e D 1; : : : ; eN Kjt
(4.72) (4.73)
t 2T
st 8j 2 J D ; t 2 T: X TjtC X Tjt XSjt xtjt
(4.74)
Data cb e Buyer’s costs change of the previous proposal e compared to the initial solution (e D 1; : : : ; e; st denotes the starting proposal) cs e Supplier’s costs change of the previous proposal e Unit penalty costs for arbitrary deviations, small number, e.g., 0.000001 hbj Buyer’s unit costs for inventory holding of the supplied item j mjt Big number, denoting the maximum cost change per unit deviation in the supply quantities e xtjt Amount of item j supplied in period t in the previous proposal e Variables CBd Costs for the decisions of the buyer’s planning domain CSd Costs for the decisions of the supplier’s planning domain e Variables defining a convex combination of previous proposals e C Endogenously determined unit prices for positive deviations from the Kjt st starting proposal xjt of item j in period t st Kjt Unit prices for negative deviations of item j in period t from xjt st X TjtC Increase in the supply of item j in period t compared to xjt X Tjt Decrease in the supply of item j in period t Model GM-CS2B can be directly derived from CS2I and GM. For ease of exposition, (DC-B) and (DC-S) abbreviate the decentralized restrictions of GM-B and GM-S, respectively, augmented by the restrictions for an explicit determination of the supply quantities. (DC-B) comprises constraints (2.1)–(2.11) with the items and resources limited to those of the buyer as well as (2.42), (2.46), and (2.48). (DC-S) comprises constraints (2.1)–(2.3), (2.6), (2.8), (2.10), and (2.11) with the items and resources limited to those of the supplier, (2.47), and50
50 The modeling of these constraints instead of (2.43) and (2.44) – which have been introduced for clearness of exposition in Sect. 2.3.2 – is more direct and allows to reduce the number of variables used.
106
4 New Coordination Schemes
Ijt 1 C Xjt D XS jt C Ijt
8j 2 J D ; t 2 T:
Moreover, we have introduced additional variables representing the decentralized costs of parties: CBd D
X X
hj Ijt C
j 2J B t 2T
C
X X
j 2J B t 2T
lscj LSjt C
j 2J E t 2T
CSd D
X X j 2J S t 2T
X X
hj Ijt C
X X
scj Yjt C
X X
m2M B t 2T
X X
blcj BLjt C
j 2J E t 2T
hbj IBjt ;
j 2J D t 2T
X X
ocm Omt C
ocm Omt C
m2M S t 2T
X X
scj Yjt :
j 2J S t 2T
Sets J S Set of items produced by the supplier M B Set of resources of the buyer M S Set of resources of the supplier For GM-CS1B and GM-CS1S , some further explanations are necessary. For each set of central resources (4.60) and (4.61), different price variables have been introC duced. Their values Kjt , Kjt are determined by constraints (4.64), (4.65), (4.66), (4.67), (4.77), and (4.73) in analogy to the generic scheme. Thereby, the upper bounds for mjt have to be set to large numbers exceeding the potential cost changes with changes in the central resource use, i.e., shortages or excess supply. In the following, we show how mjt can be determined for GM without lost sales and zero backorders at the end of the planning interval, the basic version which we have tested computationally in Sect. 6.1. For the buyer, changes in the central resource use can affect several cost types. First, shortages in the supply may lead to backorders for end items. For determining the resulting costs, we introduce a new parameter blcj denoting the maximum costs for backorders caused by a shortage in the supply or the production of item j . We set
b
b
blcj D blcj C " for end items, with " as an arbitrarily small number, and determine this parameter recursively by maxk2Sj blck blcj D C" rjk
b
b
for intermediate items. Moreover, overtime may become necessary if production has to be increased in later periods due to shortages of supplied items. This may affect all successors of the items with shortages in their supply. We introduce a further new cum parameter, rjk , the amounts of the predecessor item j that are needed to produce cum D rjk if k is a item k (where k is not necessarily a direct successor of j ). rjk direct successor of j . If k is an indirect successor of j ,
4.3 Application to Master Planning
107
Y
cum rjk D
cum rjl ;
l2Gj k
with Gjk as the set of items that are located at one of the connecting lines between items j and k in the BOM. Assuming that overtime and backorder costs are of a greater order of magnitude than holding costs, we can set 8 <
b
mjt D max blcj ; :
X X
cum rjk amk ocm
m2MS k2J B
9 = ;
C ":
For the supplier, overtime and holding costs can be affected by changes in the supply pattern. Assuming that overtime costs are of a greater order of magnitude, we can set mjt equal to the maximum overtime costs caused by increases in the supply quantities. Hence, we can define for all j 2 J D mjt D
X X
cum rkj amk ocm C ":
m2MS k2J S
In GM-CS1S and GM-CS1B , three further features have been incorporated in order to cut off inferior solutions. First, additional penalty cost terms have been included into the objective functions. These terms consist of fixed unit penalty costs " multiplied with the absolute deviation from the starting supply pattern (expressed by variables X TjtC and X Tjt ). Such penalizations are favorable if several optimal solutions to GM-CS1S or GM-CS1B exist that differ by the supply quantities of particular items in some periods. Most probably, the other party prefers the proposal where these supply quantities come nearest to the starting proposal; hence, penalizing arbitrary deviations by very small costs helps to avoid unnecessary changes in the supply pattern that could cause increases in the other party’s costs. If " is sufficiently small, the convergence of the scheme is not affected. Note that a further favorable consequence of these penalty costs is that penalties for negative deviations for the buyer and positive deviations for the supplier and, hence, constraints (4.65) and (4.72) become redundant, given zero inventories at the beginning of the planning interval.51 Second, constraint (4.67) cuts off unfavorable solutions by keeping the decentralized costs below the costs of the systemwide best solution found so far. Note that this constraint can only be applied by the party that is informed about the other party’s cost changes. Third, in order to avoid excess inventory holding of supplied items at the buyer’s site, additional holding costs for the supplied items have been included into the objective functions of the buyer’s problems.52
51
This insight has also been used in our computational tests of Chap. 6. In this context, note that inventory holding of supplied items at the buyer’s site will become more expensive than at the supplier’s if the holding costs comprise capital costs only with identical
52
108
4 New Coordination Schemes
Recall that models GM-CS1B and GM-CS1S cannot be directly tackled by a standard solver due to their nonlinear objective functions. The linearizations of these objective functions can be carried out along the lines of Sect. 4.3.1, but have not been included in GM-CS1B and GM-CS1S for ease of exposition. Next, we exemplarily provide a formulation for GM-CS1B with an explicit linearization of the objective function. (GM-CS1LB ) min CdB C
X X p;C p; Cjt C Cjt C X Tjt C X TjtC j 2J D t 2T
s.t. .DC-B/; (4.64)–(4.71) nod NX 2 2 p;C f;C s;C s;C f;C 8j 2 J D ; t 2 T Cjt x x njt njt njt njt nD1
p; Cjt
nod NX
(4.75) 2 2 f; f; s; njt xnjt 8j 2 J D ; t 2 T s; njt xnjt
nD1
(4.76) X f;C f;C 1 C st Kjt C XBjt xtjt D njt xnjt 8j 2 J D ; t 2 T (4.77) 2 nD1 N nod
NX 1 C s;C st D Kjt XBjt C xtjt D s;C njt xnjt 8j 2 J ; t 2 T 2 nD1 nod
nod NX
f;C
njt D 1
(4.78)
8j 2 J D ; t 2 T
(4.79)
D s;C njt D 1 8j 2 J ; t 2 T
(4.80)
nD1 nod NX
nD1 s;C njt
for n D 1; : : : ; N nod SOS 2 8j 2 J D ; t 2 T
(4.81)
N nod
X f; f; 1 st Kjt XBjt C xtjt D njt xnjt 8j 2 J D ; t 2 T 2 nD1
(4.82)
nod
NX s; s; 1 st Kjt C XBjt xtjt D njt xnjt 8j 2 J D ; t 2 T 2 nD1
(4.83)
interest rates for parties and if the purchase prices for these items exceed the supplier’s production costs.
4.3 Application to Master Planning nod NX
109
D f; njt D 1 8j 2 J ; t 2 T
(4.84)
s; njt D 1
(4.85)
nD1 nod NX
8j 2 J D ; t 2 T
nD1 for n D 1; : : : ; N nod SOS 2 s; njt P;C P; ; Cjt 0 8j 2 J D ; t 2 T Cjt f; s;C s; f;C njt ; njt ; njt ; njt
0
8j 2 J D ; t 2 T
for n D 1; : : : ; N nod ; 8j 2 J D ; t 2 T :
(4.86) (4.87) (4.88)
Index n Index for the nodes used for the approximation (including the borders of the interval that is to be approximated); n D 1; : : : ; N nod Data
f;C C st xnjt Node n (x-coordinate) for the linearization of f f Kjt ; XBjt xbjt f; st Node n for the linearization of f f Kjt ; XBjt xbjt xnjt s;C C st xnjt Node n for the linearization of f s Kjt ; XBjt xbjt s; st xnjt Node n for the linearization of f s Kjt ; XBjt xbjt
Variables p;C Cjt Penalty or bonus for greater supply of item j in period t p; Cjt Penalty or bonus for less supply of item j in period t
f;C C st ; XBjt xbjt njt Weight for node n for the linearization of f f Kjt f st f; K Weight for node n for the linearization of f ; XB xb jt njt jt jt s;C C s st njt Weight for node n for the linearization of f Kjt ; XBjt xbjt s st K Weight for node n for the linearization of f ; XB xb s; jt njt jt jt
d k;C d k; , Cjt The objective function of GM-CS1LB comprises new variables Cjt denoting penalties or bonuses due to changes in the supply pattern. Their values are determined by (4.75) and (4.76), respectively.53 Constraints (4.77)–(4.81) correspond to (4.56)–(4.59) and assure a correct determination of the weights for the C st piecewise linearization of Kjt XBjt xbjt . Analogously, (4.82)–(4.86) are used st for the linearization of Kjt XBjt xbjt . (4.87) and (4.88) are nonnegativity
53
The introduction of these additional variables helps to avoid potential errors in the approximation C leading to negative values for actually positive Kjt , Kjt . Such effects are natural for the approximation applied and would cause some degradation in the solution quality in case of an alternative modeling without these variables.
110
4 New Coordination Schemes
constraints. The upper and lower bounds for the linearization are determined as described in Sect. 4.3.1. As upper bounds for the deviations XBjt xbjt and xbjt XBjt , the cumulated demand for item j in the planning interval can be taken. In case backorders of the buyer are never chosen (e.g., if backorders of the buyer are much more costly than a corresponding amount of overtime at the supplier), the performance of the coordination process can be somewhat enhanced by imposing a lower limit on the cumulated supply quantities in GM-CS1S .54 If this limit is set equal to the secondary demand for item j in period t, it can be guaranteed that the supplier’s proposals complying with this limit will not cause any backorders at the buyer.55 For this purpose, we augment GM-CS1S by t X
cum XSj ejt
8j 2 J D ; t 2 T;
D1 cum with ejt as the cumulated secondary demand for item j in period t, which can be determined in the same way as bjt .56 Since the end item demand is usually private cum information of the buyer, she has to communicate ejt additionally to the supplier (e.g., at the beginning of the scheme). However, we do not regard this exchange as critical since, in contrast to absolute costs or capacities, demand data is usually not perceived as confidential by decision makers.57 For sake of completeness, we state the models needed for the evaluation of the last proposal of the other party.
(GM-CS-EVALB ) min CBd l s.t. xtjt D XBjt .DC-B/:
8j 2 J D ; t 2 T
(GM-CS-EVALS ) min CSd l s.t. xtjt D XSjt
8j 2 J D ; t 2 T
.DC-S/: If the version with the one-shot exchange of cost information is applied, all parties run models GM-CS1i without (4.67) for proposal generation. Moreover, instead of GM-CS2I , GM-MP is run once at the end of the scheme. This model can be directly derived from MP; we state it below without further explanations. 54
Note that this extension has also been used in the scheme of Dudek and Stadtler (2005, p. 677). In our computational tests, we apply this extension in Sect. 6.3, when we compare the performance of the scheme proposed here with that of Dudek and Stadtler (2005). 55 For a further illustration of this idea, see Example 2.6 on p. 29. 56 For the determination of bjt , see p. 13 of this work. 57 See, e.g., Kersten (2003, p. 332).
4.3 Application to Master Planning
(GM-MP) min
111
eN X se cs e C be cb e eD1
s.t.
eN X
e se xsjt D
eD1 eN X
eN X
e be xbjt
8j 2 J D ; t 2 T
eD1
se D 1
eD1 eN X
be D 1
eD1 se
0; be 0
for e D 1; : : : ; e: N
4.3.3 Generic Modifications As it will be shown in our computational study of Sect. 6.1, a straightforward application of the schemes presented in the previous subsection involves two major difficulties: Fast convergence and modest computing times for models GM-CS1B and GM-CS1S can only be achieved for rather small test instances. To mitigate these difficulties, we propose a couple of modifications to the schemes. In this section, we introduce a set of generic modifications, that can be directly transferred to other decision problems and defer to the next subsection the modifications specific to Master Planning. Since the option for lost sales imposes additional complexity in the exposition of the models, we initially assume that costs for lost sales are such large that the buyer never chooses this option and defer adaptations to lost sales to Sect. 4.4.3. As the base for the modifications proposed, we present models GM-CS10 B and GM-CS10 S .58 X X C C Kjt C Kjt C XBjt C XBjt
(GM-CS10 B ) min CBd C
j 2J D t 2T
s.t. .DC-B/; (4.66); (4.67); (4.68); (4.71) X X e C st C xtjt xtjt Kjt cs e cs st 8e D 1; : : : ; eN (4.89) j 2J D t 2T
X X
st e C xtjt xtjt Kjt cs e cs st 8e D 1; : : : ; e: N (4.90)
j 2J D t 2T
58
For ease of exposition, we will limit to the formulations with nonlinear objective functions. Their linearizations can be formulated analogously to GM-CS1LB .
112
4 New Coordination Schemes
X X C C Kjt (GM-CS10 S ) min CSd C C Kjt C XSjt C XSjt j 2J D t 2T
s.t. .DC-S/; (4.70); (4.71); (4.74) X X e C st C xtjt xtjt Kjt cs st cs e 8e D 1; : : : ; eN (4.91) j 2J D t 2T
X X
st e C xtjt xtjt Kjt cs st cs e 8e D 1; : : : ; e: N (4.92)
j 2J D t 2T
The basic difference to GM-CS1B and GM-CS1S is that constraints (4.89), (4.90), C (4.91), and (4.92) imply non-negative values for Kjt , Kjt and, hence, only penalties and no bonuses for changes in the central resource use incur. Hence, only proposals with cost savings compared to the starting solution are generated, which increases the probability for the identification of a systemwide improvement. The restriction to solutions with one-sided (positive) deviations from the starting solution generates new limiting hyperplanes in the solution spaces of GM-CS10 B and GM-CS10 S . Since the positions of these hyperplanes depend on the starting solutions chosen, the same holds for the vertices of the solution spaces of GM-CS10 B and GM-CS10 S that, amongst others, are determined by these hyperplanes. These “new” vertices, however, potentially become new proposals in further iterations of the scheme and might be employed as starting solutions themselves. Hence, the number of proposals generated may grow to infinity. Therefore, the above modification renders the scheme a heuristic also for LP problems, and finite convergence cannot be guaranteed. With the buyer as the cost-reporting party, these models (and analogously further models presented in this section) are modified as follows: First, (4.67) is skipped from the buyer’s model and a constraint equivalent to (4.67) is included into the supplier’s model. Second, constraints (4.89) and (4.90) are modified to X X
e C st C xtjt xtjt Kjt cb st cb e
j 2J D t 2T
X X
st e C xtjt xtjt Kjt cb st cb e
8e D 1; : : : ; e; N 8e D 1; : : : ; e; N
j 2J D t 2T
and constraints (4.91), and (4.92) modified to X X j 2J D
t 2T
e C st C xtjt xtjt Kjt cb e cb st
X X
st e C xtjt xtjt Kjt cb e cb st
8e D 1; : : : ; e; N 8e D 1; : : : ; e: N
j 2J D t 2T
Apart from improving the convergence rate, this modification considerably reduces the time for running the scheme. The main reason is the renunciation on bonuses for unfavorable deviations in the supply quantities. Then the optimizer solving these
4.3 Application to Master Planning
113
models has a significantly greater leeway to endogenously determine feasible valC C ues for Kjt and Kjt . Difficulties can be avoided that arise if Kjt and Kjt have to account for both the correct determination of bonuses for solutions with unfavorable deviations in the supply quantities and penalties for solutions with favorable deviations. Although, in theory, there is a feasible way to correctly determine the bonuses and penalties if decentralized models only contain linear variables,59 the computational effort increases considerably with the number of iterations.60 Second, model GM-CS2B is modified to allow additional deviations from the convex combination of previous proposals of the other party at the expense of penalty costs. (GM-CS2B 0 ) min CBd C
eN X
e cs e C
X X
. C cp/ X TjtC C X Tjt
j 2J D t 2T
eD1
s.t. .DC-B/; (4.62)–(4.63), (4.71) XBjt C X Tjt D
eN X
e e xtjt C X TjtC
8j 2 J D ; t 2 T:
eD1
By allowing such deviations, the limitation on exact recombinations inherent in GM-CS2B is removed, such that the probability is increased that this model yields a new proposal. Unit penalty costs cp are defined by cp D
cs i ni t cs f C : P P f i ni t xt xt D j 2J t 2T j jt
(4.93)
Index f Superscript denoting the first proposal generated The rationale behind (4.93) is to perform a rough anticipation of the other party’s cost changes with a deviation from the convex combination. For each unit of deviation in the supply, an equal share of the cost difference between the initial solution and the first solution identified in the scheme is attributed. Note that this idea is similar to the goal programming approach, which has been used by Dudek and Stadtler (2005).61 59
See Lemma 4.2 on p. 67. See Chap. 6. A further difficulty experienced in computational tests is that the optimizer running models GM-CS1B and GM-CS1S might declare barely feasible problems as infeasible. This can be caused by a large downscaling of the matrix elements in case of huge differences between the upper and lower bounds used for the linearization. 61 See Dudek and Stadtler (2005, p. 677). The difference to our approach is that their penalty costs are not fixed, but updated by exponential smoothing in each iteration. We did not follow their approach because it did not prove significantly better in computational tests, while augmenting the complexity of the scheme. 60
114
4 New Coordination Schemes
Although in infrequent cases the deviations of new proposals from the starting solution may be greater, the denominator of (4.93) can serve as an appropriate proxy for the maximum of these deviations. Hence, we can set X X
ubd D
f i ni t xtjt xtj
C
j 2J D t 2T
when performing the linearizations of the objective functions of CS10i and extensions.62
4.3.4 Modifications for Master Planning In this subsection, we introduce further modifications of the scheme that are specific for Master Planning and both aim at speeding up the convergence rate and decreasing the computational complexity of the underlying optimization models. First, we consider deviations in terms of the cumulated supply quantities instead of deviations in absolute supply quantities.63 This can be modeled by replacing constraints (4.60) and (4.61) by t X
XBj
D1 t X D1
XBj
t X
XSj
D1 t X
XSj
8j 2 J D ; t 2 T;
(4.94)
8j 2 J D ; t 2 T:
(4.95)
D1
The benefit of this modification is that penalizations of complimentary changes in the supply pattern can be avoided. We illustrate the effect of constraints (4.94) by an example. Example 4.5 Assume that the starting proposal specifies a supply of 3[U] of an item in periods 3 and 5 (see Fig. 4.15). Further assume that the supplier has tight capacity in period 3 and that he prefers a different supply schedule with only 2[U] in period 3 and 4[U] in period 5. Hence, in period 5, the cumulated supply quantities of the schedule preferred by the supplier correspond to those of the starting proposal, which does not hold for the absolute supply quantities. Since the buyer has to make use of backlogging if she cannot fulfill the demand, the late delivery in period 5 is superior to no delivery or an even later delivery in a subsequent period. Hence, the change to 4[U] in period 5 is in line with the needs of the buyer given the shortage in period 3, and the concentration on cumulated quantities grasps this issue better.
62 63
See also p. 101. A similar idea has been used by the scheme of Dudek and Stadtler (2005, p. 677).
4.3 Application to Master Planning
115
New proposal Starting proposal 6
Absolute supply quantities 4
4
2
2
3
4
Cumulated supply quantities
t
5
3
4
5
t
Fig. 4.15 Absolute vs. cumulated supply quantities
Constraints (4.95) account for savings from earlier supply. Such savings may result, e.g., if the supplier has a lack of capacity in a period and slack capacity in an earlier one. Then he will prefer earlier supply in order to reduce his costs for inventory holding. With a direct application of (4.94) and (4.95), the savings for a solution previously found with both earlier and delayed supply compared to the starting solution could be explained in models GM-CS10 B and GM-CS10 S (endogenously) by reductions of inventory holding and of shortages (i.e., lower amounts of backorders and C overtime needed). However, for a meaningful determination of variables Kjt and Kjt , contributing savings to earlier supply is less probable because the unit costs for capacity shortages usually exceed the unit costs for inventory holding. To account for this, we apply constraints (4.95) only to proposals without any delay in supply. cum We denote the resulting model formulations by GM-CS1cum and B and GM-CS1S state them below. X X d (GM-CS1cum KjtC C XT C jt C Kjt C XT jt B ) min CB C j 2J D t 2T
(4.96) s.t. .DC-B/; (4.67); (4.70); (4.71) t X X X e C xtj xtstj KjtC cs e csst 8e 2 DS j 2J D t 2T
D1
(4.97) X X j 2J D t 2T
Kjt
t X
e xtst j xtj
C
cse csst 8e 2 NDS (4.98)
D1
XT C jt XT jt D
t X D1
XBj xtstj 8j 2 J D ; t 2 T:
(4.99)
116
4 New Coordination Schemes
X X C C d K (GM-CS1cum ) min C C C XT C K C XT jt S jt jt jt S j 2J D t 2T
s.t. .DC-S/; (4.70); (4.71) t X X X e C KjtC xtj xtstj csst cse 8e 2 NES j 2J D t 2T
D1
(4.100) X X
Kjt
j 2J D t 2T
t X
st C xtj xtej csst cse 8e 2 ES (4.101)
D1
XT C jt XT jt D
t X
XSj xtstj 8j 2 J D ; t 2 T:
(4.102)
D1
Sets NDS DS NES ES
Set of proposals without delayed supply compared to the starting proposal Set of proposals with delayed supply Set of proposals without early supply Set of proposals with early supply
Since NDS \DS D ¿ and NES \ES D ¿, we can state the linking constraints C that determine Kjt and Kjt as sets of separate constraints (4.97), (4.98) and (4.100), (4.101), respectively, and thus reduce the number of nonlinear variables. Moreover, also model GM-CS20 B is adapted to GM-CS2cum B : d (GM-CS2cum B ) min CB C
eN X
X X
e cs e C
j 2J D t 2T
eD1
s.t.
t X
XBj C XT jt D
D1
. C cp/ XT C C XT jt jt
eN t X X
e e xtj C XT C jt
8j 2 J D ; t 2 T
D1 eD1
.DC-B/; (4.62)–(4.63), (4.71); with the unit penalty costs cp defined by cp D
cs i ni t cs f C : P P Pt i ni t f xt xt D D1 j 2J t 2T j j
Again, the denominator of this term is used for determining ubd .64 Second, in order to generate a broader range of proposals, we introduce two further modifications, which are successively skipped if no improvement can be
64
See p. 101.
4.3 Application to Master Planning
117
identified for any starting proposal. The first of them is to replace constraints (4.94) and (4.95) by t XX
XBj
t 2T D1 t XX
XBj
t XX
XSj
8j 2 J D ;
XSj
8j 2 J D :
t 2T D1 t XX
t 2T D1
t 2T D1
The effect of this modification is an aggregation of the deviations in the supply pattern with respect to the time periods. Then, no attention is paid to the specific periods in which these deviations occur. Example 4.6 Consider a two-party supply chain with one item supplied. Assume that a single solution has been identified previously, with cost savings of 6[MU] for the supplier and a reduction in the cumulated supply of 3[U] in one period and a reduction of 4[U] in another. Then, a unit penalty cost of Kj D 6=7[MU/U] for any deviation in the supply of this item will be charged in the supplier’s model for proposal generation. Of course, this only provides a rough picture of the potential cost changes caused by deviations in the supply quantities. However, especially if the cost sensitivities regarding such deviations differ for particular items substantially from the sensitivities for other items, such an aggregation turns out to be effective.65 Then this modification allows concentrating on the principal question, which product should be delayed, and not in which period the delay should take place, which is of minor importance there. To integrate this new formulation of the central resource constraints into the modcum els run within the scheme, we modify GM-CS1cum as follows: B and GM-CS1S agg
(GM-CS1B ) min CBd C
X X agg;C agg; C Kj C Kj C XT jt C XT jt j 2J D t 2T
s.t. .DC-B/; (4.67); (4.71); (4.99) t X X agg;C X e C xtj xtstj cse csst 8e 2 DS (4.103) Kj j 2J D t 2T
X X
D1 agg;
Kj
j 2J D t 2T agg;C agg; Kj ; Kj
65
t X
xtstj xtej
C
cse csst 8e 2 NDS (4.104)
D1
0
See the sensitivity analyses in Sect. 6.1.3.
8j 2 J D :
(4.105)
118
4 New Coordination Schemes
X X
agg
(GM-CS1S ) min CSd C
agg;C
Kj
agg;
C Kj
C
X Tjt C X TjtC
j 2J D t 2T
s.t. .DC-S/; (4.71); (4.102); (4.105) t X X agg;C X e st C Kj cs st cs e 8e 2 NES xtj xtj j 2J D t 2T
D1
(4.106) X X
agg;
Kj
j 2J D t 2T
t X
st e xtj xtj
C
cs st cs e 8e 2 ES:
D1
(4.107) Variables Kjagg;C Endogenously determined unit penalty costs for shifts of the supply of item j to later periods compared to the starting supply pattern Kjagg; Unit penalty costs for shifts of the supply of item j to earlier periods The penalty costs are now determined by constraints (4.103), (4.104), (4.106), and (4.107). This modification reduces the number of the linking variables in large part and, hence, the number of SOS2 needed for the piecewise linear approximation. Since the modified models are often able to find new proposals, the time for running the scheme can be reduced significantly.66 As a second modification, additional penalty costs for deviations from the startagg-P agg-P and GM-CS1S for the ing supply pattern are included. We write GM-CS1B agg agg resulting models that differ from GM-CS1B and GM-CS1S merely by the new objective functions min CBd
C
X j 2J D
min CSd
C
X j 2J D
C
Kjagg;C
X t 2T
X TjtC
! X agg; C C Kj C cp X Tjt ; t 2T
! X X agg;C agg; C C Kj X Tjt C C Kj C cp X Tjt : t 2T
t 2T
The modifications introduced above form building blocks that can be freely comagg-P bined with each other. As a promising sequence, first GM-CS1i is tried for proposal generation, and if no new solution can be found for any starting proposagg als, GM-CS1i and GM-CS1cum are run instead.67 In the subsequent procedure, the i proposal generation is summarized exemplarily for the buyer.
66
See also the results of our computational study in Sects. 6.1 and 6.3. Note that the use of all levels of modifications proposed is not always advisable. E.g., for GM agg-P investigated in our computational tests (Sect. 6.1), the omittance of GM-CS1i turns out to be superior.
67
4.3 Application to Master Planning
119
Procedure proposalGenerationBuyer(iteration, ˘ E ) input : iteration, ˘ E output: new new tryModelClass(GM-CS2cum ,˘ E ) i new E if 2 ˘ and iteration > 1 then agg-P new tryModelClass(GM-CS1i ,˘ E )
/* add. penalty costs */
if new 2 ˘ E then agg new tryModelClass(GM-CS1i ,˘ E )
/* aggregation */
if new 2 ˘ E then new tryModelClass(GM-CS1cum ,˘ E ) i
/* only cumulation */
The output of this algorithm is a new proposal that is not element of ˘ E , the agg-P is not applied in the first set of proposals already found.68 Note that GM-CS1S iteration since the penalty costs cannot be determined yet in this stage of the coordination process. The successive solution of the single models for different starting solutions x st is subsumed by the procedure tryModelClass stated below.
Procedure tryModelClass(modelClass, ˘ E ) input : modelClass, ˘ E output: new ˘ ST ˘E repeat x st random() new solve(modelClass, x st ) ˘ ST ˘ ST nx st until new … ˘ E or ˘ ST D fg
The procedure solve( ) renders the proposals generated by means of a model st of the type modelClass (e.g., GM-CS1cum or x l as a further input (if i ) with x new applicable). This proposal is assigned to the vector . Finally, we state the algorithm for the whole scheme (Algorithm 7). We assume here that the initial solution is determined by upstream planning; for other choices of the initial solution, the scheme can be modified straightforwardly. The procedure ProposalGenerationSupplier( ) can be formulated analogously to ProposalGenerationBuyer( ), keeping in mind that the supplier does not run CS2cum since he is not informed about the buyer’s cost changes. The i parameter nIterations, which can be exogenously specified by decision makers using the scheme, indicates the number of iterations for which the scheme is run.
68
Since in a two-party setting only one cost-reporting party exists, we can omit the index i here (as for ˘ E ).
120
4 New Coordination Schemes
Algorithm 7: CoordinationSchemeModified input : nIterations solve(GM-B) new ˘E new for it D 1 to nIterations do /* for each iteration */ solve(GM-CS-EVALS , new ) new proposalGenerationSupplier(it,˘ E ) ˘E new [ ˘ E solve(GM-CS-EVALB , new ) new proposalGenerationBuyer(it,˘ E ) E ˘ new [ ˘ E
When implementing the scheme, a further enhancement regarding computing time is applied, which for ease of exposure has not been included in the algorithms stated above: Combinations of modifications and starting solutions that have not yielded new proposals are discarded in future iterations. Since the unit penalty costs imputed by the models do not decrease in the course of the scheme, this refinement does not cut off potentially optimal solutions. Note that by straightforward adaptations, this scheme and further modifications of it (see the next subsection) can be employed for a one-shot disclosure cost information at the end of the scheme, which is required by the mechanism presented in Sect. 5.3. Then, parties skip constraints (4.67) and rely on their own cost changes for determining the penalty costs (K; cp) in models GM-CS1i and extensions. Of course, models GM-CS2i are not run by any party then. Instead, like in the generic scheme, GM-MP is run once at the end of the coordination process.
4.4 Customizations 4.4.1 Master Planning with Lot-Sizing In principle, the scheme outlined in Sect. 4.3 can also be applied for Master Planning models that include lot-sizing decisions. However, the performance of the scheme can additionally be improved by a further optimization model CS-LOT that is able to capture specific characteristics of lot-sizing models. CS-LOT synthesizes ideas of the scheme for uncapacitated lot-sizing with those of the modified version of the generic scheme outline in Sect. 4.3. The mathematical formulation of CS-LOT is69
69
Note that we use the MLCLSP as the base model here since CS-LOT only applies for lot-sizing models and, thus, not for GM.
4.4 Customizations
121
X X X Tjt C X TjtC min CBd C .cp C /
(4.108)
j 2J D t 2T
(CS-LOT) s.t. .DC-B/; (2.14), (2.15), (4.29)–(4.31), (4.67), (4.71), (4.99): In CS-LOT, the objective function and constraints have been adapted from CS1cum and MLULSP-CS. This model identifies proposals which involve a smaller B number of orders for the supplied items and thereby anticipates potential increases in the supplier’s costs for earlier supply. The latter issue is modeled by the penalty costs cp in (4.108) and helps to make ideas of the scheme for uncapacitated lotsizing applicable for capacitated problems. Without these penalty costs, the resulting solution quality would be affected since the new proposal generated by the buyer would completely ignore the effects of early orders on the supplier’s costs. CS-LOT can be used as an add-on to the scheme outlined in the previous section, i.e., CS-LOT is tried for the identification of a new proposal before applying GMagg-P CS1B , and so on. CS-LOT should only be run if a real benefit can be expected thereof. To decide about this, we apply a criterion similar to that for the inclusion of cum constraints (4.98) and (4.100) into GM-CS1cum B and GM-CS1S , respectively. These models aim to identify proposals with reduced backlog and overtime costs first, and only after this has been achieved, to consider potential reductions in inventory holding costs. Analogously to that, we run model CS-LOT if and only if a proposal with the envisaged reduction of backlog and overtime costs has been identified. As an indicator for this, we take the existence of two proposals (e1 and e2) with the following property: One of these proposals (e1) shows earlier supply compared to the other (e2) and yields a cost increase for the buyer, which is smaller than or equal to the holding costs for the supplied items due to the early supply. I.e., we require X
e1 xtjt
t D1
X
e2 xtjt
t D1
for any j 2 J D , 2 f1; : : : ; jT jg and 0 < cb e1 cb e2
X j 2J D
hj
X
e1 e2 xtjt xtjt
C
:
t 2T
Then it can be inferred that this cost increase is due to early supply and that there is a potential benefit from reducing the buyer’s order frequency. This conclusion is supported by our computational tests, where this rule proved to be effective. In the procedure proposalGenerationBuyerLotsizing( ), we summarize the proposal generation for this extension. Along the reasoning discussed above, the function potentialLotsizing() determines whether to employ CS-LOT.
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P up Procedure proposalGenerationBuyerLotsizing( j 2J ysj , ˘ E ) output: P new up input : j 2J ysj , ˘ E if potentialLotsizing() then ˘ ST ˘ E repeat w 1 x st random() ˘ ST ˘ ST nx st repeat new solve(CS-LOT,w,x st ) w wC1 ˇ ˇ P up ˇ Dˇ until new … ˘ E or j 2J ysj w < J ST new E until ˘ D fg or …˘
/* lot-sizing */
if new 2 ˘ E then /* additional penalty costs */ agg-P tryModelClass(MLCLSP-CS1B ,˘ E ) if new 2 ˘ E then agg tryModelClass(MLCLSP-CS1B ,˘ E )
/* aggregation */
if new 2 ˘ E then E tryModelClass(MLCLSP-CS1cum B ,˘ )
/* only cumulation */
4.4.2 Voluntary Compliance If the compliance of the supplier to the buyer’s proposals is voluntary, a direct implementation of the buyer’s proposals is not assured. Then, three major changes have to be applied to the scheme. First, the buyer’s proposals are not directly evaluated in terms of a systemwide improvement; analogously to upstream planning with voluntary compliance, the supplier determines the extent of his fulfillment of the buyer’s proposal, thereby taking into account potential penalty costs for backorders and lost sales, and communicates the resulting supply quantities to the buyer. The supplier’s model for evaluating the buyer’s proposals corresponds to that used for upstream planning with voluntary compliance.70 I.e., the supplier solves GM-S with the modified objective function (2.52) and the additional set of constraints (2.53). up We abbreviate this model by GMS in the following. Define ˘ up as the set of proposals identified by this model and ˘ S , ˘ B as the proposals generated by supplier and buyer, respectively. The resulting information flow is depicted in Fig. 4.16 exemplarily for the supplier as the cost-reporting party. The second change is a consequence of the first: In each iteration, two different proposals are determined: One that is communicated by the buyer to the supplier, and a further that indicates the extent to which the supplier is willing to fulfill the buyer’s proposal.
70
See p. 30.
4.4 Customizations
123
Fig. 4.16 Information exchange with voluntary compliance of the supplier
Table 4.4 Origin of proposals used as input data for CS1S ; CS1B and CS2S ; CS2B (and modifications) with voluntary compliance Cost-reporting party S B Starting proposals for CS1B Starting proposals for CS1S Proposals for Kjt -penalty costs in CS1S ; CS1B Proposals for recombination in CS2S ; CS2B
˘S ˘S ˘S ˘S
[ ˘ up [ ˘ up [ ˘ up [ ˘ up
˘ S [ ˘ up ˘S [ ˘B all ˘S [ ˘B
Third, not all proposals previously generated are used as input for the models run within the scheme. The determination of which of them are used depends on the choice of the cost-reporting party (see Table 4.4). A careful distinction has to be made about the starting proposals for models CS1i and their modifications stated in the previous sections. With the supplier as the reporting party, cost changes are only known for the proposals generated by the supplier since the buyer’s proposals are not directly implemented. Then, only proposals among ˘ S [ ˘ up can be considered. If the buyer reports the cost changes, infeasibility of proposals does not matter since parties rely on the buyer’s cost changes, which are determined for all proposals. Here, all previous proposals could be chosen in principle. However, it proves favorable to concentrate on the proposals among ˘ S [˘ B as starting proposals for CS1S and the extensions of this model. In contrast to ˘ up , these proposals incorporate to some degree the requirements of the buyer (either due to their generation by the buyer or due to the penalty costs in CS1S ). This increases the probability of the identification of a systemwide improvement. Similar arguments hold for the choice of the starting proposals for CS1B and for the proposals for the recombination in CS2S . We summarize the single steps of the scheme by Algorithm 9.
4.4.3 Lost Sales In this section, we show how to adapt the scheme to the simultaneous presence of backorders and lost sales. Then, apart from the temporal distribution of the supply quantities, their total amounts may vary, too. This extension enlarges the scope of the scheme considerably. Since differences in the cumulated supply are allowed
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4 New Coordination Schemes
Algorithm 9: CoordinationSchemeVoluntaryCompliance input : nIterations ˘ B D fg,˘ S D fg, ˘ UP D fg /* initialization */ new solve(GM-B) ˘B new C ˘ B for it D 1 to nIterations do /* iteration */ up up solve(GMS , new ) ˘ UP up C ˘ UP new ProposalGenerationSupplier(it,˘ S [ ˘ B [ ˘ up ) ˘S new C ˘ S new ProposalGenerationBuyer(it,˘ S [ ˘ B [ ˘ up ) B ˘ new C ˘ B
then, the scheme can also be applied for the coordination of parties maximizing their contribution margins, which is required by most of the real-world instances investigated in Chap. 6.5. The extension of the models presented above to lost sales can be modeled by an additional central resource, the deviations in the supply cumulated within the planning interval. We exemplarily show how to adapt GM-CS1cum to GM-CS1cum,ls , their i i agg versions including lost sales. An adaptation of the other models like GM-CS1i can be carried out analogously. (GM-CS1cum,ls ) min CBd C B X X
(4.109) C ls ls Kjt C X TjtC C Kjt C X Tjt C Kjt Djt
j 2J D t 2T
s.t. .DC-B/; (4.67); (4.70); (4.71); (4.98) ! t X X X C C e st ls Kjt xtj xtj d lb ej C Kjt d lb ejt j 2J D t 2T
D1
8e 2 DS cs cs t t X X C st ls X Tj D XBj xtj C Djt X Tj e
st
D1
(4.110)
D1 C ls Kjt ; Kjt
ls Kjt
ls Kjt
0
8j 2 J D ; t 2 T Kjt 8j 2 J D ; t 2 T
8j 2 J D ; t 2 T :
(4.111) (4.112) (4.113)
(GM-CS1cum,ls ) min CSd C (4.114) S X X C ls ls Kjt C X TjtC C Kjt C X Tjt C Kjt Djt j 2J D t 2T
s.t .DC-S/; (4.70); (4.71); (4.100); (4.112); (4.113)
4.4 Customizations
125
X X j 2J D t 2T
Kjt
t X
st xtj
e xtj
e C d lsj
! C
ls e Kjt d lsjt
D1
cs st cs e 8e 2 ES t t X X C st ls X Tj D XSj xtj Djt X Tj D1
(4.115)
D1
8j 2 J D ; t 2 T:
(4.116)
Data e d lbjt Deviation of proposal e that is due to lost sales and relevant for the buyer e d lsjt Deviation of proposal e due to lost sales for the supplier Variables ls Difference in the supply quantity of item j in period t due to lost sales Djt ls Kjt Penalty costs for lost sales of item j in period t By the introduction of constraints (4.111) and (4.116), models GM-CS1cum,ls and B allow for cumulated deviations in the supply quantities. In order to GM-CS1cum,ls S determinate the penalty costs for these deviations, constraints (4.97) and (4.101) have been modified to (4.110) and (4.115). Thereby, the decision whether to impute the cost difference between xist and a xie to lost sales is determined endogenously. Of course, the models will preferably declare shortages occurring in early periods as lost sales since the backorder costs can be minimized that way. Otherwise, if shortages occurring in later periods were declared as lost sales, more backorders in earlier periods would incur. Assuming that the penalty costs for lost sales exceed the penalties for backorders, we have introduced the redundant constraints (4.112) e in order to sharpen the linearization applied.71 Moreover, new parameters d lbjt , e d lsjt have been used, which denote the deviation in lost sales between xist and presume that lost sales occur as early as xie . As noted above, models GM-CS1cum,ls i e possible. In Algorithm 10, we exemplarily illustrate the determination of d lbjt for e a given item j and a given proposal e. The determination of d lsjt is analogous, e st st e xjt is replaced by xjt xjt . with the difference that xjt
71
Further modifications that help to somewhat sharpen the linearization are the introduction of adp;C p; ditional variables for the costs for lost sales (analogous to Cjt ,Cjt ) and of constraints assuring that, in case of shortages for a single item, the penalty costs for lost sales are directly imputed to this item. These modifications have been considered in our computational study for the real-world planning problems in Sect. 6.5, but not in the models stated above for ease of exposition.
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4 New Coordination Schemes
e Algorithm 10: DeterminationOfd lbjt e output: d lbjt P e st diff t2T xjt xjt t 1 while diff > 0 do n
e st earlySupply max 0; xjt xjt e max fdiff; earlySupplyg d lbjt diff diff earlySupply t t C1
o
4.4.4 Multiple Suppliers The scheme proposed can be straightforwardly adapted to multiple suppliers. For the suppliers’ optimization models, no changes are necessary. Regarding the buyer’s models, we exemplarily provide the modification of GM-CS1cum B . (GM-CS1-Mcum B ) min (4.96) s.t. .DC-B/; (4.67); (4.70); (4.71); (4.99) t C X X X e;i st;i C xtj Kjt xtj csie csist j 2J D t 2T
D1
8e 2 DS; i 2 PS (4.117) t X X X C st;i e;i xtj Kjt xtj csie csist j 2J D t 2T
D1
8e 2 NDS; i 2 PS :
(4.118)
Index i Suppliers, i 2 PS , with PS as the set of suppliers Data csie Costs of proposal e for supplier i e;i xtjt Supply quantity of item j in period t delivered by supplier i and specified by proposal e The necessary modifications are limited to the constraints (4.117) and (4.118), where the knowledge about the cost changes of the single suppliers is used to deterC mine the penalty costs Kjt , Kjt more precisely.
4.4 Customizations
127
The adaptation of GM-CS2cum B to multiple suppliers is: d (GM-CS2-Mcum B ) min CB C
eN XX
i e csie C
(4.119)
i 2P eD1
X X i C cpB X TjtC C X Tjt j 2J D t 2T
s.t. .DC-B/; (4.71) t X
XBj C
X Tjt
D
eN t XX X
e;i i e xtj C X TjtC
D1 i 2P eD1
D1 D
8j 2 J ; t 2 T eN X
i e D 1 8i 2 PS
eD1
i e 0
8i 2 PS ; e D 1; : : : ; e: N
Variables i e Variable indicating for supplier i the share of previous proposals e used for the recombination Data i Penalty costs for supplier i cpB
Here, the only change is that proposals recombined originate from several suppliers. This implies that several suppliers evaluate and generate proposals and that i the penalty costs cpB can be determined more precisely for each single supplier. Apart from that, no further adaptations of the scheme are needed.
Chapter 5
New Coordination Mechanisms
Apart from the identification of improved solutions within an acceptable number of iterations, a further requirement for practicable coordination schemes is their applicability by rational, self-interested parties, which implies that the schemes can be embedded into suitable coordination mechanisms. In the following sections, we outline three contractual frameworks that form building blocks for the resulting mechanisms in combination with the schemes proposed in Chap. 4. All frameworks rely on compensation payments among parties as incentives for the implementation of coordinated solutions. First, these payments are necessary for ensuring individual rationality in the mechanisms. Often, the implementation of coordinated solutions involves cost increases for at least one party. Such increases necessarily occur if a party acts as the leader and unilaterally determines the allocation of the central resources in the default solution.1 Unless several optimal solutions exist for the leader’s problem, the implementation of a coordinated proposal will force the leader to deviate from his individually optimal solution, and, hence, to implement a solution with increased costs. Second, such payments are a straightforward way to align parties’ incentives with the actions required by the schemes.2 Apart from establishing individual rationality, these compensation payments specify the sharing of the surplus from coordination, i.e., the difference between the systemwide costs of the default solution and the systemwide costs for implementing a coordinated proposal. At the same time, the rule for surplus sharing is the one of 1 An example for this is upstream planning with forced compliance (see p. 30), where the buyer acts as the leader. 2 A renunciation of such payments – as sporadically advocated in the literature, e.g., by Gjerdrum et al. (2002, p. 592) – might only be an option if no party incurs any losses from implementing a coordinated proposal. This may occur to a limited extent in voluntary compliance settings, as indicated by our computational tests of Sect. 6.4. Even then, however, it seems difficult to ensure that parties will actually follow the rules of the underlying scheme. The main problem is that – depending on the rule for determining the proposal implemented – parties will have incentives not to accept proposals with small own savings in order to increase the probability of the acceptance of proposals that are more lucrative for these parties (though less advantageous for the other parties). If all parties pursued this strategy, the overall coordination performance would suffer strongly since a great number of favorable proposals would be declined then.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 5,
129
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5 New Coordination Mechanisms
the major differences between the mechanisms proposed. In the first mechanism, the surplus sharing is solely determined by the party which receives information about the others’ cost changes during the execution of the scheme. In the second, the surplus is shared by a previously fixed lump-sum payment, and in the third, by the outcome of a sealed bid double auction. Our descriptions and analyses of these mechanisms are structured as follows: At first, we consider a two-party setting and a one-shot application of the mechanisms. Built on that, we discuss extensions to more than two parties and to a repeated application, which seems most relevant for practice since fixed costs frequently incur for setting up coordination. All mechanisms have specific advantages, which are summarized and discussed in Sect. 5.4. In Sect. 5.5, we outline how to adapt the mechanisms for rolling schedules.
5.1 Surplus Sharing Determined by the Informed Party In this and the next section, we assume – according to the generic scheme outlined in Chap. 4 – a party that recombines the proposals of the others [party I , abbreviated by IP (informed party) in the following] and a set of cost-reporting parties [abbreviated by RPi ]. Analogously to Sect. 2.3, we assume different sexes for the IP and the RPi ; we consider the IP as female and the RPi as male. As mentioned above, we initially focus on a two-party case with one RP3 and one IP. In the section at hand, we discuss the most straightforward possibility for surplus sharing, which is to let the IP determine the allocation of the surplus.4 If we additionally postulate that the RP reports his cost changes truthfully, the generic scheme can be applied directly. Of course, the assumption of truthful information exchange is questionable, particularly from the point of view of (neoclassical) economic theory. We will discuss this issue in light of experimental investigations below. The structure of this mechanism is simple (see also Fig. 5.1): Parties exchange proposals as specified by the scheme. There, the cost reporting by the RP is the only action that influences parties’ shares of the surplus. After the termination of the scheme, the best solution found is determined by the IP and implemented. Moreover,
Fig. 5.1 Mechanism with surplus sharing determined by the IP
3 4
We omit the index i when we consider a single cost-reporting party. Note that this possibility has been mentioned by Dudek and Stadtler (2007, p. 478).
5.1 Surplus Sharing Determined by the Informed Party
131
the IP specifies compensation payments, which, apart from ensuring individual rationality of parties, allocate shares of the surplus to the parties, e.g., using a rule like 80:20 or 50:50. We illustrate this procedure by an example. Example 5.1 Using the scheme, two parties have determined an improved supply schedule with a cost increase of 7[MU] for the IP and a decrease of 11[MU] for the RP. Resulting is a surplus of 4[MU]. Its allocation is determined by the IP using the 50:50 rule. With a compensation payment of 11 2 D 9[MU] by the RP to the IP, both parties obtain gains from coordination of 2[MU] (see Table 5.1, row “Basic”). The main obstacle for this approach is that parties might have incentives for distorted statements of the cost changes. For the RP, stating higher costs than actually incurred might be advantageous. Then the RP might receive a greater compensation payment or pay less for the implementation of a coordinated solution (see the example in Table 5.1, “Case A,” where the RP exaggerates his costs by 3[MU]). A potential drawback of such behavior, however, is the risk that the IP will prefer the default solution if the costs of the IP are greater than anticipated by the RP and, hence, the cost exaggeration exceeds the potential surplus (e.g., Table 5.1, “Case B”). Then, none of the parties will receive any benefit from coordination. Moreover, also the IP might exaggerate her costs and assign a smaller share of the surplus to the RP (e.g., Table 5.1, “Case C”) or simply use another allocation rule than the announced. This is even more appealing than an exaggeration of cost changes by the RP since the IP does not incur any risk of breakdown of coordination. The IP can use her knowledge to exaggerate her costs to such an amount that she can usurp the whole surplus except for an arbitrarily small share allocated to the RP. At first glance, especially the second objection, the potential opportunistic behavior by the IP, seems to affect the applicability of this mechanism seriously. If the share allocated to the RP becomes arbitrarily small, the RP might prefer not to participate in future coordination processes or to substantially exaggerate his cost changes in order to obtain a significant share of the surplus. However, there are actually some reasons for the IP to assign a substantial share of the surplus to the RP. Apart from inciting the RP for participating in future coordination processes, the IP might concede some share of the surplus for reasons of fairness. It has been shown in numerous behavioral investigations that real subjects prefer solutions that are fair to a certain degree. Examples are dictatorial bargaining
Table 5.1 Effects of different cost reporting strategies (all data in [MU]) Cost changes Payment Gains Gains from stated from exaggeration RP IP RP to IP RP IP RP IP Basic 11 7 9 2 2 – – Case A 8 7 7.5 3.5 0.5 1.5 – Case B 8 10 – – – 2 – Case C 11 10.9 10.95 0.05 3.95 – 1.95
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5 New Coordination Mechanisms
Fig. 5.2 Dictatorial bargaining game and trust game
games and trust games. In the dictatorial bargaining game,5 one party, the dictator, decides about the allocation of a monetary amount between itself and another, less powerful party (see Fig. 5.2). The experimental outcome of this game sharply contradicts the self-interest hypothesis of neoclassical economic theory. For different experimental conditions (e.g., framing, demographic parameters, culture),6 the share allocated to the less powerful party turned out to be substantial, and about 20% of the total monetary amount on average.7 This and related insights gave even rise to the development of new economic theories such as inequality aversion.8 Even more similar to the mechanism proposed is the trust game.9 There, the dictatorial bargaining game is extended by a potential investment of the less powerful party. A multiple of this investment (e.g., a triple) is allocated by the dictator to both parties, and there is no guarantee that the investment is paid back to the investor. In experiments, and again in contrast to the self-interested behavior postulated by theory, a large part of investors choose nonzero investments, which were paid back on average. Corresponding experiments have also been conducted for repeated trust games.10 For transferring the trust game to our mechanism, the investment can be interpreted as the commitment of the RP (here: the investor) not to exaggerate his cost changes. Compared to non-truthful reporting, this commitment increases the total
5
See Kahneman et al. (1986, p. 728). See, e.g., Kagel and Roth (1995, p. 256) and Camerer (2003, p. 59). 7 See Camerer (2003, p. 113). 8 This theory has been developed by Fehr and Schmidt (1999, p. 817) and been applied to supply chain coordination by Cui et al. (2007, p. 1303). For more information about behavioral theories explaining these phenomena, see Camerer (2003, p. 101). 9 This game has initially been proposed Berg et al. (1995, p. 122) and extensively investigated in the literature afterwards. For variants of this game recently examined, see, e.g., Bracht and Feltovich (2008, p. 39) and Falk (2007, p. 1501). 10 Repeated trust games have first been investigated by Camerer and Weigelt (1988, p. 1) and recently by Engle-Warnick and Slonim (2004, p. 553) and Engle-Warnick and Slonim (2006, p. 603). 6
5.2 Surplus Sharing Determined by Lump-Sum Payments
133
surplus, which is to be allocated by the IP.11 However, the behavior of real subjects in this setting is difficult to foresee since thorough empirical analyses for trust games in a supply chain environment have not been undertaken yet. Finally, note that the mechanism presented can be straightforwardly transferred to settings with three or more parties. Then several RPi report their cost changes, and the IP allocates to each of them shares of the surplus as a bonuses for their participation.
5.2 Surplus Sharing Determined by Lump-Sum Payments Although there are some experimental insights in favor, the mechanism presented in the preceding section shows basic deficits when analyzed from the perspective of game theory. Again consider a two-party setting first. Using that mechanism, a purely rational, self-interested IP will try to keep the complete surplus. Anticipating this, a rational, self-interested RP will considerably exaggerate his cost changes (e.g., by a constant markup), with the consequence that the rules of the scheme will not be kept. This has two negative implications: First, the best solution identified by the scheme might not be recognized as an improvement of the default solution. Second, the quality of the coordination process might suffer due to the internal logic of the scheme. We illustrate the latter issue by an example. Example 5.2 Assume that decentralized problems can be modeled as LP problems. Further assume that the proposals generated have two dimensions, d1 and d2 (e.g., two items supplied and one time period). Consider the starting proposal S, proposals A and B, which have been identified in previous iterations of the scheme, and C as a potential new proposal (see Fig. 5.3). Proposals A, B, and C constitute systemwide improvements compared to S. Let A, B, and C go along with cost reductions for the IP. The real changes in the systemwide costs due to the implementation of these
Fig. 5.3 Suboptimal outcome of the scheme due to cost exaggeration
11
As an illustration for the effects of distorted reporting of cost changes, see Example 5.2 in the next section.
134 Table 5.2 Cost data for proposals A, B, and C
5 New Coordination Mechanisms
Real overall cost changes Markup by the RP Overall cost changes perceived by the IP
A
B
C
10 12 2
8 12 4
13 12 –
Fig. 5.4 Mechanism with lump-sum payment
proposals are given in the first row of Table 5.2. Let the cost changes reported by the RP comprise additional markups (see Table 5.2). Since these markups exceed the real cost changes, the IP erroneously considers S superior to A and B. When the IP runs CS1i or a modification of this model, the endogenously determined penalty costs for C would exceed her savings for this proposal. These penalty costs increase linearly with a movement from S into the direction of C, whereas the costs of the IP decrease linearly or less due to the convexity of decision problems. Hence, C could only be found with a different starting proposal (e.g., A). It is remarkable that C will not be found starting from S even though the systemwide surplus of C exceeds the markup chosen by the RP. In the following, we present a mechanism for which truth-telling is a weakly dominant strategy for the RP, such that the difficulty illustrated by Example 5.2 can be avoided. In this mechanism, the RP determines the amount of a lump-sum payment in advance (see also Fig. 5.4). This lump sum is paid by the IP to the RP in addition to the (positive or negative) payment for reestablishing parties’ costs of the default solution, provided that a solution resulting from coordination is implemented. The IP can determine whether to accept a coordinated solution and to pay the lump sum to the RP or to keep the default solution. With an exaggeration of his costs, the RP runs the risk of losing the lump sum, which he obtains if coordination has been successful; if the surplus is lower than the lump sum, the IP will prefer the default solution, of course. In Fig. 5.5, the actions specified by the mechanism are displayed on a time axis. In the first step of the mechanism (t0), the amount of the lump-sum payment is determined by the RP. The scheme is applied in the second step (t1), and last (t2), the IP decides whether to implement a proposal generated by the scheme. We illustrate the mechanism by an example.
5.2 Surplus Sharing Determined by Lump-Sum Payments
135
Fig. 5.5 Time bar of the mechanism
Example 5.3 Assume that the best proposal found by the scheme yields a cost change of 6[MU] for the RP and a cost change of 3[MU] for the IP compared to the default solution. Let the previously fixed lump sum be 2[MU]. If the RP reports his cost changes correctly, the IP will opt for the implementation of the coordinated solution. Then 6 2 D 4[MU] is paid by the RP to the IP. Hence, shares of 2[MU] and 1[MU] are obtained by the RP and the IP, respectively. If the RP exaggerated his costs too much, e.g., by stating a cost reduction of only 4[MU], the IP would reject the coordinated solution since the lump-sum payment exceeds the systemwide cost savings perceived by her. Hence, the risk of losing the lump-sum payment limits the cost exaggeration by the RP. In the following, we will analyze the properties of this mechanism for different sets of assumptions. Below, we introduce our basic set. Assumption 5.1 The RP has prior, incomplete knowledge about the surplus from coordination. Such knowledge can be derived by the general experience of decision makers or can be acquired by learning if coordination is undertaken repeatedly (e.g., once a month).12 Denote this knowledge by the density distribution f .S / over the interval Œa; b for S , the random variable denoting the surplus from coordination. Assumption 5.2 In the second step of the mechanism, parties can maximize their expected surpluses by implementing the actions as specified by the scheme. This assumption requires that the scheme used in the second step of the mechanism is the most efficient among all known schemes based on the exchange of primal information. This holds, up to our knowledge, for the schemes specified in Chap. 4 when Master Planning is to be coordinated. Note that we do not recommend schemes with an exchange of dual information for use with this mechanism. Dual information communicated to the RP may entitle him to estimate the cost changes of the IP and to increase his gains by a distorted reporting of his cost changes, such that strategy proofness would get violated. Assumption 5.3 The information exchange required by the scheme does not violate individual rationality of parties. Within supply chain management, an (iterative) exchange of order information (or forecasts) is common practice, e.g., in CPFR.13 We argue that an exchange of a 12
A discussion of potential learning effects with a repeated application of the mechanism is provided at the end of this section. 13 See e.g., Aviv (2001, p. 1327).
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5 New Coordination Mechanisms
modest number of such orders and their associated, aggregated cost effects does not prejudice the party disclosing this information, as it might do an exchange of production capacities. We model the mechanism as a dynamic game with the commitment of parties to the decisions taken in previous steps of the game. Thereby, the last two steps correspond to a Stackelberg game with the RP as the leader.14 First, consider the parties’ decision problems in the second step (t1). Obviously, the IP can maximize her gains by following the rules of the scheme since the probability for improvements is highest then. The determination of the best strategy of the RP requires a somewhat deeper analysis. The gains of the RP comprise the lump sum and his markup for the proposal implemented. Due to the information status of the RP,15 we do not consider different markups in single iterations. We denote the lump sum by L and the markup by l. Hence, the decision problem of the RP at t1 is to maximize gRP .l/, the gains of the RP from coordination subject to l: Z
b
max g RP .l/ D max l
.L C l/ f .S / dS;
l
LClCr.l/
with r .l/ as the function that maps the expected reduction of S with l. By Assumption 5.2, r .l/ 0 holds. By inspection, we get that the RP weakly prefers a lump-sum payment L D LO C lO together with l D 0 to any payment LO with a markup lO > 0. Hence, truth-telling is a weakly dominant strategy for the RP since the RP is allowed to determine L at t0. Assuming l D 0, the optimization problem of the RP at t0 becomes Z
b
max g RP .L/ D max L
L
Lf .S / dS:
(5.1)
L
To determine L , the optimal solution to (5.1), we calculate the first-order derivative of (5.1): @g RP .L/ D @L
Z
b
f .S / dS Lf .L/ :
(5.2)
L
To facilitate our analysis, we assume that F .S /, the cumulated density function of f .S /, has an increasing generalized failure rate (IGFR).16 Then, the RHS of (5.2) is decreasing in L for all ranges where the RHS of (5.2) takes a value greater
14
A Stackelberg game is a dynamic game in which one party is the (Stackelberg) leader and moves before the follower, see, e.g., Myerson (1991, p. 187). 15 See Assumption 5.1. We discuss an alternative assumption about the information status of the RP below. 16 This assumption is natural since most standard probability distributions, e.g., normal, uniform, gamma, and Weibull distributions, show this property, see Lariviere and Porteus (2001, p. 296).
5.2 Surplus Sharing Determined by Lump-Sum Payments
137
than zero.17 If L0 , the root of (5.2), is smaller than a, g RP .L/ takes its minimum at a since the RHS of (5.2) decreases for ranges where it takes positive values.18 Otherwise, if f .L0 / is defined, i.e., if L0 2 Œa; b ,19 L is equal to the root of (5.2). Then we obtain Z b (5.3) f .S / dS D L f L L
as the condition for the optimality of L . For general probability distributions f .S /, a closed-form solution to (5.3) is not available. In spite of that, some quantitative insights can be obtained assuming that the prior knowledge of the RP is uniformly distributed.20 With f .S / uniformly distributed between Œa; b , f .L / D 1= .b a/ holds. Given risk neutrality of the RP,21 we obtain L b L D ba ba as the condition for the optimality, given that L0 2 Œa; b . Hence, we get b : L D max a; 2 Hence, the expected surplus that can be realized by the mechanism is Z g
mech
D
b L
Sf .S / dS D
o n 2 b 2 max a2 ; b4 2 .b a/
bCa 3b 2 D min ; : 8 .b a/ 2
Next, we compare gmech with g max , the expected surplus of the best solution identified by the scheme: Z g
max
b
D
Sf .S / dS D a
bCa b 2 a2 D : 2 .b a/ 2
R1 R1 An IGFR for f .S/ means that Lf .L/ = L f .S/ dS is increasing in L. Since L f .S/ dS R1 R1 is decreasing in L, L f .S/ dS Lf .L/ also decreases then, provided that L f .S/ dS Lf .L/ > 0. 18 A potential increase of the RHS of (5.2) for ranges where the RHS takes negative values does not affect the optimality of L . The best value for L within such a range is the lower bound of this range, which either corresponds to a or to the upper bound of the adjacent positive range. 19 Values of L > b can be excluded since the gains of RP become zero there. 20 This assumption is often used within mechanism design, see, e.g., Chatterjee and Samuelson (1983, p. 842) and Baldenius (2000, p. 32), if – as it is the case here – otherwise no meaningful analytical results can be derived. 21 For a risk-averse RP, L is lower. Risk averseness can be modeled straightforwardly here, e.g., analogously to the modeling by Chatterjee and Samuelson (1983, p. 848) for their sealed bid double auction. 17
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5 New Coordination Mechanisms
Since g mech =g max 3b 2 = .8 .b a// = ..b C a/ =2/ 3b 2 =8b = .b=2/ D 3=4, at least 3=4 of the maximum surplus can be realized by the mechanism on average. Assuming a D 0, which will hold if also instances with arbitrarily small surpluses exist, the surplus allocated to the RP is b b b2 b D : D L f .S / dS D L f .S / b L D 2b 4 L Z
g RP
b
Then a share of the surplus of gRP =g mech D .b=4/ = 3b 2 =8b D 2=3 is allocated to the RP and 1=3 to the IP on average. We summarize our main results in the following theorem. Theorem 5.1 The coordination schemes proposed in Chap. 4 with unilateral cost reporting can be embedded into an individually rational, strategy-proof, and budgetbalanced mechanism that does not require the involvement of a third party. Given a two-party setting with uniformly distributed prior knowledge about the surplus from coordination, at least 3/4 of the systemwide surplus will be realized on average. Note that the choice which party takes the role of the IP and which party the role of the RP has to be determined before applying the mechanism. For the two-party supply chain considered, both roles have their advantages. The share of 2/3, which holds for a risk-neutral RP with prior knowledge that is uniformly distributed within the interval Œ0; b , seems to favor the RP. However, there are reasons for the RP to claim smaller shares than anticipated by our model. First, the RP will do so if he is risk-averse. Second, experimental studies22 indicate that parties (and especially the less informed ones, which holds for the RP here) tend to claim considerably smaller shares of the overall profit in sealed bid double auctions, a setting related to that analyzed here. Third, settling for smaller shares strengthens long-term collaboration. This may induce the IP to put more effort into corresponding coordination activities and result in future increases of marginal surpluses. Moreover, note that the efficiency of the mechanism enhances strongly if the RP claims a smaller share (see Example 5.4). Example 5.4 Assume that the RP settles for half of the I.e., the surplus. RP determines L such that gRP =g mech D Lf .S / .b L/ = f .S / b 2 L2 =2 D 1=2. With the resulting lump sum L D b=3, the efficiency of the mechanism rises from 3=4 to b 2 b 2 =9 =b 2 D 8=9. The real decrease in the RP’s savings, however, is only .2=3 3=4 1=2 8=9/ = .2=3 3=4/ D 1=9 of his savings for a lump sum of 1=3. Next, we consider the extension of the mechanism to settings with more than two parties, i.e., one IP and several RPi , as required by the schemes with unilateral exchange of cost information. The structure of the mechanism remains unchanged here. The IP applies the mechanism bilaterally with each RPi . If the IP opts for 22
See e.g., Rapoport et al. (1998, p. 221) and Seale et al. (2001, p. 187).
5.2 Surplus Sharing Determined by Lump-Sum Payments
139
the implementation of a solution with different resource use by an RPi compared to the default solution, this RPi will receive a lump-sum payment Li apart from reestablishing his costs of the default solution. Consequently, the IP will agree on implementing a different solution for an RPi if and only if the marginal surplus from coordination S i , i.e., the increase of the systemwide surplus that results from the participation of this RPi in the coordination process, is equal to or greater than Li . To model these decisions, we propose to introduce additional binary variables i;def i;i ni t and to augment the objective functions of CS2I and extensions by Li i;def . That way, potential incentives of an RPi for claiming higher lump sums and understating his cost changes can be avoided. Such incentives might particularly arise if the sum of the marginal surpluses from coordination goes below the surplus for the whole system.23 Then an RPi might obtain a benefit by understating his costs since an increased profitability of solutions preferred by this RPi might induce the IP to focus on the solution space preferred by this and not by other RPi . This would result in an increased probability that solutions favorable to this RPi , but not to the others, are found by the scheme. Since further possibilities to influence the savings of other RPi do not exist due to the privateness of the decentralized data of the RPi , we can assume: Assumption 5.4 In the second step of the mechanism, implementing the actions as specified by the scheme maximizes the expected marginal surpluses of all parties. Since the mechanism is applied bilaterally within subsets consisting of the IP and one RPi , we can analyze the mechanism separately for each of these subsets. In case parties’ prior knowledge is uniformly distributed within an interval ai ; b i , the characterization of the optimal lump sum derived above directly extends, i.e., ˚ Li D max ai ; b i =2 . This is also true for the minimum foregone profits for each RPi (1=4). Hence, the average systemwide efficiency is P 1
i 2P
E Si
4S sys
;
with S sys as the expected surplus for the whole system. The marginal surplus of a party’s participation in coordination can be calculated by the difference between the (maximum) systemwide surplus with and without the participation of this party. The sum of marginal surpluses is not necessarily equal to S sys ; it rather depends on whether coordinated solutions of the RPi are complementary and the prior knowledge of the RPi about that. Consider the following examples. Example 5.5 Consider two different RPi A and B, that are competing about the use of one central resource, which is expandable by the IP at maximum by 3[CU] at costs of 1[MU/CU]. Moreover, let A have offered within the scheme two proposals A1 and A2 with overuses of 1[CU] and 2[CU] of this resource by A compared to the default solution and resulting cost savings of 2[MU] and 5[MU], respectively. B1 and B2, the proposals generated by B, show overuses of 1[CU] and 3[CU] of 23
We provide an example for such a setting below (Example 5.5).
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5 New Coordination Mechanisms
this resource by B and savings of 2[MU] and 5[MU], respectively. The best solution for the whole system is the simultaneous implementation of A2 and B1 resulting in a systemwide surplus of 5 2 C 2 1 D 4[MU]. Marginal surpluses are 4 .5 3/ D 2[MU] for A (only B2 is implemented if A does not participate) and 4 .5 2/ D 1[MU] for B. Here, the sum of the marginal surpluses goes below the surplus realized by the scheme. Example 5.6 Consider two different RPi A and B and two central resources that are both expandable by the IP at costs of 1[MU/CU]. Let A have generated a proposal that additionally requires 1[CU] of the first resource by A compared to the default solution, 1[CU] less of the second, and yields a benefit of 2[MU]. Further assume a proposal of B with a benefit of 3[MU] and deviations in the use of the central resources by B complementary to the proposal of A. Then, the systemwide surplus is 2 C 3 D 5[MU] and the marginal surpluses are 5 .3 1/ D 3[MU] for A and 5 .2 1/ D 4[MU] for B. The sum of the marginal surpluses is 7[MU] and exceeds the surplus for the whole system. Finally, we explicitly discuss the consequences of a repeated application of this mechanism. For a two-party supply chain, we illustrate the resulting decision situation of the RP by an example. Example 5.7 Consider two parties that have applied the mechanism six times in previous coordination activities. The history of the lump sums required and the decisions about the acceptance of coordinated proposals by the IP is displayed in Table 5.3. The question for the RP which lump sum to choose in the current application of the mechanism (7th), cannot be answered unambiguously. Based on his previous experiences, the RP may construct a cumulative probability distribution function for the minimum surplus (see Fig. 5.6). The RP knows, e.g., that a lump sum smaller than or equal to 3[MU] has led to acceptance in more than half of all previous applications of the mechanism. Perhaps, this might induce him to choose such a lump sum for the current application of the mechanism. A completely different and likewise rational strategy, however, is to claim a much greater lump sum in order to explore the solution space further; the acceptance of a lump sum of 3[MU]
Table 5.3 Example of historical data used for learning in the mechanism Application of mechanism 1st 2nd 3rd 4th 5th 6th Lump sum (MU) Accepted
Fig. 5.6 Probability distribution of the minimum surplus
3 Yes
6 No
4 No
3 Yes
5 Yes
4 No
5.3 Surplus Sharing by a Double Auction
141
may go along with, e.g., a surplus of exactly 3[MU] or even 100[MU]! The choice of real subjects, however, is far from clear and crucially depends on the particular learning strategies adopted by them. Potential learning strategies include reinforcement learning, belief learning, and experience-weighted attraction learning, which is a synthesis of the two former ones.24 Of course, learning behavior is subject-specific and difficult to assess a priori. Experimental studies for a related setting, the sealed bid double auction,25 provide some indications. There, the strategies of subjects observed in laboratory experiments could appropriately be modeled by reinforcement learning.26 Besides learning, signaling strategies become relevant with a repeated application of the mechanism since the information learnt about the surplus from coordination depends on the lump sums previously requested by the RP as well as on the IP’s previous decisions about the acceptance of coordinated solutions. E.g., it might be beneficial for the IP to reject coordinated solutions providing her very small shares of the surplus (after subtracting the lump sum) in order to build up reputation and to induce the RP to require smaller lump sums in future applications of the mechanism. Unfortunately, meaningful analytical results about parties’ best strategies are very difficult to obtain. Even if parties’ learning behavior were known, the problem of multiple equilibria would persist. There is some empirical evidence about signaling strategies in the presence of multiple equilibria;27 like in the previous section, however, we are not aware of corresponding (analytical or empirical) research for settings similar to that considered by us. To summarize, a repeated application may influence the efficiency of the mechanism proposed; the exact effect, however, cannot be assessed without significant additional research effort. It is important to note, however, that a repeated application of the mechanism only affects the parties’ prior knowledge and, hence, the amounts of the lump sums that will be required by them. The preference for truthful reporting of cost changes by the RPi , instead, remains unchanged.
5.3 Surplus Sharing by a Double Auction In the following, we show how to apply the sealed bid double auction mechanism28 to a two-party supply chain; extensions to more than two parties are discussed afterwards. We rely on the variant of the scheme with a one-shot disclosure of
24
See, e.g., Camerer et al. (2002, p. 137). For a survey of learning theories and associated experimental results, see Camerer (2003), Chap. 6. 25 See also Sect. 3.3.1. 26 See, e.g., Rapoport et al. (1998, p. 226) and Seale et al. (2001, p. 192). 27 See Brandts and Holt (1992, p. 1350) as a reference for a seminal paper on this topic and Sect. 8 of Camerer (2003). 28 See Sect. 3.3.1.
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5 New Coordination Mechanisms
information29 in combination with this mechanism. To adapt this mechanism to this scheme, the purchasing price can be interpreted as the cost savings (optionally decreased by a markdown chosen by rational parties) of one party with a proposal generated by the scheme, and the selling price as the cost increase (optionally augmented by a markup) of the other party. One assumption of bilateral trade, however, does not fit with the scheme: Multiple proposals are generated within the scheme for each party, while only one offer is processed in the sealed bid double auction. At first glance, an appealing way to tackle this issue is to carry out sealed bid double auctions for each proposal sequentially. After a new proposal has been generated, the outcome of this auction determines whether the proposal is regarded as a systemwide improvement by parties. This approach, however, has a severe drawback: A strategy potentially adopted by parties would be then to overstate the own costs when evaluating a proposal of the other party; the knowledge about the other party’s last bid could be used to design a proposal similar to the last and to submit a bid which very little exceeds the negative value of the other party’s previous bid – with the aim to usurp the lion’s share of the surplus. Anticipating this, the other party will also considerable exaggerate its bids on new proposals, which will result in an inferior and little predictable performance of the mechanism. Instead, we propose to use the sealed bid double auction only once after the termination of the scheme, but simultaneously for all proposals generated. The resulting mechanism can be described as follows: As a first step, parties (due to the symmetry of the mechanism, the roles of parties are identical; we abbreviate them by #1 and #2 in the following) generate a set of promising proposals ˘ along the lines of the scheme with one-shot disclosure of cost information.30 After that, parties simultaneously submit a set of sealed bids containing one bid for each proposal k 2 ˘ . Each bid bk1 , bk2 for #1 and #2, respectively, is the sum of the potential savings.31 sk1 ,sk2 of parties with k and markdowns m1k , m2k .32 After breaking the seals, the parties can determine the systemwide benefits from implementing the proposals. If the best ˚ proposal, i.e., the proposal j with j 2 arg maxk bk1 C bk2 , yields systemwide gains compared to the default solution, i.e., bj1 C bj2 > 0, then this proposal is im plemented. By a (positive or negative) compensation payment of bj1 bj2 =2 by #1 to #2, an equal sharing of the surplus from coordination can be achieved (see also Fig. 5.7 for the sequence of actions in the mechanism). We illustrate the mechanism by an example.
29
See Sect. 4.1.2. Note that, like in the other mechanisms presented in this work, we assume the existence of a default solution, which is implemented without coordination. 31 Note that these “savings” can also take negative values if the implementation of a proposal involves a cost increase for a party. For ease of exposition, however, we keep this terminology also in this case. 32 Such markdowns are always chosen by rational parties under mild conditions. This result has been proven by Myerson and Satterthwaite (1983, p. 265) for the bilateral trade mechanism and is directly valid for the setting considered here if j˘ j D 1. 30
5.3 Surplus Sharing by a Double Auction
143
Fig. 5.7 Double auction mechanism Table 5.4 Data for Example 5.8 #1 #2
Bids [MU] 1 2 3
4
5
6
2 3
2 4
6 3
4 5
3 1
1 1
Resulting payment by #1 to #2 [MU] 3
Example 5.8 Assume that parties have generated six proposals using the scheme and passed sealed bids for these proposals to each other (see Table 5.4). Proposal 4 yields the greatest surplus. The mechanism implies the implementation of this proposal and a compensation payment of 3[MU] by #2 to #1, such that both parties obtain a share of the surplus of 1[MU]. As a prerequisite for analyzing the properties of this mechanism, we assume analogously to Assumption 5.3 that parties’ individual rationality is not violated by the information exchange required. If the mechanism is applied repeatedly, parties will obtain knowledge about their leeways for the single proposals, i.e., about the maximum possible markdowns which allow that the corresponding proposals are recognized as systemwide improvements. Formally, define the leeway for a proposal k as lk1 D sk1 C bk1 . For our analysis of the mechanism, we assume that parties already have this kind of knowledge when applying the mechanism: Assumption 5.5 For each proposal generated, parties know the probability distributions of their leeways.33 This prior knowledge differs from the incomplete knowledge about the other parties’ savings, which is commonly assumed for analyzing bilateral trade.34 We argue that Assumption 5.5 is more realistic in our context, since the knowledge about leeways can be induced by observing the others’ bids in previous application of the mechanism (which does not hold for the knowledge about the other parties’ savings). 33
In contrast to the previous section, a more detailed prior knowledge (i.e., for each proposal) is natural here since parties submit bids about the savings for all proposals within the auction, and these bids will become globally known afterwards. 34 E.g., Myerson and Satterthwaite (1983, p. 265).
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5 New Coordination Mechanisms
We denote the knowledge about the leeways by random variables L1j and their probability density functions lj1 L1j , which are upper bounded by lj1 . Define f1 D L1 ; : : : ; L1 with the associated probability density functhe vector L 1 k 1 Qk 1 f 1 1 e tion l L D kD1 l Lk and the lower and upper support vectors le1 D e l11 ; : : : ; lk1 and li1 D li1 ; : : : ; l 1 , respectively. Moreover, denote parties’ exk
pected gains from the mechanism by g 1 and g 2 . For analyzing the properties of the mechanism, we further introduce random variables Bk2 and Bk1 with the probability density functions fk1 Bk2 and fk2 Bk1 , respectively. Due to the symmetry of the mechanism, we limit our analysis to the best bidding strategy of #1. Of course, all resultsderived hold equally for #2. Set f1 D B 1 ; : : : ; B 1 with the associated probability k D j˘ j. Define the vector B 1 k Q 1 k f 1 1 1 e B D density function f f B and the lower and upper support veckD1 k k e tors be1 D b11 ; : : : ; b 1 and b 1 D b11 ; : : : ; b 1 , respectively. The expected gains k
k
of #1 from the mechanism can be expressed by Z be1 Z bj2 1 1 1 g b1 ; : : : ; bk D 1 be bj1
sj1
bj1 Bj2 2
!
f1 d B f1 e1 B fj1 Bj2 dBj2 f
if bj1 bj2 D 0 otherwise;
(5.4)
˚
with j 2 arg maxk bk1 C Bk2 . We apply the following transformation. First, note that for the markdown chosen by #1 for proposal j , m1j D sj1 bj1 holds. Then, (5.4) can be written as
g
1
m11 ; : : : ; m1k
Z be1 Z D
e1 b
bj2
sj1 Cm1 j
sj1 C m1j C Bj2 2
if sj1 C m1j bj2
f1 d B f1 e1 B fj1 Bj2 dBj2 f (5.5)
D 0 otherwise; ˚
with j 2 arg maxk sk1 C Bk2 m1k . Obviously, in the optimal solution to (5.5), m1j 0 holds for all j . Otherwise, if m1j < 0, the benefits for proposals which also would be accepted in case of m1j 0 will decrease, and there will be a positive probability that a proposal is implemented which yields a loss for #1 compared to the initial solution. Since the proposal j is implemented only if m1j L1j , the gains of #1 can be expressed by
5.3 Surplus Sharing by a Double Auction
145
Z e l 1 Z lj1 m1 C L1 j j 1 1 f1 d L f1 lj Lj dL1je g 1 m11 ; : : : ; m1k D l1 L 1 1 e 2 l mj D 0 otherwise;
if m1j lj1 ; (5.6)
˚
with j 2 arg maxk L1k m1k . Unfortunately, the derivation of an analytical solution to this multidimensional optimization problem seems not possible. For a special form of prior knowledge – namely a uniform distribution – we are able to determine a lower bound on the efficiency of the mechanism proposed here. A possible, not necessarily optimal choice by #1 are equal markdowns, i.e., m1i D 1 m 8i . Then, (5.6) simplifies to g m1 D
Z
l1
1
m1
m1 C L1 2
h1 L1 dL1 ;
(5.7)
˚ with L1 as the largest order statistic of L1j , i.e., L1 D max L11 ; : : : ; L1n . Then, the optimization problems to determine parties’ best response strategies, i.e., markdowns m1 , m2 , become max g 1 m1 D max m1
Z
m1
max g 2 m2 D max m2
m2
l1 m1
Z
l2
m2
m1 C L1 1 1 1 l L dL ; 2
(5.8)
m2 C L2 2 2 2 l L dL : 2
(5.9)
This insight relates the mechanism proposed here to the bilateral trade mechanism:35 Parties’ problems in the latter mechanism to (5.8) and (5.9), can betransformed too.36 In bilateral trade, however, l 1 L1 and l 2 L2 are not given directly, but only the knowledge about parties’ reservation values, which l 1 L1 and l 2 L2 depend on. Moreover, this insight allows us to derive a lower bound on the efficiency of the mechanism proposed, given uniformly distributed prior knowledge: Theorem 5.2 Consider a setting where l 1 L1 and l 2 L2 are uniformly distributed and one proposal generated corresponds to the systemwide optimum. If parties choose constant markdowns, the efficiency of the mechanism is at least 3/4. Under general bidding strategies (including unequal markups), the efficiency is at least 1/3. Proof. This proof is structured as follows: First (a), we derive a general lower bound on gmech , the sum of parties’ gains from the mechanism. Since the prior knowledge about the leeways and, hence, about the other parties’ bids is (exogenously) given, 35
See Sect. 3.3.1 for more details on this mechanism. These problems correspond to (5.4) with only one proposal generated and the reservation value of the buyer (seller) as the (negative) savings of these parties. See Chatterjee and Samuelson (1983, p. 838).
36
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5 New Coordination Mechanisms
the impact of the strategic behavior of #2 on the gains of #1 is already included there. Hence, to determine a lower bound, we can rely on any strategies of parties. Here we w.l.o.g assume that parties choose constant markdowns m1 and m2 for all proposals. Second (b), we determine a general upper bound on gmax , the systemwide surplus from implementing the best proposal generated. Comparing this to the lower bound on gmech , we obtain a lower bound on the efficiency of the mechanism. Third (c), we tighten this bound for the assumption that equal markdowns are actually parties’ best strategies. (a) Define the random variables Sk as the systemwide surpluses from implementing
˚ k and S as the corresponding largest order statistic, i.e., S D max S1 ; : : : ; Sk . Let s .S / be the probability density function of S with the support Œs; s . We, w.l.o.g., assume s D 0. Negative values for s do not affect our results regarding the efficiency of the mechanism; proposals with negative surpluses, i.e., losses, will neither lead to the systemwide optimum nor be chosen within the mechanism. For all realizations of S smaller than m1k or m2k , parties obtain zero gains from the mechanism despite a positive systemwide surplus. Hence, the maximum share of the surplus is cut off by the markdowns (and, hence, the efficiency of the mechanism is lowest) if s D 0. Note that the leeway of #1 for a proposal k corresponds to the systemwide surplus for k less the markup which is chosen by and uniquely allocated to #2, i.e., L1k D Sk m2k . This also is valid for the largest order statistic of L1k , i.e., L1 D S m2 . Since mi 0 8i , parties keep with their initial solutions in case of a negativevalued realization of L1 , which results in zero gains for both of them. Hence, (5.7) can be written as m1 C L1 01 1 1 l L dL ; (5.10) 2 m1
˚ with l 01 L1 D 1= l 1 l 01 , l 01 D max 0; l 1 , and P L1 0 as the g m1 D P L1 0
Z
l1
1
probability that L1 0. Analogously to the calculation of the optimal lump sum in Sect. 5.2, the optimal solution to (5.10) can be by the first-order condition if the cumulated determined density function of l 01 L1 has an IGFR, which holds for uniform distributions. Then the partial derivative, ! Z l 1 01 1 1 1 @g 1 m1 l L 1 1 01 dL m l m ; DP L 0 @m1 2 m1
(5.11) 0
1 is decreasing for ranges i this derivative takes a positive value. If for m , the h where 0 root to (5.11), m1 2 l 01 ; l 1 holds, we obtain for the optimal markdown m1 :
Z
l1 m1
l 01 L1 dL1 D 2m1 l 01 m1 :
(5.12)
5.3 Surplus Sharing by a Double Auction 0
147
i
h
If m1 … l 01 ; l 1 , m1 D l 01 in analogy to the preceding section. Replacing l 01 L1 D 1= l 1 l 01 and applying a simple transformation to (5.12), we get for the optimal markdown ) ( 1 1 01 l m D max l ; : 3 If m1 D l 01 , the markdown does not cut off any potential improvements and the efficiency of the mechanism is maximized. Hence, for calculating the lower bound, m1 D l 1 =3 is relevant. Consequently, Z
l1 l1 3
C L1 01 1 1 l L dL D l1 2 3 2 2 2 2 1 1 1 l1 1 l1 2 C l l 3 2 9 1 1 3 l1 : DP L 0 DP L 0 2 l 1 l 01 3 l 1 l 01 g1 P L1 0
(b) Next, we calculate an upper bound on g max . Since gmax D g m C L , and m follows. Hence, we obtain: 2
1
2
l 2,
max
is maximized for m D 2
Z
l1
g max l 2 C
l 2.
Rs s 2
s .S / dS , S D
With m D l 2 , l 1 D l 1
l 01 L1 D l 2 :
l 1
Analogously, with S D m C L , we get 1
2
g max l 1 : 1 Also the probability P L 0 decreases with m2 . The minimum is P L1 0 D 1=2 for m2 D l 2 . Since si D 0, l 01 D 0 holds. Hence, we obtain g mech l 1 =6 C l 2 =6. Since g max l 1 and g max l 2 , g max 0:5 l 1 C l 2 . Hence, the minimum efficiency of the mechanism becomes 1=6= .1=2/ D 1=3. (c) Consider the case that only one proposal has been generated. From L1 D S m2 and L2 D S m1 , we obtain L1 C m2 D L2 C m1 . Since m1 increases with the upper bound of L1 , this equality is only fulfilled for L1 D L2 and m1 D m2 . Hence, g1 D g 2 holds. Since m1 D m2 , L1 D S m1 D S l 1 =3. Since s D 0, l 01 D 0, and hence, l 1 D l 1 =3, P L1 0 D l 1 = l 1 =3 C l 1 D 3=4. With g1 D g 2 , the lower bound on the gains from the mechanism becomes l 1 =2. We compare this to Z s sCs s2 s2 : s .S / D D g max D s s 2 s
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5 New Coordination Mechanisms
With s D 0 and s D l 1 Cl 1 =3, gmax D 2l 1 =3 follows. Hence, at least .1=2/=.2=3/ D 3=4 of the surplus will be realized by the mechanism on average. This result shows that minimum profits are guaranteed for both parties, which underlines the applicability of the mechanism. These profits are substantial, especially if parties choose equal markdowns, which they will do if the probability distributions of the leeways are equal for the each proposal. Equal probability distributions seem frequently a suitable approximation; due to the privateness of the decentralized data, pronounced and unambiguous differences in the advantageousness of the single proposals are not to be expected. For unequal markups, a lower performance bound could be proven. Note, however, that the decrease in the performance is mainly due to the complexity of the underlying mechanism, which does not permit the determination of a tighter bound. We expect a considerably better performance of the mechanism when applied for real-world coordination. Next, we present the extension of the mechanism to several parties. Here the participating parties submit sealed bids on all proposals. The determination of the proposals implemented, however, is somewhat more involved and requires the solution of an additional optimization model. Like in the two-party case, this model can be solved by each party since the necessary data is globally known after revealing the bids submitted. If the decentralized problems can be modeled as LP problems, the proposals implemented can be determined by MP.37 For MIP problems, combinations of proposals are not allowed. Then the “winning” proposals have to be determined along the lines of a combinatorial auction.38 For this purpose, we present a model based on a set packing problem.39
max
kN XX
sji Zij
(5.13)
ij Zij b0
(5.14)
i 2P j D1 N
(CS-COM) s.t.
k XX i 2P j D0
Zij 2 f0; 1g
N 8i 2 P; j D 0; : : : ; k:
(5.15)
Data ij Resource use for the j th proposal generated by party i ; i 0 denotes the resource use of the default solution sji Savings of party i with proposal ij Variables Zij Binary variable (D 1 if proposal i of party j is implemented, D 0 otherwise) 37
See Sect. 4.1.2. See also Sect. 3.3.1. 39 For the definition of the set packing problem and an analysis of its properties, see, e.g., Balas and Padberg (1976, p. 710). 38
5.4 Comparison of Mechanisms and Discussion
149
The objective function (5.13) minimizes the sum of the savings for the proposals implemented. Constraint (5.14) assures that the corresponding central resource use does not exceed b0 . Constraints (5.15) define variables as binary. Fortunately, the solution of CS-COM does not impose major computational difficulties. Available procedures allow the solution of problems with up to 30; 000 variables,40 which considerably exceeds usual problems sizes for the mechanism proposed (e.g., 20 proposals generated and two parties require only 42 variables). After the implementation of the winning proposals, a share of the surplus of 1= jP c j is allocated to each party among P c , with P c as the set of parties for which proposals different to the initial solution have been chosen. If we assume prior knowledge of parties about their potential leeway for bids, parties’ best strategies can be determined like in the two-party case. Analogously to the marginal surpluses, the leeway Li for bids of party i can be defined as the difference of the systemwide surplus with and without the implementation of a proposal different to the default for party i . Since the leeway for P the whole system is equal to the systemwide sur plus, we obtain that at least 1 2 i 2P E Li = .3S sys / of the systemwide surplus can be obtained on average, provided that Li is uniformly distributed.41 A further relevant issue for this mechanism is the interplay of reputation and learning effects42 if coordination is undertaken repeatedly.43 Analogously to the preceding sections, however, neither theoretical nor empirical analyses of sealed bid double auctions have been carried out yet for this dynamic environment.
5.4 Comparison of Mechanisms and Discussion In the preceding sections, we have presented three different mechanisms for implementing the coordination schemes proposed in this work. Without further assumptions about the context of their application, none of these mechanisms can be strictly preferred over the others. The most important differences between them are summarized in Table 5.5. For the mechanism with surplus sharing determined by the IP, the total surplus from coordination can be realized, provided truthful cost reporting by the RP.
40
See, e.g., Andersson et al. (2000, p. 6). Analogously to Theorem 5.2, with constant markups chosen by parties, the lower bound is P 1 i2P E Li = .4S sys / of the systemwide surplus. Further note that the sum of parties’ leeways may exceed S sys or go below Lsys , depending on the complementarities of the central resource use in coordinated solutions; for an analogous discussion, see Sect. 5.2. 42 For literature on learning and signaling strategies, see the corresponding discussion of the preceding section. 43 For experimental studies of learning in repeated (anonymous, i.e., without signaling effects) sealed bid double auctions under the standard assumption of common knowledge about the distributions of parties’ reservation values, see Rapoport et al. (1998, p. 221), Seale et al. (2001, p. 177), and Daniel et al. (1998, p. 133). 41
150
5 New Coordination Mechanisms Table 5.5 Comparison of coordination mechanisms Surplus sharing by IP Lump sum
Double auction
Total savings realized?
No
No
Yes Rather equal
No Equal
Strategy proofness Rights in savings sharing
Yes (given truth-telling) No Unequal
A natural drawback of this mechanism, however, is the missing strategy proofness. Moreover, the inability of the RP to influence surplus sharing might constitute a further obstacle for the acceptance of this mechanism in practice. An obvious advantage of the second mechanism (lump sum) is that truth-telling is a weakly dominant strategy for parties. Moreover, parties’ rights in surplus sharing are allocated more equally despite the asymmetric procedure applied. Surplus sharing does not depend on the IP only, but on the messages of all parties sent within the mechanism. For two risk-neutral parties with uniformly distributed prior knowledge about the surplus from coordination, a share of 1/3 of the surplus is allocated to the IP and 2/3 to the RP. Moreover, we could show that with slightly less aggressive bidding by the RP, surplus sharing becomes more equal and the efficiency of the mechanism improves substantially. However, since coordinated solutions are only accepted if the surplus from coordination exceeds the lump sum, not all potentially profitable proposals can be implemented. This drawback also applies to the sealed bid double auction, where rational, selfinterested parties include markups or markdowns into their bids in order to obtain larger parts of the surplus. Since a disclosure of cost effects during the proposal generation is not possible, the performance of the underlying schemes may be somewhat affected.44 Moreover, a double auction has the advantage of equal rights in surplus sharing, i.e., the shares received by parties only depend on the actual surplus and on the parties’ skills in choosing optimal bids. A further advantage of this mechanism is that real-world decision makers are familiar with auctions since auctions are frequently used within B2B (business-to-business) relationships.45 To further illustrate the advantages and the different scopes of the mechanisms, we outline three hypothetical settings for their practical application. First, consider a supply chain consisting of an OEM (original equipment manufacturer) and one of its component suppliers. In this setting, parties’ roles are fairly natural if one of the mechanisms presented in Sects. 5.1 and 5.2 is applied. The OEM will serve as the IP determining the proposal implemented and the supplier as the cost-reporting party. Assigning these roles differently seems inferior; then a supplier would determine the proposal finally implemented (for himself and other suppliers), which might not be accepted by other suppliers. 44
See also our computational tests of Sect. 6.1, p. 168. See e.g., Elmaghraby (2004, p. 214) for examples of auctions in B2B marketplaces and Hohner et al. (2003, p. 23) for a case study about the use of procurement auctions at Mars.
45
5.5 Application with Rolling Schedules
151
Whether the lump sum payment will be preferred, depends, amongst others, on the negotiation power of the OEM. Assume a very powerful OEM that has insight into the cost calculations of the supplier, e.g., due to open-book accounting agreements.46 Then an incorrect reporting by the supplier becomes less probable, which mitigates the principal disadvantage of the mechanism presented in Sect. 5.1. Moreover, a very powerful position of the OEM is in line with the procedure for savings sharing applied in this mechanism. With less negotiation power of the OEM, the mechanism based on the lumpsum payment might be preferred. If coordination is not enforced by the OEM, but introduced as a voluntary improvement project, the suppliers might not consent to leave the allocation of the surplus completely to the OEM. In order to cover their transaction costs, they might require a minimum reward if a coordinated solution is implemented. Advantages of this mechanism compared to the double auction in this setting are the guarantee of truth-telling that supports long-term collaboration, the option for the OEM to keep his cost data completely private, and the higher performance of the underlying scheme.47 As a third setting, consider parties in a rather loose relationship with equal rights for each of them. In such a setting, equitable possibilities for obtaining benefits by coordination may be crucial such that a mechanism is perceived as fair by parties. Here we propose the use of a sealed bid double auction. It captures the independence of supply chain parties and provides each party with equal rights in information disclosure and surplus sharing.
5.5 Application with Rolling Schedules To mitigate unfavorable effects of uncertainties inherent in demand forecasts, practical supply chain planning is often based on rolling schedules.48 Hence, the applicability for rolling schedules is a major requirement for practicable coordination mechanisms. The mechanisms presented in this work do not need significant adjustments; in this section, we outline how they can be embedded in the corresponding planning processes. A natural application of these mechanisms is the identification of a systemwide near optimal plan if an operational replanning has to be undertaken. Such replanning becomes often necessary when targets from a long-term contract have to be implemented in operational planning. Due to uncertainties in the end item demand, the quantities fixed in a long-term contract may become inefficient for the whole supply chain at a later point of time. Even if they have been optimal subject to the information status at the time of their negotiation, this may change with an additional revelation of information in later periods. Consider the following example. 46
See, Agndal and Nilsson (2008, p. 154) for a compilation of approaches for open-book accounting in supply chains. 47 See our computational results of Sect. 6.1. 48 See also Sect. 2.1.2.
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Example 5.9 Assume a two-party supply chain with one item supplied, for which a long-term supply contract has been negotiated between buyer and supplier. This contract specifies the quantities that have to be delivered by the supplier in each period as well as the purchase prices for these items. Assume that we are at the end of period 2 and that the operational planning for periods 3–14 has to be undertaken. In Fig. 5.8, the supply quantities fixed in the long-term contract and those needed to fulfill the current demand forecast are displayed. In period 4, the supply quantities required exceed those agreed in the long-term contract. Hence, in this period, greater quantities might be preferred for the whole supply chain, given that the supplier holds enough inventories and that his overtime costs for temporally greater production are lower than the buyer’s backorder costs. On the contrary, if demand is lower than expected (periods 3 and 5), lower supply may be preferable if holding costs of the buyer or overtime costs of the supplier can be saved. In later periods, the flexibility of production is usually greater and forecasts are less concise, such that no changes in the supply quantities may be necessary. In this example, the supply quantities of the long-term contract are (near) optimal for the buyer in periods 6–14. If the costs for the supply plan of the current period and that specified by the long-term contract differ significantly, the buyer will communicate a new order to the supplier. The supplier’s decision whether to agree to this change depends on the leeway for the supplier’s order fulfillment fixed in contractual agreements49 and the supplier’s cost changes due to this new order. If he does not agree, the resulting conflict can be resolved by one of the coordination mechanisms proposed.50 Taking the supply quantities specified by the long-term contract as the initial proposal,
Supply quantity [MU]
80 70 60 50 40 30 20 10 0
2
4
6
8
10
12
14
16
Periods Current demand forecast
Long-term contract
Fig. 5.8 Fixed long-term contract and current deviations
49
Such leeway is often specified by a flexibility range, e.g., Tsay (1999, p. 1341). Of course, such conflicts only arise if the supplier holds too little inventory to fulfill the buyer’s orders without a significant increase in his costs. Hence, a further interpretation of the mechanisms proposed is that of instruments to reduce inventory at decentralized parties.
50
5.5 Application with Rolling Schedules
153
the mechanisms generate proposals for a systemwide improvement and establish compensation payments if one of these proposals is accepted. The same idea can be applied with rolling schedules. Here, a replanning may become beneficial for both parties; for the buyer, e.g., if demand fluctuates more than anticipated, and for the supplier, e.g., if a machine failure occurs or if urgent orders of other customers have to be preponed. If replanning results in a conflict of parties’ interests, this conflict can be resolved by the coordination mechanisms proposed. Then the initial proposal corresponds either to the quantities fixed in the long-term contract or to the outcome of an earlier operational replanning process (see the subsequent example). Example 5.10 Assume a time frame of 18 periods covered by a long-term contract and a horizon of operational planning of six periods (see Fig. 5.9).51 If the buyer initiates coordination for the first time at period 2, the initial proposal corresponds to the supply quantities of periods 2–7 as fixed in the long-term contract. Here, the outcome of coordination results in smaller supply in periods 3 and 5 and greater supply in periods 4 and 6. Assume that the next plan change takes place in period 5. Then, the default solution comprises the quantities of periods 5–7 as agreed in the previous coordination process and those of periods 8–10 as fixed in the long-term contract, and so on. The approach to use the supply quantities fixed in the last coordination process as the default solution is not affected by concerns raised by Dudek (2004)52 about the application of his coordination scheme for rolling schedules. Dudek was worried by the fact that solutions determined by coordination can be discarded in future coordination processes. First, he was concerned about the consequences on compensation payments. Since the scheme proposed by him does not allow for backorders of the buyer, he had to design an extension, which includes additional penalty costs for potential infeasibilities for the buyer arising with an evaluation of the proposals
Fig. 5.9 Application of the mechanisms with rolling schedules
51 52
For ease of exposition, the existence of a frozen horizon is omitted here. See Dudek (2004, p. 116).
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5 New Coordination Mechanisms
generated by the supplier.53 Since these penalty costs have no meaningful (monetary) explication, they obviously cause problems when determining compensation payments. This issue, however, is not relevant for the schemes proposed here since they allow for modeling backlogs and lost sales, the real consequences of uncertain demand and shortages.54 Dudek’s second concern was that discarding coordinated proposals in future periods represents a waste of coordination effort. We argue, however, that such changes are natural and simply reflect the actual information statuses of parties facing a planning problem subject to uncertainty. If, e.g., tendencies for an increase of demand are recognized in an earlier period and a plan change including compensation payments is arranged, coordination effort and payments are not wasted; they fix the new status quo which – if not directly implemented – forms the base for future replanning and coordination. For parties spending positive payments for the implementation of a coordinated proposal, this new status quo solution comes closer to the expected optimum than the previous solution. Hence, it can be expected that these parties will have to incur lower payments in future coordination processes. For the whole supply chain, in turn, less deviations of the current plan from the systemwide optimum and, hence, less coordination effort can be expected.
53 54
See Dudek (2004, p. 120). See Sect. 4.4.3.
Chapter 6
Computational Tests of Coordination Schemes
In this chapter, we analyze the performance of the schemes proposed for several classes of randomly generated test instances as well as for real-world planning problems. In Sect. 6.1, we begin with basic insights about the coordination performance for the generic Master Planning model with forced compliance by the supplier. Section 6.2 deals with the coordination of uncapacitated multi-level lotsizing. In the further sections, the model complexity is successively increased: Sect. 6.3 addresses capacitated lot-sizing and Sect. 6.4 campaign planning under voluntary compliance. For the real-world data of Sect. 6.5, we additionally include lost sales and a series of further restrictions such as inventory capacity constraints and minimum transportation lot sizes.
6.1 General Master Planning Model This section and Sects. 6.2–6.4, which deal with randomly generated test instances, are structured as follows: First, we describe the generation of the test instances unless the instances have already been proposed in the literature. Second, we present basic results regarding the coordination performance and compare them with upstream planning as the default procedure without coordination. Third, we conduct sensitivity analyses for different parameter settings and specifications of the schemes.
6.1.1 Generation of Test Instances and Performance Indicators The generation of hard test instances is crucial for obtaining reliable insights into the performance of algorithms or model formulations. Therefore, this question has received broad attention in the related literature.1 Surprisingly, the literature on 1 See, e.g., Kolisch and Sprecher (1995, p. 1693) for the generation of hard instances for the resource-constrained project scheduling problem.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 6,
155
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6 Computational Tests of Coordination Schemes
coordination mechanisms does not emphasize this issue, but is limited to specific settings derived from real data,2 seemingly arbitrary random data,3 and instances developed for monolithic optimization problems.4 However, when using such approaches for test data generation, two problems may arise which lead to test instances that are rather easy to coordinate. Both problems are due to large cost asymmetries. First, proposals by the party with the larger costs may come very near to the systemwide optimum without further need for coordination. Second, large cost asymmetries among single items may allow the scheme to concentrate on the high-cost items, which has a similar effect like reducing the problem size. We illustrate each of these problems by an example. Example 6.1 Consider a supply chain with one item supplied, which is directly sold by the buyer to an external market. Let the production of the item require the use of several capacity-constrained resources at the supplier’s site. Moreover, assume that the backorder costs for the buyer are very low compared to the overtime costs of the supplier (for the production of an equivalent amount of the item). In this setting, deviations from the orders preferred by the buyer may lead to large savings in the supplier’s overtime costs and only require small increases in the buyer’s backorder costs. Hence, upstream planning with forced compliance by the supplier may result in large suboptimalities, which can easily be removed by a proposal that does not imply overtime for the supplier. Example 6.2 Consider a supply chain with two items supplied (A, B), which are directly sold by the buyer to the market. These items are produced by the supplier on a single resource with restricted capacity. Thereby, the production coefficient of A exceeds significantly that of B. Moreover, let the backorder costs for A exceed those for B by a small amount. Further assume that the sum of the order quantities of A and B is restricted for the buyer. In this setting, upstream planning with forced compliance may lead to a suboptimal outcome since the buyer may inappropriately prefer delays of B instead of A. In the systemwide optimal solution, in turn, only delays of A, but no delays of B will be included. In case of further items with characteristics similar to that of B, the structure of the optimal solution (i.e., to produce less of A), and, hence, the coordination effort remains unchanged. That way, cost asymmetries among items act in a similar way like a reduction of the problem size. To avoid these problems, we propose the generation of test instances where the sensitivities of parties’ major cost factors with respect to central resource use are equal on average. Below, we show how this can be achieved for GM as the underlying optimization model.
2
E.g., Karabuk and Wu (2002, p. 753) and Walther et al. (2008, p. 343). E.g., Arikapuram and Veeramani (2004, p. 119). 4 E.g., Ertogral and Wu (2000, p. 937), Dudek and Stadtler (2005, p. 681), and Dudek and Stadtler (2007, p. 465), all rely on instances generated for the MLCLSP by Tempelmeier and Derstroff (1996, p. 743). 3
6.1 General Master Planning Model
157
Master Planning problems that do not comprise lot-sizing or scheduling activities can be tackled by standard solvers even for large-scale instances. This explains that few test instances specifically addressing Master Planning without discrete variables have been developed in the literature.5 This holds a fortiori for test instances for the coordination of Master Planning; the only test set we are aware of is that of Arikapuram and Veeramani (2004),6 which, however, addresses a strongly simplified setting with one item and one constrained resource per supplier and does not include considerations about coordination hardness. Hence, we had to design a completely new set of test instances for this study. Assuming Master Planning with forced compliance and omitting the option for lost sales,7 the major cost components with impact on central resource use are backorder costs for the buyer and overtime costs for the supplier. In order to avoid easy coordination, we set the cost sensitivities of parties for changes in the supply to be equal on average. I.e., we set increases in the supplier’s overtime costs, which are potentially necessary to avoid shortages in the supply, equal to the average backorder costs of the successor items at the buyer’s site. Additionally, we conduct sensitivity analyses for different relations between average overtime and backorder costs (1:5, 5:1). For this purpose, we define the cumulated capacity requirements of an item j as aj D
X
M X
rkj ak C
amj :
(6.1)
mD1
k2Rj
Backorder costs for one unit of item j in a period are set to blcj D ajK ;
(6.2)
with K as a constant of an arbitrary value (e.g., 1) used for the correct transformation of the unit of aj . To focus on the temporal distribution of the supply quantities, we assume zero backorders at the end of the planning interval. Overtime costs for one unit of capacity increase on resource m are set to P ocm D
j 2Jm
blcj amj
jJm j
;
with Jm as the set of items produced on resource m. In order to determine unit costs for inventory holding, we set the value of an end item equal to its backorder costs8 5 One of the few exceptions is Scholl (2001, p. 314), who has designed instances for testing robust optimization procedures. 6 See Arikapuram and Veeramani (2004, p. 119). 7 In order to decrease complexity, the modeling of lost sales is deferred to Sect. 6.5. 8 Of course, this relation between backorder and overtime costs of items does not always hold in practice. Here we rely on this assumption to account for the fact that items with larger backorder costs are often more valuable and, hence, require higher capital costs for their holding in inventory.
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6 Computational Tests of Coordination Schemes
and the value of a preliminary item equal to its cumulated capacity requirements in monetary units. Assuming monthly planning periods and only capital costs (10% p. a.) for inventory holding, we get hj D
aj K : jT j 10
(6.3)
Like in Sects. 6.2 and 6.3, zero initial inventories are assumed. The average capacity utilization is set to 90%. By default, we assume that overtime is not restricted and continuously available. In order to assess the sensitivity of the scheme regarding discrete decisions, we additionally consider a setting where overtime can only be taken in an integer number of shifts of one-eighth of the period capacity. For this purpose, GM is extended by declaring variables Omt as integer and multiplying these variables both in the objective function of GM and in the capacity constraint (2.3) by 1=8kmt .9 We investigate several test sets differing by their structures (see Fig. 6.1), the number of time periods, the inclusion of integer variables (for modeling overtime), |JD |=2 10
|JD|=4, J=16, 4 M=7
7
|JD |=10 1
11
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items j
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S5
5S S1; S2; S3; S4; S5 (|JD|=10)
Fig. 6.1 Structures of test instances for GM 9
See Albrecht and Stadtler (2008) for an explicit modeling of this extension.
6.1 General Master Planning Model Table 6.1 Notation for test sets Characteristic Parameter ˇ ˇ ˇ Dˇ Structure ˇJ ˇ D 2 ˇJ D ˇ D 4 ˇ ˇ ˇJ D ˇ D 10 Time jT j D np Model class LP MIP Ratio backorder 1:1 / overtime costs 5:1 1:5 Cost exchange
Modifications
Number of suppliers
Initial solution
S!B B!S 1-shot All Generic agg-P Without CS1i agg-P agg Without CS1i , CS1i 1S 2S 5S Upstream RAND
159
Comment Structure with 2 items supplied Structure with 4 items supplied Structure with 10 items supplied np periods Overtime modeled by linear variables Overtime modeled by integer variables Equally distributed backorder and overtime costs Costs for backorders are five times the costs for overtime on average Costs for backorders are one fifth of overtime costs Unilateral reporting of cost changes by S Unilateral reporting of cost changes by B One-shot disclosure of cost changes All levels of modifications Generic scheme for LP agg-P All models except for CS1i are run agg-P agg All models except for CS1i and CS1i 1 supplier producing all items supplied Items produced by 2 suppliers Items produced by 5 suppliers Upstream planning with forced compliance Random initial solution
and the exact specifications of the scheme (this includes the direction of the agg-P information exchange, the use of CS1i , etc.). The single options and their abbreviations are summarized in Table 6.1. We ˇuse combinations of them in order ˇ to describe the single settings investigated (e.g., ˇJ D ˇ D 4, jT j D 2). For each test set, demands, production coefficients, and capacities have been generated based on random numbers. All random numbers have been drawn from normal distributions with base values of 1 and different combinations of coefficients of variation.10 Thereby, negative values have been replaced by 0 and all random numbers have been divided by their averages to exclude any side-effects by this replacement. Six variations of demand data have been considered, each made up of a combination of a coefficient of variation (abbreviated by CV in the following) C Vd 2 f0:1; 0:3; 0:5g and a seasonality component. This component corresponds to a cosine (cos.) oscillation with an amplitude of 0 (no seasonality) or 0.2. For production coefficients and period capacities, coefficients of variation
10
In order to avoid side effects due to fixed profiles for demand, capacities, etc., that have been generated once and potentially comprise some extreme characteristics, we have generated new (and hence, different) random data for each test instance.
160
6 Computational Tests of Coordination Schemes Table 6.2 Input parameters for the test data generation for GM Parameter Base value Coefficient of variation Demand 1, 1, 1, 1 (cos.), 1 (cos.), 1 (cos.) 0.1, 0.3, 0.5, 0.1, 0.3, 0.5 Capacity requirements 1 0.001, 0.1, 0.2, 0.3, 0.4, 0.5 Production coefficients 1 0.2, 0.5, 0.7, 1, 1.5, 2
of CV coe 2 f0:2; 0:5; 0:7; 1; 1:5; 2g and CV cap 2 f0:001; 0:1; 0:2; 0:3; 0:4; 0:5g, respectively, have been chosen. This yields 216 instances per test set. We summarize their characteristics in Table 6.2. To assess the coordination performance of the schemes proposed, we mainly rely on three performance indicators. The first is the suboptimality of upstream planning for the test instances generated. Formally, the average gap of the uncoordinated solution11 (AGU) as be expressed by PN nD1
cunc;n cce n;n cce n;n
N
:
Index n Index of test instance Data cce n;n Costs of the solution to the centralized model for test instance n cunc;n Costs of the uncoordinated solution (here: upstream planning) for n ccor;i;n Costs after i iterations of the scheme for instance n Second, we consider the average remaining gap (AGS) after the application of the scheme: PN ccor;i;n cce n;n nD1
c
ce n;n : N The third indicator is the average gap closure (AGC) after i iterations of the scheme:
PN
cunc;n ccor;i;n nD1 cunc;n cce n;n
N
:
When displaying the performance of the scheme according to the number of iterations, we rely on the remaining gaps that are equal to 1-AGC. The abbreviations for the performance measures as well as further indicators characterizing the solutions obtained can be found in Table 6.3. As a first insight into the properties of the test instances generated, Table 6.4 provides ˇ D ˇ an overview about selected results for the AGU in the default setting (ˇJ ˇ D 4, jT j D 12, LP, 1:1, Upstream). The column headers differing from
11
Note that in general, when we define a gap in this work, we consider the percentage deviation of an upper bound (UB) to a lower bound (LB), i.e., (UB-LB)/LB.
6.1 General Master Planning Model
161
Table 6.3 Abbreviations for performance measures Abbreviation Explanation AGU Average gap of the uncoordinated solution AGS Average gap after the application of the scheme AGC Average gap closure after the application of the scheme N Number of instances analyzed per test set NOPT Number of solutions found per test set, with costs equal to or lower than those of the solution to the centralized model TC Average solution time for the centralized model (in seconds) TS Average time for running the scheme (in seconds)
Table 6.4 Suboptimality of upstream planning (AGU) jT j D 3 Default MIP jJ D j D 10 jT j D 24 5:1
1:5
10.9%
66.4%
15.1%
21.8%
28.1%
22.5%
8.9%
default indicate how the test sets have been altered (e.g., jT j D 3 means that instances with three periods instead of 12 have been generated). It can be recognized that increases in the model complexity due to a larger number of items, periods or mixed-integer variables increase the average suboptimality of upstream planning and, hence, the need for coordination. Moreover, varying the ratio among average backorder and overtime costs considerably affects the AGU. With larger values of this ratio, deviations in the costs from the systemwide optimum will lead to larger cost increases for the buyer, but smaller decreases for the supplier. This will lead to an increased probability that proposals unilaterally determined by the buyer come near to the systemwide optimum, which results in smaller AGU. Figure 6.2 depicts the cumulated probability distribution of the initial gaps for the test set jJ D j D 10, jT j D 12, LP, 1:1. The shape of this curve is representative for most instances generated in this study.12 The solutions from upstream planning are characterized by a large number of solutions with small gaps and few solutions with very large gaps. This observation is interesting for two aspects: First, it underlines that the need, and hence, the benefits of coordination may differ substantially according to the concrete choice of input parameters. Second, it reveals that the assumption of a uniform distribution of the surplus, for which analytical results could be obtained in Chap. 5, will only roughly reflect parties’ expectations. Anyhow, as argued there, parties’ behavior will additionally deviate due to learning and signaling effects. Figure 6.2 in this context provides an indication
12
Note that for some other test sets, in particular for the MLULSP (see Sect. 6.2), the inclinations of the corresponding curves differ far less for smaller and larger gaps (i.e., the curves resemble more a straight line). The overall characteristic, however, that few instances with large gaps exist is also valid there.
6 Computational Tests of Coordination Schemes
0.8 0.6 0.4 0.2 0.0
Cumulated distribution function
1.0
162
0
100
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Fig. 6.2 Distribution of the initial gaps for GM
about the general characteristics of parties’ prior expectations.13 Therefore, Fig. 6.2 constitutes a starting point for further analytical or empirical investigations about the behavior of parties using the mechanism proposed.
6.1.2 Analysis of Solutions for the Generic Scheme The generic scheme has the advantage of finding the systemwide optimal solution within a finite number of iterations, provided that all but one decentralized problems can be modeled as LP (in the version with unilateral exchange, even one problem may be of a MIP type). Since the maximum number of iterations increases exponentially with the problems size, however, there is no guarantee that larger problems can be tackled within a practicable number of iterations. To assess the solution performance and the limitations of the generic scheme, we have carried out a series of test runs for small and medium-sized problem instances. As throughout this study, the main routine for running the scheme has been implemented in the programming language Java (version jdk 1.6.0) using Eclipse (version 3.3.2). Optimization problems have been solved using Xpress-MP, release
13
Interestingly, this equally extends when considering the absolute cost increases compared to the upstream solutions; there, very similar curves can be obtained (with many instances with small cost increases and few instances with large costs increases).
6.1 General Master Planning Model Table 6.5 Solution performance of the generic scheme AGU (%) AGC (%) AGC (%) ˇ ˇ ˇJ D ˇ D 3,jT j D 2 12.6 0.4 98.7 Default 10.9 2.1 62.7 MIP 21.1 16.1 22.7 jT 21.9 14.6 3.0 ˇ 6 ˇ jD ˇJ D ˇ D 3,jT j D 2,1-shot 12.6 1.7 65.7
163
N 65 139 173 194 65
NOPT 59 20 2 0 9
TC 0.02 0.02 0.03 0.03 0.02
TS 467.9 1,550.8 474.3 1,689.3 1,176.6
2008A. The data exchange between Xpress-MP and Java has been partly managed using the Java interface of Xpress-MP. In this section, tests have been run on a single thread of an Intel SMP (single machine processor) with 1.99 GB RAM and a clock speed of 2.61 GHz. ˇ ˇ As the default setting we assume here: ˇJ D ˇ D 4, jT j D 3, LP, 1:1, S!B, generic, 1S, Upstream.14 With time limits of 30 s for all problems, all centralized models have been solved to optimality. Since we aim at exploring the maximum coordination performance, we have chosen a very large number (500) of vertices for the linearization described in Sect. 4.3.1. Throughout this section, we only investigate instances with a minimum suboptimality of 0.005%. Table 6.5 provides the results obtained after 50 iterations of the scheme. In the smallest test set with two periods and three items supplied for the version of the scheme with unilateral exchange of cost changes, the systemwide optimum could be identified for almost all instances after 30 iterations. Also for the LP instances with four items supplied and three periods (default), a large part of the suboptimality due to upstream planning has been mitigated. This, however, changes drastically if decentralized parties run MIP problems or if the model size is further increased. Then, less than one third of AGU can be mitigated. This result is even worse when considering the second indicator AGC. Since the scheme performs better for test instances with greater AGU, the ratio AGS/AGU exceeds the AGC here. The version with one-shot exchange of cost information has been tested for the smallest set. Although near optimal solutions have been obtained, too, some decrease compared to an unilateral exchange can be observed. We suggest that this decrease is due to the different choice of the starting solutions, which – due to the limitation of the information exchange – does not allow to concentrate on the most favorable regions of the solution space. However, also in this case, the significant gap closure has been possible with the identification of a small share of the vertex solutions of the decentralized problems. For both versions, the average time for running the scheme has been much greater than that needed for the centralized solution. This is intuitive since LP problems can
14
We consider here the base version of the generic scheme with finite convergence for LP and not its extension for finite convergence for one decentralized MIP due to the increased computational complexity of this extension.
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6 Computational Tests of Coordination Schemes 100
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Fig. 6.3 Convergence paths for generic scheme
be tackled very efficiently by standard solvers, whereas the linearization requires SOS2, which significantly increases the computational effort.15 Further interesting insights about the coordination performance provide the convergence paths, which comprise the remaining gaps after 1–20 iterations (i.e., 1-AGC). For the scheme with unilateral exchange of cost information, the convergence paths are displayed (see Fig. 6.3). Interestingly and in contrast to the heuristic modifications considered later on, the curves grow almost linearly with the number of iterations for the LP models. The inclinations of the curves, however, decrease considerably with increased problem sizes. The flattening of the curve for MIP after 36 iterations can be explained by the infeasibilities in CS1i that do not allow further progresses in the convergence.
6.1.3 Analysis of Solutions for the Modified Scheme As mentioned above, the modified version of the scheme shows advantages in respect of computing time and convergence speed, especially when applied to the 15
Note in this context that the observed decrease in the solution time for MIP can be explained by infeasibilities and, hence, a breakdown of the scheme for some instances. In this case, the best solution found so far has been taken.
6.1 General Master Planning Model Table 6.6 Solution performance AGU (%) ˇ ˇ ˇJ D ˇ D 3, jT j D 2 12.6 jT j D 3 10.9 jT j D 3, MIP 21.2 Default 15.1 MIP 21.8 ˇ ˇ ˇJ D ˇ D 10 28.1 ˇ ˇ ˇJ D ˇ D 10, MIP 34.2
165 of the modified scheme for GM AGS (%) AGC (%) N NOPT 0.8 91.2 65 24 1.2 82.5 139 37 3.5 63.5 173 17 4.0 54.5 200 3 6.8 47.0 208 4 8.8 47.6 212 0 12.2 40.9 212 0
TC 0.00 0.05 0.05 0.01 135.3 0.06 320.2
TS 27.8 56.2 61.7 108.8 234.6 411.6 629.9
ˇ ˇ coordination of MIP problems. As the default setting we assume here: ˇJ D ˇ D 4, agg-P jT j D 12, LP, 1:1, S!B, without CS1i , 1S, Upstream. We applied 10 nodes for the linearization of the objective functions of CS1i . Time limits have been set to 600 s for the centralized models and to 10 s for the decentralized models. With these time limits, all instances of C have been solved to optimality except of 34 instances for the MIP with jJ D j D 4 and 103 instances for the MIP with jJ D j D 10.16 There, the average suboptimalities have been 0.53% and 0.56%, respectively. Table 6.6 provides basic results obtained after 20 iterations. These results indicate that the modifications applied to the generic scheme substantially improve the coordination performance. The size of test instances turned out to be the most important driver for the performance. For the smaller instances with three periods or less, near optimal solutions could be obtained. This suggests that managers applying the scheme should focus on a small number of key products and periods in order to obtain best coordination performance. The presence of integer variables affects the performance, too, although the degradations in the solution quality have been far more modest than for the generic scheme. In general, decreases in the solution time compared to the generic scheme can be observed.17 Like in the generic scheme, the reduction of the AGS compared to the AGU (i.e., the ratio .AGU AGS/=AGU) exceeds the AGC. Further insights into the coordination process provide the convergence paths displayed in Fig. 6.4. An overall characteristic that can be recognized is that the suboptimalities can be mitigated in large part after only five iterations. This is in strong contrast to the generic scheme, where the coordination performance increases roughly linearly within the first 50 iterations (see Fig. 6.3). The modest number of iterations favors considerably the applicability of the (modified) scheme in practice since this allows decision makers to manually control the cost effects of the plans generated.
16
Note that we assume here that the optimal solution has been found if the gap between the best solution and the best bound goes below the standard solution tolerance of the optimizer of 0.01%. 17 Note that the different number of iterations, time limits and number of nodes applied for the approximation may also explain this difference in part; in additional computational tests for the modified version of the scheme with jT j D 3 and the same number of iterations, nodes, and time limits as applied for the generic scheme, the average solution time was less than half the time for running the generic scheme.
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Fig. 6.4 Convergence paths of the modified scheme Table 6.7 Sensitivities regarding the generation of demand data CVd , seasonality 0.1, 0 0.3, 0 0.5, 0 0.1, 0.2 0.3, 0.2 0.5, 0.2 AGU (%) AGS (%) AGC (%)
16.7 2.0 55.7
8.9 3.6 55.4
9.2 2.1 52.1
24.6 6.1 53.0
17.3 4.8 55.7
13.2 5.2 55.2
Next, we analyze the sensitivities in the performance of the scheme regarding changes in the input parameters and the specification of the Here we again ˇ scheme. ˇ rely on the parameter settings of the default setting (i.e., ˇJ D ˇ D 4, jT j D 12, LP, agg-P 1:1, S!B, CS1i , 1S, Upstream). We start with investigating the effects of different input parameters. In Table 6.7, we address the sensitivities regarding the generation of the demand data. Here a somewhat counterintuitive trend can be recognized: The AGU decrease with greater variations in the demand, but increase with the seasonality coefficient. Regarding the AGS, the results are somewhat worse if the seasonality component is included, which may be due to the larger AGU in that case. In general, however, the performance of the scheme seems stable with variations in the demand generation, which is underlined by the small deviations in the AGC observed. Table 6.8 provides the sensitivities according to the random generation of the production coefficients. Here no unequivocal trend can be observed. Both the AGU and AGS behave irregularly, but rather stably with changes in the coefficients of variation.
6.1 General Master Planning Model
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Table 6.8 Sensitivities regarding the generation of production coefficients
CVcoe
0.2
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15.3 3.2 48.1
8.1 3.2 58.2
15.4 5.6 52.1
13.9 3.4 53.8
24.1 5.2 56.9
15.3 3.7 58.6
Table 6.9 Sensitivities regarding the generation of capacity coefficients
CVcap AGU (%) AGS (%) AGC (%)
0.001 11.6 3.4 57.4
0.1 32.8 5.4 60.2
0.2 11.7 5.0 56.7
0.3 10.1 4.1 52.9
0.4 9.1 2.4 47.9
0.5 16.2 3.9 52.6
Table 6.10 Sensitivities regarding specifications of the test instances
Default jT j D 24 1:5 5:1 RAND
AGU (%) 15.1 22.5 66.4 8.9 37,007.7
AGS (%) 4.0 4.8 11.0 1.8 4.1
AGC (%) 54.5 54.1 55.6 59.1 99.4
Next, we consider the sensitivities to changes in the generation of the capacity coefficients (Table 6.9). Also here, the deviations in the results seem rather stable, which suggests that the performance of the scheme is not affected by variations of this parameter. In the following, we analyze the robustness of the default setting (see Table 6.10). We start with considering increases in the number of time periods. Recall that in our basic results (Table 6.6), we have observed a pronounced decrease in the performance when increasing the number of periods from 3 to 12. For further increases to jT j D 24, however, the decreases in the AGC turned out to be modest (the ratio AGU/AGS has even improved compared to the default setting), which indicates an appropriate scalability of the scheme proposed. Next, we consider variations in the ratio between the average backorder and overtime costs of parties. For both increases and decreases in this ratio, both the AGC and the ratio AGU/AGS have improved compared to the default setting. We suggest that this improvement is due to the fact that – as mentioned at the beginning of this section – instances with less equilibrated costs (and, hence, a ratio between overtime and backorder costs different to 1) are easier to coordinate. Subsequently, the choice of a starting solution different to upstream planning (RAND) did not significantly affect the AGS. Of course, the AGU and AGC increase sharply since the uncoordinated supply target does not consider the potential cost increases for the decentralized parties (due to backorders or overtime). Apart from the independence of the performance of the scheme from the starting solution chosen, these results suggest that upstream planning is a rather effective heuristic for decentralized coordination, which explains the frequent use of this procedure in practice.
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Next, we consider sensitivities regarding different designs of the scheme. We begin with different specifications on the information exchange (see Table 6.11). With the buyer as the cost-reporting party, the coordination performance even increases, which underlines the robustness of the results obtained above. The other variant of the scheme with a one-shot exchange of information leads to a slight decrease in the coordination performance, which, however, does not affect the applicability of the scheme proposed. The reason for this decrease is the missing possibility for anticipating the other parties’ cost changes, which is due to the restricted information exchange during proposal generation. A further important question is the optimal specification of the scheme. As agg-P the default setting, we applied the version with all modifications but CS1i . In 18 Table 6.12, we display results for other modifications. The default setting turned out to be significantly superior to the other modifications. As indicated in Sect. 4.3.4, the computing time could be reduced substantially compared to the omittance of agg CS1i . Finally, we study the case that the production of items is undertaken in part by several suppliers (see Table 6.13). At first glance counterintuitive, increasing the number of suppliers improves the quality of the coordination process significantly. This can be explained by the more detailed knowledge of the buyer about the supplier’s costs changes. Such knowledge allows the buyer to estimate the cost effects of deviations in the supply target more concisely and provides greater leeway for
Table 6.11 Sensitivities regarding different forms of information disclosure
Default B!S 1-shot
AGU (%)
AGS (%)
AGC (%)
15.1 15.1 15.1
4.0 2.9 4.4
54.5 58.1 50.6
Table 6.12 Sensitivity regarding the modifications of the scheme AGU (%) AGS (%) AGC (%) TS Default All agg-P agg Without CS1i , CS1i
15.1 15.1 15.1
4.0 5.1 5.7
54.5 49.4 42.8
108.8 88.4 325.3
Table 6.13 Solution performance for multiple suppliers AGU (%) AGS (%) AGC (%) TS Default 2S ˇ ˇ ˇ Dˇ ˇJ ˇ D 10 ˇJ D ˇ D 10, 2S ˇ ˇ ˇJ D ˇ D 10, 5S
18
15.1 15.1 28.1 28.1 28.1
4.0 2.9 8.8 7.3 5.0
54.5 64.3 47.6 52.9 62.8
108.8 70.1 411.6 359.0 218.6
See further Albrecht and Stadtler (2008) for a larger number of test results of the scheme without agg-P agg CS1i and CS1i .
6.2 Uncapacitated Lot-Sizing Problem
169
recombining the proposals of the single suppliers in GM-CS2-MB . The improved performance went along with a significant decrease in the computing time that is due to the smaller models of the suppliers, notwithstanding the fact that the models of two or five suppliers have to be solved here. These results underline the generic character of the scheme and let seem probable a further extendibility to different organizational structures such as supply chains with multiple buyers or more than two tiers. Summarizing, the generic scheme only achieves convergence for rather small test instances within a reasonable number of iterations. The modified version can successfully tackle much larger problems, even if they comprise integer variables. The sensitivity analysis revealed that the performance of the scheme is robust to changes in the input parameters or the application settings, such as multiple suppliers or different specifications regarding the exchange of cost information between parties.
6.2 Uncapacitated Lot-Sizing Problem 6.2.1 Generation of Test Instances For uncapacitated multi-level lot-sizing, the question of coordination hardness is far less important than in Sect. 6.1. Here, reasonable test instances developed for centralized procedures comprise balanced setup and holding costs, too; otherwise, setups would either occur in all periods or only once at the beginning of the planning interval. Most literature on centralized models for uncapacitated lot-sizing relies on arbitrarily fixed input data.19 In order to generate some additional structure useful for analyzing sensitivities with changes in the input parameters, we fix the ratio between marginal holding and setup costs here. The rationale behind this is that, along the lines of the EOQ formula, the expected TBO directly depend on this ratio.20 Here, we take the marginal holding costs and the TBO as given and determine the setup costs scj to 1 (6.4) scj D mhj .TBOj /2 e j : 2 e j is the average secondary demand defined by ej D
ejcum jT j
; (6.5) jT j with ejcum jT j as the cumulated primary and secondary demand of items as defined in Sect. 2.2.2.1.21 19
E.g., Graves (1981, p. 95), Blackburn and Millen (1982, p. 51), Afentakis et al. (1984, p. 233), Afentakis and Gavish (1986, p. 243), and Heinrich (1987, p. 174). 20 See, e.g., Derstroff (1995, p. 93) and Trigeiro et al. (1989, p. 353). 21 See p. 13.
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Fig. 6.5 Structure of test instances
Other deterministic data include the planning horizon (jT j D 12 periods) and six different BOM structures (see Fig. 6.5). Moreover, we have generated a couple of random parameters drawn from a normal distribution with an expected value of 1. Negative random values have been replaced by 0, except for the TBO, where we have chosen a value of 0.00001. Demand series have been generated with different coefficients of variation (0, 0.2, 0.5, 0.2); the fourth series has been multiplied by an additional seasonality component (generation based on a cosine oscillation with an amplitude of 0.3). Random numbers for marginal holding costs are based on coefficients of variation CV mh 2 f0:2; 0:5g. For the buyer’s items, the random numbers directly correspond to mhj , whereas for the supplier’s items, these numbers are multiplied by an additional factor (1, 1.3, 2). This factor has been identified as one of the major drivers for the suboptimality of upstream planning in two-stage supply chains.22 The TBO profiles differ by their averages (1, 2, 4) and their coefficients of variation (CV TBO 2 f0:2; 0:5g). Resulting are 6 4 6 6 D 864 test instances.
22
See Simpson (2007, p. 136).
6.2 Uncapacitated Lot-Sizing Problem
171
6.2.2 Analysis of Solutions In this section as well as in Sects. 6.3 and 6.4, computational tests have been performed on eight threads of an Intel SMP with a clock speed of 2.33 GHz and 8 GB RAM, using Xpress-MP 2008A parallel 64-bit. Details on the time limits and the resulting suboptimalities of the centralized models can be found in Table 6.14. We analyze the performance of the scheme for test instances with AGU > 0:01%.23 Among the 864 instances generated, 646 of them show this characteristic. Table 6.15 depicts the results after 20 iterations of the scheme, i.e., after 20 proposals generated. On average, more than 88% of the gap of upstream planning could be closed by the scheme. In more than half of the instances, even the optimal solution could be identified. Further remarkably is the modest time required for proposal generation. Less than 46.3% of the time necessary for solving the centralized model has been needed on average. Concerning the impact of the different BOM structures on the coordination performance, several conclusions can be drawn. First of all, the size of the test sets seems to be an important driver for the performance. The highest values for the AGC have been obtained for the smaller test sets A, C, and E. The structure of the BOM can be recognized as a further driver. The average gap closure is highest (93.8% and 91.3%) for assembly and serial structures and considerably exceed the AGC for general structures (81.7%). A reason for the favorable results in assembly structures is the limited leeway for changes in the supply pattern.
Table 6.14 Computing time and optimality gaps for the MLULSP A B C D
E
F
Centralized model: – Time limits (s) – Number of suboptimal solutions – Average optimality gaps (%) Decentralized model: time limits (s)
1,200 0 – 10
1,200 23 2.44 10
Table 6.15 Solution performance of the scheme for uncapacitated lot-sizing
23
1,200 0 – 10
A B C D E F
1,200 9 0.99 10
1,200 0 – 10
1,200 14 3.34 10
AGU (%)
AGS (%)
AGC (%)
N
NOPT
4.96 3.89 3.88 2.30 7.44 4.07
0.01 0.35 0.18 0.12 0.37 1.47
98.98 85.48 95.92 91.73 92.45 64.72
104 138 86 90 140 88
102 41 77 64 88 25
Without this limitation, less meaningful results for the AGC for structure D would be obtained. There, the AGC would rise up to 234% since one instance exists where the initial gap is outperformed by 13,202.2% due to the suboptimality of the centralized solution.
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6 Computational Tests of Coordination Schemes
Altering the setup frequencies for a small number of items of the buyer influences a larger number of dependent items of the supplier. As a consequence, a larger part of PB 24 can be covered by our procedure for proposal generation. The weaker performance of the scheme for general structures has already been indicated by the scope of our analytical results of Sect. 4.2, which are only valid for special cases of assembly and serial BOM. Further insights provide the convergence paths. Figure 6.6 shows that a large part of the savings can be reached after the generation of only 10 proposals. Except for the large general test set F, the average remaining gaps have been smaller than 75% at that stage. Noteworthy is the convergence behavior in early iterations. For the small set A and the assembly structures C and D, the gap to optimality could be reduced to less than 20% after only five proposals generated, whereas for the other instances, this gap ranged between 40% and 43%. This result underlines our hypothesis that coordination can be achieved easier for smaller problems sizes and assembly systems. Next, we present sensitivity analyses for assessing how parameter settings for test data generation influence the performance of the scheme. In Table 6.16, we consider
100
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B
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Fig. 6.6 Convergence paths for the MLULSP Table 6.16 Sensitivity analysis for demand
24
See p. 82 for the definition of this set.
CVd , seasonality AGU (%) AGS (%) AGC (%)
0, 0 5.4 0.50 86.6
0.2, 0 4.8 0.35 91.4
0.5, 0 4.1 0.40 88.8
0.2, 0.3 4.4 0.35 87.4
6.2 Uncapacitated Lot-Sizing Problem
173
the impact of different demand structures. Interestingly, level demand yields the largest absolute gaps from upstream planning,25 but also many instances with low coordination potential, which explains the modest differences in the values for the AGU. According to the performance of the scheme, no unequivocal effects could be recognized. Table 6.17 presents the sensitivities regarding the assumptions on marginal holding costs. The AGU are somewhat greater with increased coefficients of variations. The sensitivity of these gaps on the ratio between parties’ marginal holding costs, however, turned out to be low. This result contrasts the study of Simpson (2007), where this parameter had a significant impact on the observed suboptimality of upstream planning.26 We suggest that this difference is due to the ratio between setup and holding costs, which is fixed in this study and randomly chosen by Simpson (2007).27 In Table 6.18, we finally address the sensitivities regarding the TBO. Little intuitively, an increased CV for TBO leads to smaller AGU, a reversed effect compared to the increase in the CV for the marginal holding costs. Moreover, there is a trend for a better performance with large TBO. A potential explication of this phenomenon is that large TBO imply a smaller number of buyer’s setups in upstream planning. This decreases the solution space where potential improvements might be located and, hence, increases the probability that a (near) optimal solution is found. Summing up, the scheme for uncapacitated lot-sizing problems allows the identification of near optimal solutions even for BOM structures comprising up to 30 items. Apart from somewhat weaker results for large general structures, the performance is robust regarding different parameterizations of the test instances.
Table 6.17 Sensitivities regarding the ratio between marginal holding costs CVmh , ratio mhj 0.2, 1 0.2, 1.3 0.2, 2 0.5, 1 0.5, 1.3 0.5, 2 AGU (%) 4.5 4.2 5.4 5.4 4.4 4.8 AGS (%) 0.37 0.37 0.37 0.48 0.42 0.35 AGC (%) 88.7 88.4 84.7 90.5 89.2 89.9
Table 6.18 Sensitivity analysis regarding the TBO CVTBO , base value 0.2, 1 0.2, 2 0.2, 4 0.5, 1 AGU (%) 4.5 5.4 6.2 3.5 AGS (%) 0.23 0.72 0.04 0.34 AGC (%) 96.1 87.7 92.8 82.5
25
0.5, 2 4.5 0.67 82.7
0.5, 4 3.9 0.14 93.1
Note that this information is not drawn from Table 6.16, but from a separate analysis undertaken by us. The same effect has also been recognized by Simpson (2007, p. 127). 26 See Simpson (2007, p. 134). 27 More exactly, Simpson (2007) does not explicitly consider this ratio; it depends on other parameters that are randomly generated, see Simpson (2007, p. 127).
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6 Computational Tests of Coordination Schemes
6.3 Multi-level Capacitated Lot-Sizing Problem In this section, we compare the performance of the scheme presented here with the scheme by Dudek and Stadtler (2005).28 We have generated test data along the lines of Dudek (2004).29 There, several problems structures (S1, S2, S3, M1, M2, L) differing by the BOM and the number of the items supplied (2–7) have been investigated. For each of these structures, 126 instances have been randomly generated based on different demand curves, expected TBO, and capacity utilization profiles. We omit a detailed description of the generation of these instances and refer to Dudek (2004) instead.30 agg-P We have run the version of the scheme: S!B, CS1i , 1S, Upstream, with 10 nodes for the linearization of the objective functions of CS1i and extensions. Table 6.19 displays information about the computing time and the optimality gaps for the centralized models after aborting the solution process.31 As in Dudek (2004),32 we only rely on capacity feasible solutions for our analysis. In Table 6.20, we compare the performance of the scheme proposed by us with the version of the scheme of Dudek (2004)33 that requires the same level of
28
Table 6.19 Computing time and optimality gaps for different structures S1 S2 S3 M1 M2
L
Centralized model: – Time limits (s) – TC – Number of capacity feasible solutions – Number of suboptimal solutions – Average optimality gaps (%) Decentralized model: time limits (s)
600 601.5 101 101 5.8 15
200 11.7 113 1 1.3 5
200 23.2 116 5 1.3 5
200 8.7 117 0 0.0 5
400 168.7 121 25 3.6 10
400 166.4 110 22 3.9 10
See Dudek and Stadtler (2005, p. 681). Instead of the more aggregated presentation of the computational tests in this paper, we rely on the more detailed description of Dudek (2004, p. 165) in the following. 29 See Dudek (2004, p. 168). 30 Note that we have adapted the test instance generation by Dudek (2004) in a minor degree. In order to exclude potential distortions due to fixed demand profiles, the demand data has been generated randomly for each combination of problem parameters; however, both for this specification and for another with fixed demand profiles used in preliminary test runs, the gaps after upstream planning turned out as significantly larger than those reported by Dudek (2004, p. 180) (see also Table 6.20). 31 Surprisingly, for the centralized model L, the average solution time observed exceeded the time limit by a minor degree. A potential reason for that may lie in time lags due to the communication of this data to the Java routine used. 32 See Dudek (2004, p. 178). 33 See Dudek (2004, p. 192).
6.3 Multi-level Capacitated Lot-Sizing Problem Table 6.20 Comparison with Dudek (2004) AGU AGS AGS/AGU It AGU (%) (%) (%) (%) Dudek (2004) S1 S2 S3 M1 M2 L
25.6 33.9 18.3 18.8 15.3 26.6
2.7 1.6 1.2 1.5* 1.6* 2.0*
175
AGS4 (%)
AGS10 (%)
AGC10 (%)
AGS4 /AGU (%)
2.0 1.9 1.3 1.8 1.7 1.9
67.1 46.4 61.0 71.1 64.6 79.4
6 7 4 9 7 8
Albrecht (2008) 11 5 7 8* 10* 8*
3.8 3.2 3.7 4.7* 4.3* 5.1*
39.1 35.4 33.3 25.1 36.0 34.0
2.4 2.6 1.5 2.2 2.5 2.7
information exchange.34 Unfortunately, Dudek (2004) has only tested the sets S1, S2, and S3 with this version. Although not directly comparable, we additionally display for instances M1, M2, and L the results of a version of the scheme of Dudek (2004) that includes a more demanding requirement on information exchange – bilateral exchange of cost changes (we denote this by an “*” behind the corresponding results).35 Similarly to the scheme of Dudek (2004), the scheme presented here was able to reduce most of the gaps of the uncoordinated setting. Unfortunately, the performance of both schemes is not directly comparable since the gaps of upstream planning turned out to be substantially greater in the present study. As a resort, we rely on the relative average reductions of AGU, the ratios AGS/AGU. In the scheme proposed, AGS4 /AGU, the relative gap closure after four iterations – which is somewhat below the average number of iterations the scheme of Dudek (2004) is run36 – is similar to that of Dudek (2004); it has been larger in half of the test sets.37 After 10 iterations, the scheme proposed here clearly outperforms the results reported by Dudek (2004) with four iterations; unfortunately, results for running the scheme by Dudek (2004) for 10 iterations are not available.38 Like in Sect. 6.2, we only consider test instances with AGU 0:01% when determining the AGC in order to obtain meaningful results for this indicator. 35 Since the results of Dudek (2004) for M1, M2, and L with bilateral reporting of cost changes are most probably superior to a limitation of the information exchange (as it has been the case for S1, S2, and S3), the conclusion drawn below – that our results are at least comparable to those of Dudek (2004) – is not affected. Moreover, note that we limit to the AGS here since the AGC are not reported by Dudek (2004) for unilateral information exchange. 36 The number of iterations is determined in Dudek (2004) by a random stopping criterion along the lines of simulated annealing, see Dudek (2004, p. 97). 37 Note that the different time limits chosen by Dudek (2004) (due to the different hard- and software used) do not affect this conclusion. The limits chosen here are rather unfavorable compared to those of Dudek (2004); the ratio between the time limits for the decentralized and centralized models has been chosen as 1:40 here instead of 1:10, and the average gaps to the optimal solution to the centralized MLCLSP have been substantially lower (see Table 6.19). 38 Our preliminary computational experience with the scheme by Dudek (2004) has shown that the convergence rate of this scheme does not improve significantly with a larger number of iterations due to the limited capabilities for generating proposals different to those previously found; a comprehensive computational study, however, has not been carried out on this subject. 34
176
6 Computational Tests of Coordination Schemes
A further commonness to the scheme by Dudek (2004) is the relative low value of AGC(10/ compared to the ratio .AGUAGS10 /=AGU, which is far more accentuated than for GM and the MLULSP.39 This and the rather small AGS can be explained by that fact that Dudek (2004) has generated instances that are relatively easy to coordinate.40 One share of instances shows large initial gaps and involves elevated overtime costs for the supplier after upstream planning. These overtime costs, however, are of a greater order of magnitude compared to the setup and holding costs and can be saved rather easily by delayed orders of the buyer, which only require some minor increases in the setup and holding costs. For the other instances comprising only setup and holding costs, the gaps (which are far smaller there) have been closed more slowly and to a smaller extent. These insights are sustained by our sensitivity analysis in Table 6.21 and the convergence paths displayed in Fig. 6.7. Table 6.21 shows that the presence of overtime has a huge influence on the AGU; the AGS, instead, are not significantly affected by that. This is also the reason why a large part of the AGU could be closed after only one iteration (the AGC ranged between 33% and 52% there, see Fig. 6.7). The convergence paths for further iterations of the scheme turned out as highly dependent on the size of test instances. While for the smaller instances S1 and S2, almost the complete savings have been reached after 3 iterations, for the larger instances (especially L), considerable shares could be realized in later iterations. This result is intuitive. Larger instances are likely to show greater difficulties for coordination since manifold interdependencies between items have to be considered. Moreover, the large gap closure for set L indicates a favorable scalability of the scheme. Note that this result has been favored by suboptimalities in the solutions to the centralized model (see also Table 6.19) due to the time limits applied. In Table 6.22, we compare the performance for different specifications of the scheme after 10 iterations.41 With the exception of S3, the AGS are lowest for the default version of the scheme, which includes all levels of modifications. The Table 6.21 Sensitivities regarding the need for overtime at the supplier with upstream planning
39
S1 S2 S3 M1 M2 L
Without overtime AGU (%) AGS10 (%)
With overtime AGU (%) AGS10 (%)
5.4 1.8 2.7 4.4 3.6 3.6
103.2 113.2 75.8 65.4 94.9 84.3
2.2 1.0 1.4 2.2 1.6 1.4
1.5 4.0 1.1 1.5 1.8 2.7
See Sects. 6.1 and 6.2. See Sect. 6.1.1 for a discussion on coordination hardness. 41 For the analysis of sensitivities regarding the input data for this test set, we refer to Dudek (2004, p. 186). 40
6.3 Multi-level Capacitated Lot-Sizing Problem
177
100
Remaining gap (%)
80
60
40
20
0
0
2
4
6
8
10
Iteration S1
S2
S3
M1
M2
L
Fig. 6.7 Convergence paths for the MLCLSP
agg-P
exception occurred for the version without CS1i , which showed the best results for GM in Sect. 6.1. In spite of this exception, the hypothesis that the default specifiagg-P cation yields the same AGS as the scheme without CS1i can be rejected in favor of assuming smaller AGS for the default at a significance level of 99% based on a Wilcoxon matched pairs signed rank test simultaneously performed over all test instances. Surprisingly, however, the second indicator considered behaves contrariagg-P wise to AGS. AGC is mostly even greater if CS1i is not used. Anyway, the use agg-P of CS1i has only minor impact on the convergence. For the other modifications, the differences are far more pronounced. The hypothesis that the default specification yields the same AGS as the scheme without CS-LOT can be rejected in favor of assuming smaller AGS for the default at a significance level of 95% for each single test set based on the Wilcoxon matched pairs signed rank test. The only benefits of not running CS-LOT are modest reductions in TS. This, however, seems of minor relevance compared to the differences in the solution quality. Finally, without any additional modifications that are successively skipped if no agg agg-P new solution can be obtained (i.e., without CS1i , CS1i , and CS-LOT), the degradation of the convergence rate is substantially greater. Again applying the Wilcoxon matched pairs signed rank test, the hypothesis that this specification yields the same AGS as the default can be rejected in favor of assuming smaller AGS for the default at a significance level of 99% for each single test set. Summarizing, the performance of the scheme presented here is at least comparable to that developed by Dudek and Stadtler (2005), while showing a series of
S1 S2 S3 M1 M2 L
AGS (%) 2.0 1.9 1.3 1.8 1.7 1.9
AGC (%) 67.1 46.4 61.0 71.1 64.6 79.4
TS 26 24 28 140 68 320
AGS (%) 2.1 2.2 1.2 2.2 2.0 2.1
AGC (%) 69.7 56.7 66.3 61.4 64.3 89.5
TS 24 20 22 101 54 385
AGS (%) 2.0 2.0 1.5 2.5 2.2 2.6
AGC (%) 62.8 42.4 56.7 56.6 54.9 54.2
Table 6.22 Sensitivities regarding different specifications of the scheme applied to the MLCLSP agg-P Default Without CS1i Without CS-LOT TS 15 12 19 120 50 328
agg-P
CS1i , CS-LOT AGS (%) AGC (%) 2.9 62.3 4.1 41.9 1.6 56.9 2.8 54.9 2.4 54.7 2.9 62.7
agg
Without CS1i , TS 57 45 44 158 148 466
178 6 Computational Tests of Coordination Schemes
6.4 Models for Campaign Planning
179
advantages: It can coordinate settings with voluntary compliance by the suppliers, it can be directly applied for planning based on rolling schedules, and it permits the decision makers to determine the duration of the coordination process, instead of relying on a random stopping criterion.
6.4 Models for Campaign Planning 6.4.1 Generation of Test Instances Again, no comprehensive test sets for multi-level campaign planning have been reported in the literature.42 This holds a fortiori for decentralized instances that are hard to coordinate. Hence, we have developed new test sets, thereby meshing ideas from Sects. 6.1 and 6.2. We start with a description of the underlying methodology and provide the relations between the input parameters that have been fixed. We define aj by (6.1) and blcj by (6.2). Overtime costs for one unit of capacity on resource m are set to P ocm D
blcj j 2Jm amj
jJm j conoc
;
with Jm as the set of items produced on resource m and conoc as a constant stating the average ratio between backorder and overtime costs. Assuming that backorders are more expensive than overtime, we set conoc D 2. In order to reduce the number of necessary backlogs, we have introduced nonnegative initial inventories inij : inij D 2ej : capmt , the capacity of resource m in period t, is determined to capmt D
fmt
P j 2J
ut
amj dj
T 2 ; T
(6.6)
with fmt as a randomly generated factor (generation see below), dj as the average demand for item j , and ut as the average capacity utilization (here: 0.9). The last factor of (6.6) accounts for the fact that the demand is partly covered by the initial inventories and, hence, does not need to be produced completely. Holding cost factors hj are determined according to (6.3) and setup costs scj according to (6.4).
42 Related test sets include those developed by Trigeiro et al. (1989, p. 358), which have been extended by Suerie (2005a, p. 59) to the modeling of (single-level) campaigns, and those by Tempelmeier and Buschk¨uhl (2008) for the MLCLSPL without campaign restrictions.
180
6 Computational Tests of Coordination Schemes
mhj , the marginal holding costs, are a dependent factor here. They are recursively determined by X mhj D hj rkj hk : k2Rj prev
Finally, for all items j , the initial setup states w0j and inflows ijt have been set equal to zero and lead times equal to 1. We rely on the BOM structures S1 and M1 of Dudek and Stadtler (2005). We varied the remaining input parameters within a broad range. Demand data has been generated randomly with coefficients of variation CV 2 f0:1; 0:3; 0:5; 0:3g; for the last series, the random data has additionally been multiplied by a seasonality component consisting of a cosine oscillation with an amplitude of 0.3. All random numbers have been taken from a normal distribution with an expected value of 1. As in the previous sections, random values smaller than zero have been replaced by 0 here. Random numbers for capacity requirements amj have been generated based on coefficients of variation of 0.3 and 0.5. Four coefficients of variation have been used for generating production coefficients rjk : CV coe 2 f0:2; 0:5; 1; 1:5g. Similarly to Dudek (2004),43 three TBO profiles have been generated, which determine the relation between inventory holding and setup costs. The average TBO for the buyer’s items are 4, 3, and 5, respectively, and 4, 5, and 3 for the supplier’s items. All of these profiles have been multiplied by random numbers generated based on a coefficient of variation of 0.5. Analogously to the literature,44 the setup times have been set to fractions of period capacities. We have chosen fractions of 10% multiplied with random numbers based on a coefficient of variation of 0.5. Finally, the batch sizes have been varied. Apart from test instances without any batch size restrictions (NOBATCH),45 three profiles for batch sizes have been specified (all of them are subsumed under “BATCH”). The average batch sizes for the buyer’s items are the amounts resulting from production during 1, 1, and 2 period lengths (assuming full capacity utilization), respectively, and for the supplier 1, 2, and 1. Again, these values have been multiplied by random numbers based on a coefficient of variation of 0.5. By combining the profiles for all parameters, we obtain 768 test instances. Table 6.23 summarizes the input parameters chosen.
6.4.2 Analysis of Solutions For the computational study of this section, the same hard- and software and the same specification of the scheme as in Sect. 6.3 have been used. Due to the elevated
43
See Dudek (2004, p. 172). See, e.g., Porkka et al. (2003, p. 1139) and Suerie (2006, p. 882). 45 For a better comparability of the results for instances with and without batch size restrictions, all tests have been run based on the formulation for the campaign planning model provided in Sect. 2.2.2.2, but with arbitrary small values for the minimum batch sizes if no batch restrictions have been present. 44
6.4 Models for Campaign Planning
181
Table 6.23 Parametrization of test instances for the MLCLSPL-C Parameter Base value Coefficient of variation Demand 1, 1, 1, 1 (cos.) 0.1, 0.3, 0.5, 0.3 Capacity requirements 1 0.3, 0.5 Minimum batch sizes 0 (NOBATCH), 1-1, 1-2, 2-1 0.5 Production coefficients 1 0.2, 0.5, 1, 1.5 0.5 Setup times 10% kmt Structure S1, M1 – TBO 3-3, 3-5, 5-3 0.5
Table 6.24 Time limits and outcomes of the centralized models S1 M1 NOBATCH BATCH NOBATCH Centralized – Time limits 1,200 1,200 2,400 – Number of instances 96 288 96 – Number of suboptimal solutions 72 282 91 – Average optimality gaps (%) 16.7 46.2 26.7 Decentralized time limits (s) 10 10 20
BATCH 2,400 288 283 57.7 20
Table 6.25 Solution performance for the coordination of campaign planning AGU (%) AGS (%) AGC (%) NOPT TS Win-win (%) S1, NOBATCH S1, BATCH M1, NOBATCH M1, BATCH
23.3 38.2 523.7 629.3
12.7 2.4 10.5 0.4
34.6 172.5 79.7 109.9
6 127 19 157
225.7 608.0 842.5 1171.5
11.9 35.2 20.7 28.9
problem complexity, we only chose five nodes for the linearization of the objective functions. The time limits and details about the centralized solutions for these instances are summarized in Table 6.24.46 An overview of the results is given in Table 6.25. Apart from the performance indicators used in the previous sections, we provide the percentage of test instances for which the coordinated proposals lead to cost reductions for both parties (win-win).47 The results differ significantly with respect to the sizes of instances and the presence of batch restrictions. The AGU for S1 turned out to be significantly lower than the AGU for M1, whereas the AGS and AGC of S1 and M1 come much closer to each other. The explanation for this is that the small structure S1 only comprises two Like in Sect. 6.2, we only consider test instances with AGU 0:01% when determining the AGC in order to obtain meaningful results for this indicator. AGC > 1 means that more than the (complete) suboptimality from upstream planning can be mitigated, which holds if a solution identified by the scheme is superior to the centralized solution. 47 In contrast to the test settings of the preceding sections, win-win situations may arise here due to assumption of voluntary compliance. 46
182
6 Computational Tests of Coordination Schemes
production stages, whereas M1 comprises three stages. With two stages, the initial inventories were sufficient to avoid backorders in large part in upstream planning. This, however, does not extend to M1, where the necessary amount of backorders and, hence, the AGU, increase sharply. This is due to the positive production lead times causing shortages of items produced at the lower levels, an effect, which is substantially aggravated by the three-level structure in M1. Somewhat surprisingly, the suboptimalities due to these changes only affect the AGU in major degree; the rather equal values of the AGS for S1 and M1 suggest that, irrespectively of the problem structure, these suboptimalities can be mitigated almost completely by the scheme. A further counterintuitive issue is the decrease in the performance for instances without batch size restrictions. As shown by the optimality gaps displayed in Table 6.24, the presence of campaign restrictions augments the computational difficulties for the centralized models considerably. This, however, at the same time decreases the difficulties for the scheme to identify solutions that are equivalent or superior to those resulting from the centralized model. This is because campaign restrictions increase the computational difficulties less for the scheme than for the centralized model, which substantially improves the relative performance of the scheme. For both S1 and M1 with campaign restrictions, the best solutions identified by the scheme outperformed the solutions found by the standard solver on average. Thereby, less than half the computing time for solving the centralized model was needed, which suggests that the scheme can be applied as an effective heuristic for large-scale multi-level campaign planning problems. Some insights into problem structures where this heuristic is most effective are provided by our sensitivity analysis below. Moreover, we have evaluated the extent in which coordinated solutions lead to improvements for both parties. The share of such solutions ranged between 11.9% and 35.2%. On the one hand, this result underlines that even with voluntary compliance, coordination payments are needed if parties want to extract the maximum benefits from coordination. On the other hand, the fact that coordinated solutions can be implemented without recurring on compensation payments in some cases – if this would be an issue48 – widens the scope of the scheme proposed. Further insights provide the convergence paths (see Fig. 6.8). For M1, most savings are obtained within the two first iterations, whereas for S1, the savings realized in later iterations are substantial. This supports our hypothesis that there are two levels of coordination difficulties: The mitigation of the (major) suboptimalities due to insufficient initial inventories and nonzero lead time requires less coordination effort and pays most (see also the differences in the AGU between S1 and M1). The alignment of the remaining (minor) suboptimalities, which are prevailing in S1, requires greater coordination effort, instead. This also explains why larger relative savings could be obtained in later iterations for S1 – in contrast to M1 and Sect. 6.3, where almost all savings have been identified after two iterations due to the comparatively easy mitigation of the lion’s share of the suboptimalities. 48
See our discussion at the beginning of Chap. 5.
6.4 Models for Campaign Planning
183
100 80
Remaining gap (%)
60 40 20 0 −20 −40 −60 −80
0
2
4
6
8
10
Iteration S1, NOBATCH
S1, BATCH
M1, NOBATCH
M1, BATCH
Fig. 6.8 Convergence paths for the coordination of campaign planning Table 6.26 Sensitivities regarding demand
CVd , seasonality
0, 0
0.2, 0
0.5, 0
0.2, 0.3
AGU (%) AGS (%) AGC (%)
321.7 2.0 119.5
307.2 1.4 117.5
308.3 1.3 118.2
399.4 0.6 203.1
Next, we consider the sensitivities regarding changes in the input parameters for test instance generation. Since the focus in this section is on the instances with batch restrictions (BATCH), we skip the other instances for this analysis. We start with changes in the demand (see Table 6.26). Here, no unequivocal results can be recognized. Somewhat counter-intuitively, both the AGS and AGC are largest for the seasonal cycle; however, the absolute differences in the AGS have been minor for all parameterizations considered. Since only two variations for capacities have been included, we omit an evaluation of their effects and consider the variations in the production coefficients instead (Table 6.27). Here, a pronounced trend can be recognized: Both the AGU and AGS increase with variations in the production coefficients, whereas the AGC decreases. A reason for this is that greater variations favor inappropriate relations among the quantities of the single items supplied ordered by the buyer. This increases the AGU, which in turn favors increases in the AGS. The lower values for the AGC indicate that the difficulties for coordinating such structures are somewhat greater in general.
184 Table 6.27 Sensitivities regarding production coefficients
6 Computational Tests of Coordination Schemes CVcoe , base value
0.2
0.5
1
1.5
AGU (%) AGS (%) AGC (%)
233.5 3.1 201.0
351.1 4.1 154.6
370.3 0.2 106.0
381.2 1.8 96.1
Table 6.28 Sensitivities regarding TBO
TBOB , TBOS AGU (%) AGS (%) AGC (%)
4, 4 382.8 2.2 161.5
3, 5 358.4 0.6 110.0
5, 3 260.4 1.1 145.1
Table 6.29 Sensitivities regarding batch sizes
Batch size B, batch size S
1, 1
1, 2
2, 1
AGU (%) AGS (%) AGC (%)
277.4 1.7 109.0
412.5 4.7 128.7
312.1 1.7 180.1
Although near optimal solutions could be obtained for all parameter settings, the results suggest that it is most beneficial to apply the scheme as a heuristic if the variations in the production coefficients are low. Next, we consider the sensitivities regarding the TBO (Table 6.28). The AGU are largest for equal TBO. Moreover, a trend for a decreasing coordination performance with greater differences in the parties’ TBO can be recognized. A potential explanation for this is that coordination may become more difficult if the optimal TBO and, hence, the expected production patterns of buyer and supplier, are less aligned. Finally, Table 6.29 shows the sensitivities regarding different batch sizes. The observed increase of the AGU with greater differences between the parties’ batch sizes is intuitive since it seems less probable that upstream planning yields wellaligned plans in that case. For greater batch sizes of the buyer, both AGU and AGS are largest; the difference in the AGS between the other choices for the batch sizes is somewhat counterintuitive, but minor, and might be due to stochastic influences. Summarizing the results for the MLCLSPL-C, the scheme is able to significantly mitigate the gap resulting from upstream planning on average. If minimum batch sizes come into play, the results are even superior to the centralized solution generated by the solver, with little more than half of the computational time needed. This suggests a further potential application of the scheme as a heuristic for multi-level batch production problems.
6.5 Real-World Supply Chain Planning Problems To verify the applicability of the scheme for real-world supply chain planning, we have tested it for selected problems from clients of the SAP AG, Walldorf. In Sect. 6.5.1, we present the characteristics of these problems and the associated
6.5 Real-World Supply Chain Planning Problems
185
(centralized) model formulation. Moreover, we discuss the consequences of decentralization on the planning processes for these instances and their modeling. The results of our tests and a numerical example are presented in Sect. 6.5.2.
6.5.1 Planning Problems and Model Formulation We evaluated three real-world test instances (abbreviated by #1, #2, #3 in the following) provided by SAP. These data have been used as input for modeling customer problems in the module SNP (Supply Network Planning) of SAP APO (Advanced Planner and Optimizer).49 Table 6.30 gives an overview of the dimensions of these instances. A new term introduced here is PPM, the abbreviation for “production process model.” A PPM is a modeling feature used in SAP SNP to describe the characteristics of production processes (e.g., inflows and outflows of items, resource consumption). Modeling production planning problems using PPM is an alternative to the model formulations presented in Sect. 2.2. We rely on the modeling by PPM here because joint production, which is relevant for #3, can be modeled straightforwardly that way and additional adaptations of the real-world data can be avoided. It is remarkable that even the smallest test instance, #1, is considerably larger than the theoretical instances considered in the previous sections due to the large number of locations and time periods. The structures of the supply chains that underlie these test instances are sketched in Fig. 6.9. In Table 6.31, we list the characteristics of the instances with impact on the model formulation used.50 Based on the MLCLSPL-C,51 we have developed a model formulation that covers all the characteristics mentioned in Table 6.31. Although differing from the general model implemented in SAP SNP, this formulation is able to capture the decisions and restrictions relevant for the real-world problems considered.
49
Table 6.30 Overview of real-world instances Number of. . . Locations Items PPM Production resources
Periods
#1 #2 #3
32 35 104
12 7 8
4 8 69
6 18 62
6 12 13
For a description of the functionalities of this module, see Meyr et al. (2007a, p. 362). Note that we have “cleaned” the original data in order to remove infeasibilities and to reduce the problem sizes for some instances, such that they can be tackled by a standard solver. Amongst others, we have investigated two variants of #3 that differ by the inclusion of binary variables (#3-MIP and #3-LP). See P¨uttmann et al. (2007, p. 23) for details. 51 See Sect. 2.2.2.2. 50
186
6 Computational Tests of Coordination Schemes
Fig. 6.9 Structures of real-world test instances
min
XX
cvp Xpt C
p2P t 2T
C
j 2JlE
XX
s2S t 2T
X
csslj SSljt
a2A l2ABa
XT ajtaj D
a2A l2AE a ;t > a
XT ajt C
lscljs LSljst (6.7)
s2S t 2T
b e mf jp Xpt C X ptp C p
X
X
LSljst C
X
p2P
j 2JlE
p2P P 2Ol ;t > p
s2S
hlj Iljt
l2L j 2J t 2T
s2S
X
XXX
BLljst C Iljt1 C
X
X X XX l2L
csel ICElt C
XXX l2L j 2J t 2T
blcljs BLljst C
l2L t 2T
(RW-C) s.t.
ctaj X Tajt C
a2A j 2J t 2T
X X XX l2L
C
XXX
X
BLljst1 C Iljt
X
dljst C
(6.8)
s2S
8l 2 L; j 2 J; t 2 T
s2S
X amp Xptb C Xpte C stp Ypt kmt 8m 2 M; t 2 T p2P amp >0
(6.9)
6.5 Real-World Supply Chain Planning Problems
187
Table 6.31 Characteristics of real-world instances #1 Production Variable production costs x Variable capacity consumption x Production lead times x Setup times Production batch sizes Minimum production lot sizes Storage Holding costs x Storage limits x Penalty costs for increase of storage capacity x Safety stocks Penalty costs for shortages of safety stocks Transportation Transportation costs x Transportation lead times x Minimum transportation lot sizes x Maximum transportation lot sizes x Demand fulfillment Backlogs Lost sales x Maximum lateness
X
Iljt iclj icmax C ICElt l
#2
#3
x x x
x x x x x x
x
x x x x
x
x x
x x x
x x x
8l 2 L; t 2 T
(6.10)
j 2J
ICElt icemax l
8l 2 L; t 2 T
SSljt ssljt Iljt
XT ajt
8l 2 L; j 2 J; t 2 T
t X
BLljst
dljst2 8l 2 L; j 2 JlE ; s 2 S; t 2 T
t 2Dt lmaxljs C1 xtmax 8a 2 A; j ajt
Xpte
bpt Ypt
Xptb
bpt Wpt1
(6.11)
2 J; t 2 T
(6.12) (6.13) (6.14)
8p 2 P; t 2 T
(6.15)
8p 2 P; t 2 T
(6.16)
Wpt Ypt C Wpt1 8p 2 P; t 2 T n fjT jg X Wpt 1 8m 2 M; t 2 T n fjT jg
(6.17) (6.18)
p2P amp >0
Wpt1 C Wpt Ypt C Yp0 t 2 8m 2 M; p 2 P; p 0 2 P; p ¤ p 0 ; amp > 0; amp0 > 0; t 2 T n fjT jg
(6.19)
CAM pt CAM pt1 C
Xptb
C bpt Ypt
8p 2 P; t 2 T
(6.20)
CAM pt CAM pt1 C
Xptb
bpt Yjt
8p 2 P; t 2 T
(6.21)
188
6 Computational Tests of Coordination Schemes
CAM p0 minlotp CAM pt
Xpte
CAM pt
Xptb
CAM pt1 C
C bpt
Xptb
8p 2 P 1 Ypt
(6.22) 8p 2 P; t 2 T
(6.23)
8p 2 P; t 2 T
(6.24)
minlotp YI mt
8m 2 M; p 2 P; amp > 0; t 2 T (6.25)
CAM pt1 C Xptb D bsp Rpt C Spt 8p 2 P; t 2 T n f1g Spt bsp .1 Y Imt / 8m 2 M; p 2 P; amp > 0; t 2 T n f1g
(6.26) (6.27)
YI mt Ypt 8m 2 M; p 2 P; amp > 0; t 2 T X YI mt Ypt 8m 2 M; t 2 T
(6.28) (6.29)
p2P amp >0
Wp0 D 0 8p 2 P Rpt 2 N0 8p 2 P; t 2 T
(6.30) (6.31)
CAM pt 0 8p 2 P; t 2 T Wpt 2 f0; 1g 8p 2 P; t 2 T n f1g
(6.32) (6.33)
Spt 0; 8p 2 P; t 2 T n f1g YI mt 0 8m 2 M; t 2 T
(6.34) (6.35)
BLljs0 D 0; BLljsjT j D 0; LSljs0 D 0 Ilj0 D inilj 8l 2 L; j 2 J BLljst 0
8l 2 L; j 2 JlE ; s 2 S
8l 2 L; j 2 JlE ; s 2 S; t D 0; : : : ; jT j
52
(6.38)
Iljst 0 8l 2 L; j 2 J; s 2 S; t D 0; : : : ; jT j LSljst 0 8l 2 L; j 2 J; s 2 S; t 2 T
(6.39) (6.40)
Xpt 0 8p 2 P; t 2 T 8a 2 A; j 2 J; t 2 T: XT ajt 2 f0g [ xtmin ajt ; 1
(6.41) (6.42)
Indices and Index Sets a Arc linking two locations, a 2 A; ABa and AEa are the locations at the beginning and the end of arc a, respectively JlE Subset of items sold at location l l Location, l 2 L p PPM, p 2 P s Customer class, s 2 S . Data amp bpt bsp blcljs
(6.36) (6.37)
Capacity needed on resource m for one unit of PPM p Big number indicating the maximum quantity of PPM p in period t 52 Batch size for PPM p Backorder costs for one unit of item j of customer class s in a period at location l
This quantity can be calculated analogously to bjt , see p. 13.
6.5 Real-World Supply Chain Planning Problems
csslj
Penalty costs for one unit of stock below the required safety stock of item j at location l Costs for one unit of storage capacity increase at location l csel Transportation costs for one unit of item j along arc a ctaj Variable production costs of PPM p cvp Primary, gross demand for item j of customer class s in period t at dljst location l Consumption of storage capacity at location l by one unit of item j iclj Maximum storage capacity at location l icmax l Maximum extension of storage capacity at location l icemax l Inventory of item j at location l at the beginning of the planning inilj interval Holding cost for one unit of item j at location l in a period hlj lmaxljs Maximum lateness for demand fulfillment of item j of customer class s at location l Costs for lost sales of item j of customer class s at location l lscljs Material flow of item j from PPM p (a positive value corresponds mf jp to production, a negative value to consumption) minlotp Minimal lot size for item p Required level for the safety stock of item j at location l in period t ssljt p Production lead time of PPM p aj Transportation lead time for item j along arc a xtajmi n Minimum transportation quantity of item j along arc a in a period xtajmax Maximum transportation quantity of item j along arc a in a period Variables BLljst Amount of backorders of item j of customer class s at location l in period t CAM pt Campaign variable for PPM p in period t (quantity of the current campaign up to period t) Amount of inventory of item j at location l at the end of period t Iljt ICElt Increase of storage capacity at location l in period t LSljst Amount of lost sales of item j of customer class s at location l in period t Integer number of full batches produced in the current campaign of Rpt PPM p up to period t Quantity of the last batch of PPM p in period t which is not Spt finished in t SSljt Undershot of safety stock of item j at location l in period t Setup state indicator variable (D 1 if PPM p is set up at the end of Wpt period t, D 0 otherwise) Production quantity of PPM p at the beginning of period t Xptb
189
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6 Computational Tests of Coordination Schemes
Xpte XT ajt Ypt
Production quantity of PPM p that is not produced at the beginning of period t Transportation quantity of item j along arc a in period t occurs in n ( xt mi aj if item j is transported on arc a in period t; D 0 otherwise). Binary setup variable (D 1, if PPM p is produced in period t, D 0 otherwise)
The objective function (6.7) minimizes the sum of the variable production costs, transportation costs as well as the costs for inventory holding, backlogging, lost sales, and the expansion of storage capacity. Constraints (6.8) are inventory balance constraints specifying equal inflows and outflows for each item in each (storage) location. Apart from the inventories of the preceding period, the inflows consist of unfulfilled demand in form of backorders or lost sales, the production output of the PPM, and the quantities transported to the location considered. The outflows are made up by the demand, the new inventory levels, the backorders taken over from the preceding period, and the quantities transported to other locations. Constraints (6.9) are standard restrictions for production capacities. Constraints (6.10) force the utilization of the storage capacity below the sum of the available storage capacity and the expansion of the storage capacity chosen. For the maximum expansion of storage capacity, a limit is imposed by (6.11). If the lower bound on the safety stock is violated, constraints (6.12) determine the number of units that fall below this lower bound. Constraints (6.13) restrict the maximum lateness for backorders. Campaigns are modeled through restrictions (6.15)–(6.35). For a detailed description of these constraints, we refer to Sect. 2.2.2.2. Constraints (6.36)–(6.37) are used for initialization purposes and constraints (6.38)–(6.42) determine the scope of the decision variables. Thereby, constraints (6.42) define transportation quantities using semicontinuous variables, i.e., if an item is transported along an arc, its transportation quantity has to exceed a prespecified lower bound. A prerequisite for the application of the scheme is a decentralized structure of the organization considered. In the original modeling of the real-world instances, the existence of a unique centralized decision maker is assumed. In order to generate a decentralized structure in the test data, we have divided the test instances into data for two different entities, a supplier and a buyer, which comprise a subset of the relevant locations each. The splits applied are stated in Table 6.32. Note that for instance #3, only two different splits have been relevant due to the asymmetric distribution of items and PPM among the locations.
Table 6.32 Splits of test instances Split 1 Split 2 #1 #2 #3
Supplier (S) A–D A, B A, B
Buyer (B) E–L C–G D–H
S C, D A–C B
Split 3 B A, B, E–L D–G A, C–H
S A, B B –
Split 4 B C–L A, C–G –
S A–F, H A–E –
B G, I–L F, G –
6.5 Real-World Supply Chain Planning Problems
191
The models used in the scheme can be formulated analogously to those for voluntary compliance53 with the additional option for lost sales.54 Two minor issues have additionally to be taken into account here. First, the option for transporting the items supplied has to be modeled. For this purpose, dummy locations indicating the destinations of the items supplied have been included into the supplier’s model formulation. Second, with the presence of tight storage capacities, infeasibilities may be caused by the restriction that the buyer – when evaluating the supplier’s proposals – must not use less than the quantities specified by these proposals. To avoid such infeasibilities, we allowed the buyer to supply smaller quantities than specified in the supplier’s proposals, but charged high penalty costs to exclude that such proposals are considered as a systemwide improvement. All input data for the decentralized models could be directly derived from the real-world data, except for the costs for lost sales and backlogs for the supplier. Analogously to the well-known dilemma of determining optimal transfer prices,55 the values for these cost factors that are optimal for the whole system cannot be identified without the knowledge of the dual prices for the optimal solution to the centralized model. Since this solution is not known a priori, these costs have to be determined by some rules of thumb (in practice). These rules, of course, do not guarantee that the optimal values for these costs are determined.56 In the following, we apply a straightforward rule of thumb: We determine the costs for backorders and lost sales for intermediary items based on their potential impact on the corresponding costs for the end items. Hence, we recursively set blcj D min
k2Sj
blck pre nk rjk
! ;
(6.43)
with npre as the number of items preceding item k. The assumptions underlying k (6.43) are the following: For BOM structures with more than one predecessor item, the backorders costs of the successor item should be partitioned evenly among the predecessor items. For structures with more than one successor item, it is favorable in case of shortages of a preliminary item to resort on shortages of the successor items with the lowest backorder costs. Therefore, the minimum of the costs of the successor items has to be taken in (6.43). The costs for lost sales are determined analogously.
53
See Sect. 4.4.2. See Sect. 4.4.3. 55 See, e.g., Schweitzer and K¨upper (1998, p. 475). 56 In other words, the outcome of these rules is affected by the driver Q for the suboptimality of the default setting. 54
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6 Computational Tests of Coordination Schemes
6.5.2 Analysis of Solutions For the tests of this section, we rely on the same specification of the scheme that has been applied in Sects. 6.3 and 6.4 (augmented by the extensions to voluntary compliance and lost sales presented in Sects. 4.4.2 and 4.4.3).57 We have run the scheme for 10 iterations. We applied time limits of 3,600 s for the centralized models and 60 s for the decentralized models. These time limits have been defined such that they are only valid after a feasible solution has been identified. All centralized models could be solved to optimality, except of those for the variant of #3 including the campaign restrictions (#3-MIP). We display our main results in Table 6.33. In contrast to the preceding sections, we state the gaps of the uncoordinated solution (GU) and the gaps after coordination (GC) separately for all instances. The results differ substantially among the instances and splits performed. For the first three splits (abbreviated by “spl.”) for the smallest instance, #1, near optimal solutions could be obtained. For split 4, however, no improvements have been found. In principle, this is not really surprising; due to the elevated model sizes, the probability that no systemwide improvement can be found is much more elevated than for our smaller academic test problems. However, we additionally could identify a particular reason for this result. Split 4 is characterized by both a large number of transportation arcs for the items supplied and by the simultaneous presence of tight storage capacity restrictions and minimum transportation lot sizes. This reduces the number of systemwide feasible solutions considerably since supply quantities complying with the minimum transportation lot sizes may lead to infeasibilities due
Table 6.33 Solution performance of the scheme for the real-world data
57
Instances #1, spl. 1 #1, spl. 2 #1, spl. 3 #1, spl. 4
GU (%) 56.32 4.37 4.27 144.20
GS (%) 2.35 0.00 1.29 144.20
TC 1.0 1.0 1.0 1.0
TS 43.8 65.1 75.8 26.5
#2, spl. 1 #2, spl. 2 #2, spl. 3 #2, spl. 4
5.48 7.31 6.20 43.74
5.41 7.31 3.47 23.98
0.7 0.7 0.7 0.7
35.6 21.1 18.2 120.7
#3-LP, spl. 1 #3-LP, spl. 2 #3-MIP, spl. 1 #3-MIP, spl. 2
19.53 19.17 60.39 74.49
16.89 9.02 9.03 15.93
215.1 210.1 3,600 3,600
1,891.5 1,793.2 1,853.1 1,744.2
Note when carrying out these tests within the EU-project InCoCo-S, we have included further minor modifications like a somewhat different determination of the penalty costs factor cp in agg-P CSB . Preliminary computational tests showed that the effect of these modifications is negligible. Hence, we did not consider it as necessary to rerun this study using the exact specifications presented in Chap. 4.
6.5 Real-World Supply Chain Planning Problems
193
to excess inventory holding. In fact, when testing a modified model without the restrictions on the storage capacities for split 4, the resulting gap from upstream planning (144.9%) could be mitigated almost completely by the scheme (gap after coordination: 3.11%). For #2, the improvements for the two first splits have been low or non-existing. Also the initial gaps after upstream planning have been relatively small here, which usually makes the identification of further improvements more difficult. For the larger GU in split 4, however, a considerable fraction could be mitigated. For #3, a large part of the suboptimality from upstream planning has been mitigated on average. This result is somewhat surprising in light of the huge size of this instance. A potential explanation is that the costs factors (e.g., the unit costs for lost sales) differ substantially among items in #3, whereas for #1 and #2, the variations of these costs and of other factors like production coefficients have been rather low. We suppose that greater variations may increase the leeway for the identification of improvements. Further note that for #3, the difference between the time for running the centralized model and the scheme is substantially smaller than for the other instances. This is due to the disproportionally increasing computational effort for the solution of larger optimization models, an effect which has already been taken into account when setting the time limits for the centralized and the decentralized models. In Fig. 6.10, we display the convergence paths for the test instances with a substantial improvement by the scheme, i.e., GS 5%. Interestingly, the convergence
Remaining gap (GS/GU) (%)
100
80
60
40
20
0 0
2
4
6
8
Iteration 1,spl.1
#2, spl. 3
#3-LP, spl. 2
#1, spl. 2
#2, spl. 4
#3-MIP, spl. 1
#1, spl. 3
#3-LP, spl. 1
#3-MIP, spl. 2
Fig. 6.10 Convergence paths for the real-world instances
10
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6 Computational Tests of Coordination Schemes
Fig. 6.11 Data of #1, spl. 1
speed differs substantially among instances and splits. For some of them, e.g., #1, spl. 2, almost the entire benefits from coordination have been identified after only one iteration. For others (e.g., #2, spl. 4), substantial improvements have been found even in the tenth iteration. All in all, most of the potential coordination benefits have been obtained after five iterations, which indicates a modest effort for the corresponding coordination activities. To illustrate how the real-world problems are tackled by the scheme, we provide a numerical example for the smallest instance, #1. Figure 6.11 displays the structure of #1 including the locations of the PPM, their inflows and outflows, and the locations with end item demand, which, for ease of exposition, have been aggregated into one client location (H). We consider split 1, i.e., we assume that locations A, B, C, and D pertain to the supplier, and the other locations to the buyer. Note that there are several possibilities to produce single items in #1, e.g., item 1 can be produced by the supplier at location A or by the buyer at location F. The supply plans for the first 16 periods of the solution from upstream planning are displayed in Fig. 6.12.58 The column “initialization buyer” displays the buyer’s orders for the supplied items and the column “evaluation supplier” the quantities the supplier is willing (and able) to deliver. Here, the complete orders of the buyer cannot be fulfilled. Due to positive production and transportation lead times, the orders for items 1 and 4 cannot be delivered on time. The order for item 2 is not fulfilled at all. The minimum transportation lot size for the arc between the locations B and F (5 units) exceeds the quantities required per period. Since backlogging is not allowed in #1, a supply of this item has not been possible. As a consequence, the buyer had to incur additional costs for lost sales resulting in a large suboptimality of upstream planning (56.32%). The proposal generation within the scheme is illustrated by Fig. 6.13. In his first proposal, the supplier offers to deliver item 2, but different quantities than required
58
Note that their absolute values have been altered for reasons of privacy.
6.5 Real-World Supply Chain Planning Problems
195 evaluation supplier item 1 [units]
item 1 [units]
initialization buyer 10 5 0
10 5 0 2 4 6 8 10 12 14 16 periods
item 2 [units]
item 2 [units]
2 4 6 8 10 12 14 16 periods 10 5 0
10 5 0
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16 periods item 3 [units]
item 3 [units]
periods 10 5 0
10 5 0
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16 periods item 4 [units]
item 4 [units]
periods 10 5 0
10 5 0
2 4 6 8 10 12 14 16 periods
2 4 6 8 10 12 14 16 periods
item 1 [units]
Fig. 6.12 Supply quantities with upstream planning (#1, spl. 1)
1st proposal supplier 10 5 0
1st proposal buyer 2nd proposal supplier 2nd proposal buyer 10
10
10
5
5
5
5
0
item 2 [units]
4 8 12 16
item 3 [units]
0 4 8 12 16
0 4 8 12 16
0 4 8 12 16
4 8 12 16
10
10
10
10
10
5
5
5
5
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0
0 4 8 12 16
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0 4 8 12 16
item 4 [units]
3rd proposal supplier
10
0 4 8 12 16
0 4 8 12 16
0 4 8 12 16
4 8 12 16
10
10
10
10
10
5
5
5
5
5
0
0
0
0
0
4 8 12 16
4 8 12 16
4 8 12 16
4 8 12 16
4 8 12 16
periods
periods
periods
periods
periods
Fig. 6.13 Proposal generation by the scheme (#1, spl. 1)
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6 Computational Tests of Coordination Schemes
by the buyer due to the minimum transportation lot sizes. This proposal realizes a considerable percentage (85.5%) of the total surplus achieved by the scheme. At first glance, the underlying idea of this proposal, i.e., to propose greater delivery quantities and thereby to assume that the buyer will do some additional inventory holding, seems trivial. In fact, we admit that this specific proposal or a similar one could also have been found by manual negotiations. However, note that this idea does not always work since larger supplies than needed might not always be feasible for the supplier due to the tight restrictions on storage capacity in some locations. Moreover, as we will show in the following, the scheme is able to identify further and rather unobvious improvements in later iterations. In her first proposal, the buyer complies with the reduction of the supply for items 1 and 3 in earlier periods proposed by the supplier, but asks again for a more even supply of item 2. The rationale behind this is that a supply of item 2 in early periods would substantially reduce the buyer’s costs for lost sales. Although a previous proposal with this characteristic (i.e., the initial solution) turned out to be inferior, the buyer running the scheme is not aware of the exact reason for this. The potential reasons of the inferiority of upstream planning compared to the supplier’s first proposal can be attributed to the changes in the supply of any of the items 1, 2, and 4 (compare the right column of Fig. 6.12 with the left column of Fig. 6.13). In the scheme, the buyer charges internally determined penalty costs for a subset of these changes to prevent that the entire changes are repeated in the new proposal generated. Here, the inferiority of upstream planning has been attributed to the changes for item 1, but not for items 2 and 4. This choice, however, did not result in a favorable new proposal. In his second proposal, the supplier tries different supply quantities for item 2, but without success. In his next proposal, the buyer internally attributes some penalty costs for changes in the supply of item 2. The resulting proposal addresses another region of the solution space, i.e., the delivery of items 1 and 2 is partially replaced by the delivery of item 3. This proposal did not lead to a direct improvement. The knowledge, however, that also the supply of item 3 could be useful for the buyer served as the base for the third proposal of the supplier. There, additional percentage cost savings of 9.9% are obtained by incorporating the supply of this item. The rest of the savings are found in later iterations, which are not illustrated here. Summarizing, the results obtained in this section underscore the practical applicability of the scheme proposed. Although all real-world test instances comprise different supply chain structures, decisions and restrictions, a substantial part of the suboptimalities due to upstream planning could be mitigated at least for most of the splits considered. As expected, the gap closures have been somewhat lower than for the smaller-sized academic instances studied in the previous sections. The favorable results for #3, however, suggest that the scheme can also be successfully applied for problems comprising a huge number of decision variables and a series of difficult restrictions.
Chapter 7
Summary and Outlook
This thesis has proposed new supply chain coordination mechanisms, which incorporate a number of characteristics favoring their applicability in practice. The mechanisms can coordinate plans generated on the basis of complex mathematical programming models on behalf of self-interested parties holding private information, and do not require the participation of a third party. In the literature, all of these characteristics have been regarded as important, but they have never been covered within a single approach. An introduction to the research problem tackled by this thesis is given in Chap. 2. There, we have presented mathematical models for (intra-organizational) mid-term supply chain planning (Master Planning) as well as basic definitions that are used throughout this work. Moreover, we have shown how to model inter-organizational planning processes formally. Last, we have determined drivers for suboptimalities without coordination, which, e.g., result from the application of myopic procedures such as upstream planning. Chapter 3 provides a review of the state-of-the-art in supply chain coordination. In contrast to existing surveys, which focus on specific types of mechanisms, our review covers the whole spectrum of the related literature. The mechanisms and the underlying ideas are described separately according to the assumptions on parties’ information statuses (no, unilateral, or multilateral information asymmetry) and the concepts from game theory the mechanisms rely on. The coordination mechanisms presented in Chaps. 4 and 5 constitute the principal contribution of this work. Two main tasks have to be resolved by them: The identification of coordinated solutions, i.e., systemwide improvements compared to a given initial solution, and the determination of incentives for their implementation. In Chap. 4, several coordination schemes have been devised that identify coordinated solutions for a broad range of Master Planning problems and cover different requirements on the information exchange. At first, we have developed two variants of a generic scheme, which can be applied for coordinating linear programming (LP) models in arbitrarily structured decentralized organizations. In this scheme, parties iteratively exchange proposals about the use of the central resources (i.e., supply quantities in supply chains). Depending on the mechanism the scheme is embedded in, the cost effects to these proposals are either reported iteratively by all but one decentralized party or M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-02833-5 7,
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7 Summary and Outlook
simultaneously disclosed by all parties. We have shown by analytical proofs that both variants are able to identify the systemwide optimum in a finite number of iterations, provided that the decentralized problems can be formulated as LP models. The variant with an iterative exchange of cost changes is even able to identify the systemwide optimum if one decentralized model is of a mixed-integer programming (MIP) type. The second scheme developed in this work aims to coordinate supply chains consisting of one buyer and one or multiple suppliers planning based on multi-level uncapacitated lot-sizing problems (MLULSP). This scheme also relies on the exchange of supply proposals, but – in contrast to the generic scheme – does not require specific assumptions on the exchange of cost changes. We can show that the maximum number of proposals generated increases linearly with the number of periods and the number of the items supplied. For the special case of time-invariant end item demand, zero initial inventories, and the production portfolio of the buyer limited to one item, we have derived an upper bound on the maximum gap between the systemwide optimum and the best solution identified by this scheme. Moreover, we have shown how to adapt the schemes for an effective coordination of Master Planning in two-tier supply chains. First, the generic scheme for LP has been customized for coordinating Master Planning among two parties. Second, we have presented modifications that improve the convergence and allow an application of this scheme to MIP models. Third, we have devised extensions that effectively coordinate decentralized models with capacitated lot-sizing, voluntary compliance by the supplier, lost sales, and multiple suppliers involved in the coordination. For each of the schemes, extensive numerical tests have been conducted based on randomly generated test instances. The results of these tests have been reported in Chap. 6. In all settings considered, the systemwide costs have been significantly reduced compared to upstream planning after 10 or 20 iterations. For small problem sizes or specific model classes (e.g., the MLULSP), near optimal solutions can be obtained. For the campaign planning models, the scheme can even identify improvements over the solutions to the centralized model that have been found after a given time limit, which is two times the limit for running the scheme on average. In addition, we have tested the performance of the scheme for real-world data originating from customers of the SAP AG. Although these data comprise additional constraints like storage capacity and transportation lot size restrictions and involve huge model sizes, the scheme was able to identify significant improvements. Apart from the identification of improved solutions, the second task to be tackled by a coordination mechanism is the establishment of incentives for the implementation of these solutions. For this purpose, we have presented in Chap. 5 three different contractual frameworks, where the schemes can be embedded. First, the sharing of the surplus can solely be determined by the party that does not report its cost changes during the execution of the scheme. For an efficient coordination, fair and truthful behavior by all parties is required then. Fairness, however, has only been observed to a certain degree in behavioral experiments. In spite of that, we believe that this mechanism can be employed in organizational structures where one dominant party carries out auditing measures like open-book accounting.
7 Summary and Outlook
199
Second, we have devised a contractual framework where truthful cost reporting is a Nash equilibrium for parties. There, the surplus is shared such that the costreporting parties obtain lump-sum payments, while one party (which pays these lump sums) receives the remaining share of the surplus. Analyzing parties’ best strategies in this mechanism under the assumption of prior incomplete knowledge about the surplus, a lower bound on the efficiency of this framework has been derived: For two parties and uniformly distributed prior knowledge, the efficiency exceeds 75% on average. Third, the surplus can be shared by a sealed bid double auction. Here, parties simultaneously submit their costs changes in form of sealed bids for all proposals generated and implement the proposals with the lowest systemwide costs. The surplus is shared equally among parties. Again, a lower bound on the efficiency of this auction can be derived. Finally, we have outlined how the mechanisms can be adapted if production planning is based on rolling schedules. Summarizing, the schemes and mechanisms proposed form generic, innovative concepts for collaborative supply chain planning and include a series of features favoring practical applications. Given the existence of an effective scheme, the mechanisms can be applied for coordinating any decisions in decentralized systems. The generic scheme is able to identify the systemwide optimum for any decentralized LP models within a finite number of steps. The applicability of (a modified version) of this scheme to Master Planning problems of one buyer and one or several suppliers including binary variables has been shown by the computational study of this work. An investigation of the transferability of the schemes to other economic decision problems seems worthwhile. This way has been pursued by P¨uttmann (2007),1 who adapts the generic scheme to the coordination of intermodal freight transportation. Apart from transportation, other potential applications are in areas like production scheduling and controlling, where the cost allocation for several agents sharing a common resource has to be resolved, as well as different organizational structures like three-tier supply chains. Furthermore of interest is the determination of the problem classes for which the scheme proposed constitutes an effective heuristic (like for multi-level campaign planning problem with batch size restrictions). Challenges for future research are analyses of the actions of parties using repeatedly the mechanisms proposed. As mentioned in Chap. 5, learning and signaling strategies have to be taken into account then, which considerably augments the complexity of the resulting modeling. Both laboratory research to evaluate the empirical behavior of parties and analytical characterizations of parties’ best strategies constitute promising ways to advance this area of research.
1
See P¨uttmann (2007, p. 63) for the outline of her application setting.
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