Robust Discrete Optimization and Its Applications

N onconvex Optimization and Its Applications

Volume 14

Managing Editors:

Panos Pardalos University of Florida, U.S.A.

Reiner Horst University o/Trier, Germany

Advisory Board:

Ding-Zhu Du University of Minnesota, U.S.A.

c.A. Floudas Princeton University, U.S.A.

G.Infanger Stanford University, U.S.A.

J. Mockus Lithuanian Academy of Sciences, Lithuania

P.D. Panagiotopoulos Aristotle University, Greece

H.D. Sherali Virginia Polytechnic Institute and State University, U.S.A.

The titles published in this series are listed at the end o/this volume.

Robust Discrete Optimization and Its Applications

by

Panos Kouvelis Washington University at St. Louis, Olin School of Business, St. Louis, Missouri, U.S.A.

and

Gang Yu The University of Texas, Center for Cybernetic Studies, Austin, Texas, U.SA.

Springer-Science+Business Media, B.Y.

Library of Congress Cataloging-in-Publication Data

Kouvelis, Panos. Robust discrete optimization and its applications I by Panos

Kouvelis, Gang Yu. p. cm. -- (Nonconvex optimization and its applications; v.

14) Includes bibliographical references and index.

1. Mathematical optimization. III. Ser ies. OA402.5.K668 1996 003' .56--dc20

I. Yu, Gang. II. Title.

ISBN 978-1-4419-4764-2 ISBN 978-1-4757-2620-6 (eBook) DOI 10.1007/978-1-4757-2620-6

Printed on acid-free paper

All Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by K1uwer Academic Publishers in 1997.

Softcover reprint of the hardcover 1 st edition 1997

96-43291

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

To the memory of my father, Vaios K ouveiis, who taught me the meaning of honor, decency, excellence and uncompromising values

P. K.

To my newborn son, Ray Yu, for touching my heart and soul with peace, life, wonder, pride, and happi;uss

G. Y.

CONTENTS

DEDICATION v

PREFACE xi

ACKNOWLEDGMENTS xv

1 APPROACHES FOR HANDLING UNCERTAINTY IN DECISION MAKING 1 1.1 Traditional Approaches for Handling Uncertainty in Decision

Making 1 1.2 A Formal Definition of the Robustness Approach 8 1.3 Robust Decision Making Framework 11 1.4 Motivate the Robustness Approach Through International Sourc-

ing Applications 17 1.5 A Brief Guide Through Related Literature 23

REFERENCES u

2 A ROBUST DISCRETE OPTIMIZATION FRAMEWORK 26 2.1 The Robust Discrete Optimization Problem 26

2.2 Efficiency and Expected Performance of Robust Solutions 59

2.3 A Brief Guide Through Related Literature 69

REFERENCES ro

Vlll ROBUST OPTIMIZATION AND ApPLICATIONS

3 COMPUTATIONAL COMPLEXITY RESULTS OF ROBUST DISCRETE OPTIMIZATION PROBLEMS 74

3.1 Complexity Results for the Robust Assignment Problem 76

3.2 Complexity Results for the Robust Shortest Path Problem 77

3.3 Complexity Results for the Robust Minimum Spanning Tree Problem 85

3.4 Complexity Results for the Robust Resource Allocation Problem 90

3.5 Complexity Results for the Robust Machine Scheduling Problem 95

3.6 Complexity Results for the Robust Multi-period Production Planning Problem 100

3.7 Complexity Results for the Robust Knapsack Problem 103

3.8 Complexity Results for the Robust Multi-Item Newsvendor Problem 107

3.9 A Brief Guide Through Related Literature 111

REFERENCES 113

4 EASILY SOLVABLE CASES OF ROBUST DISCRETE OPTIMIZATION PROBLEMS 116 4.1 Robust I-Median Location Problem on a Tree 116

4.2 Robust Multi-period Production Planning with Demand Un-certainty 122

4.3 Robust Economic Order Quantity (EOQ) Model 124

4.4 Robust Newsvendor Problems 137

4.5 Robust Multi-Item Newsvendor Models with a Budget Con-straint and Interval Demand Data 142

4.6 Parameter Robust Distribution Free Newsvendor Models 147

4.7 A Brief Guide Through Related Literature 150

REFERENCES 151

5 ALGORITHMIC DEVELOPMENTS FOR DIFFICULT ROBUST DISCRETE OPTIMIZATION PROBLEMS 153

5.1 A Surrogate Relaxation Based Branch-and-Bound Method 153

5.2 An Approximation Algorithm 160

Contents

5.3 Computational Results 5.4 A Brief Guide Through Related Literature

REFERENCES

6 ROBUST I-MEDIAN LOCATION PROBLEMS:

IX

168 191

192

DYNAMIC ASPECTS AND UNCERTAINTY 193 6.1 Notation, Problem Formulation and Basic Results 197 6.2 Robust I-Median with Linear Node Demands and Edge Dis-

tances 204

6.3 Robust I-Median with Linear Node Demands 6.4 Robust I-Median with Linear Edge Distances 6.5 Observations on Uncertain Node Demands and Edge Distances,

209 215

and Conclusions on Robust I-Median with Discrete Scenarios 221 6.6 Robust I-Median Problem on a Tree with Interval Input Data 223

6.7 Robust I-Median on a Tree with Mixed Scenarios 6.8 A Brief Guide Through Related Literature

REFERENCES

7 ROBUST SCHEDULING PROBLEMS 7.1 Properties of Robust Schedules for Single Machine Scheduling

235 237

239

241

with Interval Processing Time Data 243 7.2 Properties of Robust Schedules for Two Machine Flowshop

Scheduling with Interval Processing Time Data 249 7.3 Algorithms for the Robust Single Machine Scheduling Problem

with Interval Processing Time Data 254 7.4 Algorithms for the Robust Two Machine Flowshop Scheduling

Problem with Interval Processing Time Data 266 7.5 Algorithms for the Robust Two Machine Flowshop Scheduling

Problem with Discrete Processing Time Data 277

7.6 A Brief Guide Through Related Literature 286

REFERENCES 289

x ROBUST OPTIMIZATION AND ApPLICATIONS

8 ROBUST UNCAPACITATED NETWORK DESIGN AND INTERNATIONAL SOURCING PROBLEMS 290 8.1 Notation and Problem Formulation of Uncapacitated Network

Design Problems 292 8.2 Adaptation of the Benders Decomposition Methodology to the

Generation of Robust Network Designs 294

8.3 A Multi-Master Benders Algorithm For Robust Uncapacitated Network Design Problems 299

8.4 Robust Network Designs and the Expected Cost Uncapaci-tated Network Design Problem 305

8.5 Computational Results 306 8.6 Notation and Formulation of Robust International Sourcing

Problem 315 8.7 An Algorithm to Generate the N Best Robust Solutions to the

International Sourcing Problem 318 8.8 Computational Performance of the Robust International Sourc-

ing Algorithm 322

8.9 Managerial Uses of the Robust International Sourcing Model 326 8.10 A Brief Guide Through Related Literature 329

REFERENCES 331

9 ROBUST DISCRETE OPTIMIZATION: PAST SUCCESSES AND FUTURE CHALLENGES 333 9.1 Summary of Main Results 334 9.2 Implementation Considerations of the Robustness Approach 343 9.3 Future Research Directions 351

PREFACE

This book deals with decision making in environments of significant data un­certainty, with particular emphasis on operations and production management applications. For such environments, we suggest the use of the robustness ap­proach to decision making, which assumes inadequate knowledge of the decision maker about the random state of nature and develops a decision that hedges against the worst contingency that may arise. The main motivating factors for a decision maker to use the robustness approach are:

• It does not ignore uncertainty and takes a proactive step in response to the fact that forecasted values of uncertain parameters will not occur in most environments;

• It applies to decisions of unique, non-repetitive nature, which are common in many fast and dynamically changing environments;

• It accounts for the risk averse nature of decision makers; and

• It recognizes that even though decision environments are fraught with data uncertainties, decisions are evaluated ex post with the realized data.

For all of the above reasons, robust decisions are dear to the heart of opera­tional decision makers. This book takes a giant first step in presenting decision support tools and solution methods for generating robust decisions in a variety of interesting application environments.

Robust Discrete Optimization is a comprehensive mathematical programming framework for robust decision making. Our robust optimization framework applies minimax regret criteria to differentiate the performance of the various solutions over the given set of realizable scenarios, and it is mostly developed for models with discrete decision variables using state of the art convex and combinatorial optimization techniques. We demonstrate the applicability of the framework to a variety of decision making environments such as

Xl

xii ROBUST OPTIMIZATION AND ApPLICATIONS

• linear programming

• assignment problems

• shortest paths

• minimum spanning trees

• knapsack problems

• resource allocation

• scheduling

• production planning

• location

• inventory

• layout planning

• network design

• international sourcing.

The book is based on our recent research results (in the last five years) and those of a few colleagues. Among the most noteworthy results in this book are:

• the characterization of the algorithmic complexity for a large class of robust discrete optimization problems;

• discussion of polynomial algorithms for interesting applications such as: robust 1-median location on a tree, robust economic order quantity and newsvendor models, and robust multi-period production planning with demand uncertainty;

• development of a Surrogate Relaxation based Branch-and-Bound approach with impressive computational performance for a class of NP-hard robust discrete optimization problems;

• development of a general approximation algorithm for a class of robust discrete optimization problems;

• specialized algorithmic developments for robust discrete optimization ap­plications with interval input data in 1-median location and scheduling;

• innovative adaptations of Benders decomposition methodology to robust network design generation; and

Preface xiii

• characterizing properties of worst case scenarios and robust solutions in many application environments.

Beyond theoretical results, the book provides many suggestions and useful ad­vice to the practitioner of the robustness approach. Emphasis is placed upon the

• assessment of the decision environment for applicability of the approach;

• structuring of data uncertainty and the scenario generation process;

• choice of appropriate robustness criteria; and

• formulation and solution of robust decision problems.

The structure of the book is as follows. In Chapter 1, we provide a conceptual treatment of the robustness approach to decision making, clearly illustrate its differences from other approaches to decision making under uncertainty, and strongly motivate the reasons for its wide applicability in operational decision environments. Chapter 2 provides a rigorous treatment of the previously intro­duced robust decision making ideas and progressively leads to the formulation of the robust discrete optimization framework. We then present a vast array of problems for which the framework applies. We subsequently restrict our attention to the class of robust discrete optimization problems with scenario independent feasibility constraints, and discuss in Chapter 3 complexity results for many interesting application problems. As one would expect, robust dis­crete optimization problems are, in general, difficult to solve, but as Chapter 4 points out there are still some interesting polynomially solvable problems. De­tailed discussion of polynomial procedures for these problems follows. Chapter 5 restricts its attention to robust discrete optimization problems with equiva­lent single scenario (deterministic) problems that can be efficiently solved with polynomial or pseudo-polynomial procedures. For these problems, a surrogate relaxation based branch-and-bound algorithm is presented, and extensive com­putational results on various applications substantiate the computational effi­ciency of the algorithm. Chapter 6 demonstrates how dynamic and uncertain aspects of location decisions can be incorporated with the use of the robustness approach, and then proceeds to discuss in detail the robust I-median loca­tion problem on a tree. Another important application area of the robustness approach is scheduling, and Chapter 7 devotes its attention to the detailed dis­cussion of robust scheduling problems. Special algorithmic results are presented for robust single machine and two machine flowshop scheduling environments.

xiv ROBUST OPTIMIZATION AND ApPLICATIONS

Robust network design formulations cover a wide range of applications from material handling design to plant location and international sourcing. Chapter 8 presents innovative ways to adapt the Benders decomposition methodology for the solution of these problems. Finally, Chapter 9 concludes with a brief account of the main results in the book, suggestions to the practitioners on how to face the implementation challenges of the approach, and an outline of future research directions for those researchers who are intrigued by the subject.

ACKNOWLEDGMENTS

Panos Kouvelis became interested in the topic of robust decision making after reading the paper of Rosenblatt and Lee (1987) "A Robustness Approach to Facilities Design," International Journal of Production Research, 25, 479-486. Hau Lee, his thesis advisor, contributed to his thinking not just on this topic but more generally on ways to pursue quality research, and hopefully some of these lessons are reflected in the contents of this book. Meir Rosenblatt has influenced his research agenda in many areas beyond robust optimization. Both of them also have been extremely helpful professional friends in critical stages of his career. Among the researchers he has done joint research with, he would like to acknowledge the strong influence of Genaro Gutierrez on his early thinking on robustness and the development of the first results. Many interesting results came out of the research collaborations with Rich Daniels and George Vairaktarakis, two colleagues who embraced the robustness idea with the same excitement and intellectual passion as the authors of this book did.

Gang Yu would, first of all, like to thank his parents, Deqian Yu and Junxiu Zhang, who taught him integrity, honesty, principles, and values. They did all they could for their son's education, career, and happiness. Hopefully, Gang's successes will reward their endeavors and sacrifices. Gang is grateful to his wife, Xiaomei Song. Her encouragement, confidence, support, love, and pas­sion throughout the years relieve his pressure, purify his soul, color his life, and energize his spirit to face all challenges. Under the influence and advice of his friend, Jiang Wang, Gang made his career transition ten years ago from the Physics Department at Cornell University to the Decision Sciences Department of The Wharton School, University of Pennsylvania. During his three years of study at Wharton, he was fortunate to have Marshall Fisher serve as his the­sis advisor. Marshall's earnest interest in applied research, perpetual effort in striving for excellence, and ingenious approaches to real world complex prob­lems has greatly impacted his career ever since. He is also grateful to have been influenced and trained by such prominent researchers as Moris Cohen, Monique Guignard-Spielberg, Yusheng Zheng, and Patrick Harker. Their ef­forts are clearly reflected in his research output. Gang would like to express his

xv

xvi

gratitude to his research collaborators, especially Olivier Goldschmidt, Dorit Hochbaum, Leon Lasdon, Boaz Golany, Quanling Wei, Patrick J aillet, Kavindra Malik, and Jonathan Bard. Through working with them, he has not only en­joyed fruitful research results, but also many interesting and intellectual brain­storm sessions, shared learning and exploring experiences, and the gratifying feeling of conquering difficulties. Gang would like to thank his colleagues, es­pecially Patrick Brockett, James Dyer, and William Cooper, for their many years of support and advice. Their guidance and tremendous help have made his research and teaching environment a pleasant one. Gang has been lucky to supervise a talented group of students, including David Nehme, Songjun Luo, Ahmad Jarrah, Gao Song, Alex Takvorian, Guo Wei, Mike Arguello, and Li Zhou. Hopefully, his input will benefit them in their careers; through their hard work they deserve a bright future. Finally, Gang would like to extend his appreciation to Caryn Cluiss for carefully and patiently proofreading many of his research papers. Those articles would not be nearly as smooth without her magic touch.

Needless to say, we have enjoyed our research collaboration of the last few years. We are equally responsible, as well as deserve equal credit if any, for the final product. The usual disclaimer applies. We are responsible for any errors and/or omissions in the book, and our co-authors in various research papers on the topic deserve no blame for any of our oversights. Finally, we would like to thank the Series Editor, Panos Pardalos, for encouraging us to publish this book. We were fortunate in, and have enormously enjoyed, working with a patient, pleasant, conscientious and supportive editor - John Martindale. Thanks, Panos and John, and sorry for the many delays in submitting the final manuscript. Still, we have not learned how to manage bookwriting tasks of significant uncertainty.