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xiii

TABLE OF CONTENTS

PREFACE xxiii

CHAPTER 1Introduction 1

1.1 The Origins of Operations Research 11.2 The Nature of Operations Research 21.3 The Impact of Operations Research 31.4 Algorithms and OR Courseware 5Problems 6

CHAPTER 2Overview of the Operations Research Modeling Approach 7

2.1 Defining the Problem and Gathering Data 72.2 Formulating a Mathematical Model 102.3 Deriving Solutions from the Model 142.4 Testing the Model 162.5 Preparing to Apply the Model 182.6 Implementation 202.7 Conclusions 21Selected References 22Problems 22

CHAPTER 3Introduction to Linear Programming 24

3.1 Prototype Example 253.2 The Linear Programming Model 313.3 Assumptions of Linear Programming 363.4 Additional Examples 443.5 Some Case Studies 613.6 Displaying and Solving Linear Programming Models on a Spreadsheet 673.7 Formulating Very Large Linear Programming Models 733.8 Conclusions 79Appendix 3.1 The LINGO Modeling Language 79

Selected References 89Learning Aids for This Chapter in Your OR Courseware 90Problems 90Case 3.1 Auto Assembly 103Case 3.2 Cutting Cafeteria Costs 104Case 3.3 Staffing a Call Center 106

CHAPTER 4Solving Linear Programming Problems: The Simplex Method 109

4.1 The Essence of the Simplex Method 1094.2 Setting Up the Simplex Method 1144.3 The Algebra of the Simplex Method 1184.4 The Simplex Method in Tabular Form 1234.5 Tie Breaking in the Simplex Method 1284.6 Adapting to Other Model Forms 1324.7 Postoptimality Analysis 1524.8 Computer Implementation 1604.9 The Interior-Point Approach to Solving Linear Programming Problems 1634.10 Conclusions 168Appendix 4.1 An Introduction to Using LINDO 169Selected References 171Learning Aids for This Chapter in Your OR Courseware 172Problems 172Case 4.1 Fabrics and Fall Fashions 182Case 4.2 New Frontiers 185Case 4.3 Assigning Students to Schools 188

CHAPTER 5The Theory of the Simplex Method 190

5.1 Foundations of the Simplex Method 1905.2 The Revised Simplex Method 2025.3 A Fundamental Insight 2125.4 Conclusions 220Selected References 220Learning Aids for This Chapter in Your OR Courseware 221Problems 221

CHAPTER 6Duality Theory and Sensitivity Analysis 230

6.1 The Essence of Duality Theory 2316.2 Economic Interpretation of Duality 2396.3 Primal-Dual Relationships 2426.4 Adapting to Other Primal Forms 2476.5 The Role of Duality Theory in Sensitivity Analysis 2526.6 The Essence of Sensitivity Analysis 254

xiv TABLE OF CONTENTS

6.7 Applying Sensitivity Analysis 2626.8 Conclusions 284Selected References 284Learning Aids for This Chapter in Your OR Courseware 285Problems 285Case 6.1 Controlling Air Pollution 302Case 6.2 Farm Management 304Case 6.3 Assigning Students to Schools (Revisited) 307

CHAPTER 7Other Algorithms for Linear Programming 309

7.1 The Dual Simplex Method 3097.2 Parametric Linear Programming 3127.3 The Upper Bound Technique 3177.4 An Interior-Point Algorithm 3207.5 Linear Goal Programming and Its Solution Procedures 3327.6 Conclusions 339Selected References 340Learning Aids for This Chapter in Your OR Courseware 340Problems 341Case 7.1 A Cure for Cuba 347

CHAPTER 8The Transportation and Assignment Problems 350

8.1 The Transportation Problem 3518.2 A Streamlined Simplex Method for the Transportation Problem 3658.3 The Assignment Problem 3818.4 Conclusions 391Selected References 391Learning Aids for This Chapter in Your OR Courseware 392Problems 392Case 8.1 Shipping Wood to Market 401Case 8.2 Project Pickings 402

CHAPTER 9Network Optimization Models 405

9.1 Prototype Example 4069.2 The Terminology of Networks 4079.3 The Shortest-Path Problem 4119.4 The Minimum Spanning Tree Problem 4159.5 The Maximum Flow Problem 4209.6 The Minimum Cost Flow Problem 4299.7 The Network Simplex Method 4389.8 Conclusions 448Selected References 449

TABLE OF CONTENTS xv

Learning Aids for This Chapter in Your OR Courseware 449Problems 450Case 9.1 Aiding Allies 458Case 9.2 Money in Motion 464

CHAPTER 10Project Management with PERT/CPM 468

10.1 A Prototype ExampleThe Reliable Construction Co. Project 46910.2 Using a Network to Visually Display a Project 47010.3 Scheduling a Project with PERT/CPM 47510.4 Dealing with Uncertain Activity Durations 48510.5 Considering Time-Cost Trade-Offs 49210.6 Scheduling and Controlling Project Costs 50210.7 An Evaluation of PERT/CPM 50810.8 Conclusions 512Selected References 513Learning Aids for This Chapter in Your OR Courseware 514Problems 514Case 10.1 Steps to Success 524Case 10.2 Schools out forever . . . 527

CHAPTER 11Dynamic Programming 533

11.1 A Prototype Example for Dynamic Programming 53311.2 Characteristics of Dynamic Programming Problems 53811.3 Deterministic Dynamic Programming 54111.4 Probabilistic Dynamic Programming 56211.5 Conclusions 568Selected References 568Learning Aids for This Chapter in Your OR Courseware 568Problems 569

CHAPTER 12Integer Programming 576

12.1 Prototype Example 57712.2 Some BIP Applications 58012.3 Innovative Uses of Binary Variables in Model Formulation 58512.4 Some Formulation Examples 59112.5 Some Perspectives on Solving Integer Programming Problems 60012.6 The Branch-and-Bound Technique and Its Application to Binary Integer

Programming 60412.7 A Branch-and-Bound Algorithm for Mixed Integer Programming 61612.8 Other Developments in Solving BIP Problems 62212.9 Conclusions 630Selected References 631

xvi TABLE OF CONTENTS

Learning Aids for This Chapter in Your OR Courseware 631Problems 632Case 12.1 Capacity Concerns 642Case 12.2 Assigning Art 645Case 12.3 Stocking Sets 649Case 12.4 Assigning Students to Schools (Revisited Again) 653

CHAPTER 13Nonlinear Programming 654

13.1 Sample Applications 65513.2 Graphical Illustration of Nonlinear Programming Problems 65913.3 Types of Nonlinear Programming Problems 66413.4 One-Variable Unconstrained Optimization 67013.5 Multivariable Unconstrained Optimization 67313.6 The Karush-Kuhn-Tucker (KKT) Conditions for Constrained Optimization 67913.7 Quadratic Programming 68313.8 Separable Programming 69013.9 Convex Programming 69713.10 Nonconvex Programming 70213.11 Conclusions 706Selected References 706Learning Aids for This Chapter in Your OR Courseware 707Problems 708Case 13.1 Savvy Stock Selection 720

CHAPTER 14Game Theory 726

14.1 The Formulation of Two-Person, Zero-Sum Games 72614.2 Solving Simple GamesA Prototype Example 72814.3 Games with Mixed Strategies 73314.4 Graphical Solution Procedure 73514.5 Solving by Linear Programming 73814.6 Extensions 74114.7 Conclusions 742Selected References 743Learning Aids for This Chapter in Your OR Courseware 743Problems 743

CHAPTER 15Decision Analysis 749

15.1 A Prototype Example 75015.2 Decision Making without Experimentation 75115.3 Decision Making with Experimentation 75815.4 Decision Trees 76415.5 Utility Theory 770

TABLE OF CONTENTS xvii

15.6 The Practical Application of Decision Analysis 77815.7 Conclusions 781Selected References 781Learning Aids for This Chapter in Your OR Courseware 782Problems 782Case 15.1 Brainy Business 795Case 15.2 Smart Steering Support 798

CHAPTER 16Markov Chains 802

16.1 Stochastic Processes 80216.2 Markov Chains 80316.3 Chapman-Kolmogorov Equations 80816.4 Classification of States of a Markov Chain 81016.5 Long-Run Properties of Markov Chains 81216.6 First Passage Times 81816.7 Absorbing States 82016.8 Continuous Time Markov Chains 822Selected References 827Learning Aids for This Chapter in Your OR Courseware 828Problems 828

CHAPTER 17Queueing Theory 834

17.1 Prototype Example 83517.2 Basic Structure of Queueing Models 83517.3 Examples of Real Queueing Systems 84017.4 The Role of the Exponential Distribution 84117.5 The Birth-and-Death Process 84817.6 Queueing Models Based on the Birth-and-Death Process 85217.7 Queueing Models Involving Nonexponential Distributions 87117.8 Priority-Discipline Queueing Models 87917.9 Queueing Networks 88517.10 Conclusions 889Selected References 890Learning Aids for This Chapter in Your OR Courseware 890Problems 891Case 17.1 Reducing In-Process Inventory 905

CHAPTER 18The Application of Queueing Theory 907

18.1 Examples 90718.2 Decision Making 90918.3 Formulation of Waiting-Cost Functions 912

xviii TABLE OF CONTENTS

18.4 Decision Models 91718.5 Some Award-Winning Applications of Queueing Theory 92318.6 Conclusions 926Selected References 926Learning Aids for This Chapter in Your OR Courseware 926Problems 927Case 18.1 Queueing Quandary 932

CHAPTER 19Inventory Theory 935

19.1 Examples 93619.2 Components of Inventory Models 93819.3 Deterministic Continuous-Review Models 94119.4 A Deterministic Periodic-Review Model 95119.5 A Stochastic Continuous-Review Model 95619.6 A Stochastic Single-Period Model for Perishable Products 96119.7 Stochastic Periodic-Review Models 97519.8 Larger Inventory Systems in Practice 98319.9 Conclusions 987Selected References 987Learning Aids for This Chapter in Your OR Courseware 987Problems 988Case 19.1 Brushing Up on Inventory Control 1000Case 19.2 TNT: Tackling Newsboys Teachings 1002Case 19.3 Jettisoning Surplus Stock 1004

CHAPTER 20Forecasting 1009

20.1 Some Applications of Forecasting 101020.2 Judgmental Forecasting Methods 101320.3 Time Series 101420.4 Forecasting Methods for a Constant-Level Model 101620.5 Incorporating Seasonal Effects into Forecasting Methods 101820.6 An Exponential Smoothing Method for a Linear Trend Model 102120.7 Forecasting Errors 102520.8 Box-Jenkins Method 102620.9 Causal Forecasting with Linear Regression 102820.10 Forecasting in Practice 103620.11 Conclusions 1038Selected References 1038Learning Aids for This Chapter in Your OR Courseware 1038Problems 1039Case 20.1 Finagling the Forecasts 1048

TABLE OF CONTENTS xix

CHAPTER 21Markov Decision Processes 1053

21.1 A Prototype Example 105321.2 A Model for Markov Decision Processes 105621.3 Linear Programming and Optimal Policies 105921.4 Policy Improvement Algorithm for Finding Optimal Policies 106421.5 Discounted Cost Criterion 106921.6 ConclusionsSelected References 1077Learning Aids for This Chapter in Your OR Courseware 1078Problems 1078

CHAPTER 22Simulation 1084

22.1 The Essence of Simulation 108422.2 Some Common Types of Applications of Simulation 109722.3 Generation of Random Numbers 110122.4 Generation of Random Observations from a Probability Distribution 110522.5 Outline of a Major Simulation Study 111022.6 Performing Simulations on Spreadsheets 111522.7 Variance-Reducing Techniques 112622.8 Regenerative Method of Statistical Analysis 113122.9 Conclusions 1138Selected References 1140Learning Aids for This Chapter in Your OR Courseware 1140Problems 1141Case 22.1 Planning Planers 1151Case 22.2 Pricing under Pressure 1153

APPENDIXES1. Documentation for the OR Courseware 11562. Convexity 11593. Classical Optimization Methods 11654. Matrices and Matrix Operations 11695. Tables 1174

PARTIAL ANSWERS TO SELECTED PROBLEMS 1176

INDEXESAuthor Index 1195Subject Index 1199

xx TABLE OF CONTENTS

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ADVANCE PRAISE FOR INTRODUCTION TO

OPERATIONS RESEARCH,SEVENTH EDITION

Reviewers seem to agree that this is clearly the best edition yet. Here is a sampling ofcomments:

The new edition seems to contain the most current information available.The new edition of Hillier/Lieberman is very well done and greatly enhances this clas-sic text.

The authors have done an admirable job of rewriting and reorganizing to reflect mod-ern management practices and the latest software developments.It is a complete package.

Hillier/Lieberman has recaptured any advantage it may have lost (to other competitors)in the past.

The changes in this new edition make Hillier/Lieberman the preeminent book for oper-ations research and I would highly recommend it.

INTRODUCTION TOOPERATIONS RESEARCH

McGraw-Hill Series in Industrial Engineering and Management Science

CONSULTING EDITORS

Kenneth E. Case, Department of Industrial Engineering and Management, Oklahoma State UniversityPhilip M. Wolfe, Department of Industrial and Management Systems Engineering, Arizona State University

BarnesStatistical Analysis for Engineers and Scientists: A Computer-Based ApproachBedworth, Henderson, and WolfeComputer-Integrated Design and ManufacturingBlank and TarquinEngineering EconomyEbelingReliability and Maintainability EngineeringGrant and LeavenworthStatistical Quality ControlHarrell, Ghosh, and BowdenSimulation Using PROMODELHillier and LiebermanIntroduction to Operations ResearchGrynaQuality Planning and Analysis: From Product Development through UseKelton, Sadowski, and SadowskiSimulation with ARENAKhalilManagement of TechnologyKolarikCreating Quality: Concepts, Systems, Strategies, and ToolsCreating Quality: Process Design for ResultsLaw and KeltonSimulation Modeling and AnalysisNash and SoferLinear and Nonlinear ProgrammingNelsonStochastic Modeling: Analysis and SimulationNiebel and FreivaldsMethods, Standards, and Work DesignPegdenIntroduction to Simulation Using SIMANRiggs, Bedworth, and RandhawaEngineering EconomicsSipper and BulfinProduction: Planning, Control, and IntegrationSteinerEngineering Economics Principles

INTRODUCTION TOOPERATIONS RESEARCH

Seventh Edition

FREDERICK S. HILLIER,Stanford University

GERALD J. LIEBERMAN,Late of Stanford University

Cases developed by Karl Schmedders and Molly Stephens

Tutorial software developed by Mark Hillier and Michael OSullivan

Boston Burr Ridge, IL Dubuque, IA Madison, WI New YorkSan Francisco St. Louis Bangkok Bogot Caracas Lisbon LondonMadrid Mexico City Milan New Delhi Seoul Singapore Sydney

Taipei Toronto

INTRODUCTION TO OPERATIONS RESEARCHPublished by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,New York, NY, 10020. Copyright 2001, 1995, 1990, 1986, 1980, 1974, 1967, by The McGraw-Hill Com-panies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form orby any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission,or broadcast for distance learning.Some ancillaries, including electronic and print components, may not be available to customers outside theUnited States.

This book is printed on acid-free paper.

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ISBN 0072321695

Vice president/Editor-in-chief: Kevin KanePublisher: Thomas CassonExecutive editor: Eric M. MunsonDevelopmental editor: Maja LorkovicMarketing manager: John WannemacherProject manager: Christine A. VaughanManager, new book production: Melonie SalvatiCoordinator, freelance design: Gino CieslikSupplement coordinator: Cathy TepperMedia technology producer: Judi DavidCover design: Gino CieslikCover Illustration: Paul TurnbaughCompositor: York Graphic Services, Inc.Typeface: 10/12 TimesPrinter: R. R. Donnelley & Sons Company

Library of Congress Cataloging-in-Publication DataHillier, Frederick S.

Introduction to operations research/Frederick S. Hillier, Gerald J. Lieberman; cases developed by Karl Schmedders and Molly Stephens; tutorial software developed by Mark Hillier and Michael OSullivan.7th ed.

p. cm.ISBN 0-07-232169-5

1. Operations research. I. Lieberman, Gerald J. II. Title.T57.6. H53 2001658.4034dc21 00-025683

www.mhhe.com

McGraw-Hill Higher EducationA Division of The McGraw-Hill Companies

vii

ABOUT THE AUTHORS

Frederick S. Hillier was born and raised in Aberdeen, Washington, where he was an awardwinner in statewide high school contests in essay writing, mathematics, debate, and mu-sic. As an undergraduate at Stanford University he ranked first in his engineering class ofover 300 students. He also won the McKinsey Prize for technical writing, won the Out-standing Sophomore Debater award, played in the Stanford Woodwind Quintet, and wonthe Hamilton Award for combining excellence in engineering with notable achievementsin the humanities and social sciences. Upon his graduation with a B.S. degree in IndustrialEngineering, he was awarded three national fellowships (National Science Foundation, TauBeta Pi, and Danforth) for graduate study at Stanford with specialization in operations re-search. After receiving his Ph.D. degree, he joined the faculty of Stanford University, andalso received visiting appointments at Cornell University, Carnegie-Mellon University, theTechnical University of Denmark, the University of Canterbury (New Zealand), and theUniversity of Cambridge (England). After 35 years on the Stanford faculty, he took earlyretirement from his faculty responsibilities in 1996 in order to focus full time on textbookwriting, and so now is Professor Emeritus of Operations Research at Stanford.

Dr. Hilliers research has extended into a variety of areas, including integer program-ming, queueing theory and its application, statistical quality control, and the application ofoperations research to the design of production systems and to capital budgeting. He has pub-lished widely, and his seminal papers have been selected for republication in books of se-lected readings at least ten times. He was the first-prize winner of a research contest on Cap-ital Budgeting of Interrelated Projects sponsored by The Institute of Management Sciences(TIMS) and the U.S. Office of Naval Research. He and Dr. Lieberman also received the hon-orable mention award for the 1995 Lanchester Prize (best English-language publication ofany kind in the field of operations research), which was awarded by the Institute of Opera-tions Research and the Management Sciences (INFORMS) for the 6th edition of this book.

Dr. Hillier has held many leadership positions with the professional societies in hisfield. For example, he has served as Treasurer of the Operations Research Society of Amer-ica (ORSA), Vice President for Meetings of TIMS, Co-General Chairman of the 1989 TIMSInternational Meeting in Osaka, Japan, Chair of the TIMS Publications Committee, Chairof the ORSA Search Committee for Editor of Operations Research, Chair of the ORSAResources Planning Committee, Chair of the ORSA/TIMS Combined Meetings Commit-tee, and Chair of the John von Neumann Theory Prize Selection Committee for INFORMS.

He currently is serving as the Series Editor for the International Series in Operations Re-search and Management Science being published by Kluwer Academic Publishers.

In addition to Introduction to Operations Research and the two companion volumes,Introduction to Mathematical Programming and Introduction to Stochastic Models in Op-erations Research, his books are The Evaluation of Risky Interrelated Investments (North-Holland, 1969), Queueing Tables and Graphs (Elsevier North-Holland, 1981, co-authoredby O. S. Yu, with D. M. Avis, L. D. Fossett, F. D. Lo, and M. I. Reiman), and Introduc-tion to Management Science: A Modeling and Case Studies Approach with Spreadsheets(Irwin/McGraw-Hill, co-authored by M. S. Hillier and G. J. Lieberman).

The late Gerald J. Lieberman sadly passed away shortly before the completion of this edi-tion. He had been Professor Emeritus of Operations Research and Statistics at Stanford Uni-versity, where he was the founding chair of the Department of Operations Research. He wasboth an engineer (having received an undergraduate degree in mechanical engineering fromCooper Union) and an operations research statistician (with an A.M. from Columbia Uni-versity in mathematical statistics, and a Ph.D. from Stanford University in statistics).

Dr. Lieberman was one of Stanfords most eminent leaders in recent decades. Afterchairing the Department of Operations Research, he served as Associate Dean of the Schoolof Humanities and Sciences, Vice Provost and Dean of Research, Vice Provost and Deanof Graduate Studies, Chair of the Faculty Senate, member of the University AdvisoryBoard, and Chair of the Centennial Celebration Committee. He also served as Provost orActing Provost under three different Stanford presidents.

Throughout these years of university leadership, he also remained active profession-ally. His research was in the stochastic areas of operations research, often at the interfaceof applied probability and statistics. He published extensively in the areas of reliabilityand quality control, and in the modeling of complex systems, including their optimal de-sign, when resources are limited.

Highly respected as a senior statesman of the field of operations research, Dr. Liebermanserved in numerous leadership roles, including as the elected President of The Institute ofManagement Sciences. His professional honors included being elected to the National Acad-emy of Engineering, receiving the Shewhart Medal of the American Society for Quality Con-trol, receiving the Cuthbertson Award for exceptional service to Stanford University, and serv-ing as a fellow at the Center for Advanced Study in the Behavioral Sciences. In addition, theInstitute of Operations Research and the Management Sciences (INFORMS) awarded himand Dr. Hillier the honorable mention award for the 1995 Lanchester Prize for the 6th edi-tion of this book. In 1996, INFORMS also awarded him the prestigious Kimball Medal forhis exceptional contributions to the field of operations research and management science.

In addition to Introduction to Operations Research and the two companion volumes,Introduction to Mathematical Programming and Introduction to Stochastic Models in Op-erations Research, his books are Handbook of Industrial Statistics (Prentice-Hall, 1955,co-authored by A. H. Bowker), Tables of the Non-Central t-Distribution (Stanford Uni-versity Press, 1957, co-authored by G. J. Resnikoff), Tables of the Hypergeometric Prob-ability Distribution (Stanford University Press, 1961, co-authored by D. Owen), Engi-neering Statistics, Second Edition (Prentice-Hall, 1972, co-authored by A. H. Bowker),and Introduction to Management Science: A Modeling and Case Studies Approach withSpreadsheets (Irwin/McGraw-Hill, 2000, co-authored by F. S. Hillier and M. S. Hillier).

viii ABOUT THE AUTHORS

ix

ABOUT THE CASE WRITERS

Karl Schmedders is assistant professor in the Department of Managerial Economics andDecision Sciences at the Kellogg Graduate School of Management (Northwestern Uni-versity), where he teaches quantitative methods for managerial decision making. His re-search interests include applications of operations research in economic theory, generalequilibrium theory with incomplete markets, asset pricing, and computational economics.Dr. Schmedders received his doctorate in operations research from Stanford University,where he taught both undergraduate and graduate classes in operations research. Amongthe classes taught was a case studies course in operations research, and he subsequentlywas invited to speak at a conference sponsored by the Institute of Operations Researchand the Management Sciences (INFORMS) about his successful experience with thiscourse. He received several teaching awards at Stanford, including the universitys pres-tigious Walter J. Gores Teaching Award.

Molly Stephens is currently pursuing a J.D. degree with a concentration in technologyand law. She graduated from Stanford University with a B.S. in Industrial Engineeringand an M.S. in Operations Research. A champion debater in both high school and col-lege, and president of the Stanford Debating Society, Ms. Stephens taught public speak-ing in Stanfords School of Engineering and served as a teaching assistant for a case stud-ies course in operations research. As a teaching assistant, she analyzed operations researchproblems encountered in the real world and the transformation of these problems intoclassroom case studies. Her research was rewarded when she won an undergraduate re-search grant from Stanford to continue her work and was invited to speak at an INFORMSconference to present her conclusions regarding successful classroom case studies. Fol-lowing graduation, Ms. Stephens worked at Andersen Consulting as a systems integrator,experiencing real cases from the inside, before resuming her graduate studies.

xi

DEDICATION

To the memory of our parents

and

To the memory of one of the true giants of our field, Jerry Lieberman,whose recent passing prevented him

from seeing the publication of this edition

xxiii

PREFACE

It now is 33 years since the first edition of this book was published in 1967. We have beenhumbled by having had both the privilege and the responsibility of introducing so manystudents around the world to our field over such a long span of time. With each new edi-tion, we have worked toward the goal of meeting the changing needs of new generationsof students by helping to define the modern approach to teaching the current status of op-erations research effectively at the introductory level. Over 33 years, much has changedin both the field and the pedagogical needs of the students being introduced to the field.These changes have been reflected in the substantial revisions of successive editions ofthis book. We believe that this is true for the current 7th edition as well.

The enthusiastic response to our first six editions has been most gratifying. It was aparticular pleasure to have the 6th edition receive honorable mention for the 1995 IN-FORMS Lanchester Prize (the prize awarded for the years most outstanding English-language publication of any kind in the field of operations research), including receivingthe following citation. This is the latest edition of the textbook that has introduced ap-proximately one-half million students to the methods and models of Operations Research.While adding material on a variety of new topics, the sixth edition maintains the highstandard of clarity and expositional excellence for which the authors have long been known.In honoring this work, the prize committee noted the enormous cumulative impact thatthe Hillier-Lieberman text has had on the development of our field, not only in the UnitedStates but also around the world through its many foreign-language editions.

As we enter a new millennium, the particular challenge for this new edition was torevise a book with deep roots in the 20th century so thoroughly that it would become fullysuited for the 21st century. We made a special effort to meet this challenge, especially inregard to the software and pedagogy in the book.

The new CD-ROM that accompanies the book provides an exciting array of software op-tions that reflect current practice.

One option is to use the increasingly popular spreadsheet approach with Excel andits Solver. Using spreadsheets as a key medium of instruction clearly is one new wave in

A WEALTH OF SOFTWARE OPTIONS

the teaching of operations research. The new Sec. 3.6 describes and illustrates how to useExcel and its Solver to formulate and solve linear programming models on a spreadsheet.Similar discussions and examples also are included in several subsequent chapters forother kinds of models. In addition, the CD-ROM provides an Excel file for many of thechapters that displays the spreadsheet formulation and solution for the relevant examplesin the chapter. Several of the Excel files also include a number of Excel templates forsolving the models in the chapter. Another key resource is a collection of Excel add-inson the CD-ROM (Premium Solver, TreePlan, SensIt, and RiskSim) that are integrated intothe corresponding chapters. In addition, Sec. 22.6 describes how some simulations can beperformed efficiently on spreadsheets by using another popular Excel add-in (@RISK)that can be downloaded temporarily from a website.

Practitioners of operations research now usually use a modeling language to formu-late and manage models of the very large size commonly encountered in practice. A mod-eling language system also will support one or more sophisticated software packages thatcan be called to solve a model once it has been formulated appropriately. The new Sec.3.7 discusses the application of modeling languages and illustrates it with one modelinglanguage (MPL) that is relatively amenable to student use. The student version of MPLis provided on the CD-ROM, along with an extensive MPL tutorial. Accompanying MPLas its primary solver is the student version of the renowned state-of-the-art software pack-age, CPLEX. The student version of CONOPT also is provided as the solver for nonlin-ear programming. We are extremely pleased to be able to provide such powerful and pop-ular software to students using this book. To further assist students, many of the chaptersinclude an MPL/CPLEX file (or MPL/CPLEX/CONOPT file in the case of the nonlinearprogramming chapter) on the CD-ROM that shows how MPL and CPLEX would formu-late and solve the relevant examples in the chapter. These files also illustrate how MPLand CPLEX can be integrated with spreadsheets.

As described in the appendix to Chaps. 3 and 4, a third attractive option is to employthe student version of the popular and student-friendly software package LINDO and itsmodeling language companion LINGO. Both packages can be downloaded free from theLINDO Systems website. Associated tutorial material is included on the CD-ROM, alongwith a LINDO/LINGO file for many of the chapters showing how LINDO and LINGOwould formulate and solve the relevant examples in the chapter. Once again, integrationwith spreadsheets also is illustrated.

Complementing all these options on the CD-ROM is an updated version of the tuto-rial software that many instructors have found so useful for their students with the 5th and6th editions. A program called OR Tutor provides 16 demonstration examples from the6th edition, but now with an attractive new design based on JavaScript. These demosvividly demonstrate the evolution of an algorithm in ways that cannot be duplicated onthe printed page. Most of the interactive routines from the 6th edition also are includedon the CD-ROM, but again with an attractive new design. This design features a spread-sheet format based on VisualBasic. Each of the interactive routines enables the student tointeractively execute one of the algorithms of operations research, making the needed de-cision at each step while the computer does the needed arithmetic. By enabling the stu-dent to focus on concepts rather than mindless number crunching when doing homeworkto learn an algorithm, we have found that these interactive routines make the learningprocess far more efficient and effective as well as more stimulating. In addition to these

xxiv PREFACE

routines, the CD-ROM includes a few of the automatic routines from the 6th edition (againredesigned with VisualBasic) for those cases that are not covered by the software optionsdescribed above. We were very fortunate to have the services of Michael OSullivan, atalented programmer and an advanced Ph.D. student in operations research at Stanford,to do all this updating of the software that had been developed by Mark S. Hillier for the5th and 6th editions.

Microsoft Project is introduced in Chap. 10 as a useful tool for project management.This software package also is included on the CD-ROM.

PREFACE xxv

Todays students in introductory operations research courses tend to be very interested inlearning more about the relevance of the material being covered, including how it is ac-tually being used in practice. Therefore, without diluting any of the features of the 6thedition, the focus of the revision for this edition has been on increasing the motivationand excitement of the students by making the book considerably more real world ori-ented and accessible. The new emphasis on the kinds of software that practitioners use isone thrust in this direction. Other major new features are outlined below.

Twenty-five elaborate new cases, embedded in a realistic setting and employing astimulating storytelling approach, have been added at the end of the problem sections. Allbut one of these cases were developed jointly by two talented case writers, Karl Schmed-ders (a faculty member at the Kellogg Graduate School of Management at NorthwesternUniversity) and Molly Stephens (recently an operations research consultant with Ander-sen Consulting). We also have further fleshed out six cases that were in the 6th edition.The cases generally require relatively challenging and comprehensive analyses with sub-stantial use of the computer. Therefore, they are suitable for student projects, working ei-ther individually or in teams, and can then lead to class discussion of the analysis.

A complementary new feature is that many new problems embedded in a realistic set-ting have been added to the problem section of many chapters. Some of the current prob-lems also have been fleshed out in a more interesting way.

This edition also places much more emphasis on providing perspective in terms ofwhat is actually happening in the practice of operations research. What kinds of applica-tions are occurring? What sizes of problems are being solved? Which models and tech-niques are being used most widely? What are their shortcomings and what new develop-ments are beginning to address these shortcomings? These kinds of questions are beingaddressed to convey the relevance of the techniques under discussion. Eight new sections(Secs. 10.7, 12.2, 15.6, 18.5, 19.8, 20.1, 20.10, and 22.2) are fully devoted to discussingthe practice of operations research in such ways, along with briefer mentions elsewhere.

The new emphases described above benefited greatly from our work in developingour recent new textbook with Mark S. Hillier (Introduction to Management Science: AModeling and Case Studies Approach with Spreadsheets, Irwin/McGraw-Hill, 2000). Thatbook has a very different orientation from this one. It is aimed directly at business stu-dents rather than students who may be in engineering and the mathematical sciences, andit provides almost no coverage of the mathematics and algorithms of operations research.Nevertheless, its applied orientation enabled us to adapt some excellent material devel-oped for that book to provide a more well-rounded coverage in this edition.

NEW EMPHASES

xxvi PREFACE

In addition to all the new software and new emphases just described, this edition receiveda considerable number of other enhancements as well.

The previous section on project planning and control with PERT/CPM has been re-placed by a complete new chapter (Chap. 10) with an applied orientation. Using the ac-tivity-on-node (AON) convention, this chapter provides an extensive modern treatment ofthe topic in a very accessible way.

Other new topics not yet mentioned include the SOB mnemonic device for deter-mining the form of constraints in the dual problem (in Sec. 6.4), 100 percent rules for si-multaneous changes when conducting sensitivity analysis (in Sec. 6.7), sensitivity analy-sis with Bayes decision rule (in Sec. 15.2), a probability tree diagram for calculatingposterior probabilities (in Sec. 15.3), a single-server variation of the nonpreemptive pri-orities model where the service for different priority classes of customers now have dif-ferent mean service rates (in Sec. 17.8), a new simpler analysis of a stochastic continu-ous-review inventory model (Sec. 19.5), the mean absolute deviation as a measure ofperformance for forecasting methods (in Sec. 20.7), and the elements of a major simula-tion study (Sec. 22.5).

We also have added much supplementary text material on the books new website,www.mhhe.com/hillier. Some of these supplements are password protected, but are avail-able to all instructors who adopt this textbook. For the most part, this material appearedin previous editions of this book and then was subsequently deleted (for space reasons),to the disappointment of some instructors. Some also appeared in our Introduction to Math-ematical Programming textbook. As delineated in the table of contents, this supplemen-tary material includes a chapter on additional special types of linear programming prob-lems, a review or primer chapter on probability theory, and a chapter on reliability, alongwith supplements to a few chapters in the book.

In addition to providing this supplementary text material, the website will give up-dates about the book, including an errata, as the need arises.

We made two changes in the order of the chapters. The decision analysis chapter hasbeen moved forward to Chap. 15 in front of the stochastic chapters. The game theorychapter has been moved backward to Chap. 14 to place it next to the related decisionanalysis chapter. We believe that these changes provide a better transition from topics thatare mainly deterministic to those that are mainly stochastic.

Every chapter has received significant revision and updating, ranging from modestrefining to extensive rewriting. Chapters receiving a particularly major revision and reor-ganization included Chaps. 15 (Decision Analysis), 19 (Inventory Theory), 20 (Forecast-ing), and 22 (Simulation). Many sections in the linear programming and mathematicalprogramming chapters also received major revisions and updating.

The overall thrust of all the revision efforts has been to build upon the strengths ofprevious editions while thoroughly updating and clarifying the material in a contempo-rary setting to fully meet the needs of todays students.

We think that the net effect has been to make this edition even more of a studentsbookclear, interesting, and well-organized with lots of helpful examples and illustra-tions, good motivation and perspective, easy-to-find important material, and enjoyablehomework, without too much notation, terminology, and dense mathematics. We believe

OTHER FEATURES

and trust that the numerous instructors who have used previous editions will agree thatthis is the best edition yet. This feeling has been reinforced by the generally enthusiasticreviews of drafts of this edition.

The prerequisites for a course using this book can be relatively modest. As with pre-vious editions, the mathematics has been kept at a relatively elementary level. Most ofChaps. 1 to 14 (introduction, linear programming, and mathematical programming) re-quire no mathematics beyond high school algebra. Calculus is used only in Chaps. 13(Nonlinear Programming) and in one example in Chap. 11 (Dynamic Programming). Ma-trix notation is used in Chap. 5 (The Theory of the Simplex Method), Chap. 6 (DualityTheory and Sensitivity Analysis), Sec. 7.4 (An Interior-Point Algorithm), and Chap. 13,but the only background needed for this is presented in Appendix 4. For Chaps. 15 to 22(probabilistic models), a previous introduction to probability theory is assumed, and cal-culus is used in a few places. In general terms, the mathematical maturity that a studentachieves through taking an elementary calculus course is useful throughout Chaps. 15 to22 and for the more advanced material in the preceding chapters.

The content of the book is aimed largely at the upper-division undergraduate level(including well-prepared sophomores) and at first-year (masters level) graduate students.Because of the books great flexibility, there are many ways to package the material intoa course. Chapters 1 and 2 give an introduction to the subject of operations research. Chap-ters 3 to 14 (on linear programming and on mathematical programming) may essentiallybe covered independently of Chaps. 15 to 22 (on probabilistic models), and vice versa.Furthermore, the individual chapters among Chaps. 3 to 14 are almost independent, ex-cept that they all use basic material presented in Chap. 3 and perhaps in Chap. 4. Chap-ter 6 and Sec. 7.2 also draw upon Chap. 5. Sections 7.1 and 7.2 use parts of Chap. 6. Sec-tion 9.6 assumes an acquaintance with the problem formulations in Secs. 8.1 and 8.3,while prior exposure to Secs. 7.3 and 8.2 is helpful (but not essential) in Sec. 9.7. WithinChaps. 15 to 22, there is considerable flexibility of coverage, although some integrationof the material is available.

An elementary survey course covering linear programming, mathematical program-ming, and some probabilistic models can be presented in a quarter (40 hours) or semes-ter by selectively drawing from material throughout the book. For example, a good sur-vey of the field can be obtained from Chaps. 1, 2, 3, 4, 15, 17, 19, 20, and 22, along withparts of Chaps. 9, 11, 12, and 13. A more extensive elementary survey course can be com-pleted in two quarters (60 to 80 hours) by excluding just a few chapters, for example,Chaps. 7, 14, and 21. Chapters 1 to 8 (and perhaps part of Chap. 9) form an excellent ba-sis for a (one-quarter) course in linear programming. The material in Chaps. 9 to 14 cov-ers topics for another (one-quarter) course in other deterministic models. Finally, the ma-terial in Chaps. 15 to 22 covers the probabilistic (stochastic) models of operations researchsuitable for presentation in a (one-quarter) course. In fact, these latter three courses (thematerial in the entire text) can be viewed as a basic one-year sequence in the techniquesof operations research, forming the core of a masters degree program. Each course out-lined has been presented at either the undergraduate or the graduate level at Stanford Uni-versity, and this text has been used in the manner suggested.

To assist the instructor who will be covering only a portion of the chapters and whoprefers a slimmer book containing only those chapters, all the material (including the sup-plementary text material on the books website) has been placed in McGraw-Hills PRIMIS

PREFACE xxvii

system. This system enables an instructor to pick and choose precisely which material toinclude in a self-designed book, and then to order copies for the students at an econom-ical price. For example, this enables instructors who previously used our Introduction toMathematical Programming or Introduction to Stochastic Models in Operations Researchtextbooks to obtain updated versions of the same material from the PRIMIS system. Forthis reason, we will not be publishing new separate editions of these other books.

Again, as in previous editions, we thank our wives, Ann and Helen, for their en-couragement and support during the long process of preparing this 7th edition. Our chil-dren, David, John, and Mark Hillier, Janet Lieberman Argyres, and Joanne, Michael, andDiana Lieberman, have literally grown up with the book and our periodic hibernations toprepare a new edition. Now, most of them have used the book as a text in their own col-lege courses, given considerable advice, and even (in the case of Mark Hillier) become asoftware collaborator. It is a joy to see them and (we trust) the book reach maturity to-gether.

And now I must add a very sad note. My close friend and co-author, Jerry Lieber-man, passed away on May 18, 1999, while this edition was in preparation, so I am writ-ing this preface on behalf of both of us. Jerry was one of the great leaders of our fieldand he had a profound influence on my life. More than a third of a century ago, we em-barked on a mission together to attempt to develop a path-breaking book for teaching op-erations research at the introductory level. Ever since, we have striven to meet and extendthe same high standards for each new edition. Having worked so closely with Jerry forso many years, I believe I understand well how he would want the book to evolve to meetthe needs of each new generation of students. As the substantially younger co-author, Iam grateful that I am able to carry on our joint mission to continue to update and improvethe book, both with this edition and with future editions as well. It is the least I can doto honor Jerry.

I welcome your comments, suggestions, and errata to help me improve the book inthe future.

xxviii PREFACE

We are indebted to an excellent group of reviewers who provided sage advice throughoutthe revision process. This group included Jeffery Cochran, Arizona State University; YahyaFathi, North Carolina State University; Yasser Hosni and Charles Reilly, University ofCentral Florida; Cerry Klein, University of MissouriColumbia; Robert Lipset, Ohio Uni-versity; Mark Parker, United States Air Force Academy; Christopher Rump, State Uni-versity of New York at Buffalo; and Ahmad Seifoddini, California Polytechnic State Uni-versitySan Luis Obispo. We also received helpful advice from Judith Liebman, SiegfriedSchaible, David Sloan, and Arthur F. Veinott, Jr., as well as many instructors who sent usletters or e-mail messages. In addition, we also thank many dozens of Stanford studentsand many students at other universities who gave us helpful written suggestions.

This edition was very much of a team effort. Our case writers, Karl Schmedders andMolly Stephens (both graduates of our department), made a vital contribution. One of ourdepartments current Ph.D. students, Roberto Szechtman, did an excellent job in prepar-ing the solutions manual. Another Ph.D. student, Michael OSullivan, was very skillful inupdating the software that Mark Hillier had developed for the 5th and 6th editions. Mark

ACKNOWLEDGMENTS

(who was born the same year as the first edition and now is a tenured faculty member inthe Management Science Department at the University of Washington) helped to overseethis updating and also provided both the spreadsheets and the Excel files (including manyExcel templates) for this edition. Linus Schrage of the University of Chicago and LINDOSystems (and who took an introductory operations research course from me 37 years ago)supervised the development of LINGO/LINDO files for the various chapters as well asproviding tutorial material for the CD-ROM. Another long-time friend, Bjarni Kristjans-son (who heads Maximal Software), did the same thing for the MPL/CPLEX files andMPL tutorial material, as well as arranging to provide student versions of MPL, CPLEX,CONOPT, and OptiMax 2000 for the CD-ROM. One of our departments Ph.D. gradu-ates, Irv Lustig, was the ILOG project manager for providing CPLEX. Linus, Bjarni, andIrv all were helpful in checking material going into this edition regarding their software.Ann Hillier devoted numerous long days and nights to sitting with a Macintosh, doingword processing and constructing many figures and tables, in addition to endless cuttingand pasting, photocopying, and FedExing of material. Helen Lieberman also carried aheavy burden in supporting Jerry. They all were vital members of the team.

The inside back cover lists the various companies and individuals who have providedsoftware for the CD-ROM. We greatly appreciate their key contributions.

It was a real pleasure working with McGraw-Hills thoroughly professional editorialand production staff, including Eric Munson (executive editor), Maja Lorkovic (develop-mental editor), and Christine Vaughan (project manager).

Frederick S. HillierStanford University (fhillier@Leland.Stanford.edu) January 2000

PREFACE xxix

11Introduction

Since the advent of the industrial revolution, the world has seen a remarkable growth inthe size and complexity of organizations. The artisans small shops of an earlier era haveevolved into the billion-dollar corporations of today. An integral part of this revolution-ary change has been a tremendous increase in the division of labor and segmentation ofmanagement responsibilities in these organizations. The results have been spectacular.However, along with its blessings, this increasing specialization has created new prob-lems, problems that are still occurring in many organizations. One problem is a tendencyfor the many components of an organization to grow into relatively autonomous empireswith their own goals and value systems, thereby losing sight of how their activities andobjectives mesh with those of the overall organization. What is best for one componentfrequently is detrimental to another, so the components may end up working at cross pur-poses. A related problem is that as the complexity and specialization in an organizationincrease, it becomes more and more difficult to allocate the available resources to the var-ious activities in a way that is most effective for the organization as a whole. These kindsof problems and the need to find a better way to solve them provided the environment forthe emergence of operations research (commonly referred to as OR).

The roots of OR can be traced back many decades, when early attempts were madeto use a scientific approach in the management of organizations. However, the beginningof the activity called operations research has generally been attributed to the military ser-vices early in World War II. Because of the war effort, there was an urgent need to allo-cate scarce resources to the various military operations and to the activities within eachoperation in an effective manner. Therefore, the British and then the U.S. military man-agement called upon a large number of scientists to apply a scientific approach to deal-ing with this and other strategic and tactical problems. In effect, they were asked to doresearch on (military) operations. These teams of scientists were the first OR teams. Bydeveloping effective methods of using the new tool of radar, these teams were instrumentalin winning the Air Battle of Britain. Through their research on how to better manage con-voy and antisubmarine operations, they also played a major role in winning the Battle ofthe North Atlantic. Similar efforts assisted the Island Campaign in the Pacific.

When the war ended, the success of OR in the war effort spurred interest in apply-ing OR outside the military as well. As the industrial boom following the war was run-

1.1 THE ORIGINS OF OPERATIONS RESEARCH

ning its course, the problems caused by the increasing complexity and specialization inorganizations were again coming to the forefront. It was becoming apparent to a growingnumber of people, including business consultants who had served on or with the OR teamsduring the war, that these were basically the same problems that had been faced by themilitary but in a different context. By the early 1950s, these individuals had introducedthe use of OR to a variety of organizations in business, industry, and government. Therapid spread of OR soon followed.

At least two other factors that played a key role in the rapid growth of OR duringthis period can be identified. One was the substantial progress that was made early in im-proving the techniques of OR. After the war, many of the scientists who had participatedon OR teams or who had heard about this work were motivated to pursue research rele-vant to the field; important advancements in the state of the art resulted. A prime exam-ple is the simplex method for solving linear programming problems, developed by GeorgeDantzig in 1947. Many of the standard tools of OR, such as linear programming, dynamicprogramming, queueing theory, and inventory theory, were relatively well developed be-fore the end of the 1950s.

A second factor that gave great impetus to the growth of the field was the onslaughtof the computer revolution. A large amount of computation is usually required to dealmost effectively with the complex problems typically considered by OR. Doing this byhand would often be out of the question. Therefore, the development of electronic digitalcomputers, with their ability to perform arithmetic calculations thousands or even millionsof times faster than a human being can, was a tremendous boon to OR. A further boostcame in the 1980s with the development of increasingly powerful personal computers ac-companied by good software packages for doing OR. This brought the use of OR withinthe easy reach of much larger numbers of people. Today, literally millions of individualshave ready access to OR software. Consequently, a whole range of computers from main-frames to laptops now are being routinely used to solve OR problems.

2 1 INTRODUCTION

1.2 THE NATURE OF OPERATIONS RESEARCH

As its name implies, operations research involves research on operations. Thus, opera-tions research is applied to problems that concern how to conduct and coordinate the op-erations (i.e., the activities) within an organization. The nature of the organization is es-sentially immaterial, and, in fact, OR has been applied extensively in such diverse areasas manufacturing, transportation, construction, telecommunications, financial planning,health care, the military, and public services, to name just a few. Therefore, the breadthof application is unusually wide.

The research part of the name means that operations research uses an approach thatresembles the way research is conducted in established scientific fields. To a considerableextent, the scientific method is used to investigate the problem of concern. (In fact, theterm management science sometimes is used as a synonym for operations research.) Inparticular, the process begins by carefully observing and formulating the problem, in-cluding gathering all relevant data. The next step is to construct a scientific (typicallymathematical) model that attempts to abstract the essence of the real problem. It is thenhypothesized that this model is a sufficiently precise representation of the essential fea-tures of the situation that the conclusions (solutions) obtained from the model are also

valid for the real problem. Next, suitable experiments are conducted to test this hypothe-sis, modify it as needed, and eventually verify some form of the hypothesis. (This step isfrequently referred to as model validation.) Thus, in a certain sense, operations researchinvolves creative scientific research into the fundamental properties of operations. How-ever, there is more to it than this. Specifically, OR is also concerned with the practicalmanagement of the organization. Therefore, to be successful, OR must also provide pos-itive, understandable conclusions to the decision maker(s) when they are needed.

Still another characteristic of OR is its broad viewpoint. As implied in the precedingsection, OR adopts an organizational point of view. Thus, it attempts to resolve the con-flicts of interest among the components of the organization in a way that is best for theorganization as a whole. This does not imply that the study of each problem must giveexplicit consideration to all aspects of the organization; rather, the objectives being soughtmust be consistent with those of the overall organization.

An additional characteristic is that OR frequently attempts to find a best solution (re-ferred to as an optimal solution) for the problem under consideration. (We say a best in-stead of the best solution because there may be multiple solutions tied as best.) Ratherthan simply improving the status quo, the goal is to identify a best possible course of ac-tion. Although it must be interpreted carefully in terms of the practical needs of manage-ment, this search for optimality is an important theme in OR.

All these characteristics lead quite naturally to still another one. It is evident that nosingle individual should be expected to be an expert on all the many aspects of OR workor the problems typically considered; this would require a group of individuals having di-verse backgrounds and skills. Therefore, when a full-fledged OR study of a new problemis undertaken, it is usually necessary to use a team approach. Such an OR team typicallyneeds to include individuals who collectively are highly trained in mathematics, statisticsand probability theory, economics, business administration, computer science, engineeringand the physical sciences, the behavioral sciences, and the special techniques of OR. Theteam also needs to have the necessary experience and variety of skills to give appropriateconsideration to the many ramifications of the problem throughout the organization.

1.3 THE IMPACT OF OPERATIONS RESEARCH 3

1.3 THE IMPACT OF OPERATIONS RESEARCH

Operations research has had an impressive impact on improving the efficiency of numer-ous organizations around the world. In the process, OR has made a significant contribu-tion to increasing the productivity of the economies of various countries. There now area few dozen member countries in the International Federation of Operational ResearchSocieties (IFORS), with each country having a national OR society. Both Europe and Asiahave federations of OR societies to coordinate holding international conferences and pub-lishing international journals in those continents.

It appears that the impact of OR will continue to grow. For example, according to theU.S. Bureau of Labor Statistics, OR currently is one of the fastest-growing career areasfor U.S. college graduates.

To give you a better notion of the wide applicability of OR, we list some actual award-winning applications in Table 1.1. Note the diversity of organizations and applications inthe first two columns. The curious reader can find a complete article describing each ap-plication in the JanuaryFebruary issue of Interfaces for the year cited in the third col-

TABLE 1.1 Some applications of operations research

Year of Related AnnualOrganization Nature of Application Publication* Chapters Savings

The Netherlands Develop national water management 1985 28, 13, 22 $15 millionRijkswaterstaat policy, including mix of new facilities,

operating procedures, and pricing.Monsanto Corp. Optimize production operations in 1985 2, 12 $2 million

chemical plants to meet production targetswith minimum cost.

United Airlines Schedule shift work at reservation offices 1986 29, 12, 17, $6 millionand airports to meet customer needs with 18, 20minimum cost.

Citgo Petroleum Optimize refinery operations and the supply, 1987 29, 20 $70 millionCorp. distribution, and marketing of products.

San Francisco Optimally schedule and deploy police 1989 24, 12, 20 $11 millionPolice Department patrol officers with a computerized system.

Texaco, Inc. Optimally blend available ingredients into 1989 2, 13 $30 milliongasoline products to meet quality andsales requirements.

IBM Integrate a national network of spare parts 1990 2, 19, 22 $20 millioninventories to improve service support. $250 million

less inventoryYellow Freight Optimize the design of a national trucking 1992 2, 9, 13, 20, $17.3 millionSystem, Inc. network and the routing of shipments. 22

New Haven Health Design an effective needle exchange 1993 2 33% lessDepartment program to combat the spread of HIV/AIDS. HIV/AIDS

AT&T Develop a PC-based system to guide 1993 17, 18, 22 $750 millionbusiness customers in designing their callcenters.

Delta Airlines Maximize the profit from assigning 1994 12 $100 millionairplane types to over 2500 domesticflights.

Digital Equipment Restructure the global supply chain of 1995 12 $800 millionCorp. suppliers, plants, distribution centers,

potential sites, and market areas.China Optimally select and schedule massive 1995 12 $425 million

projects for meeting the countrys futureenergy needs.

South African Optimally redesign the size and shape of 1997 12 $1.1 billiondefense force the defense force and its weapons systems.

Proctor and Gamble Redesign the North American production 1997 8 $200 millionand distribution system to reduce costsand improve speed to market.

Taco Bell Optimally schedule employees to provide 1998 12, 20, 22 $13 milliondesired customer service at a minimumcost.

Hewlett-Packard Redesign the sizes and locations of 1998 17, 18 $280 millionbuffers in a printer production line to meet more revenueproduction goals.

*Pertains to a JanuaryFebruary issue of Interfaces in which a complete article can be found describing the application.Refers to chapters in this book that describe the kinds of OR techniques used in the application.

4 1 INTRODUCTION

1.4 ALGORITHMS AND OR COURSEWARE 5

umn of the table. The fourth column lists the chapters in this book that describe the kindsof OR techniques that were used in the application. (Note that many of the applicationscombine a variety of techniques.) The last column indicates that these applications typi-cally resulted in annual savings in the millions (or even tens of millions) of dollars. Fur-thermore, additional benefits not recorded in the table (e.g., improved service to customersand better managerial control) sometimes were considered to be even more important thanthese financial benefits. (You will have an opportunity to investigate these less tangiblebenefits further in Probs. 1.3-1 and 1.3-2.)

Although most routine OR studies provide considerably more modest benefits thanthese award-winning applications, the figures in the rightmost column of Table 1.1 do ac-curately reflect the dramatic impact that large, well-designed OR studies occasionally canhave.

We will briefly describe some of these applications in the next chapter, and then wepresent two in greater detail as case studies in Sec. 3.5.

1.4 ALGORITHMS AND OR COURSEWARE

An important part of this book is the presentation of the major algorithms (systematicsolution procedures) of OR for solving certain types of problems. Some of these algo-rithms are amazingly efficient and are routinely used on problems involving hundreds orthousands of variables. You will be introduced to how these algorithms work and whatmakes them so efficient. You then will use these algorithms to solve a variety of problemson a computer. The CD-ROM called OR Courseware that accompanies the book will bea key tool for doing all this.

One special feature in your OR Courseware is a program called OR Tutor. This pro-gram is intended to be your personal tutor to help you learn the algorithms. It consists ofmany demonstration examples that display and explain the algorithms in action. Thesedemos supplement the examples in the book.

In addition, your OR Courseware includes many interactive routines for executingthe algorithms interactively in a convenient spreadsheet format. The computer does all theroutine calculations while you focus on learning and executing the logic of the algorithm.You should find these interactive routines a very efficient and enlightening way of doingmany of your homework problems.

In practice, the algorithms normally are executed by commercial software packages.We feel that it is important to acquaint students with the nature of these packages thatthey will be using after graduation. Therefore, your OR Courseware includes a wealth ofmaterial to introduce you to three particularly popular software packages described be-low. Together, these packages will enable you to solve nearly all the OR models encoun-tered in this book very efficiently. We have added our own automatic routines to the ORCourseware only in a few cases where these packages are not applicable.

A very popular approach now is to use todays premier spreadsheet package, Mi-crosoft Excel, to formulate small OR models in a spreadsheet format. The Excel Solverthen is used to solve the models. Your OR Courseware includes a separate Excel file fornearly every chapter in this book. Each time a chapter presents an example that can besolved using Excel, the complete spreadsheet formulation and solution is given in thatchapters Excel file. For many of the models in the book, an Excel template also is pro-

vided that already includes all the equations necessary to solve the model. Some Exceladd-ins also are included on the CD-ROM.

After many years, LINDO (and its companion modeling language LINGO) contin-ues to be a dominant OR software package. Student versions of LINDO and LINGO nowcan be downloaded free from the Web. As for Excel, each time an example can be solvedwith this package, all the details are given in a LINGO/LINDO file for that chapter inyour OR Courseware.

CPLEX is an elite state-of-the-art software package that is widely used for solvinglarge and challenging OR problems. When dealing with such problems, it is common toalso use a modeling system to efficiently formulate the mathematical model and enter itinto the computer. MPL is a user-friendly modeling system that uses CPLEX as its mainsolver. A student version of MPL and CPLEX is available free by downloading it fromthe Web. For your convenience, we also have included this student version in your ORCourseware. Once again, all the examples that can be solved with this package are de-tailed in MPL/CPLEX files for the corresponding chapters in your OR Courseware.

We will further describe these three software packages and how to use them later (es-pecially near the end of Chaps. 3 and 4). Appendix 1 also provides documentation for theOR Courseware, including OR Tutor.

To alert you to relevant material in OR Courseware, the end of each chapter fromChap. 3 onward has a list entitled Learning Aids for This Chapter in Your OR Course-ware. As explained at the beginning of the problem section for each of these chapters,symbols also are placed to the left of each problem number or part where any of this ma-terial (including demonstration examples and interactive routines) can be helpful.

6 1 INTRODUCTION

PROBLEMS

1.3-1. Select one of the applications of operations research listedin Table 1.1. Read the article describing the application in the JanuaryFebruary issue of Interfaces for the year indicated in thethird column. Write a two-page summary of the application andthe benefits (including nonfinancial benefits) it provided.

1.3-2. Select three of the applications of operations research listedin Table 1.1. Read the articles describing the applications in the Jan-uaryFebruary issue of Interfaces for the years indicated in the thirdcolumn. For each one, write a one-page summary of the applica-tion and the benefits (including nonfinancial benefits) it provided.

72Overview of the Operations Research Modeling ApproachThe bulk of this book is devoted to the mathematical methods of operations research (OR).This is quite appropriate because these quantitative techniques form the main part of whatis known about OR. However, it does not imply that practical OR studies are primarilymathematical exercises. As a matter of fact, the mathematical analysis often represents onlya relatively small part of the total effort required. The purpose of this chapter is to placethings into better perspective by describing all the major phases of a typical OR study.

One way of summarizing the usual (overlapping) phases of an OR study is the following:

1. Define the problem of interest and gather relevant data.2. Formulate a mathematical model to represent the problem.3. Develop a computer-based procedure for deriving solutions to the problem from the

model.4. Test the model and refine it as needed.5. Prepare for the ongoing application of the model as prescribed by management.6. Implement.

Each of these phases will be discussed in turn in the following sections.Most of the award-winning OR studies introduced in Table 1.1 provide excellent ex-

amples of how to execute these phases well. We will intersperse snippets from these ex-amples throughout the chapter, with references to invite your further reading.

2.1 DEFINING THE PROBLEM AND GATHERING DATA

In contrast to textbook examples, most practical problems encountered by OR teams areinitially described to them in a vague, imprecise way. Therefore, the first order of busi-ness is to study the relevant system and develop a well-defined statement of the problemto be considered. This includes determining such things as the appropriate objectives, con-straints on what can be done, interrelationships between the area to be studied and otherareas of the organization, possible alternative courses of action, time limits for making adecision, and so on. This process of problem definition is a crucial one because it greatlyaffects how relevant the conclusions of the study will be. It is difficult to extract a rightanswer from the wrong problem!

The first thing to recognize is that an OR team is normally working in an advisory ca-pacity. The team members are not just given a problem and told to solve it however theysee fit. Instead, they are advising management (often one key decision maker). The teamperforms a detailed technical analysis of the problem and then presents recommendationsto management. Frequently, the report to management will identify a number of alterna-tives that are particularly attractive under different assumptions or over a different range ofvalues of some policy parameter that can be evaluated only by management (e.g., the trade-off between cost and benefits). Management evaluates the study and its recommendations,takes into account a variety of intangible factors, and makes the final decision based on itsbest judgment. Consequently, it is vital for the OR team to get on the same wavelength asmanagement, including identifying the right problem from managements viewpoint, andto build the support of management for the course that the study is taking.

Ascertaining the appropriate objectives is a very important aspect of problem defini-tion. To do this, it is necessary first to identify the member (or members) of managementwho actually will be making the decisions concerning the system under study and then toprobe into this individuals thinking regarding the pertinent objectives. (Involving the de-cision maker from the outset also is essential to build her or his support for the imple-mentation of the study.)

By its nature, OR is concerned with the welfare of the entire organization rather thanthat of only certain of its components. An OR study seeks solutions that are optimal forthe overall organization rather than suboptimal solutions that are best for only one com-ponent. Therefore, the objectives that are formulated ideally should be those of the entireorganization. However, this is not always convenient. Many problems primarily concernonly a portion of the organization, so the analysis would become unwieldy if the stated ob-jectives were too general and if explicit consideration were given to all side effects on therest of the organization. Instead, the objectives used in the study should be as specific asthey can be while still encompassing the main goals of the decision maker and maintain-ing a reasonable degree of consistency with the higher-level objectives of the organization.

For profit-making organizations, one possible approach to circumventing the prob-lem of suboptimization is to use long-run profit maximization (considering the time valueof money) as the sole objective. The adjective long-run indicates that this objective pro-vides the flexibility to consider activities that do not translate into profits immediately(e.g., research and development projects) but need to do so eventually in order to be worth-while. This approach has considerable merit. This objective is specific enough to be usedconveniently, and yet it seems to be broad enough to encompass the basic goal of profit-making organizations. In fact, some people believe that all other legitimate objectives canbe translated into this one.

However, in actual practice, many profit-making organizations do not use this ap-proach. A number of studies of U.S. corporations have found that management tends toadopt the goal of satisfactory profits, combined with other objectives, instead of focusingon long-run profit maximization. Typically, some of these other objectives might be tomaintain stable profits, increase (or maintain) ones share of the market, provide for prod-uct diversification, maintain stable prices, improve worker morale, maintain family con-trol of the business, and increase company prestige. Fulfilling these objectives mightachieve long-run profit maximization, but the relationship may be sufficiently obscure thatit may not be convenient to incorporate them all into this one objective.

8 2 OVERVIEW OF THE OPERATIONS RESEARCH MODELING APPROACH

Furthermore, there are additional considerations involving social responsibilities thatare distinct from the profit motive. The five parties generally affected by a business firmlocated in a single country are (1) the owners (stockholders, etc.), who desire profits (div-idends, stock appreciation, and so on); (2) the employees, who desire steady employmentat reasonable wages; (3) the customers, who desire a reliable product at a reasonable price;(4) the suppliers, who desire integrity and a reasonable selling price for their goods; and(5) the government and hence the nation, which desire payment of fair taxes and consid-eration of the national interest. All five parties make essential contributions to the firm,and the firm should not be viewed as the exclusive servant of any one party for the ex-ploitation of others. By the same token, international corporations acquire additional obli-gations to follow socially responsible practices. Therefore, while granting that manage-ments prime responsibility is to make profits (which ultimately benefits all five parties),we note that its broader social responsibilities also must be recognized.

OR teams typically spend a surprisingly large amount of time gathering relevant dataabout the problem. Much data usually are needed both to gain an accurate understandingof the problem and to provide the needed input for the mathematical model being formu-lated in the next phase of study. Frequently, much of the needed data will not be availablewhen the study begins, either because the information never has been kept or because whatwas kept is outdated or in the wrong form. Therefore, it often is necessary to install a newcomputer-based management information system to collect the necessary data on an on-going basis and in the needed form. The OR team normally needs to enlist the assistanceof various other key individuals in the organization to track down all the vital data. Evenwith this effort, much of the data may be quite soft, i.e., rough estimates based only oneducated guesses. Typically, an OR team will spend considerable time trying to improvethe precision of the data and then will make do with the best that can be obtained.

Examples. An OR study done for the San Francisco Police Department1 resulted inthe development of a computerized system for optimally scheduling and deploying policepatrol officers. The new system provided annual savings of $11 million, an annual $3 mil-lion increase in traffic citation revenues, and a 20 percent improvement in response times.In assessing the appropriate objectives for this study, three fundamental objectives wereidentified:

1. Maintain a high level of citizen safety.2. Maintain a high level of officer morale.3. Minimize the cost of operations.

To satisfy the first objective, the police department and city government jointly establisheda desired level of protection. The mathematical model then imposed the requirement thatthis level of protection be achieved. Similarly, the model imposed the requirement of bal-ancing the workload equitably among officers in order to work toward the second objec-tive. Finally, the third objective was incorporated by adopting the long-term goal of min-imizing the number of officers needed to meet the first two objectives.

2.1 DEFINING THE PROBLEM AND GATHERING DATA 9

1P. E. Taylor and S. J. Huxley, A Break from Tradition for the San Francisco Police: Patrol Officer Schedul-ing Using an Optimization-Based Decision Support System, Interfaces, 19(1): 424, Jan.Feb. 1989. See es-pecially pp. 411.

The Health Department of New Haven, Connecticut used an OR team1 to de-sign an effective needle exchange program to combat the spread of the virus that causesAIDS (HIV), and succeeded in reducing the HIV infection rate among program clientsby 33 percent. The key part of this study was an innovative data collection programto obtain the needed input for mathematical models of HIV transmission. This programinvolved complete tracking of each needle (and syringe), including the identity, loca-tion, and date for each person receiving the needle and each person returning the needle during an exchange, as well as testing whether the returned needle was HIV-positive or HIV-negative.

An OR study done for the Citgo Petroleum Corporation2 optimized both refineryoperations and the supply, distribution, and marketing of its products, thereby achievinga profit improvement of approximately $70 million per year. Data collection also playeda key role in this study. The OR team held data requirement meetings with top Citgo man-agement to ensure the eventual and continual quality of data. A state-of-the-art manage-ment database system was developed and installed on a mainframe computer. In caseswhere needed data did not exist, LOTUS 1-2-3 screens were created to help operationspersonnel input the data, and then the data from the personal computers (PCs) were up-loaded to the mainframe computer. Before data was inputted to the mathematical model,a preloader program was used to check for data errors and inconsistencies. Initially, thepreloader generated a paper log of error messages 1 inch thick! Eventually, the numberof error and warning messages (indicating bad or questionable numbers) was reduced toless than 10 for each new run.

We will describe the overall Citgo study in much more detail in Sec. 3.5.

10 2 OVERVIEW OF THE OPERATIONS RESEARCH MODELING APPROACH

2.2 FORMULATING A MATHEMATICAL MODEL

After the decision makers problem is defined, the next phase is to reformulate this prob-lem in a form that is convenient for analysis. The conventional OR approach for doingthis is to construct a mathematical model that represents the essence of the problem. Be-fore discussing how to formulate such a model, we first explore the nature of models ingeneral and of mathematical models in particular.

Models, or idealized representations, are an integral part of everyday life. Commonexamples include model airplanes, portraits, globes, and so on. Similarly, models play animportant role in science and business, as illustrated by models of the atom, models ofgenetic structure, mathematical equations describing physical laws of motion or chemicalreactions, graphs, organizational charts, and industrial accounting systems. Such modelsare invaluable for abstracting the essence of the subject of inquiry, showing interrelation-ships, and facilitating analysis.

1E. H. Kaplan and E. OKeefe, Let the Needles Do the Talking! Evaluating the New Haven Needle Exchange,Interfaces, 23(1): 726, Jan.Feb. 1993. See especial