OPERATIONS RESEARCH AN INTRODUCTION

E I G H T H E D I T I O N

1IAMDYA. IAIIA

Upper Saddle River, New Jersey 07458Upper Saddle River, New Jersey 07458Upper Saddle River, New Jersey 07458Upper Saddle River, New Jersey 07458

Operations Research:

An Introduction

Eighth Edition

H amdy A . Tah a University of Arkansas, Fayetteville

PEARSONPEARSONPEARSONPEARSON

To Karen

Los rios no llevan agua, Los rios no llevan agua, Los rios no llevan agua, Los rios no llevan agua, el sol las fuentes el sol las fuentes el sol las fuentes el sol las fuentes seed... seed... seed... seed... jYo jYo jYo jYo se donde hay una se donde hay una se donde hay una se donde hay una fuente que no ha de fuente que no ha de fuente que no ha de fuente que no ha de secar el sol! La fuente secar el sol! La fuente secar el sol! La fuente secar el sol! La fuente que no se agota es mi que no se agota es mi que no se agota es mi que no se agota es mi propio propio propio propio corazon...corazon...corazon...corazon...

—V Ruiz Aguilera (1862)

Library of Congress Cataloging_in-PubHcation Data

Taha, Hamdy A. Operations research: an introduction / Hamdy A. Taha.—8th ed. p.

cm. Includes bibliographical references and index. ISBN 0-13-188923-0 1. Operations research. 2. Programming (Mathematics) 1. Title.

T57.6.T3 1997 96-37160 003 -dc21 96-37160

Vice President and Editorial Director, ECS: Marcia J. Horton Senior Editor: Holly Stark Executive Managing Editor: Vince O'Brieti Managing Editor: David A. George Production Editor: Craig Little Director of Creative Services: Paul B elf and Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Art Editor: Greg Dulles Manufacturing Manager: Alexis Heydt-Long Manufacturing Buyer: Lisa McDowell

© 2007 by Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458

All rights reserved. No part of ihis book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Pearson Prentice Hall™ is a trademark of Pearson Education, Tnc.

Preliminary edition, first, and second editions © 1968,1971 and 1976，respectively, by Hamdy A. Taha. Third, fourth, and fifth editions © 1982，1987, and 1992, respectively, by Macmillan Publishing Company. Sixth and seventh editions © 1997 and 2003, respectively, by Pearson Education，Inc.

The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs.

Printed in the United States of America 10 9 8 7 6 5 4 3 2

ISBN ISBN ISBN ISBN 0000----13131313----lfifilfifilfifilfifi 犯犯犯犯 3333----0000

Pearson Education Ltd., London Pearson Education Australia Pty. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educaci6n de Mexico, S.A. de C.V. Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Inc., Upper Saddle River, New Jersey

Chapter 1Chapter 1Chapter 1Chapter 1

Contents

Chapter 2Chapter 2Chapter 2Chapter 2

Chapter 3Chapter 3Chapter 3Chapter 3

Preface xvii About the Author xix Trademarks xx What Is Operations Preface xvii About the Author xix Trademarks xx What Is Operations Preface xvii About the Author xix Trademarks xx What Is Operations Preface xvii About the Author xix Trademarks xx What Is Operations

Research? 1Research? 1Research? 1Research? 1

1.1 Operations Research Models 1 1.2 Solving the OR Model 4

1.3 Queuing and Simulation Models 5 1.4 Art of Modeling 5 1.5 More Than Just Mathematics 7 1.6 Phases of an OR Study 8

1.7 About This Book 10 References 10

Modeling with Linear Programming 1iModeling with Linear Programming 1iModeling with Linear Programming 1iModeling with Linear Programming 1i

2.1 Two-Variable LP Model 12 2.2 Graphical LP Solution 75

2.2.1 Solution of a Maximization Model 16 2.2.2 Solution of a Minimization Model 23

12.3 Selected LP Applications 27 2.3.1 Urban Planning 27 23.2 Currency Arbitrage 32 23.3 Investment 37

2.3.4 Production Planning and Inventory Control 42 2.3.5 Blending and Refining 57 2.3.6 Manpower Planning 57 2.3.7 Additional Applications 60

2.4 Computer Solution with Excel Solver and AMPL 68 2.4.1 LP Solution with Excel Solver 69

2.4.2 LP Solution with AMPL 73 References 80

The Simplex Method and Sensitivity Analysis 81The Simplex Method and Sensitivity Analysis 81The Simplex Method and Sensitivity Analysis 81The Simplex Method and Sensitivity Analysis 81

3.1 LP Model in Equation Form 82 3.1- 1 Converting Inequalities into Equations with Nonnegative

Right-Hand Side 82 3.1.2 Dealing with Unrestricted Variables 84

3.2 Transition from Graphical to Algebraic Solution 85

viii Contents

Chapter 4Chapter 4Chapter 4Chapter 4

Chapter 5Chapter 5Chapter 5Chapter 5

3.3 The Simplex Method 90 3.3.1 Iterative Nature of the Simplex Method 90 3.3.2 Computational Details of the Simplex Algorithm 93 3.3.3 Summary of the Simplex Method 99

3.4 Artificial Starting Solution 103 3.4.1 /W-Method 104 3.4.2 Two-Phase Method 108

3.5 Special Cases in the Simplex Method 113 3.5.1 Degeneracy 113 3.5.2 Alternative Optima 116 3.5.3 Unbounded Solution 119 3.5.4 Infeasible Solution 121

3.6 Sensitivity Analysis 123 3.6.1 Graphical Sensitivity Analysis 123 3.6.2 Algebraic Sensitivity Analysis 一Changes in the Right- Hand Side 129 3.6.3 Algebraic Sensitivity Analysis—Objective Function 139 3.6.4 Sensitivity Analysis with TORA, Solver, and AMPL 146 References 150

Duality and PostDuality and PostDuality and PostDuality and Post----Optimal Analysis 151Optimal Analysis 151Optimal Analysis 151Optimal Analysis 151

4.1 Definition of the Dual Problem 151 4.2 Primal-Dual Relationships 156

4.2.1 Review of Simple Matrix Operations 156 4.2.2 Simplex Tableau Layout 158 4.2.3 Optimal Dual Solution 159 4.2.4 Simplex Tableau Computations 165

4.3 Economic Interpretation of Duality 169 4.3.1 Economic Interpretation of Dual Variables 170 4.3.2 Economic Interpretation of Dual Constraints 172

4.4 Additional Simplex Algorithms 174 4.4.1 Dual Simplex Method 174 4.4.2 Generalized Simplex Algorithm 180

4.5 Post-Optimal Analysis 181 4.5.1 Changes Affecting Feasibility 182 4.5.2 Changes Affecting Optimality 187 References 191

TVansportation Model and Its Variants 193TVansportation Model and Its Variants 193TVansportation Model and Its Variants 193TVansportation Model and Its Variants 193

5.1 Definition of the Transportation Model 194 5.2 Nontraditional Transportation Models 201 5.3 The Transportation Algorithm 206

5.3.1 Determination of the Starting Solution 207 5.3.2 Iterative Computations of the Transportation Algorithm 211

666.

.4

本页已使用福昕阅读器进行编辑。 福昕软件(O2005-2007,版权所有 , 仅供试用。

Contents ix

5.3.3 Simplex Method Explanation of the Method of Multipliers 220

5.4 The Assignment Model 221 5A1 The Hungarian Method 222 5.4.2 Simplex Explanation of the Hungarian Method 228

5.5 The Transshipment Mode 丨 229 References 234

Chapter 6 Network Models 235Chapter 6 Network Models 235Chapter 6 Network Models 235Chapter 6 Network Models 235

Scope and Definition of Network Models 236 Minimal Spanning Tree Algorithm 239 Shortest-Route Problem 243 6.3.1 Examples of the Shortest-Route Applications 243 6.3.2 Shortest-Route Algorithms 248 6.3.3 Linear Programming Formulation

of the Shortest-Route Problem 257 Maximal flow model 263 6.4.1 Enumeratron of Cuts 263 6.4.2 Maximal-Flow Algorithm 264 6.4.3 Linear Programming Formulation of Maximal Flow

Mode 273 CPM and PERT 275 6.5.1 Network Representation 277 6.5.2 Critical Path (CPM) Computations 282 6.5.3 Construction of the Time Schedule 285 6.5.4 Linear Programming Formulation of CPM 292 6.5.5 PERT Calculations 293 References 296

Chapter 7 Advanced Linear Programming 297Chapter 7 Advanced Linear Programming 297Chapter 7 Advanced Linear Programming 297Chapter 7 Advanced Linear Programming 297

7.1 Simplex Method Fundamentals 298 7.1.1 From Extreme Points to Basic Solutions 300

7.1.2 Generalized Simplex Tableau in Matrix Form 303 7.2 Revised Simplex Method 306

7.2.1 Development of the Optimality and Feasibility Conditions 307

7.2.2 Revised Simplex Algorithm 309 7.3 Bounded-Variables Algorithm 315 7.4 Duality 321

7.4.1 Matrix Definition of the Dual Problem 322 1A2 Optimal Dual Solution 322

7.5 Parametric Linear Programming 326 7.5.1 Parametric Changes in C 327

7.5.2 Parametric Changes in b 330 References 332

Chapter 8 Goal Programming 333

Contents

亮页 g 使畀福昕阅读器进行编辑。 炉盟骜件 (c)2005-2007 , 版 权 所 有 仅供试用。

Chapter 9Chapter 9Chapter 9Chapter 9

Chapter 10Chapter 10Chapter 10Chapter 10

Chapter 11Chapter 11Chapter 11Chapter 11

8.1 A Goal Programming Formulation 334 8.2 Goal Programming Algorithms 338

8.2.1 The Weights Method 338 8.2.2 The Preemptive Method 341 References 348

Integer Linear Programming 349Integer Linear Programming 349Integer Linear Programming 349Integer Linear Programming 349

9.1 Illustrative Applications 350 9.1.1 Capital Budgeting 350 9.1.2 Set-Covering Problem 354 9.1.3 Fixed-Charge Problem 360 9.1.4 Either-Or and If-Then Constraints 364

9.2 Integer Programming Algorithms 369 9.2.1 Branch-and-Bound (B&B) Algorithm 370

9.2.2 Cutting-Plane Algorithm 379 9.2.3 Computational Considerations in ILP 384

9.3 Traveling Salesperson Problem (TSP) 385 9.3.1 Heuristic Algorithms 389 9.3.2 B&B Solution Algorithm 392 9.3.3 Cutting-Plane Algorithm 395 References 397

Deterministic Dynamic Programming 399Deterministic Dynamic Programming 399Deterministic Dynamic Programming 399Deterministic Dynamic Programming 399

10.1 Recursive Nature of Computations in DP 400 10.2 Forward and Backward Recursion 403 10.3 Selected DP Applications 405

10.3.1 Knapsack/Fly-Away/Cargo-Loading Model 405 10.3.2 Work-Force Size Model 413 10.3.3 Equipment Replacement Model 416 10.3.4 Investment Model 420 10.3.5 Inventory Models 423

10.4 Problem of Dimensionality 424 References 426

Deterministic Inventory Models 427Deterministic Inventory Models 427Deterministic Inventory Models 427Deterministic Inventory Models 427

11.1 General Inventory Model 427 11.2 Role of Demand in the Development of Inventory Models 428 11.3 Static Economic-Order-Quantity (EOQ) Models 430

11.3.1 Classic EOQ model 430 11.3.2 EOQ with Price Breaks 436

11.3.3 Multi-Item EOQ with Storage Limitation 440 11.4 Dynamic EOQ Models 443

11.4.1 No-Setup Model 445 11.4.2 Setup Model 449 References 461

Chapter 12 Review of Basic Probability 463

Contents xi

12.1

12.2 12.3

12.4

12.5

539

5 5

5

5

11

11

Laws of Probability 3 12.1.1 Addition Law of Probability 12.1.2 Conditional Law of Probability 5 Random Variables and Probability Distributions Expectation of a Random Variable 9 12.3.1 Mean and Variance (Standard Deviation) of a Random Variable 12.3.2 Mean and Variance of Joint Random Variables Four Common Probability Distributions 475 12.4.1 Binomial Distribution 5 12.4.2 Poisson Distribution 12.4.3 Negative Exponential Distribution 477 12.4.4 Normal Distribution Empirical Distributions References

Chapter 13 Decision Analysis and Games 4B9Chapter 13 Decision Analysis and Games 4B9Chapter 13 Decision Analysis and Games 4B9Chapter 13 Decision Analysis and Games 4B9

13.1 Decision Making under Certainty—Analytic Hierarchy Process (AHP) 9

13.2 Decision Making under Risk 5 13.2.1 Decision Tree-Based Expected Value Criterion 5 13.2.2 Variations of the Expected Value Criterion 5

13.3 Decision under Uncertainty 575 13.4 Game Theory 5

13.4.1 Optimal Solution of Two-Person Zero-Sum Games 5 13.4.2 Solution of Mixed Strategy Games 5 References 53

Chapter 14 Probabilistic Inventory Models 531Chapter 14 Probabilistic Inventory Models 531Chapter 14 Probabilistic Inventory Models 531Chapter 14 Probabilistic Inventory Models 531

14.1 Continuous Review Models 53 14.1.1 "Probabilitized" EOQ Model 53

14.1.2 Probabilistic EOQ Model 53 14.2 Single-Period Models 539

14.2.1 No-Setup Model (Newsvendor Model) 14.2.2 Setup Model (s-S Policy) 5 3

14.3 Multiperiod Model 5 5 References 5

Chapter 15 Chapter 15 Chapter 15 Chapter 15 Queuing Systems 549Queuing Systems 549Queuing Systems 549Queuing Systems 549

Why Study Queues? 55 Elements of a Queuing Model 55

Role of Exponential Distribution 553 Pure Birth and Death Models (Relationship Between

the Exponential and Poisson Distributions) 55 15.4.1 Pure Birth Model 55 15.4.2 Pure Death Model 5

Table of Contents

16.1 16.2 16.3

16.4 16.5

16.6

16.7

17.1 17.2 17.3 17.4

17. 17.

15.5 Generalized Poisson Queuing Model 563 15.6 Specialized Poisson Queues 568

15.6.1 Steady-State Measures of Performance 569 15.6.2 Single-Server Models 573

15.6.3 Multiple-Server Models 582 15.6.4 Machine Servicing Model— K 592

15.7 (M/G/1):(GD/ oo/ oo)—Pollaczek-Khintchine (P-K) Formula 595 15.8 Other Queuing Models 597 15.9 Queuing Decision Models 597

15.9.1 Cost Models 598 15.9.2 Aspiration Level Model 602 References 604

Chapter 16 Simulation Modeling 605Chapter 16 Simulation Modeling 605Chapter 16 Simulation Modeling 605Chapter 16 Simulation Modeling 605

Monte Carlo Simulation 605 Types of Simulation 610 Elements of Discrete-Event Simulation 611 16.3.1 Generic Definition of Events 611 16.3.2 Sampling from Probability Distributions 613 Generation of Random Numbers 622 Mechanics of Discrete Simulation 624 16.5.1 Manual Simulation of a Single-Server Model 624 16.5.2 Spreadsheet-Based Simulation of the Single-Server

Model 630 Methods for Gathering Statistical Observations 633 16.6.1 Subinterval Method 634 16.6.2 Replication Method 635 16.6.3 Regenerative (Cycle) Method 636 Simulation Languages 638 References 640

Chapter 17 Markov Chains 641Chapter 17 Markov Chains 641Chapter 17 Markov Chains 641Chapter 17 Markov Chains 641

Definition of a Markov Chain 641 Absolute and n-Step Transition Probabilities 644 Classification of the States in a Markov Chain 646 Steady-State Probabilities and Mean Return Times of Ergodic Chains 649 First Passage Time 654 Analysis of Absorbing States 658 References 663

Chapter 18 Classical Optimization Theory 665Chapter 18 Classical Optimization Theory 665Chapter 18 Classical Optimization Theory 665Chapter 18 Classical Optimization Theory 665

18.1 Unconstrained Problems 665 18.1.1 Necessary and Sufficient Conditions 676

18.1.2 The Newton-Raphson Method 670

Table of Contents

AppenAppenAppenAppen

A.1 A.2 A.3 A.4 A.5

A.6 A.7

A.8

18.2 Constrained Problems 672

18.2.1 Equality Constraints 673 18.2.2 Inequality Constraints—Karush-Kuhn-Tucker

(KKT) Conditions 685 References 690

Chapter 19 Nonlinear Programming Algorithms 691Chapter 19 Nonlinear Programming Algorithms 691Chapter 19 Nonlinear Programming Algorithms 691Chapter 19 Nonlinear Programming Algorithms 691

19.1 Unconstrained Algorithms 691 19.1.1 Direct Search Method 691

19.1.2 Gradient Method 695 19.2 Constrained Algorithms 699

19.2.1 Separable Programming 699 19.2.2 Quadratic Programming 708 19.2.3 Chance-Constrained Programming 713 19.2.4 Linear Combinations Method 718 19.2.5 SUMT Algorithm 721 References 722

,L Modeling Language 723,L Modeling Language 723,L Modeling Language 723,L Modeling Language 723

A Rudimentary AMPL Model 723 Components of AMPL Model 724 Mathematical Expressions and Computed Parameters 732 Subsets and Indexed Sets 735 Accessing External Files 737 A.5.1 Simple Read Files 737 A.5.2 Using Print and Printf to Retrieve Output 739 A.5.3 Input Table Files 739 A.5.4 Output Table Files 742 A.5.5 Spreadsheet Input/Output Tables 744 Interactive Commands 744 Iterative and Conditional Execution of AWIPL Commands 746 Sensitivity Analysis Using AMPl 748 References 748

Appendix B Statistical T^mes 749 Appendix C ParAppendix B Statistical T^mes 749 Appendix C ParAppendix B Statistical T^mes 749 Appendix C ParAppendix B Statistical T^mes 749 Appendix C Partial Solutions to tial Solutions to tial Solutions to tial Solutions to Selected Problems 753Selected Problems 753Selected Problems 753Selected Problems 753

Index 803Index 803Index 803Index 803

Table of Contents

xv

23. 23. 23.

•4

Chapter 24 Case Analysis CDChapter 24 Case Analysis CDChapter 24 Case Analysis CDChapter 24 Case Analysis CD----79797979

Case 1: Airline Fuel Allocation Using Optimum Tankering CD-79 Case 2: Optimization of Heart Valves Production CD-86 Case 3: Scheduling Appointments at Australian Tourist Commission Trade Events CD-89

On the CD-ROM

Chapter 20 Additional Network and LP Algorithms CDChapter 20 Additional Network and LP Algorithms CDChapter 20 Additional Network and LP Algorithms CDChapter 20 Additional Network and LP Algorithms CD----IIII

20.1 Minimum-Cost Capacitated Flow Problem CD-1 20.1.1 Network Representation CD-2 20.1.2 Linear Programming Formulation CD-4 20.1.3 Capacitated Network Simplex Algorithm CD~9

20.2 Decomposition Algorithm CD-16 20.3 Karmarkar Interior-Point Method CD-25

20.3.1 Basic Idea of the Interior-Point Algorithm CD-25 20.3.2 Interior-Point Algorithm CD-27 References CD-36

Chapter 21 Forecasting Models CDChapter 21 Forecasting Models CDChapter 21 Forecasting Models CDChapter 21 Forecasting Models CD----37373737

21.1 Moving Average Technique CD-37 21.2 Exponential Smoothing CD-41 21.3 Regression CD-42 References CD-46

Chapter 22 Probabilistic Dynamic Programming CDChapter 22 Probabilistic Dynamic Programming CDChapter 22 Probabilistic Dynamic Programming CDChapter 22 Probabilistic Dynamic Programming CD----47474747

22.1 A Game of Chance CD-47 22.2 Investment Problem CD-50 22.3 Maximization of the Event of Achieving a Goal CD-54

References CD-57

Chapter 23 Markovian Decision Process Chapter 23 Markovian Decision Process Chapter 23 Markovian Decision Process Chapter 23 Markovian Decision Process CDCDCDCD----58585858

Scope of the Markovian Decision Problem CD-58 Finite-Stage Dynamic Programming Model CD-60 Infinite-Stage Model CD-64 23.3.1 Exhaustive Enumeration Method CD-64 23.3.2 Policy Iteration Method Without Discounting CD-67 23.3.3 Policy Iteration Method with Discounting CD-70 Linear Programming Solution CD-73 References CD-77

xvi On the CD-ROM

Appendix D Review of ors and Matrices CDAppendix D Review of ors and Matrices CDAppendix D Review of ors and Matrices CDAppendix D Review of ors and Matrices CD----145145145145

Case 4: Saving Federal Travel Dollars CD-93 Case 5: Optima! Ship Routing and Personnel Assignments for Nava! Recruitment in Thailand CD-97 Case 6: Allocation of Operating Room Time in Mount Sinai Hospital CD-103 Case 7: Optimizing Trailer Payloads at PFG Building Glass 107 Case 8: Optimization of Crosscutting and Log Allocation at Weyerhauser 113 Case 9: Layout Planning for a Computer Integrated Manufacturing (CIM) Facility CD-118 Case 10: Booking Limits in Hotel Reservations CD-125 Case 11: Case/s Problem: Interpreting and Evaluating a New Test CD-128 Case 12: Ordering Golfers on the Final Day of Ryder Cup Matches CD-131 Case 13: Inventory Decisions in Dell*s Supply Chain CD-133 Case 14: Analysis of Internal Transport System in a Manufacturing Plant CD-136 Case 15: Telephone Sales Manpower Planning at Qantas

CD-139 D.1 Vectors 145

D.1.1

Definition of a Vector CD-145 D.1.2 Addition (Subtraction) of Vectors CD-145 D.1.3 Multiplication of Vectors by Scalars CD-146 D.1.4 Linearly Independent Vectors CD-I46 D.2 Matrices CD-146

D.2.1 Definition of a Matrix CD-146 D.2.2 Types of Matrices CD-146 D.2.3 Matrix Arithmetic Operations CD-147 D.2.4 Determinant of a Square Matrix CD-148 D.2.5 Nonsingular Matrix CD-150 D.2.6 Inverse of a Nonsingular Matrix CD-150 D.2.7 Methods of Computing the Inverse of Matrix CD-151 D.2.8 Matrix Manipulations with Excel CD-156 D.3 Quadratic Forms CD-157 D.4 Convex and Concave Functions CD-159 Problems 159 Selected References 160

Appendix E Case Studies CDAppendix E Case Studies CDAppendix E Case Studies CDAppendix E Case Studies CD----161161161161

Preface

The eighth edition is a major revision that streamlines the presentation of the text The eighth edition is a major revision that streamlines the presentation of the text The eighth edition is a major revision that streamlines the presentation of the text The eighth edition is a major revision that streamlines the presentation of the text mamamamaterial with emphasis on the applications and computations in operations research:terial with emphasis on the applications and computations in operations research:terial with emphasis on the applications and computations in operations research:terial with emphasis on the applications and computations in operations research:

• Chapter 2 is dedicatedChapter 2 is dedicatedChapter 2 is dedicatedChapter 2 is dedicated to linear programming modeling, with applications in the to linear programming modeling, with applications in the to linear programming modeling, with applications in the to linear programming modeling, with applications in the areas of urban renewal, currency arbitrage, investment, production planning, areas of urban renewal, currency arbitrage, investment, production planning, areas of urban renewal, currency arbitrage, investment, production planning, areas of urban renewal, currency arbitrage, investment, production planning, blending, scheduling, and trim loss. New endblending, scheduling, and trim loss. New endblending, scheduling, and trim loss. New endblending, scheduling, and trim loss. New end----ofofofof----section problems deal with section problems deal with section problems deal with section problems deal with topics ranging from water quality management atopics ranging from water quality management atopics ranging from water quality management atopics ranging from water quality management and traffic control to warfare.nd traffic control to warfare.nd traffic control to warfare.nd traffic control to warfare.

• Chapter 3 presents the general LP sensitivity analysis, including dual prices and Chapter 3 presents the general LP sensitivity analysis, including dual prices and Chapter 3 presents the general LP sensitivity analysis, including dual prices and Chapter 3 presents the general LP sensitivity analysis, including dual prices and reduced costs, in a simple and straightforward manner as a direct extension of reduced costs, in a simple and straightforward manner as a direct extension of reduced costs, in a simple and straightforward manner as a direct extension of reduced costs, in a simple and straightforward manner as a direct extension of

the simplex tableau computations.the simplex tableau computations.the simplex tableau computations.the simplex tableau computations. • Chapter 4 is now dedicated to Chapter 4 is now dedicated to Chapter 4 is now dedicated to Chapter 4 is now dedicated to LP postLP postLP postLP post----optimal analysis based on duality.optimal analysis based on duality.optimal analysis based on duality.optimal analysis based on duality. • An ExcelAn ExcelAn ExcelAn Excel----based combined nearest neighborbased combined nearest neighborbased combined nearest neighborbased combined nearest neighbor----reversal heuristic is presented for reversal heuristic is presented for reversal heuristic is presented for reversal heuristic is presented for

the traveling salesperson problem.the traveling salesperson problem.the traveling salesperson problem.the traveling salesperson problem. • Markov chains treatment has been expanded into new Chapter 17.Markov chains treatment has been expanded into new Chapter 17.Markov chains treatment has been expanded into new Chapter 17.Markov chains treatment has been expanded into new Chapter 17.

• The totally new Chapter The totally new Chapter The totally new Chapter The totally new Chapter 24*24*24*24* presents 1 presents 1 presents 1 presents 15 fully developed real5 fully developed real5 fully developed real5 fully developed real----life applications. life applications. life applications. life applications. The analysis, which often cuts across more than one OR technique (e.g., The analysis, which often cuts across more than one OR technique (e.g., The analysis, which often cuts across more than one OR technique (e.g., The analysis, which often cuts across more than one OR technique (e.g., heuristics and LPheuristics and LPheuristics and LPheuristics and LP，，，，or ILP and queuing), deals with the modeling, data collection, or ILP and queuing), deals with the modeling, data collection, or ILP and queuing), deals with the modeling, data collection, or ILP and queuing), deals with the modeling, data collection,

and comand comand comand computational aspects of solving the problem. These applputational aspects of solving the problem. These applputational aspects of solving the problem. These applputational aspects of solving the problem. These applications are ications are ications are ications are crosscrosscrosscross----referenced in pertinent chapters to provide an appreciation of the use of referenced in pertinent chapters to provide an appreciation of the use of referenced in pertinent chapters to provide an appreciation of the use of referenced in pertinent chapters to provide an appreciation of the use of OR techniques in practice.OR techniques in practice.OR techniques in practice.OR techniques in practice.

• The new Appendix E* includes approximately 50 mini cases of realThe new Appendix E* includes approximately 50 mini cases of realThe new Appendix E* includes approximately 50 mini cases of realThe new Appendix E* includes approximately 50 mini cases of real----life life life life situations categorized by chapters.situations categorized by chapters.situations categorized by chapters.situations categorized by chapters.

• More than 1000 endMore than 1000 endMore than 1000 endMore than 1000 end----ofofofof----section section section section problem are included in the book.problem are included in the book.problem are included in the book.problem are included in the book. • Each chapter starts with a Each chapter starts with a Each chapter starts with a Each chapter starts with a study guidestudy guidestudy guidestudy guide that facilitates the understanding of the that facilitates the understanding of the that facilitates the understanding of the that facilitates the understanding of the

material and the effective use of the accompanying software.material and the effective use of the accompanying software.material and the effective use of the accompanying software.material and the effective use of the accompanying software.

• The integration of software in the text allows testing concepts that otherwise The integration of software in the text allows testing concepts that otherwise The integration of software in the text allows testing concepts that otherwise The integration of software in the text allows testing concepts that otherwise ccccould not be presented effectively:ould not be presented effectively:ould not be presented effectively:ould not be presented effectively: 1. Excel spreadsheet implementations are used throughout the book, includExcel spreadsheet implementations are used throughout the book, includExcel spreadsheet implementations are used throughout the book, includExcel spreadsheet implementations are used throughout the book, including ing ing ing

dynamic programming, traveling salesperson, inventory, AHP, Bayesdynamic programming, traveling salesperson, inventory, AHP, Bayesdynamic programming, traveling salesperson, inventory, AHP, Bayesdynamic programming, traveling salesperson, inventory, AHP, Bayes 1111 probabilities, probabilities, probabilities, probabilities, ““““electronicelectronicelectronicelectronic”””” statistical tables, queuing, simulation, Markov statistical tables, queuing, simulation, Markov statistical tables, queuing, simulation, Markov statistical tables, queuing, simulation, Markov chachachachains, and nonlinear programming. The interactive user input in some ins, and nonlinear programming. The interactive user input in some ins, and nonlinear programming. The interactive user input in some ins, and nonlinear programming. The interactive user input in some

spreadsheets promotes a better understanding of the underlying spreadsheets promotes a better understanding of the underlying spreadsheets promotes a better understanding of the underlying spreadsheets promotes a better understanding of the underlying techniques.techniques.techniques.techniques.

2. The use of Excel Solver has been expanded throughout the book, particuThe use of Excel Solver has been expanded throughout the book, particuThe use of Excel Solver has been expanded throughout the book, particuThe use of Excel Solver has been expanded throughout the book, particu----larly in the areas of linear, network, intelarly in the areas of linear, network, intelarly in the areas of linear, network, intelarly in the areas of linear, network, integer, and nonlinear programming.ger, and nonlinear programming.ger, and nonlinear programming.ger, and nonlinear programming.

3. The powerful commercial modeling language, AMPLThe powerful commercial modeling language, AMPLThe powerful commercial modeling language, AMPLThe powerful commercial modeling language, AMPL®, has been integrated , has been integrated , has been integrated , has been integrated in the book using numerous examples ranging from linear and network toin the book using numerous examples ranging from linear and network toin the book using numerous examples ranging from linear and network toin the book using numerous examples ranging from linear and network to

•Contained on the CD-ROM.

xviii Preface

integer and nonlinear programming. Tlie syntax of AMPL is given in Apinteger and nonlinear programming. Tlie syntax of AMPL is given in Apinteger and nonlinear programming. Tlie syntax of AMPL is given in Apinteger and nonlinear programming. Tlie syntax of AMPL is given in Appendix pendix pendix pendix

A and its material crossA and its material crossA and its material crossA and its material cross----referenced within the examples in the book.referenced within the examples in the book.referenced within the examples in the book.referenced within the examples in the book. 4. TORA continue to play the key role of tutorial software.TORA continue to play the key role of tutorial software.TORA continue to play the key role of tutorial software.TORA continue to play the key role of tutorial software.

• All computerAll computerAll computerAll computer----related material has been deliberately compartmentalized either in related material has been deliberately compartmentalized either in related material has been deliberately compartmentalized either in related material has been deliberately compartmentalized either in separate sections or as subsection titled separate sections or as subsection titled separate sections or as subsection titled separate sections or as subsection titled AAAA MPL/Excel/Solver/TORA moment MPL/Excel/Solver/TORA moment MPL/Excel/Solver/TORA moment MPL/Excel/Solver/TORA moment to to to to minimize disruptions in the main presentation in the book.minimize disruptions in the main presentation in the book.minimize disruptions in the main presentation in the book.minimize disruptions in the main presentation in the book.

To keep the page count at a manageable level, some sections, complete chapters, To keep the page count at a manageable level, some sections, complete chapters, To keep the page count at a manageable level, some sections, complete chapters, To keep the page count at a manageable level, some sections, complete chapters, and two appendixes have been moved to the accompanying CD. The selection of the and two appendixes have been moved to the accompanying CD. The selection of the and two appendixes have been moved to the accompanying CD. The selection of the and two appendixes have been moved to the accompanying CD. The selection of the

exciexciexciexcised material is based on the authorsed material is based on the authorsed material is based on the authorsed material is based on the author’’’’s judgment regarding frequency of use in s judgment regarding frequency of use in s judgment regarding frequency of use in s judgment regarding frequency of use in introintrointrointroductory OR classes.ductory OR classes.ductory OR classes.ductory OR classes.

ACKNOWLEDGMENTS I wish to acknowledge the importance of the seventh edition reviews provided by I wish to acknowledge the importance of the seventh edition reviews provided by I wish to acknowledge the importance of the seventh edition reviews provided by I wish to acknowledge the importance of the seventh edition reviews provided by

Layek L. AbdelLayek L. AbdelLayek L. AbdelLayek L. Abdel----Malek, New Jersey Institute of Technology,Malek, New Jersey Institute of Technology,Malek, New Jersey Institute of Technology,Malek, New Jersey Institute of Technology, Evangelos Triantaphyllou, Evangelos Triantaphyllou, Evangelos Triantaphyllou, Evangelos Triantaphyllou, Louisiana State University, Michael Branson, Oklahoma State University, Charles H. Louisiana State University, Michael Branson, Oklahoma State University, Charles H. Louisiana State University, Michael Branson, Oklahoma State University, Charles H. Louisiana State University, Michael Branson, Oklahoma State University, Charles H. Reilly, University of Central Florida, and Mazen Arafeh, Virginia Polytechnic Institute Reilly, University of Central Florida, and Mazen Arafeh, Virginia Polytechnic Institute Reilly, University of Central Florida, and Mazen Arafeh, Virginia Polytechnic Institute Reilly, University of Central Florida, and Mazen Arafeh, Virginia Polytechnic Institute

and State University. In particular, I owe special thanks and State University. In particular, I owe special thanks and State University. In particular, I owe special thanks and State University. In particular, I owe special thanks to two individuals who have to two individuals who have to two individuals who have to two individuals who have ininininfluenced my thinking during the preparation of the eighth edition: R. Michael Harnett fluenced my thinking during the preparation of the eighth edition: R. Michael Harnett fluenced my thinking during the preparation of the eighth edition: R. Michael Harnett fluenced my thinking during the preparation of the eighth edition: R. Michael Harnett (Kansas State University), who over the years has provided me with valuable (Kansas State University), who over the years has provided me with valuable (Kansas State University), who over the years has provided me with valuable (Kansas State University), who over the years has provided me with valuable

feedback regarding the organization and the contents of the boofeedback regarding the organization and the contents of the boofeedback regarding the organization and the contents of the boofeedback regarding the organization and the contents of the book, and Richard H. k, and Richard H. k, and Richard H. k, and Richard H. Bernhard (North Carolina State University), whose detailed critique of the seventh Bernhard (North Carolina State University), whose detailed critique of the seventh Bernhard (North Carolina State University), whose detailed critique of the seventh Bernhard (North Carolina State University), whose detailed critique of the seventh edition prompted a reorganization of the opening chapters in this edition.edition prompted a reorganization of the opening chapters in this edition.edition prompted a reorganization of the opening chapters in this edition.edition prompted a reorganization of the opening chapters in this edition.

Robert Fourer (Northwestern University) patiently provided me with valuable Robert Fourer (Northwestern University) patiently provided me with valuable Robert Fourer (Northwestern University) patiently provided me with valuable Robert Fourer (Northwestern University) patiently provided me with valuable feedfeedfeedfeedback about the AMPL material presented in this edition. I appreciate his help in back about the AMPL material presented in this edition. I appreciate his help in back about the AMPL material presented in this edition. I appreciate his help in back about the AMPL material presented in this edition. I appreciate his help in editing the material and for suggesting changes that made the presentation more editing the material and for suggesting changes that made the presentation more editing the material and for suggesting changes that made the presentation more editing the material and for suggesting changes that made the presentation more

readreadreadreadable. I also wish to acknowledge his help in securing permissions to include the able. I also wish to acknowledge his help in securing permissions to include the able. I also wish to acknowledge his help in securing permissions to include the able. I also wish to acknowledge his help in securing permissions to include the AMPL studAMPL studAMPL studAMPL student version and the solvers CPLEX, KNITRO, LPSOLVE, LOQOent version and the solvers CPLEX, KNITRO, LPSOLVE, LOQOent version and the solvers CPLEX, KNITRO, LPSOLVE, LOQOent version and the solvers CPLEX, KNITRO, LPSOLVE, LOQO，，，，and and and and MINOS on the accompanying CD.MINOS on the accompanying CD.MINOS on the accompanying CD.MINOS on the accompanying CD.

As always, I remain indebted to my colleagues and to hundreds of students for As always, I remain indebted to my colleagues and to hundreds of students for As always, I remain indebted to my colleagues and to hundreds of students for As always, I remain indebted to my colleagues and to hundreds of students for their comments and their encouragement. In particular, I wish to acknowledge the their comments and their encouragement. In particular, I wish to acknowledge the their comments and their encouragement. In particular, I wish to acknowledge the their comments and their encouragement. In particular, I wish to acknowledge the supporsupporsupporsupport I ret I ret I ret I receive from Yuhceive from Yuhceive from Yuhceive from Yuh----Wen Chen (DaWen Chen (DaWen Chen (DaWen Chen (Da----Yeh University, Taiwan), Miguel Crispin Yeh University, Taiwan), Miguel Crispin Yeh University, Taiwan), Miguel Crispin Yeh University, Taiwan), Miguel Crispin

(University of Texas, El Paso), David Elizandro (Tennessee Tech University), Rafael (University of Texas, El Paso), David Elizandro (Tennessee Tech University), Rafael (University of Texas, El Paso), David Elizandro (Tennessee Tech University), Rafael (University of Texas, El Paso), David Elizandro (Tennessee Tech University), Rafael Gutierrez (UniverGutierrez (UniverGutierrez (UniverGutierrez (University of Texas, El Paso), Yasser Hosni (University of Central Florida), sity of Texas, El Paso), Yasser Hosni (University of Central Florida), sity of Texas, El Paso), Yasser Hosni (University of Central Florida), sity of Texas, El Paso), Yasser Hosni (University of Central Florida), Erhan KuErhan KuErhan KuErhan Kutanoglu (University of Texas, Austin), Robert E. Lewis (United States Army tanoglu (University of Texas, Austin), Robert E. Lewis (United States Army tanoglu (University of Texas, Austin), Robert E. Lewis (United States Army tanoglu (University of Texas, Austin), Robert E. Lewis (United States Army

Management EngiManagement EngiManagement EngiManagement Engineering College), Gino Lim (University of Houston), Scott Mason neering College), Gino Lim (University of Houston), Scott Mason neering College), Gino Lim (University of Houston), Scott Mason neering College), Gino Lim (University of Houston), Scott Mason (University of Arkansas), Heather Nachtman (University of Arkanas), Manuel Rossetti (University of Arkansas), Heather Nachtman (University of Arkanas), Manuel Rossetti (University of Arkansas), Heather Nachtman (University of Arkanas), Manuel Rossetti (University of Arkansas), Heather Nachtman (University of Arkanas), Manuel Rossetti (University of Ar(University of Ar(University of Ar(University of Arkansas), Tarek Taha (JB Hunt, Inc.), and NabeelYousef (University of kansas), Tarek Taha (JB Hunt, Inc.), and NabeelYousef (University of kansas), Tarek Taha (JB Hunt, Inc.), and NabeelYousef (University of kansas), Tarek Taha (JB Hunt, Inc.), and NabeelYousef (University of

Central Florida).Central Florida).Central Florida).Central Florida). I wish to express my sincere appreciation to the Pearson Prentice Hall editorial I wish to express my sincere appreciation to the Pearson Prentice Hall editorial I wish to express my sincere appreciation to the Pearson Prentice Hall editorial I wish to express my sincere appreciation to the Pearson Prentice Hall editorial

and production teams for their superb assistance during the production of the book: and production teams for their superb assistance during the production of the book: and production teams for their superb assistance during the production of the book: and production teams for their superb assistance during the production of the book:

Dee Dee Dee Dee Bernhard (Associate Editor), David George (Production Manager Bernhard (Associate Editor), David George (Production Manager Bernhard (Associate Editor), David George (Production Manager Bernhard (Associate Editor), David George (Production Manager ---- Engineering), Engineering), Engineering), Engineering), Bob Lentz (Copy Editor), Craig Little (Production Editor), and Holly Stark (Senior Bob Lentz (Copy Editor), Craig Little (Production Editor), and Holly Stark (Senior Bob Lentz (Copy Editor), Craig Little (Production Editor), and Holly Stark (Senior Bob Lentz (Copy Editor), Craig Little (Production Editor), and Holly Stark (Senior Acquisitions Editor).Acquisitions Editor).Acquisitions Editor).Acquisitions Editor).

HHHHAMDY AMDY AMDY AMDY A.A.A.A. TTTTAHA AHA AHA AHA hat@uark.eduhat@uark.eduhat@uark.eduhat@uark.edu http://ineg.uark.edu/TahaORbook/http://ineg.uark.edu/TahaORbook/http://ineg.uark.edu/TahaORbook/http://ineg.uark.edu/TahaORbook/

About the Author

Hamdy A. Taha is a University Professor Emeritus Hamdy A. Taha is a University Professor Emeritus Hamdy A. Taha is a University Professor Emeritus Hamdy A. Taha is a University Professor Emeritus of of of of Industrial Engineering with the University of Industrial Engineering with the University of Industrial Engineering with the University of Industrial Engineering with the University of

Arkansas, where he taught and conducted research Arkansas, where he taught and conducted research Arkansas, where he taught and conducted research Arkansas, where he taught and conducted research in in in in operations reseoperations reseoperations reseoperations research and simulation. He is the author arch and simulation. He is the author arch and simulation. He is the author arch and simulation. He is the author of of of of three other books on integer programming and three other books on integer programming and three other books on integer programming and three other books on integer programming and

simulation, and his works have been translated into simulation, and his works have been translated into simulation, and his works have been translated into simulation, and his works have been translated into Malay, Chinese, Korean, Spanish, Japanese, Russian, Malay, Chinese, Korean, Spanish, Japanese, Russian, Malay, Chinese, Korean, Spanish, Japanese, Russian, Malay, Chinese, Korean, Spanish, Japanese, Russian, Turkish, and Indonesian. He is also the author of Turkish, and Indonesian. He is also the author of Turkish, and Indonesian. He is also the author of Turkish, and Indonesian. He is also the author of several book chaseveral book chaseveral book chaseveral book chapters, and his technical articles pters, and his technical articles pters, and his technical articles pters, and his technical articles have appeared in have appeared in have appeared in have appeared in European Journal of Operations European Journal of Operations European Journal of Operations European Journal of Operations Research, IEEE Transactions on Reliability, HE Research, IEEE Transactions on Reliability, HE Research, IEEE Transactions on Reliability, HE Research, IEEE Transactions on Reliability, HE

Transactions, InterfacesTransactions, InterfacesTransactions, InterfacesTransactions, Interfaces，，，，Management Science, Naval Research Logistics Quarterly, Operations Management Science, Naval Research Logistics Quarterly, Operations Management Science, Naval Research Logistics Quarterly, Operations Management Science, Naval Research Logistics Quarterly, Operations Research,Research,Research,Research, and and and and Simulation.Simulation.Simulation.Simulation. ProfessorProfessorProfessorProfessor Taha was the recipient of the Alumni Award for excellence in research and the Taha was the recipient of the Alumni Award for excellence in research and the Taha was the recipient of the Alumni Award for excellence in research and the Taha was the recipient of the Alumni Award for excellence in research and the universityuniversityuniversityuniversity----wide Nadine Baum Award for excellence in teaching, both from the University of wide Nadine Baum Award for excellence in teaching, both from the University of wide Nadine Baum Award for excellence in teaching, both from the University of wide Nadine Baum Award for excellence in teaching, both from the University of Arkansas, and numerous other research and teaching awards from the College of EngineeriArkansas, and numerous other research and teaching awards from the College of EngineeriArkansas, and numerous other research and teaching awards from the College of EngineeriArkansas, and numerous other research and teaching awards from the College of Engineering, ng, ng, ng, University of Arkansas. He was also named a Senior Fulbright Scholar to Carlos III University, University of Arkansas. He was also named a Senior Fulbright Scholar to Carlos III University, University of Arkansas. He was also named a Senior Fulbright Scholar to Carlos III University, University of Arkansas. He was also named a Senior Fulbright Scholar to Carlos III University, Madrid, Spain. He is fluent in three languages and has held teaching and consulting positions in Madrid, Spain. He is fluent in three languages and has held teaching and consulting positions in Madrid, Spain. He is fluent in three languages and has held teaching and consulting positions in Madrid, Spain. He is fluent in three languages and has held teaching and consulting positions in Europe, Mexico, and the Middle East.Europe, Mexico, and the Middle East.Europe, Mexico, and the Middle East.Europe, Mexico, and the Middle East.

Trademarks

AMPL is a reAMPL is a reAMPL is a reAMPL is a registered trademark of AMPL Optimization, LLC, 900 Sierra PI. SE, gistered trademark of AMPL Optimization, LLC, 900 Sierra PI. SE, gistered trademark of AMPL Optimization, LLC, 900 Sierra PI. SE, gistered trademark of AMPL Optimization, LLC, 900 Sierra PI. SE, Albuquerque, NM 87108Albuquerque, NM 87108Albuquerque, NM 87108Albuquerque, NM 87108----3379.3379.3379.3379. CPLEX is a registered trademark of ILOG, Inc., 1080 Linda Vista Ave.CPLEX is a registered trademark of ILOG, Inc., 1080 Linda Vista Ave.CPLEX is a registered trademark of ILOG, Inc., 1080 Linda Vista Ave.CPLEX is a registered trademark of ILOG, Inc., 1080 Linda Vista Ave.，，，，Mountain Mountain Mountain Mountain View, CA 94043.View, CA 94043.View, CA 94043.View, CA 94043. KNITRO is a registered trademark of Ziena Optimization Inc., 1801 MapKNITRO is a registered trademark of Ziena Optimization Inc., 1801 MapKNITRO is a registered trademark of Ziena Optimization Inc., 1801 MapKNITRO is a registered trademark of Ziena Optimization Inc., 1801 Maple Ave, le Ave, le Ave, le Ave, Evanston, IL 60201.Evanston, IL 60201.Evanston, IL 60201.Evanston, IL 60201. LOQO is a trademark of Princeton University, Princeton, NJ 08544.LOQO is a trademark of Princeton University, Princeton, NJ 08544.LOQO is a trademark of Princeton University, Princeton, NJ 08544.LOQO is a trademark of Princeton University, Princeton, NJ 08544. Microsoft, Windows, and Excel registered trademarks of Microsoft Corporation in Microsoft, Windows, and Excel registered trademarks of Microsoft Corporation in Microsoft, Windows, and Excel registered trademarks of Microsoft Corporation in Microsoft, Windows, and Excel registered trademarks of Microsoft Corporation in the United States and/or other countries.the United States and/or other countries.the United States and/or other countries.the United States and/or other countries. MINOS is a trademark of Stanford UniversMINOS is a trademark of Stanford UniversMINOS is a trademark of Stanford UniversMINOS is a trademark of Stanford University, Stanford, CA 94305.ity, Stanford, CA 94305.ity, Stanford, CA 94305.ity, Stanford, CA 94305. Solver is a trademark of Frontline Systems, Inc.Solver is a trademark of Frontline Systems, Inc.Solver is a trademark of Frontline Systems, Inc.Solver is a trademark of Frontline Systems, Inc.，，，，7617 Little River Turnpike, Suite 7617 Little River Turnpike, Suite 7617 Little River Turnpike, Suite 7617 Little River Turnpike, Suite 960, Annandale960, Annandale960, Annandale960, Annandale，，，，VA 22003.VA 22003.VA 22003.VA 22003. TORA is a trademark of SimTec, Inc., P.O. Box TORA is a trademark of SimTec, Inc., P.O. Box TORA is a trademark of SimTec, Inc., P.O. Box TORA is a trademark of SimTec, Inc., P.O. Box 3492349234923492，，，，Fayetteville, AR 72702Fayetteville, AR 72702Fayetteville, AR 72702Fayetteville, AR 72702

Note:Note:Note:Note: Other product and company names that are mentioned herein may be trade Other product and company names that are mentioned herein may be trade Other product and company names that are mentioned herein may be trade Other product and company names that are mentioned herein may be trade----marks or registered trademarks of their respemarks or registered trademarks of their respemarks or registered trademarks of their respemarks or registered trademarks of their respective owners in the United States ctive owners in the United States ctive owners in the United States ctive owners in the United States and/or other countries.and/or other countries.and/or other countries.and/or other countries.

What Is Operations Research?

Chapter Guide*Chapter Guide*Chapter Guide*Chapter Guide* The first formal activities of Operations Research (OR) were initiated in The first formal activities of Operations Research (OR) were initiated in The first formal activities of Operations Research (OR) were initiated in The first formal activities of Operations Research (OR) were initiated in England during World War England during World War England during World War England during World War IIIIIIII，，，，when a team of British scientists set out to make scwhen a team of British scientists set out to make scwhen a team of British scientists set out to make scwhen a team of British scientists set out to make sciiii----entifically based decisions regarding the best utilization of war materiel. After the war, entifically based decisions regarding the best utilization of war materiel. After the war, entifically based decisions regarding the best utilization of war materiel. After the war, entifically based decisions regarding the best utilization of war materiel. After the war, the ideas advanced in military operations were adapted to improve efficiency and prothe ideas advanced in military operations were adapted to improve efficiency and prothe ideas advanced in military operations were adapted to improve efficiency and prothe ideas advanced in military operations were adapted to improve efficiency and pro----ductivity in the civilian sector.ductivity in the civilian sector.ductivity in the civilian sector.ductivity in the civilian sector.

This chapter will familiarize you with the bThis chapter will familiarize you with the bThis chapter will familiarize you with the bThis chapter will familiarize you with the basic terminology of operations reasic terminology of operations reasic terminology of operations reasic terminology of operations re----search, including mathematical modeling, feasible solutions, optimization, and search, including mathematical modeling, feasible solutions, optimization, and search, including mathematical modeling, feasible solutions, optimization, and search, including mathematical modeling, feasible solutions, optimization, and iterative computations. You will learn that defining the problem correctly is the most iterative computations. You will learn that defining the problem correctly is the most iterative computations. You will learn that defining the problem correctly is the most iterative computations. You will learn that defining the problem correctly is the most important (and most difficult) phase of practicing OR. Timportant (and most difficult) phase of practicing OR. Timportant (and most difficult) phase of practicing OR. Timportant (and most difficult) phase of practicing OR. The chapter also emphasizes he chapter also emphasizes he chapter also emphasizes he chapter also emphasizes that, while mathematical modeling is a cornerstone of OR, intangible (unquantifiable) that, while mathematical modeling is a cornerstone of OR, intangible (unquantifiable) that, while mathematical modeling is a cornerstone of OR, intangible (unquantifiable) that, while mathematical modeling is a cornerstone of OR, intangible (unquantifiable) factors (such as human behavior) must be accounted for in the final decision. As you factors (such as human behavior) must be accounted for in the final decision. As you factors (such as human behavior) must be accounted for in the final decision. As you factors (such as human behavior) must be accounted for in the final decision. As you proceed through the book, you will be presented with a varproceed through the book, you will be presented with a varproceed through the book, you will be presented with a varproceed through the book, you will be presented with a variety of applications through iety of applications through iety of applications through iety of applications through solved examples and chapter problems. In particular, Chapter 24 (on the CD) is solved examples and chapter problems. In particular, Chapter 24 (on the CD) is solved examples and chapter problems. In particular, Chapter 24 (on the CD) is solved examples and chapter problems. In particular, Chapter 24 (on the CD) is entirely deentirely deentirely deentirely devoted to the presentation of fully developed case analyses. Chapter voted to the presentation of fully developed case analyses. Chapter voted to the presentation of fully developed case analyses. Chapter voted to the presentation of fully developed case analyses. Chapter materials are crossmaterials are crossmaterials are crossmaterials are cross---- referenced with the cases to provide an appreci referenced with the cases to provide an appreci referenced with the cases to provide an appreci referenced with the cases to provide an appreciation of the use ation of the use ation of the use ation of the use of OR in practice.of OR in practice.of OR in practice.of OR in practice.

OPERATIONS RESEARCH MODELS Imagine that you have a 5Imagine that you have a 5Imagine that you have a 5Imagine that you have a 5----week business commitment between Fayetteville (FYV) week business commitment between Fayetteville (FYV) week business commitment between Fayetteville (FYV) week business commitment between Fayetteville (FYV) and Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesand Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesand Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesand Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesdays. days. days. days. A regular roundA regular roundA regular roundA regular round----trip ticket costrip ticket costrip ticket costrip ticket costs $400, but a 20% discount is granted if the dates of ts $400, but a 20% discount is granted if the dates of ts $400, but a 20% discount is granted if the dates of ts $400, but a 20% discount is granted if the dates of

the ticket span a weekend. A onethe ticket span a weekend. A onethe ticket span a weekend. A onethe ticket span a weekend. A one----way ticket in either direction costs 75% of the regway ticket in either direction costs 75% of the regway ticket in either direction costs 75% of the regway ticket in either direction costs 75% of the reg----ular price. How should you buy the tickets for the 5ular price. How should you buy the tickets for the 5ular price. How should you buy the tickets for the 5ular price. How should you buy the tickets for the 5----week period?week period?week period?week period?

We can look at the situation as a decisionWe can look at the situation as a decisionWe can look at the situation as a decisionWe can look at the situation as a decision----making making making making problem whose solution reproblem whose solution reproblem whose solution reproblem whose solution re----

quires answering three questions:quires answering three questions:quires answering three questions:quires answering three questions:

1. What What What What are the decision decision decision decision alternatives? 2. Under what Under what Under what Under what restrictions restrictions restrictions restrictions is the decision made?is the decision made?is the decision made?is the decision made? 3. What What What What is an is an is an is an appropriate appropriate appropriate appropriate objective criterion for objective criterion for objective criterion for objective criterion for evaluating evaluating evaluating evaluating the the the the alternatives?alternatives?alternatives?alternatives?

2 Chapter 1 What Is Operations Research?

Three alternatives are considered:Three alternatives are considered:Three alternatives are considered:Three alternatives are considered:

1.1.1.1. BuyBuyBuyBuy five regular FYV five regular FYV five regular FYV five regular FYV----DENDENDENDEN----FYV for departure on Monday and return on FYV for departure on Monday and return on FYV for departure on Monday and return on FYV for departure on Monday and return on WednesWednesWednesWednesday of the same week.day of the same week.day of the same week.day of the same week.

2.2.2.2. Buy one FYVBuy one FYVBuy one FYVBuy one FYV----DEN, four DENDEN, four DENDEN, four DENDEN, four DEN----FYVFYVFYVFYV----DEN that span weekends, and one DENDEN that span weekends, and one DENDEN that span weekends, and one DENDEN that span weekends, and one DEN---- FYV.FYV.FYV.FYV.

3.3.3.3. Buy one FYVBuy one FYVBuy one FYVBuy one FYV----DENDENDENDEN----FYV to cover Monday of the first week and Wednesday of the FYV to cover Monday of the first week and Wednesday of the FYV to cover Monday of the first week and Wednesday of the FYV to cover Monday of the first week and Wednesday of the last week and folast week and folast week and folast week and four DENur DENur DENur DEN----FYVFYVFYVFYV----DEN to cover the remaining legs. All tickets in DEN to cover the remaining legs. All tickets in DEN to cover the remaining legs. All tickets in DEN to cover the remaining legs. All tickets in

this alternative span at least one weekend.this alternative span at least one weekend.this alternative span at least one weekend.this alternative span at least one weekend.

The restriction on these options is that you should be able to leave FYV on Monday The restriction on these options is that you should be able to leave FYV on Monday The restriction on these options is that you should be able to leave FYV on Monday The restriction on these options is that you should be able to leave FYV on Monday

and return on Wednesday of the same week.and return on Wednesday of the same week.and return on Wednesday of the same week.and return on Wednesday of the same week.

An obvious objective criteriAn obvious objective criteriAn obvious objective criteriAn obvious objective criterion for evaluating the proposed alternative is the on for evaluating the proposed alternative is the on for evaluating the proposed alternative is the on for evaluating the proposed alternative is the price of the tickets. The alternative that yields the smallest cost is the best price of the tickets. The alternative that yields the smallest cost is the best price of the tickets. The alternative that yields the smallest cost is the best price of the tickets. The alternative that yields the smallest cost is the best

Specifically, we haveSpecifically, we haveSpecifically, we haveSpecifically, we have

Alternative 1 cost = 5 Alternative 1 cost = 5 Alternative 1 cost = 5 Alternative 1 cost = 5 X X X X 400 = $2000400 = $2000400 = $2000400 = $2000

Alternative 2 cost = .75 Alternative 2 cost = .75 Alternative 2 cost = .75 Alternative 2 cost = .75 X X X X 400 400 400 400 十十十十 4 4 4 4 X X X X (.8 (.8 (.8 (.8 X X X X 400) + .75 400) + .75 400) + .75 400) + .75 X X X X 400 = $1880400 = $1880400 = $1880400 = $1880

Alternative 3 cost Alternative 3 cost Alternative 3 cost Alternative 3 cost = = = = 5 5 5 5 X X X X (.8 (.8 (.8 (.8 X X X X 400) 400) 400) 400) = $1600= $1600= $1600= $1600 Thus, you should choose alternative 3.Thus, you should choose alternative 3.Thus, you should choose alternative 3.Thus, you should choose alternative 3.

Though the preceding example illustrates the three main components of an OR Though the preceding example illustrates the three main components of an OR Though the preceding example illustrates the three main components of an OR Though the preceding example illustrates the three main components of an OR

modelmodelmodelmodel————alternatives, objective criterion, and constraintsalternatives, objective criterion, and constraintsalternatives, objective criterion, and constraintsalternatives, objective criterion, and constraints————sitsitsitsituations differ in the uations differ in the uations differ in the uations differ in the dedededetails of how each component is developed and constructed. To illustrate this point, tails of how each component is developed and constructed. To illustrate this point, tails of how each component is developed and constructed. To illustrate this point, tails of how each component is developed and constructed. To illustrate this point, conconconconsider forming a maximumsider forming a maximumsider forming a maximumsider forming a maximum----area rectangle out of a piece of wire of length area rectangle out of a piece of wire of length area rectangle out of a piece of wire of length area rectangle out of a piece of wire of length LLLL inches. inches. inches. inches. What should be the width and height of the rectangle?What should be the width and height of the rectangle?What should be the width and height of the rectangle?What should be the width and height of the rectangle?

In contIn contIn contIn contrast with the tickets example, the number of alternatives in the present rast with the tickets example, the number of alternatives in the present rast with the tickets example, the number of alternatives in the present rast with the tickets example, the number of alternatives in the present exexexexample is not finite; namely, the width and height of the rectangle can assume an ample is not finite; namely, the width and height of the rectangle can assume an ample is not finite; namely, the width and height of the rectangle can assume an ample is not finite; namely, the width and height of the rectangle can assume an

infinite number of values. To formalize this observation, the alternatives of the infinite number of values. To formalize this observation, the alternatives of the infinite number of values. To formalize this observation, the alternatives of the infinite number of values. To formalize this observation, the alternatives of the problem are identproblem are identproblem are identproblem are identified by defining the width and height as continuous (algebraic) ified by defining the width and height as continuous (algebraic) ified by defining the width and height as continuous (algebraic) ified by defining the width and height as continuous (algebraic) variables.variables.variables.variables.

LetLetLetLet

wwww = width of the rectangle in inches = width of the rectangle in inches = width of the rectangle in inches = width of the rectangle in inches h =h =h =h = height of the rectangle in height of the rectangle in height of the rectangle in height of the rectangle in

inches Based on these definitionsinches Based on these definitionsinches Based on these definitionsinches Based on these definitions，，，，the restrictions of the situation can be expressed the restrictions of the situation can be expressed the restrictions of the situation can be expressed the restrictions of the situation can be expressed

verbally asverbally asverbally asverbally as

1_ Width of1_ Width of1_ Width of1_ Width of rectangle + Height of rectangle = Half the length of the wire rectangle + Height of rectangle = Half the length of the wire rectangle + Height of rectangle = Half the length of the wire rectangle + Height of rectangle = Half the length of the wire

2.2.2.2. Width and height cannot be negativeWidth and height cannot be negativeWidth and height cannot be negativeWidth and height cannot be negative

These restrictions are translated algebraically asThese restrictions are translated algebraically asThese restrictions are translated algebraically asThese restrictions are translated algebraically as The only remaining component now is the objective of the problem; namely, The only remaining component now is the objective of the problem; namely, The only remaining component now is the objective of the problem; namely, The only remaining component now is the objective of the problem; namely,

maximization of the area of the maximization of the area of the maximization of the area of the maximization of the area of the rectangle. Let rectangle. Let rectangle. Let rectangle. Let zzzz be the area of the rectangle, then the be the area of the rectangle, then the be the area of the rectangle, then the be the area of the rectangle, then the complete model becomescomplete model becomescomplete model becomescomplete model becomes

Maximize Maximize Maximize Maximize zzzz = = = = whwhwhwh

1.1 Operations Research Models 3

subject tosubject tosubject tosubject to

2(w + h) = L w , h > 0

Hie optimal solution of this model Hie optimal solution of this model Hie optimal solution of this model Hie optimal solution of this model i s w ~ h =i s w ~ h =i s w ~ h =i s w ~ h = which calls for constructing a square which calls for constructing a square which calls for constructing a square which calls for constructing a square shape.shape.shape.shape.

Based on the preceding two examples, theBased on the preceding two examples, theBased on the preceding two examples, theBased on the preceding two examples, the general OR model can be organized in general OR model can be organized in general OR model can be organized in general OR model can be organized in the following general format:the following general format:the following general format:the following general format:

Maximize or minimize Objective Function

subject to

Constraints

A solution of the mode is A solution of the mode is A solution of the mode is A solution of the mode is feasible feasible feasible feasible if it satisfies all the constraints. It is if it satisfies all the constraints. It is if it satisfies all the constraints. It is if it satisfies all the constraints. It is optimal optimal optimal optimal if, if, if, if, in addition to being feasible, it yin addition to being feasible, it yin addition to being feasible, it yin addition to being feasible, it yields the best (maximum or minimum) value of the obields the best (maximum or minimum) value of the obields the best (maximum or minimum) value of the obields the best (maximum or minimum) value of the ob----jective function. In the tickets example, the problem presents three feasible alternajective function. In the tickets example, the problem presents three feasible alternajective function. In the tickets example, the problem presents three feasible alternajective function. In the tickets example, the problem presents three feasible alterna----

tives, with the third alternative yielding the optimal solution. In the rectangle problem, tives, with the third alternative yielding the optimal solution. In the rectangle problem, tives, with the third alternative yielding the optimal solution. In the rectangle problem, tives, with the third alternative yielding the optimal solution. In the rectangle problem, a feasible alternative musta feasible alternative musta feasible alternative musta feasible alternative must satisfy the condition satisfy the condition satisfy the condition satisfy the condition w w w w ••••¥ h = ~ h = ~ h = ~ h = ~ with w and with w and with w and with w and hhhh assuming assuming assuming assuming nonnegative values. This leads to an infinite number of feasible solutions and, unlike nonnegative values. This leads to an infinite number of feasible solutions and, unlike nonnegative values. This leads to an infinite number of feasible solutions and, unlike nonnegative values. This leads to an infinite number of feasible solutions and, unlike

the tickets problem, the optimum solution is determined by an appropriate the tickets problem, the optimum solution is determined by an appropriate the tickets problem, the optimum solution is determined by an appropriate the tickets problem, the optimum solution is determined by an appropriate mathematmathematmathematmathematical tool (in this case, diffeical tool (in this case, diffeical tool (in this case, diffeical tool (in this case, differential calculus).rential calculus).rential calculus).rential calculus).

Though OR models are designed to Though OR models are designed to Though OR models are designed to Though OR models are designed to ““““optimizeoptimizeoptimizeoptimize”””” a specific objective criterion sub a specific objective criterion sub a specific objective criterion sub a specific objective criterion sub----

ject to a set of constraints, the quality of the resulting solution depends on the comject to a set of constraints, the quality of the resulting solution depends on the comject to a set of constraints, the quality of the resulting solution depends on the comject to a set of constraints, the quality of the resulting solution depends on the com----pleteness pleteness pleteness pleteness ofofofof the model in representing the real system. Take the model in representing the real system. Take the model in representing the real system. Take the model in representing the real system. Take，，，，for example, the tickets for example, the tickets for example, the tickets for example, the tickets model. If one is not able to identify all the dominant alternatives for purchasing the model. If one is not able to identify all the dominant alternatives for purchasing the model. If one is not able to identify all the dominant alternatives for purchasing the model. If one is not able to identify all the dominant alternatives for purchasing the

ticktickticktickets, then the resulting solution is optimum only relative to the choices represented ets, then the resulting solution is optimum only relative to the choices represented ets, then the resulting solution is optimum only relative to the choices represented ets, then the resulting solution is optimum only relative to the choices represented in the model. To be specific, if alternative 3 is lefin the model. To be specific, if alternative 3 is lefin the model. To be specific, if alternative 3 is lefin the model. To be specific, if alternative 3 is left out of the model, then the resulting t out of the model, then the resulting t out of the model, then the resulting t out of the model, then the resulting "opti"opti"opti"opti---- mum mum mum mum”””” solution would call for purchasing the tickets for $1880, which is a solution would call for purchasing the tickets for $1880, which is a solution would call for purchasing the tickets for $1880, which is a solution would call for purchasing the tickets for $1880, which is a

suboptimal sosuboptimal sosuboptimal sosuboptimal solution. Hie conclusion is that lution. Hie conclusion is that lution. Hie conclusion is that lution. Hie conclusion is that ““““thethethethe”””” optimum solution of a model is best optimum solution of a model is best optimum solution of a model is best optimum solution of a model is best only for only for only for only for that that that that model. If the model happens to reprmodel. If the model happens to reprmodel. If the model happens to reprmodel. If the model happens to represent the real system reasonably well, esent the real system reasonably well, esent the real system reasonably well, esent the real system reasonably well, then its soluthen its soluthen its soluthen its solution is optimum also for the real situation.tion is optimum also for the real situation.tion is optimum also for the real situation.tion is optimum also for the real situation.

PROBLEM SET1.1A L In the tickets example, identify a fourth feasible alternative.

2.2.2.2. In the rectangle problem, identify two feasible solutions and determine whicIn the rectangle problem, identify two feasible solutions and determine whicIn the rectangle problem, identify two feasible solutions and determine whicIn the rectangle problem, identify two feasible solutions and determine which h h h one is better.one is better.one is better.one is better.

3.3.3.3. Determine the optimal solution of the rectangle problem. Determine the optimal solution of the rectangle problem. Determine the optimal solution of the rectangle problem. Determine the optimal solution of the rectangle problem. {Hint:{Hint:{Hint:{Hint: Use the Use the Use the Use the constraint to exconstraint to exconstraint to exconstraint to express the objective function in terms of one variable, then use press the objective function in terms of one variable, then use press the objective function in terms of one variable, then use press the objective function in terms of one variable, then use differential calcufus.)differential calcufus.)differential calcufus.)differential calcufus.)

1.1.1.1.::::

1.4

4 Chapter 1 What Is Operations Research?

4.4.4.4. Amy, Jim, John, andAmy, Jim, John, andAmy, Jim, John, andAmy, Jim, John, and Kelly are standing on the east bank of a river and wish to Kelly are standing on the east bank of a river and wish to Kelly are standing on the east bank of a river and wish to Kelly are standing on the east bank of a river and wish to cross to the west side using a canoe. The canoe can hold at most two people at cross to the west side using a canoe. The canoe can hold at most two people at cross to the west side using a canoe. The canoe can hold at most two people at cross to the west side using a canoe. The canoe can hold at most two people at a time. Amy, being a time. Amy, being a time. Amy, being a time. Amy, being the most athletic, can row the most athletic, can row the most athletic, can row the most athletic, can row across across across across the river in 1 minute. Jim, John, the river in 1 minute. Jim, John, the river in 1 minute. Jim, John, the river in 1 minute. Jim, John, and Kelly would take and Kelly would take and Kelly would take and Kelly would take 2222, , , , 5555，，，，and 10 minutes, respectively. If two people are in the and 10 minutes, respectively. If two people are in the and 10 minutes, respectively. If two people are in the and 10 minutes, respectively. If two people are in the canoe, the slower person dictates the crossing time. The objective is for all canoe, the slower person dictates the crossing time. The objective is for all canoe, the slower person dictates the crossing time. The objective is for all canoe, the slower person dictates the crossing time. The objective is for all four people to be on the other side of the river in the shortest time possible.four people to be on the other side of the river in the shortest time possible.four people to be on the other side of the river in the shortest time possible.four people to be on the other side of the river in the shortest time possible. (a)(a)(a)(a) Identify at least two feasible plans for cIdentify at least two feasible plans for cIdentify at least two feasible plans for cIdentify at least two feasible plans for crossing the river (remember, the rossing the river (remember, the rossing the river (remember, the rossing the river (remember, the

canoe is the only mode of transportation and it cannot be shuttled empty).canoe is the only mode of transportation and it cannot be shuttled empty).canoe is the only mode of transportation and it cannot be shuttled empty).canoe is the only mode of transportation and it cannot be shuttled empty). (b)(b)(b)(b) Define the the the the criterion for evaluating the alternatives. ^(c)^(c)^(c)^(c)1111 What is the smallest time for moving all four people to the other side of the What is the smallest time for moving all four people to the other side of the What is the smallest time for moving all four people to the other side of the What is the smallest time for moving all four people to the other side of the

river?river?river?river?

*5.*5.*5.*5. In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can throw either a fast or a curve ball at random. If Joe correctly predicts a curve throw either a fast or a curve ball at random. If Joe correctly predicts a curve throw either a fast or a curve ball at random. If Joe correctly predicts a curve throw either a fast or a curve ball at random. If Joe correctly predicts a curve ball, he can mainball, he can mainball, he can mainball, he can maintain a .500 batting average, else if Jim throws a curve ball and tain a .500 batting average, else if Jim throws a curve ball and tain a .500 batting average, else if Jim throws a curve ball and tain a .500 batting average, else if Jim throws a curve ball and Joe prepJoe prepJoe prepJoe prepares for a fast ball, his batting average is kept down to .200. On the ares for a fast ball, his batting average is kept down to .200. On the ares for a fast ball, his batting average is kept down to .200. On the ares for a fast ball, his batting average is kept down to .200. On the other hand, if Joe correctly predicts a fast ball, he gets a .300 batting average, other hand, if Joe correctly predicts a fast ball, he gets a .300 batting average, other hand, if Joe correctly predicts a fast ball, he gets a .300 batting average, other hand, if Joe correctly predicts a fast ball, he gets a .300 batting average, else his batting average is only .100.else his batting average is only .100.else his batting average is only .100.else his batting average is only .100. (a)(a)(a)(a) Define the alternatives for this situation. (b)(b)(b)(b) Define the objective function for the problem and discuss how it differs from the

familiar optimization (maximization or minimization) of a criterion.familiar optimization (maximization or minimization) of a criterion.familiar optimization (maximization or minimization) of a criterion.familiar optimization (maximization or minimization) of a criterion. 6. During the construction of a house, six joists of 24 feet each must be trimmed to the cor-

rect length of 23 feet. The operations for cutting a joist involve the following sequence:

Three persons are involved: Two loaders must work simultaneously on Three persons are involved: Two loaders must work simultaneously on Three persons are involved: Two loaders must work simultaneously on Three persons are involved: Two loaders must work simultaneously on operations 1,2, and 5, and one cutter handles operations 3 and 4. There are two operations 1,2, and 5, and one cutter handles operations 3 and 4. There are two operations 1,2, and 5, and one cutter handles operations 3 and 4. There are two operations 1,2, and 5, and one cutter handles operations 3 and 4. There are two pairs of saw horses on which untrimmed joists are placed in preparation for pairs of saw horses on which untrimmed joists are placed in preparation for pairs of saw horses on which untrimmed joists are placed in preparation for pairs of saw horses on which untrimmed joists are placed in preparation for cucucucutting, and each pair can hold up to three sidetting, and each pair can hold up to three sidetting, and each pair can hold up to three sidetting, and each pair can hold up to three side----bybybyby----side joists. Suggest a good side joists. Suggest a good side joists. Suggest a good side joists. Suggest a good schedule for trimming the six joists.schedule for trimming the six joists.schedule for trimming the six joists.schedule for trimming the six joists.

1.2 SOLVING THE OR MODEL I

In OR, we do not have a single general technique to solve all mathematical models In OR, we do not have a single general technique to solve all mathematical models In OR, we do not have a single general technique to solve all mathematical models In OR, we do not have a single general technique to solve all mathematical models that that that that ；；；； can. arise in practice. Instcan. arise in practice. Instcan. arise in practice. Instcan. arise in practice. Instead, the type and complexity of the mathematical model ead, the type and complexity of the mathematical model ead, the type and complexity of the mathematical model ead, the type and complexity of the mathematical model dicdicdicdic---- !!!! tate the nature of the solution method. For example, in Section 1.1 the solution of thetate the nature of the solution method. For example, in Section 1.1 the solution of thetate the nature of the solution method. For example, in Section 1.1 the solution of thetate the nature of the solution method. For example, in Section 1.1 the solution of the

tickets problem requires simple ranking of alternatives based on the total tickets problem requires simple ranking of alternatives based on the total tickets problem requires simple ranking of alternatives based on the total tickets problem requires simple ranking of alternatives based on the total purchasing price, whereas the spurchasing price, whereas the spurchasing price, whereas the spurchasing price, whereas the solution of the rectangle problem utilizes differential olution of the rectangle problem utilizes differential olution of the rectangle problem utilizes differential olution of the rectangle problem utilizes differential calculus to decalculus to decalculus to decalculus to determine the maximum area.termine the maximum area.termine the maximum area.termine the maximum area.

The most prominent OR technique is The most prominent OR technique is The most prominent OR technique is The most prominent OR technique is linear programming. linear programming. linear programming. linear programming. It is designed for It is designed for It is designed for It is designed for models with linear objective and constraint functions. Other techniques include models with linear objective and constraint functions. Other techniques include models with linear objective and constraint functions. Other techniques include models with linear objective and constraint functions. Other techniques include integeintegeintegeinteger programming r programming r programming r programming (in which the variables assume integer vaiues(in which the variables assume integer vaiues(in which the variables assume integer vaiues(in which the variables assume integer vaiues))))，，，，dynamic dynamic dynamic dynamic programmingprogrammingprogrammingprogramming

Operation Time (seconds)

1. Place joist on saw horses 15

2. Measure correct length (23 feet) 5 3. Mark cutting line for circular saw 5 4. Trim joist to correct length 20 5. Stack trimmed joist in a designated area 20

1.4 Art of Modeling 5

asterisk (*) designates problems whose solution is provided in Appendix C‘

1.4 Art of Modeling 6

(in which the original model can be decomposed into more manageable subproblems), (in which the original model can be decomposed into more manageable subproblems), (in which the original model can be decomposed into more manageable subproblems), (in which the original model can be decomposed into more manageable subproblems), network network network network programmiprogrammiprogrammiprogramming ng ng ng (in which the problem can be modeled as a network), and (in which the problem can be modeled as a network), and (in which the problem can be modeled as a network), and (in which the problem can be modeled as a network), and nonlinear programming nonlinear programming nonlinear programming nonlinear programming (in which (in which (in which (in which functions of the model are nonlinear). These are only a few among many available OR tools.functions of the model are nonlinear). These are only a few among many available OR tools.functions of the model are nonlinear). These are only a few among many available OR tools.functions of the model are nonlinear). These are only a few among many available OR tools.

A peculiarity of most OR techniques A peculiarity of most OR techniques A peculiarity of most OR techniques A peculiarity of most OR techniques isisisis that solutions are not generally ob that solutions are not generally ob that solutions are not generally ob that solutions are not generally obtained tained tained tained in in in in (formulalike) closed forms. Instead, they are determined by (formulalike) closed forms. Instead, they are determined by (formulalike) closed forms. Instead, they are determined by (formulalike) closed forms. Instead, they are determined by algorithms. An algorithms. An algorithms. An algorithms. An algoalgoalgoalgorithm provides fixed rithm provides fixed rithm provides fixed rithm provides fixed computational rules that are applied repetitively to the problem, with each repetition (called computational rules that are applied repetitively to the problem, with each repe