Operations Research, an Introduction

838 Pages·2007·22.62 MB·English

by  Hamdy A. Taha

OPERATIONS
RESEARCH
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Operations
Research:
An Introduction
Eighth Edition
Hamdy A. Taha
University of Arkansas, Fayetteville
Upper Saddle River, New Jersey 07458
Library of Congress Calaloging.in-Publicalion Data
Taha, Hamdy A.
Operations research: an introduction I Hamdy A. Taha.~8th ed.
p. ern.
Includes bibliographical references and index_
ISBN 0-13-188923·0
1. Operations research. 2. Programming (Mathematics) 1. Title.
T57.6.T3 199796-37160
003 -dc21
96-37160
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_ © 2007 by Pearson Education, Inc.
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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.
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To Karen
Los rios no llevan agua,
el sallas fuentes sec6 ...
jYo se donde hay una fuente
que no ha de secar el sol!
La fuente que no se agota
es mi propio coraz6n ...
-li: RuizAguilera (1862)
Contents
Preface xvii
About the Author xix
Trademarks xx
Chapter 1 What Is Operations Research? 1
1.1 Operations Research Models 1
1.2 Solving the OR Model 4
1.3 Queuing and Simulation Mode!s 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
Chapter 2 Modeling with Linear Programming 11
2.1 Two-Variable LP Model 12
2.2 Graphical LP Solution 15
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
2.3.2 Currency Arbitrage 32
2.3.3 Investment 37
2.3.4 Production Planning and Inventory Control 42
2.3.5 Blending and Refining 51
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
Chapter 3 The 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
vii
viii Contents
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 M-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-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
Chapter 4 Duality and Post-Optimal 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
Chapter 5 Transportation 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
Contents ix
5.3.3 Simplex Method Explanation of the Method
of Multipliers 220
5.4 The Assignment Model 221
5.4.1 The Hungarian Method 222
5.4.2 Simplex Explanation of the Hungarian Method 228
5.5 The Transshipment Model 229
References 234
Chapter 6 Network Models 235
6.1 Scope and Definition of Network Models 236
6.2 Minimal Spanning Tree Algorithm 239
6.3 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
6.4 Maximal flow model 263
6.4.1 Enumeration of Cuts 263
6.4.2 Maximal-Flow Algorithm 264
6.4.3 Linear Programming Formulation of Maximal
Flow Mode 273
6.5 (PM 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 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
1.4.2 Optimal Dual Solution 322
7.5 Parametric Linear Programming 326
7.5.1 Parametric Changes in ( 327
7.5.2 Parametric Changes in b 330
References 332
x Contents
Chapter 8 Goal Programming 333
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
Chapter 9 Integer 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
Chapter 10 Deterministic 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 KnapsackJFly-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
Chapter 11 Deterministic Inventory Models 427
11.1 General Inventory Model 427
11.2 Role of Demand in the Development of Inventory
Models 428
11.3 Static Ecol1omic-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