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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 Models 5

1.4 Art of Modeling 5

1.5 More Than Just Mathematics 6

1.6 Phases of an OR Study 8

1.7 About This Book 9

Chapter 2 Introduction to Linear Programming 11

2.1 Two-Variable LP Model 11

2.2 Graphical LP Solution 14

2.2.1 Solution of a Maximization Model 15

2.2.2 Solution of a Minimization Model 18

2.2.3 Graphical Solution with TORA 20

2.3 Graphical Sensitivity Analysis 23

2.3.1 Changes in the Objective Function Coefficients 24

2.3.2 Change in Availability of Resources 27

2.3.3 Unit Worth of a Resource 28

2.4 Computer Solution of LP Problems 33

2.4.1 LP Solution with TORA 33

2.4.2 LP Solution with Excel Solver 36

2.4.3 LP Solution with LINGO and AMPL 38

2.5 Analysis of Selected LP Models 47

Selected References 66

Comprehensive Problems 67

Chapter 3 The SimplexMethod 71

3.1 LP Solution Space in Equation Form 71

3.1.1 Converting Inequalities into Equations 71

3.1.2 Dealing with Unrestricted Variables 73

3.2 Transition from Graphical to Algebraic Solution 75

3.3 The Simplex Method 80

3.3.1 Iterative Nature of the Simplex Method 80

3.3.2 Computational Details of the Simplex Algorithm 83

3.3.3 Simplex iterations with TORA 92

3.4 Artificial Starting Solution 94

3.4.1 M-Method 94

3.4.2 Two-Phase Method 98

3.5 Special Cases in Simplex Method Application 103

3.5.1 Degeneracy 103

3.5.2 Alternative Optima 106

3.5.3 Unbounded Solution 109

3.5.4 Infeasible Solution 110

Selected References 112

Comprehensive Problems 112

Chapter 4 Duality and Sensitivity Analysis 115

4.1 Definition of the Dual Problem 115

4.2 Primal-Dual Relationships 120

4.2.1 Review of Simple Matrix Operations 120

4.2.2 Simplex Tableau Layout 122

4.2.3 Optimal Dual Solution 122

4.2.4 Simplex Tableau Computations 126

4.2.5 Primal and Dual Objective Value 130

4.3 Economic Interpretation of Duality 132

4.3.1 Economic Interpretation of Dual Variables 132

4.3.2 Economic Interpretation of Dual Constraints 135

4.4 Additional Simplex Algorithms for LP 137

4.4.1 Dual Simplex Method 137

4.4.2 Generalized Simplex Algorithm 143

4.5 Postoptimal or Sensitivity Analysis 144

4.5.1 Changes Affecting Feasibility 145

4.5.2 Changes Affecting Optimality 155

Selected References 161

Comprehensive Problems 162

Chapter 5 Transportation Model and Its Variants 165

5.1 Definition of the Transportation Model 165

5.2 Nontraditional Transportation Models 172

5.3 The Transportation Algorithm 177

5.3.1 Determination of the Starting Solution 178

5.3.2 Iterative Computations of the Transportation Algorithm 182

5.3.3 Solution of the Transportation Model with TORA 187

5.3.4 Simplex Method Explanation of the Method of Multipliers 195

5.4 The Assignment Model 196

5.4.1 The Hungarian Method 197

5.4.2 Simplex Explanation of the Hungarian Method 202

5.5 The Transshipment Model 203

Selected References 208

Comprehensive Problems 208

Chapter 6 Network Models 213

6.1 Network Definitions 214

6.2 Minimal Spanning Tree Algorithm 215

6.3 Shortest-Route Problem 220

6.3.1 Examples of the Shortest-Route Applications 220

6.3.2 Shortest-Route Algorithms 224

6.3.3 Linear Programming Formulation of the Shortest-Route Problem 234

6.3.4 Excel Spreadsheet Solution of the Shortest-Route Problem 237

6.4 Maximal Flow Model 239

6.4.1 Enumeration of Cuts 240

6.4.2 Maximal Flow Algorithm 241

6.4.3 Linear Programming Formulation of the Maximum Flow Model 250

6.4.4 Excel Spreadsheet Solution of the Maximum Flow Model 250

6.5 Minimum-Cost Capacitated Flow Problem 252

6.5.1 Network Representation 252

6.5.2 Linear Programming Formulation 254

6.5.3 Capacitated Network Simplex Algorithm 259

6.5.4 Excel Spreadsheet Solution of the Minimum-Cost Capacitated Flow Model 265

6.6 CPM and PERT 266

6.6.1 Network Representation 267

6.6.2 Critical Path （CPM） Computations 272

6.6.3 Construction of the Time Schedule 275

6.6.4 Linear Programming Formulation of CPM 281

6.6.5 PERT Networks 283

Selected References 286

Comprehensive Problems 286

Chapter 7 Advanced Linear Programming 289

7.1 Simplex Method Fundamentals 289

7.1.1 From Extreme Points to Basic Solutions 290

7.1.2 Generalized Simplex Tableau in Matrix Form 294

7.2 Revised Simplex Method 297

7.2.1 Development of the Optimality and Feasibility Conditions 298

7.2.2 Revised Simplex Algorithm 300

7.3 Bounded Variables Algorithm 305

7.4 Decomposition Algorithm 312

7.5 Duality 322

7.5.1 Matrix Definition of the Dual Problem 322

7.5.2 Optimal Dual Solution 322

7.6 Parametric Linear Programming 326

7.6.1 Parametric Changes in C 327

7.6.2 Parametric Changes in b 329

7.7 Karmarkar Interior-Point Method 332

7.7.1 Basic Idea of the Interior-Point Algorithm 332

7.7.2 Interior-Point Algorithm 334

Selected References 344

Comprehensive Problems 344

Chapter 8 Goal Programming 347

8.1 A Goal Programming Formulation 347

8.2 Goal Programming Algorithms 352

8.2.1 The Weights Method 352

8.2.2 The Preemptive Method 354

Selected References 359

Comprehensive Problems 359

Chapter 9 Integer Linear Programming 361

9.1 Illustrative Applications 361

9.2 Integer Programming Algorithms 372

9.2.1 Branch-and-Bound （B&B） Algorithm 373

9.2.2 TORA-Generated B&B Tree 379

9.2.3 Cutting Plane Algorithm 384

9.2.4 Computational Considerations in ILP 389

9.3 Solution of the Traveling Salesperson Problem 390

9.3.1 B&B Solution Algorithm 393

9.3.2 Cutting Plane Algorithm 396

Selected References 397

Comprehensive Problems 397

Chapter 10 Deterministic Dynamic Programming 401

10.1 Recursive Nature of Computations in DP 401

10.2 Forward and Backward Recursion 404

10.3 Selected DP Applications 406

10.3.1 Knapsack/Flyaway Kit/Cargo-Loading Model 407

10.3.2 Workforce Size Model 415

10.3.3 Equipment Replacement Model 418

10.3.4 Investment Model 421

10.3.5 Inventory Models 425

10.4 Problem of Dimensionality 425

Selected References 428

Comprehensive Problems 428

Chapter 11 Deterministic InventoryModels 429

11.1 General Inventory Model 429

11.2 Static Economic Order Quantity （EOQ） Models 430

11.2.1 Classic EOQ Model 430

11.2.2 EOQ with Price Breaks 435

11.2.3 Multi-Item EOQ with Storage Limitation 439

11.3 Dynamic EOQ Models 443

11.3.1 No-Setup Model 444

11.3.2 Setup Model 448

Selected References 460

Comprehensive Problems 460

Chapter 12 Review of Basic Probability 463

12.1 Laws of Probability 463

12.1.1 Addition Law of Probability 464

12.1.2 Conditional Law of Probability 465

12.2 Random Variables and Probability Distributions 467

12.3 Expectation of a Random Variable 469

12.3.1 Mean and Variance of a Random Variable 470

12.3.2 Mean and Variance of Joint Random Variables 471

12.4 Four Common Probability Distributions 474

12.4.1 Binomial Distribution 474

12.4.2 Poisson Distribution 476

12.4.3 Negative Exponential Distribution 477

12.4.4 Normal Distribution 478

12.5 Empirical Distributions 480

Selected References 489

Chapter 13 Forecasting Models 491

13.1 Moving Average Technique 491

13.2 Exponential Smoothing 495

13.3 Regression 497

Selected References 501

Comprehensive Problem 502

Chapter 14 Decision Analysis and Games 503

14.1 Decision Making under Certainty-Analytic Hierarchy Process（AHP） 503

14.2 Decision Making under Risk 513

14.2.1 Expected Value Criterion 514

14.2.2 Variations of the Expected Value Criterion 519

14.3 Decision under Uncertainty 527

14.4 Game Theory 532

14.4.1 Optimal Solution of Two-Person Zero-Sum Games 532

14.4.2 Solution of Mixed Strategy Games 536

Selected References 543

Comprehensive Problems 543

Chapter 15 Probabilistic Dynamic Programming 547

15.1 A Game of Chance 547

15.2 Investment Problem 550

15.3 Maximization of the Event of Achieving a Goal 554

Selected References 558

Comprehensive Problem 558

Chapter 16 Probabilistic Inventory Models 559

16.1 Continuous Review Models 559

16.1.1 “Probabilitized” EOQ Model 559

16.1.2 Probabilistic EOQ Model 562

16.2 Single-Period Models 567

16.2.1 No Setup Model 567

16.2.2 Setup Model （s-S Policy） 571

16.3 Multiperiod Model 573

Selected References 576

Comprehensive Problems 576

Chapter 17 Queuing Systems 579

17.1 Why Study Queues？ 579

17.2 Elements of a Queuing Model 581

17.3 Role of Exponential Distribution 582

17.4 Pure Birth and Death Models （Relationship Between the Exponential and Poisson Distributions） 585

17.4.1 Pure Birth Model 586

17.4.2 Pure Death Model 590

17.5 Generalized Poisson Queuing Model 593

17.6 Specialized Poisson Queues 597

17.6.1 Steady-State Measures of Performance 599

17.6.2 Single-Server Models 602

17.6.3 Multiple-Server Models 611

17.6.4 Machine Servicing Model—（M/M/R）：（GDIK/K），R<K 621

17.7 （M/G/1）：（GD/∞/∞）—Pollaczek-Khintchine （P-K） Formula 624

17.8 Other Queuing Models 627

17.9 Queuing Decision Models 627

17.9.1 Cost Models 627

17.9.2 Aspiration Level Model 632

Selected References 634

Comprehensive Problems 634

Chapter 18 Simulation Modeling 639

18.1 Monte Carlo Simulation 639

18.2 Types of Simulation 644

18.3 Elements of Discrete Event Simulation 645

18.3.1 Generic Definition of Events 645

18.3.2 Sampling from Probability Distributions 647

18.4 Generation of Random Numbers 656

18.5 Mechanics of Discrete Simulation 657

18.5.1 Manual Simulation of a Single-Server Model 657

18.5.2 Spreadsheet-Based Simulation of the Single-Server Model 663

18.6 Methods for Gathering Statistical Observations 666

18.6.1 Subinterval Method 667

18.6.2 Replication Method 669

18.6.3 Regenerative （Cycle） Method 669

18.7 Simulation Languages 672

Selected References 674

Chapter 19 Markovian Decision Process 675

19.1 Scope of the Markovian Decision Problem：The Gardener Problem 675

19.2 Finite-Stage Dynamic Programming Model 677

19.3 Infinite-Stage Model 681

19.3.1 Exhaustive Enumeration Method 681

19.3.2 Policy Iteration Method Without Discounting 684

19.3.3 Policy Iteration Method with Discounting 687

19.4 Linear Programming Solution 690

19.5 Appendix： Review of Markov Chains 693

19.5.1 Markov Processes 694

19.5.2 Markov Chains 694

Selected References 700

Chapter 20 Classical Optimization Theory 701

20.1 Unconstrained Problems 701

20.1.1 Necessary and Sufficient Conditions 702

20.1.2 The Newton-Raphson Method 706

20.2 Constrained Problems 708

20.2.1 Equality Constraints 708

20.2.2 Inequality Constraints 723

Selected References 730

Chapter 21 Nonlinear Programming Algorithms 731

21.1 Unconstrained Algorithms 731

21.1.1 Direct Search Method 731

21.1.2 Gradient Method 735

21.2 Constrained Algorithms 738

21.2.1 Separable Programming 739

21.2.2 Quadratic Programming 747

21.2.3 Geometric Programming 752

21.2.4 Stochastic Programming 757

21.2.5 Linear Combinations Method 761

21.2.6 SUMT Algorithm 763

Selected References 764

Appendix A Review of Vectors and Matrices 765

A.1 Vectors 765

A.1.1 Definition of a Vector 765

A.1.2 Addition （Subtraction） of Vectors 765

A.1.3 Multiplication of Vectors by Scalars 766

A.1.4 Linearly Independent Vectors 766

A.2 Matrices 766

A.2.1 Definition of a Matrix 766

A.2.2 Types of Matrices 766

A.2.3 Matrix Arithmetic Operations 767

A.2.4 Determinant of a Square Matrix 768

A.2.5 Nonsingular Matrix 770

A.2.6 Inverse of a Nonsingular Matrix 770

A.2.7 Methods of Computing the Inverse of Matrix 771

A.3 Quadratic Forms 775

A.4 Convex and Concave Functions 777

Selected References 777

Problems 777

Appendix B TORA Primer 779

B.1 Main Menu 779

B.2 Input Mode and Format 780

B.3 Input Data Screen 780

B.4 Solve/Modify Menu 781

B.5 Output Format 782

B.6 Output Results 782

Appendix C Statistical Tables 785

Appendix D Partial Answers to Selected Problems 789

Index 825