Operations Research

745 Pages·2014·18.544 MB·English

by  A. M. Natarajan, P. Balasubramanie, A. Tamilarasi

Operations Research
A. M. Natarajan
Chief Executive Officer
Bannari Amman Institute of Technology
Sathyamangalam
Tamil Nadu
P. Balasubramanie
Professor, Department of Computer Science and Engineering
Kongu Engineering College
Perundurai, Erode
Tamil Nadu
A. Tamilarasi
Professor, Department of Computer Science and Engineering
Kongu Engineering College
Perundurai, Erode
Tamil Nadu
Chennai • Delhi
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Contents
Preface xi
About the Authors xv
1 Basics of Operations Research 1
1.1 Development of Operations Research 1
1.2 Definition of Operations Research 2
1.3 Necessity of Operations Research in Industry 3
1.4 Scope/Applications of Operations Research 4
1.5 Operations Research and Decision-Making 5
1.6 Operations Research in Modern Management 7
1.7 Phases of Operations Research 7
1.8 Models in Operations Research 8
1.8.1 Classification of Models 8
1.8.2 Characteristics of a Good Model 10
1.8.3 Principles of Modelling 10
1.8.4 General Methods for Solving Operations Research Models 10
1.9 Role of Operations Research in Engineering 10
1.10 Limitations of Operations Research 11
Exercises 11
2 Linear Programming Problem (LPP) 13
2.1 Introduction 13
2.2 Mathematical Formulation of Linear Programming Problem 14
2.3 Statements of Basic Theorems and Properties 22
2.3.1 General Formulation of Linear Programming Problems 22
2.3.2 Standard Form of Linear Programming Problem 23
2.3.3 Matrix Form of Linear Programming Problem 30
2.3.4 Definitions 32
2.3.5 Basic Assumptions 33
2.3.6 Fundamental Theorem of Linear Programming 34
2.3.7 Fundamental Properties of Solutions 35
2.4 Graphical Solutions to a Linear Programming Problem 43
2.4.1 Some Exceptional Cases 50
2.5 Simplex Method 53
2.6 Artifical Variable Techniques 61
2.6.1 The Big M Method 62
2.6.2 The Two-phase Simplex Method 67
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iv Contents
2.7 Variants of Simplex Method 74
2.8 Solution of Simultaneous Equations by Simplex Method 83
2.9 Inverse of a Matrix by Simplex Method 83
Exercises 85
3 Advanced Topics in Linear Programming 96
3.1 Duality Theory 96
3.1.1 The Dual Problem 96
3.1.2 Duality Theorems 100
3.1.3 Duality and Simplex Method 104
3.2 Dual Simplex Method 108
3.2.1 Introduction 108
3.2.2 Difference between Regular Simplex Method and Dual Simplex Method 108
3.2.3 Dual Simplex Algorithm 108
3.3 Revised Simplex Method 113
3.3.1 Introduction 113
3.3.2 Revised Simplex Algorithm 113
3.3.3 Advantages of the Revised Simplex Method Over the
Regular Simplex Method 119
3.4 Sensitivity Analysis 119
3.4.1 Introduction 119
3.4.2 Changes Affecting Optimality 124
3.5 Parametric Programming 130
3.5.1 Introduction 130
3.5.2 Parametric Cost Problem 130
3.5.3 Parametric Right Hand Side Problem 134
3.6 Goal Programming 137
3.6.1 Introduction 137
3.6.2 Concepts of Goal Programming 137
3.6.3 Formulation of Goal Programming Problem 138
3.6.4 Graphical Method of Goal Programming 145
3.7 Integer Programming 148
3.7.1 Introduction 148
3.7.2 Formulation of Integer Programming Problem 149
3.7.3 Gomory’s all IPP Method (or) Cutting Plane Method or
Gomory’s Fractional Cut Method 151
3.7.4 Branch and Bound Technique 158
3.8 Zero-one Programming 161
3.8.1 Introduction 161
3.8.2 Examples of Zero-one Programming Problems 162
3.8.3 Importance of Zero-one Integer Programming 162
3.8.4 Solution Methodologies 162
3.9 Limitations of Linear Programming Problem 166
3.9.1 Advantages of Linear Programming Problem 166
3.9.2 Limitations of Linear Programming 167
Exercises 167
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Contents v
4 The Transportation Problem 179
4.1 Introduction 179
4.2 Mathematical Formulation 179
4.2.1 Definitions 180
4.2.2 Theorem (Existence of Feasible Solution) 181
4.2.3 Theorem (Basic Feasible Solution) 182
4.2.4 Definition: Triangular Basis 183
4.2.5 Theorem 183
4.3 Methods for Finding Initial Basic Feasible Solution 184
4.3.1 First Method—North-West Corner Rule 184
4.3.2 Second Method—Least Cost or Matrix Minima Method 184
4.3.3 Third Method—Vogel’s Approximation Method (VAM) or
Unit Cost Penalty Method 185
4.3.4 Fourth Method—Row Minima Method 185
4.3.5 Fifth Method—Column Minima Method 185
4.4 Optimum Solution of a Transportation Problem 193
4.4.1 The Stepping-stone Method 195
4.4.2 MODI Method or the u -v Method 195
4.5 Degeneracy in Transportation Problem 203
4.5.1 Resolution of Degeneracy in the Initial Stage 203
4.5.2 Resolution of Degeneracy during Solution Stages 203
4.6 Unbalanced Transporation Problems 208
4.7 Maximisation in Transportation Problems 216
4.8 The Trans-shipment Problem 220
4.9 Sensitivity Analysis in Transportation Problem 222
4.10 Applications 224
Exercises 226
5 Assignment Problem 233
5.1 Introduction and Formulation 233
5.2 Hungarian Assignment Algorithm 234
5.2.1 Theorem 234
5.2.2 Theorem 235
5.2.3 Hungarian Assignment Algorithm 235
5.3 Variations of the Assignment Problem 242
5.4 Travelling Salesman Problem 249
5.4.1 Formulation of a Travelling-Salesman Problem as
Assignment Problem 250
Exercises 254
6 Dynamic Programming 259
6.1 Introduction 259
6.1.1 Need for Dynamic Programming Problem 261
6.1.2 Application of Dynamic Programming Problem 261
6.1.3 Characteristics of Dynamic Programming 262
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6.1.4 Definition 262
6.1.5 Dynamic Programming Algorithm 263
6.2 Some Dynamic Programming Techniques 265
6.2.1 Single Additive Constraint, Multiplicatively Separable Return 265
6.2.2 Single Additive Constraint, Additively Separable Return 266
6.2.3 Single Multiplicative Constraint, Additively Separable Return 267
6.2.4 Systems Involving More than One Constraint 267
6.2.5 Problems 268
6.3 Capital Budgeting Problem 284
6.4 Reliability Problem 286
6.5 Stage Coach Problem (Shortest-route Problem) 289
6.6 Solution of Linear Programming Problem by Dynamic Programming 292
Exercises 295
7 Decision Theory and Introduction to Quantitative Methods 301
7.1 Introduction to Decision Analysis 301
7.1.1 Steps in Decision Theory Approach 301
7.1.2 Decision-making Environments 303
7.2 Decision Under Uncertainty 303
7.2.1 Criterion of Pessimism (Minimax or Maximin) 303
7.2.2 Criterion of Optimism (Maximax or Minimin) 304
7.2.3 Laplace Criterion or Equally Likely Decision Criterion 304
7.2.4 Criterion of Realism (Hurwicz Criterion) 304
7.2.5 Criterion of Regret (Savage Criterion) or Minimax Regret Criterion 305
7.3 Decision Under Certainty 308
7.4 Decision-making Under Risk 308
7.4.1 Expected Monetary Value (EMV) Criterion 308
7.4.2 Expected Opportunity Loss (EOL) Criterion 310
7.4.3 Expected Value of Perfect Information (EVPI) 311
7.5 Decision Trees 316
7.6 Introduction to Quantitative Methods 319
7.6.1 Definition and Classification 319
7.6.2 Role of Quantitative Methods in Business and Industry 321
7.6.3 Quantitative Techniques and Business Management 321
7.6.4 Limitations of Quantitative Techniques 322
Exercises 322
8 Theory of Games 331
8.1 Introduction to Games 331
8.1.1 Some Basic Terminologies 331
8.2 Two-person Zero-sum Game 332
8.2.1 Games with Saddle Point 333
8.2.2 Games without Saddle Point: Mixed Strategies 336
8.2.3 Matrix Method 342
8.3 Graphical Method (for 2 × n or for m × 2 Games) 345
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Contents vii
8.4 Solution of m × n Size Games 351
8.5 n-Person Zero-sum Game 355
Exercises 357
9 Sequencing Models 367
9.1 Introduction and Basic Assumption 367
9.1.1 Definition 367
9.1.2 Terminology and Notations 368
9.1.3 Assumptions 368
9.1.4 Solution of Sequencing Problems 368
9.2 Flow Shop Scheduling 369
9.2.1 Characteristics of Flow Shop Scheduling Problem 369
9.2.2 Processing n Jobs Through Two Machines 369
9.2.3 Processing n Jobs Through 3 Machines 372
9.2.4 Processing n Jobs Through m Machines 374
9.3 Job Shop Scheduling 376
9.3.1 Difference between Flow Shop Scheduling and Job Shop Scheduling 376
9.3.2 Processing Two Jobs on n Machines 377
9.4 Gantt Chart 380
9.5 Shortest Cyclic Route Models (Travelling Salesmen Problem) 382
9.6 Shortest Acyclic Route Models (Minimal Path Problem) 383
Exercises 386
10 Replacement Models 396
10.1 Introduction 396
10.2 Replacement of Items that Deteriorates Gradually 397
10.2.1 Replacement Policy When Value of Money Does Not Change
with Time 397
10.2.2 Replacement Policy When Value of Money Changes with Time 403
10.3 Replacement of Items that Fail Completely and Suddenly 410
10.3.1 Theorem (Mortality) 410
10.3.2 Theorem (Group Replacement Policy) 413
10.4 Other Replacement Problems 418
10.4.1 Recruitment and Promotion Problems 418
Exercises 422
11 Inventory Models 427
11.1 Introduction 427
11.2 Cost Involved in Inventory Problems 430
11.3 EOQ Models 432
11.3.1 Economic Order Quantity (EOQ) 432
11.3.2 Determination of EOQ by Tabular Method 433
11.3.3 Determination of EOQ by Graphical Method 433
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11.3.4 Model I: Purchasing Model with No Shortages 434
11.3.5 Model II: Manufacturing Model with No Shortages 437
11.3.6 Model III: Purchasing Model with Shortages 441
11.3.7 Model IV: Manufacturing Model with Shortages 447
11.4 EOQ Problems with Price Breaks 451
11.5 Reorder Level and Optimum Buffer Stock 457
11.6 Probabilistic Inventory Models 463
11.6.1 Model V: Instantaneous Demand, No Setup Cost,
Stock in Discrete Units 463
11.6.2 Model VI: Instantaneous Demand and Continuous Units 466
11.6.3 Model VII: Uniform Demand, No Setup cost 468
11.6.4 Model VIII: Uniform Demand and Continuous Units 471
11.7 Selection Inventory Control Techniques 473
Exercises 478
12 Queuing Models 484
12.1 Characteristics of Queuing Models 485
12.2 Transient and Steady States 487
12.3 Role of Exponential Distribution 488
12.4 Kendall’s Notation for Representing Queuing Models 488
12.5 Classification of Queuing Models 488
12.6 Pure Birth and Death Models 489
12.6.1 Pure Birth Model 489
12.6.2 Pure Death Model 491
12.7 Model I: (M|M|1): (∞|FIFO) (Birth and Death Model) 491
12.7.1 Measures of Model I 492
12.7.2 Little’s Formula 493
12.8 Model II: Multi-service Model (M|M|s): (∞|FIFO) 498
12.8.1 Measures of Model II 500
12.9 Model III: (M/M/1): (N/FIFO) 505
12.10 Model IV: (M/M/s): (N/FIFO) 508
12.11 Non-Poisson Queues 510
12.12 Queuing Control 517
Exercises 517
13 Network Models 524
13.1 Introduction 524
13.1.1 Phases of Project Management 524
13.1.2 Differences Between PERT and CPM 525
13.2 Network Construction 525
13.2.1 Some Basic Definitions 525
13.2.2 Rules of Network Construction 528
13.2.3 Fulkerson’s Rule (i–j rule) of Numbering Events 529
13.3 Critical Path Method (CPM) 532
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13.3.1 Forward Pass Computation (for Earliest Event Time) 532
13.3.2 Backward Pass Computation (for Latest Allowable Time) 533
13.3.3 Computation of Float and Slack Time 533
13.3.4 Critical Path 535
13.4 Project Evaluation and Review Technique (PERT) 544
13.4.1 PERT Procedure 545
13.5 Resource Analysis in Network Scheduling 554
13.5.1 Time Cost Optimisation Algorithm (or) Time Cost Trade-off
Algorithm (or) Least Cost Schedule Algorithm 555
13.6 Resource Allocation and Scheduling 564
13.7 Application and Disadvantages of Networks 570
13.7.1 Application Areas of PERT/CPM Techniques 570
13.7.2 Uses of PERT/CPM for Management 571
13.7.3 Disadvantages of Network Techniques 571
13.8 Network Flow Problems 571
13.8.1 Introduction 571
13.8.2 Max-flow Min-cut Theorem 573
13.8.3 Enumeration of Cuts 573
13.8.4 Ford–Fulkerson Algorithm 574
13.8.5 Maximal Flow Algorithm 575
13.8.6 Linear Programming Modelling of Maximal Flow Problem 582
13.9 Spanning Tree Algorithms 583
13.9.1 Basic Terminologies 583
13.9.2 Some Applications of the Spanning Tree Algorithms 584
13.9.3 Algorithm for Minimum Spanning Tree 584
13.10 Shortest Route Problem 588
13.10.1 Shortest Path Model 588
Exercises 596
14 Simulation 630
14.1 Introduction 630
14.1.1 What is Simulation 631
14.1.2 Definitions of Simulation 631
14.1.3 Types of Simulation 632
14.1.4 Why to Use Simulation 632
14.1.5 Limitations of Simulation 633
14.1.6 Advantages of Simulation 634
14.1.7 Phases of Simulation Model 634
14.2 Event Type Simulation 634
14.3 Generation of Random Numbers (or) Digits 637
14.4 Monte-Carlo Method of Simulation 641
14.5 Applications to Queueing Problems 645
14.6 Applications to Inventory Problems 647
14.7 Applications to Capital Budgeting Problem 650
14.8 Applications to PERT Problems 652
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