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Operations Research Nita H. Shah Ravi M. Gor Hardik Soni Operations Research NITA H. SHAH Associate Professor, Department of Mathematics Gujarat University, Ahmedabad RAVI M. GOR Professor and Dean (Academics) St. Kabir Institute of Professional Studies, Ahmedabad HARDIK SONI Assistant Professor Chimanbhai Patel Institute of Computer Applications Gujarat University, Ahmedabad New Delhi-110001 2010 OPERATIONS RESEARCH Nita H. Shah, Ravi M. Gor, and Hardik Soni © 2007 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-3128-0 The export rights of this book are vested solely with the publisher. Fourth Printing ººººº ººººº ººººº February, 2010 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Mudrak, 30-A, Patparganj, Delhi-110091. Contents Preface xi 1. WHY OPERATIONS RESEARCH? 1–16 1.1 Introduction 1 1.2 Origin of Operations Research 2 1.3 Definitions of Operations Research 2 1.4 Characteristics of Operations Research 4 1.5 Models in Operations Research 5 1.5.1 What are Models? 5 1.5.2 Classifications of Models 6 1.5.3 Characteristics of a Good Model 8 1.5.4 Steps in Constructing a Model 8 1.5.5 Points to Remember while Building a Model 8 1.5.6 Advantages of a Good Model 9 1.5.7 Limitations of a Model 9 1.5.8 Quantitative Methods in Practice 9 1.6 Operations Research—An Approach to Decision-Making 11 1.7 Role of Operations Research in Decision-Making 12 1.8 Methods of Solving Operations Research Problems 12 1.9 Phases in Solving Operations Research Problems 13 1.10 Typical Problems in Operations Research 13 1.11 Scope of Operations Research 14 1.12 Why To Study Operations Research? 15 Review Exercises 16 iii iv ®®®®® Contents 2. PREREQUISITE FOR OPERATIONS RESEARCH 17–32 2.1 Introduction 17 2.2 Matrices and Determinants 17 2.2.1 Definitions 17 2.2.2 Algebra of Matrices 19 2.2.3 Determinant of Square Matrix 20 2.2.4 Adjoint of Matrix 21 2.2.5 Inverse of Matrix 21 2.2.6 Rank of Matrix 22 2.3 System of Linear Equations and Consistency 23 2.4 Vectors and Convexity 24 2.4.1 Definitions 24 2.4.2 Convex Sets 25 2.4.3 Constructing a Convex Set 27 2.4.4 Hyperplanes 29 2.4.5 Supporting and Separating Hyperplane 30 2.5 Probability and Its Fundamentals 31 2.5.1 Definitions 31 3. LINEAR PROGRAMMING 33–124 3.1 Introduction 33 3.1.1 Model Components 34 3.1.2 Properties of Linear Programming Models 34 3.2 Steps of Formulating Linear Programming Problem (LPP) 35 3.3 General Form of LPP 45 3.3.1 LPP in Canonical Form 46 3.4 Graphical Method 50 3.4.1 Extreme Point Approach 50 3.4.2 Iso-profit (cost) Function Line Approach 51 3.5 Special Cases in LP 55 3.5.1 Alternative (or Multiple) Optimal Solution 55 3.5.2 An Unbounded Solution 56 3.5.3 Infeasible Solution 57 3.5.4 Redundant Constraint 58 3.6 Simplex Method 58 3.7 Minimization Case 71 3.7.1 Two-Phase Method 71 3.7.2 Big-M Method 79 3.8 Degeneracy in LP 85 3.9 Duality in LPP 89 3.9.1 Duality Theorems 98 3.9.2 Advantages of Duality 100 3.9.3 Dual Simplex Method 100 3.10 Revised Simplex Method 106 3.10.1 Standard Form I 107 3.10.2 Standard Form II 109 3.11 Post Optimality Analysis 113 3.11.1 Variations in b 113 3.11.2 Variations in c 115 Review Exercises 117 Contents ®®®®® v 4. INTEGER PROGRAMMING 125–147 4.1 Introduction 125 4.2 Forms of Integer Programming Problems (IPP) 126 4.3 Gomory’s Cutting Plane 126 4.3.1 Gomory’s All–Integer Cutting Plane Method 127 4.3.2 Gomory’s Mixed—Integer Cutting Plane Method 136 4.4 Branch and Bound (B&B) Method 140 Review Exercises 147 5. GOAL PROGRAMMING 148–155 5.1 Introduction 148 5.2 Goal Programming Model Formulation 149 5.2.1 Single Goal with Multiple Sub-goals 149 5.2.2 Multiple Goals with Equal Weightage 150 5.2.3 Goal Programming with Weighted Goals 152 Review Exercises 154 6. NON-LINEAR PROGRAMMING 156–187 6.1 Introduction 156 6.2 Prerequisites 157 6.2.1 Maxima and Minima of Functions and Their Solutions 157 6.2.2 Quadratic Forms 158 6.2.3 Convex and Concave Functions 161 6.3 Non-Linear Programming Problems 163 6.4 Unconstrained Optimization 165 6.4.1 Functions with Single Variables 165 6.4.2 Multi Variable Functions 166 6.5 Constrained Optimization 168 6.5.1 Equality Constraints 168 6.5.2 Inequality Constraints 171 6.6 Quadratic Programming 179 6.7 Wolfe’s Method 181 Review Exercises 187 7. GEOMETRIC PROGRAMMING 188–194 7.1 Geometric Programming 188 7.2 Primal Geometric Programming with Equality Constraints 192 Review Exercises 194 8. TRANSPORTATION PROBLEM 195–240 8.1 Introduction 195 8.2 Formulation of a General Transportation Problem 196 8.2.1 Matrix Form of a TP 197 8.3 Types of Transportation Problem 198 8.4 Some Theorems 199 8.4.1 Triangular Basis 202 vi ®®®®® Contents 8.5 Solving the Transportation Problem (Finding Initial Basic Feasible Solution) 203 8.5.1 Why Using Simplex Method to Solve a TP is Unwise? 204 8.5.2 The North-West Corner Method (NWCM) 204 8.5.3 The Least-Cost (Matrix Minimum) Method (LCM) 208 8.5.4 Vogel’s Approximation Method (VAM)—Penalty Method 211 8.6 Loops in a Transportation Method 214 8.7 Optimality in a Transportation Problem 215 8.7.1 Dual of a Transportation Problem 216 8.8 Transportation Algorithm: Modified Distribution (MODI) Method 219 8.9 Stepping Stone Method 224 8.10 Variations of a Transportation Problem 226 8.10.1 Maximization Transportation Problem 226 8.10.2 Alternative Optimal Solutions 226 8.10.3 Infeasible Transportation Problem 226 8.10.4 Degeneracy in a Transportation Problem 226 8.11 Trans-shipment Problem 232 8.11.1 Sources and Destinations Acting as Transient Nodes 232 8.11.2 Some Transient Nodes between Sources and Destinations 234 Review Exercises 236 9. ASSIGNMENT PROBLEM 241–259 9.1 Introduction 241 9.2 Mathematical Formulation of the AP 242 9.3 Solution Methods of AP 243 9.3.1 Enumeration Method 243 9.3.2 Simplex Method 244 9.3.3 Transportation Method 244 9.3.4 The Hungarian Method 244 9.4 Variations of the Assignment Problem 249 9.4.1 Multiple Optimal Solutions 249 9.4.2 Unbalanced Assignment Problems 249 9.4.3 Problem with Infeasible (Restricted) Assignment 249 9.4.4 Maximization Case in Assignment Problems 249 Review Exercises 256 10. DECISION ANALYSIS 260–299 10.1 Introduction 260 10.2 Characteristics of a Decision Problem 261 10.3 Pay-Off Table 262 10.4 The Different Environments in Which Decisions are Made 262 10.5 Constructing a Regret Table from Profit Table 266 10.6 Expected Value 267 10.7 Expected Value Criterion for Decision Making under Risk 267 10.8 Method of Marginal Probabilities 273 10.9 Decision Trees 275 10.10 Valuing Imperfect Information (Use of Bayes’ Theorem) 286 Review Exercises 293 Contents ®®®®® vii 11. INVENTORY PROBLEMS 300–334 11.1 Introduction 300 11.2 Types of Inventory 300 11.3 Costs Involved in Inventory Problems 301 11.4 Notations 302 11.5 Economic Order Quantity (EOQ) Model with Constant Rate of Demand 303 11.6 Limitations of the EOQ Formula 306 11.7 EOQ Model with Finite Replenishment Rate 308 11.8 EOQ Model with Shortages 311 11.9 Order-Level, Lot-Size System 313 11.10 Order-Level Lot-Size System with Finite Replenishment Rate 316 11.11 Several Items Inventory Model with Constraints 318 11.11.1 EOQ Model with Floor Space Constraint 319 11.11.2 EOQ Model with Average Inventory Level Constraint 320 11.11.3 EOQ Model with Investment Constraint 321 11.12 EOQ Model with Quantity Discounts 322 11.12.1 EOQ with One-Price Break 323 11.12.2 EOQ with Two-Price Breaks 324 11.13 Probabilistic Order-Level System 326 11.14 Probabilistic Order-Level System with Instantaneous Demand 328 Review Exercises 331 12. QUEUING THEORY 335–371 12.1 Introduction 335 12.2 Queuing System 335 12.3 Classification of Queuing Models 337 12.4 Distribution of Arrivals (The Poisson Process): Pure Birth Process 338 12.5 Distribution of Inter-arrival Time 341 12.6 Distribution of Departures (Pure Death Process) 342 12.7 Distribution of Service Time 343 12.8 Solution of Queuing Models 344 12.9 Model 1 (M/M/1): (•/FCFS): Birth and Death Model 344 12.10 Model 2 (M/M/1): (N/FCFS) 351 12.11 Model 3 (M/M/C): (•/FCFS) 354 12.12 Model 4 (M/M/C): (N/FCFS) 357 12.13 Model 5 (M/M/1): (R/GD) Single Server, Finite Source of Arrivals 360 12.14 Model 6 (M/M/C): (R/GD): Multi Server–Finite Input Source 361 12.15 Model 7 (M/E /1): (•/FCFS) Erlang Service Time Distribution with K k-phases 363 Review Exercises 367 13. REPLACEMENT MODELS 372–400 13.1 Introduction 372 13.2 Failure of Items 372 13.3 Replacement of Items That Deteriorate 373 viii ®®®®® Contents 13.4 Replacement of Items with Increasing Running Cost 380 13.5 Replacement of Items That Fail Completely 386 13.6 Group Replacement Policy 388 13.7 Recruitment and Promotional Problems 392 13.8 Equipment Renewal Problem 395 Review Exercises 398 14. DYNAMIC PROGRAMMING 401–423 14.1 Introduction 401 14.2 Components of Dynamic Programming 401 14.3 Computational Algorithm 402 14.4 Shortest Route Problem 402 14.5 Single Additive Constraint, Multiplicative Separable Return Function 405 14.6 Single Additive Constraint, Additive Separable Return Function 408 14.7 Single Multiplicative Constraint, Additive Separable Return Function 415 14.8 Solution of Linear Programming Problem 415 14.9 Some Applications 417 Review Exercises 420 15. PROJECT MANAGEMENT 424–469 15.1 Introduction 424 15.1.1 Project Planning 425 15.1.2 Project Scheduling 425 15.1.3 Project Controlling 426 15.2 Origin and Use of PERT 426 15.3 Origin and Use of CPM 426 15.4 Applications of PERT and CPM 427 15.5 Framework of PERT and CPM 427 15.6 Constructing the Project Network 428 15.7 Dummy Activities and Events 430 15.8 Rules for Network Construction 431 15.9 Finding the Critical Path 432 15.9.1 Floats 438 15.10 Project Evaluation and Review Technique (PERT) 444 15.11 PERT/Cost Analysis 450 15.12 Cost and Networks—Basic Definitions 452 15.13 Least Cost Scheduling Rules 452 Review Exercises 463 16. SEQUENCING 470–485 16.1 Introduction 470 16.2 Notations and Terminologies 470 16.2.1 Notations 470 16.2.2 Terminologies 470 Contents ®®®®® ix 16.3 Principal Assumptions 471 16.4 Sequencing Rules 471 16.5 Sequencing Jobs Through One Process 472 16.6 Sequencing Jobs Through Two Serial Process 474 16.7 Johnson’s Algorithm 475 16.8 Processing n Jobs Through Three Machines 478 16.9 Processing n Jobs Through m Machines 480 16.10 Scope of Sequencing 481 16.10.1 What is Scheduling? 482 Review Exercises 482 17. SIMULATION 486–504 17.1 Introduction 486 17.2 Steps Involved in Simulation 486 17.3 Advantages and Disadvantages of Simulation 487 17.4 Monte Carlo Simulation 488 17.5 Applications of Simulation 488 Review Exercises 503 18. GAME THEORY 505–553 18.1 Introduction 505 18.2 Two Person Zero-Sum Games 507 18.3 Maximin and Minimax Principles 508 18.4 Mixed Strategies, Expected Pay-Off 511 18.5 Solution of 2 ¥ 2 Mixed Strategy Game 515 18.6 Solution of 2 ¥ 2 Mixed Strategy Game by the Method of Oddments 517 18.7 Dominance Principle 519 18.8 Solution of Game by Matrix Method 522 18.9 Solution of a Two Person Zero-Sum 2 ¥ n Game 524 18.10 Graphical Method for Solving a 2 ¥ n or m ¥ 2 Game 525 18.11 Linear Programming Method for the Solutions of Game 532 18.12 Algebraic Method for Solving a Game 539 18.13 Solution of 3 ¥ 3 Games with Mixed Strategy by the Method of Oddments 542 18.14 Iterative Method for Approximate Solution 543 18.15 Summary of the Procedure to Solve a Game 546 Review Exercises 548 Appendix: STATISTICAL TABLES 555–559 INDEX 561–563