OPERATIONS RESEARCH SECOND EDITION R. PANNEERSELVAM Professor and Head Department of Management Studies School of Management Pondicherry University NEW DELHI-110001 2009
OPERATIONS RESEARCH, Second Edition R. Panneerselvam 2006 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-978-81-203-2928-7 The export rights of this book are vested solely with the publisher. Fifteenth Printing (Second Edition)......... July 2009 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Syndicate Binders, A-20, Hosiery Complex, Noida, Phase-II Extension, Noida-201305 (N.C.R. Delhi).
To My Grandparents
CONTENTS Preface xi 1. OVERVIEW OF OPERATIONS RESEARCH 1 8 1.1 Introduction 1 1.2 Concept of a Model 2 1.2.1 Steps of Modelling 3 1.3 Important Topics of Operations Research 3 1.4 Scope of Operations Research 4 1.5 Operations Research A Tool for Decision Support System 6 1.6 Operations Research A Productivity Improvement Tool 7 1.6.1 Increased Output for the Same Input 7 1.6.2 Decreased Input for the Same Output 7 1.6.3 Increase in the Output is more than the Increase in the Input 7 1.6.4 Decrease in the Input is more than the Decrease in the Output 8 1.6.5 Increase in the Output with Decrease in the Input 8 Questions 8 2. LINEAR PROGRAMMING 9 70 2.1 Introduction 9 2.2 Concept of Linear Programming Model 9 2.2.1 Product Mix Problem 9 2.2.2 Assumptions in Linear Programming 12 2.2.3 Properties of Linear Programming Solution 12 2.3 Development of LP Models 13 2.4 Graphical Method 21 2.5 Linear Programming Methods 25 2.5.1 Simplex Method 25 2.5.2 Big M Method 30 2.5.3 Dual Simplex Method 34 2.5.4 Two-phase Method 38 2.6 Special Cases of Linear Programming 40 2.6.1 Identification of Special Cases from Simplex Table 46 2.7 Duality 47 2.7.1 Formulation of Dual Problem 47 2.7.2 Application of Duality 53 2.8 Sensitivity Analysis 56 2.8.1 Changes in the Right-hand Side Constants of Constraints 57 v
vi Contents 2.8.2 Changes in the Objective Function Coefficients 59 2.8.3 Adding a New Constraint 61 2.8.4 Adding a New Variable 63 Questions 65 3. TRANSPORTATION PROBLEM 71 126 3.1 Introduction 71 3.2 Mathematical Model for Transportation Problem 72 3.3 Types of Transportation Problem 74 3.3.1 Balanced Transportation Problem 74 3.3.2 Unbalanced Transportation Problem 74 3.4 Methods to Solve Transportation Problem 76 3.4.1 Finding the Initial Basic Solution 77 3.4.2 Optimizing the Basic Feasible Solution Applying U V Method 78 3.5 Transshipment Model 106 3.5.1 Transshipment Problem with Sources and Destinations Acting as Transient Nodes 106 3.5.2 Transportation Problem with some Transient Nodes between Sources and Destinations 111 3.6 Modelling the Transportation Problem with Quantity Discounts 115 3.6.1 Model for AQDS 117 3.6.2 Model for IQDS 118 Questions 122 4. ASSIGNMENT PROBLEM 127 153 4.1 Introduction 127 4.2 Zero-One Programming Model for Assignment Problem 128 4.3 Types of Assignment Problem 130 4.4 Hungerian Method 130 4.5 Branch-and-Bound Technique for Assignment Problem 147 Questions 151 5. NETWORK TECHNIQUES 154 195 5.1 Introduction 154 5.2 Shortest-path Model 154 5.2.1 Systematic Method 154 5.2.2 Dijkstra s Algorithm 163 5.2.3 Floyd s Algorithm 171 5.3 Minimum Spanning Tree Problem 174 5.3.1 PRIM Algorithm 175 5.3.2 Kruskal s Algorithm 181 5.4 Maximal Flow Problem 186 5.4.1 Linear Programming Modelling of Maximal Flow Problem 186 5.4.2 Maximal Flow Problem (MFP) Algorithm 188 Questions 193
Contents vii 6. INTEGER PROGRAMMING 196 229 6.1 Introduction 196 6.2 Integer Programming Formulations 197 6.3 The Cutting-plane Algorithm 201 6.4 Branch-and-Bound Technique 209 6.5 Zero-One Implicit Enumeration Algorithm 219 6.5.1 Generalized 0-1 Programming Problem 221 6.5.2 Zero-One Implicit Enumeration Technique 221 Questions 228 7. INVENTORY CONTROL 230 274 7.1 Introduction 230 7.2 Models of Inventory 230 7.2.1 Purchase Model with Instantaneous Replenishment and without Shortages 231 7.2.2 Manufacturing Model without Shortages 236 7.2.3 Purchase Model with Instantaneous Replenishment and with Shortages 238 7.2.4 Manufacturing Model with Shortages 241 7.3 Operation of Inventory System 245 7.4 Quantity Discount 247 7.5 Implementation of Purchase Inventory Model 250 7.5.1 Fixed Order Quantity System (Q System) 250 7.5.2 Periodic Review System (P System) 251 7.6 Multiple-item Model with Shortage Limitation 254 7.7 Purchase Model of Inventory for Multi-item with Inventory Carrying Cost Constraint 256 7.8 EOQ Model for Multi-item Joint Replenishment 258 7.8.1 Purchase Model of Inventory for Multi-item Joint Replenishment without Shortages 259 7.8.2 Manufacturing Model of Inventory with Multi-item Joint Replenishment without Shortages 262 7.9 EOQ for the Purchase Model of Inventory for Multi-item Joint Replenishment with Space Constraint 265 7.10 Determination of Stock Level of Perishable Items under Probabilistic Condition 269 Questions 272 8. DYNAMIC PROGRAMMING 275 297 8.1 Introduction 275 8.2 Application of Dynamic Programming 276 8.2.1 Capital Budgeting Problem 276 8.2.2 Reliability Improvement Problem 278 8.2.3 Stage-coach Problem (Shortest-path Problem) 281 8.2.4 Cargo Leading Problem 284 8.2.5 Minimizing Total Tardiness in Single Machine Scheduling Problem 286 8.2.6 Optimal Subdividing Problem 290
viii Contents 8.2.7 Solution of Linear Programming Problem through Dynamic Programming 292 Questions 295 9. QUEUEING THEORY 298 354 9.1 Introduction 298 9.2 Terminologies of Queueing System 299 9.3 Empirical Queueing Models 300 9.3.1 (M/M/1) : (GD/ / ) Model 301 9.3.2 (M/M/C) : (GD/ / ) Model 305 9.3.3 (M/M/1) : (GD/N/ ) Model 309 9.3.4 (M/M/C) : (GD/N/ ) Model (for C N) 313 9.3.5 (M/M/C) : (GD/N/N) Model (for C < N) 318 9.3.6 (M/M/1) : (GD/N/N) Model (for N > 1) 322 9.4 Simulation 325 9.4.1 Need for Simulation 325 9.4.2 Types of Simulation 326 9.4.3 Major Steps of Simulation 327 9.4.4 Simulation using High-level Languages 327 9.4.5 General Purpose Simulation System (GPSS) 337 Questions 352 10. PROJECT MANAGEMENT 355 408 10.1 Introduction 355 10.2 Phases of Project Management 358 10.3 Guidelines for Network Construction 359 10.4 Critical Path Method (CPM) 359 10.5 Gantt Chart (Time Chart) 365 10.6 Project Evaluation and Review Technique (PERT) 368 10.7 Crashing of Project Network 375 10.7.1 General Guidelines for Network Crashing 376 10.7.2 Crashing of Project Network with Cost Trade-off 377 10.8 Project Scheduling with Constrained Resources 390 10.8.1 Resource Levelling Technique 390 10.8.2 Resource Allocation Technique 398 Questions 404 11. DECISION THEORY 409 423 11.1 Introduction 409 11.2 Decision under Certainty (Deterministic Decision) 409 11.3 Decision under Risk 409 11.3.1 Expected Value Criterion 410 11.3.2 Expected Value Combined with Variance Criterion 411 11.4 Decision under Uncertainty 411 11.4.1 Laplace Criterion 412 11.4.2 Maximin Criterion 413
Contents ix 11.4.3 Minimax Criterion 414 11.4.4 Savage Minimax Regret Criterion 414 11.4.5 Hurwicz Criterion 416 11.5 Decision Tree 417 Questions 421 12. GAME THEORY 424 470 12.1 Introduction 424 12.1.1 Terminologies of Game Theory 424 12.2 Game with Pure Strategies 426 12.3 Game with Mixed Strategies 428 12.4 Dominance Property 430 12.5 Graphical Method for 2 n or m 2 Games 440 12.6 Linear Programming Approach for Game Theory 453 Questions 467 13. REPLACEMENT AND MAINTENANCE ANALYSIS 471 493 13.1 Introduction 471 13.2 Types of Maintenance 471 13.3 Types of Replacement Problem 472 13.4 Determination of Economic Life of an Asset 472 13.4.1 Basics of Interest Formulae 473 13.4.2 Examples of Determination of Economic Life of an Asset 475 13.5 Simple Probabilistic Model for Items which Completely Fail 485 Questions 492 14. PRODUCTION SCHEDULING 494 537 14.1 Introduction 494 14.2 Single-machine Scheduling 494 14.2.1 Measures of Performance 495 14.2.2 Shortest Processing Time (SPT) Rule to Minimize Mean Flow Time 496 14.2.3 Weighted Shortest Processing Time (WSPT) Rule to Minimize Weighted Mean Flow Time 497 14.2.4 Earliest Due Date (EDD) Rule to Minimize Maximum Lateness 498 14.2.5 Model to Minimize Total Tardiness 499 14.2.6 Introduction to Branch-and-Bound Technique to Minimize Mean Tardiness 502 14.2.7 Model to Minimize Sum of Weighted Number of Early and Tardy Jobs 512 14.3 Flow Shop Scheduling 515 14.3.1 Johnson s Algorithm for n Jobs and Two Machines Problem 517 14.3.2 Extension of Johnson s Algorithm for n Jobs and Three Machines Problem 519 14.3.3 Branch-and-Bound Method for n Jobs and m Machines 521
x Contents 14.4 Job Shop Scheduling 532 14.4.1 Two Jobs and m Machines Job Shop Scheduling 533 Questions 535 15. GOAL PROGRAMMING 538 548 15.1 Introduction 538 15.2 Simplex Method for Solving Goal Programming 541 Questions 547 16. PARAMETRIC LINEAR PROGRAMMING 549 561 16.1 Introduction 549 16.2 Changes in Objective Function Coefficients (C j Values) 550 16.3 Changes in Right-hand Side Constants (B i Values) of Constraints 554 16.4 Introduction to Changes in Resource Requirements Vector(s), P j 559 Questions 560 17. NONLINEAR PROGRAMMING 562 590 17.1 Introduction 562 17.2 Lagrangean Method 562 17.3 Kuhn Tucker Conditions 569 17.4 Quadratic Programming 572 17.5 Separable Programming 581 17.6 Chance-constrained Programming or Stochastic Programming 585 Questions 589 Appendix 591 Suggested Further Reading 593 594 Answers to Questions 595 604 Index 605 608
PREFACE An organizational system consists of various subsystems. The most ideal approach to optimize the performance of a system is to consider different subsystems as an integrated single unit. In some reality, integrating all the subsystems as a single unit will make the problem-solving process more complex, because of its size and different constraints. Under such situation, it is inevitable to optimize the performance of each subsystem. Operations research consists of topics to achieve each of these objectives depending on the reality. Based on the feedback from academicians, I have revised this book in the following lines. l l l l Inclusion of quantity discount models for transportation problem. Inclusion of more worked-out examples in many chapters. This will help the students to have enhanced understanding of the concepts and techniques, which are discussed in different chapters. Inclusion of additional topics in dynamic programming and inventory control. Inclusion of chapter-end questions for the additional topics, which are included in this edition. The quantity discount in transportation problem can be classified into all quantity discount scheme (AQDS) and incremental quantity discount scheme (IQDS). A mathematical model and a numerical illustration for each of these two quantity discount schemes are presented at the end of the chapter on transportation problem. The chapter on dynamic programming contains an additional topic on minimizing total tardiness in single machine scheduling problem. Here, the single machine scheduling problem is mapped in such a way that the dynamic programming technique is applied to it for minimizing the total tardiness. Under inventory control, the following topics have been included: l l l l Multiple-item model with storage limitation. Purchase model of inventory for multi-item with inventory carrying cost constrains. EOQ model for multi-item joint replenishment without shortages for purchase model of inventory as well as for manufacturing model of inventory. EOQ for the purchase model of inventory for multi-item joint replenishment with space constraint. The methods/models in each of the additional topics are illustrated with suitable worked-out examples. xi
xii Preface I express my profound gratitude and appreciation to academic colleagues who gave valuable feedback about the first edition of this text which helped me to improve its content. My heartfelt thanks are due to the editorial and production teams of Prentice-Hall of India for their meticulous processing of the manuscript of this second edition. Any suggestions to further improve the contents of this edition would be warmly appreciated. R. PANNEERSELVAM
OVERVIEW OF OPERATIONS RESEARCH 1 1.1 INTRODUCTION Operations research is a scientific approach to problem solving for executive decision making which requires the formulation of mathematical, economic and statistical models for decision and control problems to deal with situations arising out of risk and uncertainty. In fact, decision and control problems in any organization are more often related to certain daily operations such as inventory control, production scheduling, manpower planning and distribution, and maintenance. According to Operations Research Society of America (ORSA), it is a tool which is concerned with the design and operation of the man-machine system scientifically, usually under conditions requiring the optimum allocation of limited resources. As per the Operations Research Society of Great Britain, operations research is the application of the scientific methods to complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business and government. The origin and development of operations research can be studied under the following classification: 1. Pre-World War II developments 2. Developments during World War II 3. Post-World War II developments 4. Computer era 5. Inclusion of uncertainty models. Pre-World War II developments Many of the techniques of today s operations research have been actually developed and used even before the term operations research was coined. Some of the techniques are: inventory control, queueing theory, and statistical quality control. In 1915, Ford Harris developed a simple EOQ (economic order quantity) model to optimize the total cost of inventory system, which was eventually analyzed in 1934 by R.H. Wilson. Around the same time (1916), A.K. Erlang, a Danish telephone engineer, was responsible for many of the early theoretical developments in the area of queueing theory. In the early 1900s, routine quality checks conducted by inspectors were not found to be satisfactory for some companies. The problem was analyzed in the inspection engineering department of Western Electric s Bell Laboratory by Shewhart who ultimately designed control charts in 1924. These are called as the first Shewhart control charts. During the period 1925 26, the Western Electric Company defined various terminologies associated with acceptance sampling of 1
2 Operations Research quality control that was used as a tool for controlling attributes of raw materials/components/ finished products. The terminologies include consumer s risk, producer s risk, probability of acceptance, operating characteristics (OC) curve, lot tolerance percent defective (LTPD), double sampling plan, type I error, type II error and so on. In 1925, Dodge introduced the basic concept of sampling inspection. Ten years later, Pearson developed the British Standard Institution Number 600, entitled Application of statistical method to international standardization and quality control. In 1939, H. Roming presented his work on variable sampling plan in his Ph.D. dissertation. Developments during World War II During the World War II, the effective utilization of scarce resources was the top-most concern of the military in Britain. So, in Britain, scientists from different fields were jointly directed to do research on military operations for improving its effectiveness with the limited resources. Later on, this scientific and interdisciplinary approach became an important problem-solving aspect of operations research methodologies. Post-World War II developments After the World War II, the industries in America and Britain concentrated in applying the operations research methodologies to industrial problems for maximizing the profitability with limited resources. In 1947, Dantzig, developed simplex method to solve linear programming problem. Thereafter the Operations Research Society of America, and the Institute of Management Science were founded in 1952 and 1953, respectively. Computer era Many of the operations research techniques involve complex computations and hence they take longer time for providing solutions to real life problems. The developments of high speed digital computers made it possible to successfully apply some of the operations research techniques to large size problems. The developments of recent interactive computers make the job of solving large size problems even more simple because of human intervention towards sensitivity analysis. Inclusion of uncertainty models The use of probability theory and statistics to tackle undeterministic situations made the operations research techniques more realistic. 1.2 CONCEPT OF A MODEL Model is an abstraction of reality. Some examples of models are road map of a city to trace the shortest route from a given source to a given destination, three-dimensional view of a factory to plan the material movements in its shop floor, electrical network to compute the current flow in a particular arc, and linear equation to forecast the demand of a product. An operations research model is defined as an idealized (simplified) representation of a real-life system. Operations research uses a number of models to obtain solutions of various realistic problems.
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