OPERATIONS RESEARCH SECOND EDITION. R. PANNEERSELVAM Professor and Head Department of Management Studies School of Management Pondicherry University

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2 OPERATIONS RESEARCH SECOND EDITION R. PANNEERSELVAM Professor and Head Department of Management Studies School of Management Pondicherry University NEW DELHI

3 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 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 and Printed by Syndicate Binders, A-20, Hosiery Complex, Noida, Phase-II Extension, Noida (N.C.R. Delhi).

4 To My Grandparents

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6 CONTENTS Preface xi 1. OVERVIEW OF OPERATIONS RESEARCH Introduction Concept of a Model Steps of Modelling Important Topics of Operations Research Scope of Operations Research Operations Research A Tool for Decision Support System Operations Research A Productivity Improvement Tool Increased Output for the Same Input Decreased Input for the Same Output Increase in the Output is more than the Increase in the Input Decrease in the Input is more than the Decrease in the Output Increase in the Output with Decrease in the Input 8 Questions 8 2. LINEAR PROGRAMMING Introduction Concept of Linear Programming Model Product Mix Problem Assumptions in Linear Programming Properties of Linear Programming Solution Development of LP Models Graphical Method Linear Programming Methods Simplex Method Big M Method Dual Simplex Method Two-phase Method Special Cases of Linear Programming Identification of Special Cases from Simplex Table Duality Formulation of Dual Problem Application of Duality Sensitivity Analysis Changes in the Right-hand Side Constants of Constraints 57 v

7 vi Contents Changes in the Objective Function Coefficients Adding a New Constraint Adding a New Variable 63 Questions TRANSPORTATION PROBLEM Introduction Mathematical Model for Transportation Problem Types of Transportation Problem Balanced Transportation Problem Unbalanced Transportation Problem Methods to Solve Transportation Problem Finding the Initial Basic Solution Optimizing the Basic Feasible Solution Applying U V Method Transshipment Model Transshipment Problem with Sources and Destinations Acting as Transient Nodes Transportation Problem with some Transient Nodes between Sources and Destinations Modelling the Transportation Problem with Quantity Discounts Model for AQDS Model for IQDS 118 Questions ASSIGNMENT PROBLEM Introduction Zero-One Programming Model for Assignment Problem Types of Assignment Problem Hungerian Method Branch-and-Bound Technique for Assignment Problem 147 Questions NETWORK TECHNIQUES Introduction Shortest-path Model Systematic Method Dijkstra s Algorithm Floyd s Algorithm Minimum Spanning Tree Problem PRIM Algorithm Kruskal s Algorithm Maximal Flow Problem Linear Programming Modelling of Maximal Flow Problem Maximal Flow Problem (MFP) Algorithm 188 Questions 193

8 Contents vii 6. INTEGER PROGRAMMING Introduction Integer Programming Formulations The Cutting-plane Algorithm Branch-and-Bound Technique Zero-One Implicit Enumeration Algorithm Generalized 0-1 Programming Problem Zero-One Implicit Enumeration Technique 221 Questions INVENTORY CONTROL Introduction Models of Inventory Purchase Model with Instantaneous Replenishment and without Shortages Manufacturing Model without Shortages Purchase Model with Instantaneous Replenishment and with Shortages Manufacturing Model with Shortages Operation of Inventory System Quantity Discount Implementation of Purchase Inventory Model Fixed Order Quantity System (Q System) Periodic Review System (P System) Multiple-item Model with Shortage Limitation Purchase Model of Inventory for Multi-item with Inventory Carrying Cost Constraint EOQ Model for Multi-item Joint Replenishment Purchase Model of Inventory for Multi-item Joint Replenishment without Shortages Manufacturing Model of Inventory with Multi-item Joint Replenishment without Shortages EOQ for the Purchase Model of Inventory for Multi-item Joint Replenishment with Space Constraint Determination of Stock Level of Perishable Items under Probabilistic Condition 269 Questions DYNAMIC PROGRAMMING Introduction Application of Dynamic Programming Capital Budgeting Problem Reliability Improvement Problem Stage-coach Problem (Shortest-path Problem) Cargo Leading Problem Minimizing Total Tardiness in Single Machine Scheduling Problem Optimal Subdividing Problem 290

9 viii Contents Solution of Linear Programming Problem through Dynamic Programming 292 Questions QUEUEING THEORY Introduction Terminologies of Queueing System Empirical Queueing Models (M/M/1) : (GD/ / ) Model (M/M/C) : (GD/ / ) Model (M/M/1) : (GD/N/ ) Model (M/M/C) : (GD/N/ ) Model (for C N) (M/M/C) : (GD/N/N) Model (for C < N) (M/M/1) : (GD/N/N) Model (for N > 1) Simulation Need for Simulation Types of Simulation Major Steps of Simulation Simulation using High-level Languages General Purpose Simulation System (GPSS) 337 Questions PROJECT MANAGEMENT Introduction Phases of Project Management Guidelines for Network Construction Critical Path Method (CPM) Gantt Chart (Time Chart) Project Evaluation and Review Technique (PERT) Crashing of Project Network General Guidelines for Network Crashing Crashing of Project Network with Cost Trade-off Project Scheduling with Constrained Resources Resource Levelling Technique Resource Allocation Technique 398 Questions DECISION THEORY Introduction Decision under Certainty (Deterministic Decision) Decision under Risk Expected Value Criterion Expected Value Combined with Variance Criterion Decision under Uncertainty Laplace Criterion Maximin Criterion 413

10 Contents ix Minimax Criterion Savage Minimax Regret Criterion Hurwicz Criterion Decision Tree 417 Questions GAME THEORY Introduction Terminologies of Game Theory Game with Pure Strategies Game with Mixed Strategies Dominance Property Graphical Method for 2 n or m 2 Games Linear Programming Approach for Game Theory 453 Questions REPLACEMENT AND MAINTENANCE ANALYSIS Introduction Types of Maintenance Types of Replacement Problem Determination of Economic Life of an Asset Basics of Interest Formulae Examples of Determination of Economic Life of an Asset Simple Probabilistic Model for Items which Completely Fail 485 Questions PRODUCTION SCHEDULING Introduction Single-machine Scheduling Measures of Performance Shortest Processing Time (SPT) Rule to Minimize Mean Flow Time Weighted Shortest Processing Time (WSPT) Rule to Minimize Weighted Mean Flow Time Earliest Due Date (EDD) Rule to Minimize Maximum Lateness Model to Minimize Total Tardiness Introduction to Branch-and-Bound Technique to Minimize Mean Tardiness Model to Minimize Sum of Weighted Number of Early and Tardy Jobs Flow Shop Scheduling Johnson s Algorithm for n Jobs and Two Machines Problem Extension of Johnson s Algorithm for n Jobs and Three Machines Problem Branch-and-Bound Method for n Jobs and m Machines 521

11 x Contents 14.4 Job Shop Scheduling Two Jobs and m Machines Job Shop Scheduling 533 Questions GOAL PROGRAMMING Introduction Simplex Method for Solving Goal Programming 541 Questions PARAMETRIC LINEAR PROGRAMMING Introduction Changes in Objective Function Coefficients (C j Values) Changes in Right-hand Side Constants (B i Values) of Constraints Introduction to Changes in Resource Requirements Vector(s), P j 559 Questions NONLINEAR PROGRAMMING Introduction Lagrangean Method Kuhn Tucker Conditions Quadratic Programming Separable Programming Chance-constrained Programming or Stochastic Programming 585 Questions 589 Appendix 591 Suggested Further Reading Answers to Questions Index

12 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

13 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

14 OVERVIEW OF OPERATIONS RESEARCH 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 These are called as the first Shewhart control charts. During the period , the Western Electric Company defined various terminologies associated with acceptance sampling of 1

15 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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