## Description

This book provides a clear understanding of the basic principles of the various principles of Operations Research. It would serve as a comprehensive textbook on the subject, which would fill the gap being experienced by the students on the subject at present. The contents of the book are so designed that they will satisfy the syllabus requirements for the B.Tech. and MBA programmes of all Indian Universities.

## Table of Content

Foreword

Preface

Acknowledgements

**CHAPTER 1 INTRODUCTION TO OPERATIONS RESEARCH**

1.1 Origin and Development of Operations Research

1.2 Meaning of Operations Research

1.3 Various Definitions of Operations Research

1.4 Scope of Operations Research

1.4.1 Agriculture

1.4.2 Finance

1.4.3 Industry

1.4.4 Production Management

1.4.5 Marketing Management

1.4.6 Personnel Management

1.5 Principal Characteristics of Operations Research

1.5.1 Inter-disciplinary Approach

1.5.2 Holistic Approach

1.5.3 Imperfectness of Solutions

1.5.4 Use of Scientific Research

1.5.5 Optimization of the Total Output

1.6 Models in Operations Research

1.7 Classification of Operations Research Models

1.7.1 Iconic (Physical) Models

1.7.2 Analogue Models

1.7.3 Mathematical (Symbolic) Models

1.7.4 Combined Analogue and Mathematical Models

1.8 Phases of Operations Research

1.8.1 Formulating the Problem

1.8.2 Constituting the Model

1.8.3 Deriving the Solution from the Model

1.8.4 Testing and Updating the Model

1.8.5 Controlling the Solution

1.8.6 Implementing the Solution

1.9 Advantages of Operations Research

1.9.1 Optimal Use of the Production Factors

1.9.2 Improved Quality of the Solution

1.9.3 Preparation of Future Managers

1.9.4 Scope for Modification of Solution

1.9.5 Possibility of Alternate Solutions

1.10 Limitations of Operations Research

1.10.1 Limited Practical Application

1.10.2 Reliability of the Proposed Solution

1.10.3 Cost of Time and Money

1.10.4 Lack of Feasibility of Combining Two or More Objectives

Summary

Review Questions

Objective Questions

Descriptive Questions

**CHAPTER 2 ALLOCATION—LINEAR PROGRAMMING**

2.1 Allocation

2.1.1 Additive Property

2.1.2 Multiplicative Property

2.2 Formulation of Problems in Linear Programming Model

2.2.1 LP Formulation for Product Mix Problem

2.2.2 LP Formulation Problem on Transportation

2.2.3 LP Formulation Problem on Inventory

2.2.4 LP Formulation Problem on Marketing

2.2.5 LP Formulation Problem on Advertising

2.2.6 LP Formulation Problem on Agriculture

2.3 Solution of LP Problems

2.3.1 Solution by Graphical Method

2.3.2 Special Cases in LP Problems

2.3.2.1 Degenerate solution

2.3.2.2 Unbounded solution

2.3.2.3 Multiple (alternate) optimal solutions

2.3.2.4 Inconsistent constraints

2.4 Simplex Method

2.4.1 General Form of LP Problem

2.4.2 Some Important Definitions

2.4.3 Solution of LP Problems by Simplex Method

2.4.3.1 Steps in the simplex procedure

2.4.3.2 Starting simplex table

2.4.3.3 Optimality test

2.5 Artificial Variables Technique

2.5.1 Two-Phase Method

2.5.1.1 Phase I: Elimination of the artificial variables

2.5.1.2 Phase II: Application of simplex method

2.5.2 Big-M Method

2.6 Unrestricted Variables

2.7 Unbounded Solution

2.8 Degeneracy in Linear Programming

2.8.1 Resolution of Degeneracy by Generalized Simplex Method

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 3 DUALITY IN LINEAR PROGRAMMING**

3.1 General Rules for Converting the Primal into Dual

3.2 Fundamental Duality Theorems

3.3 Problems to Write the Dual from the Given Primal Problem

3.4 Problems to Solve the Dual to Find the Optimal Solution of the Primal

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 4 ASSIGNMENT MODEL**

4.1 Statement of the Problem

4.2 Mathematical Formulation of the Assignment Problem

4.3 Reduction Theorem

4.4 Solution Procedure of the Assignment Problem

4.5 Travelling Salesman Problem

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 5 SEQUENCING**

5.1 Introduction

5.2 Johnson’s Algorithm

5.2.1 Processing of n Jobs through m Machines

5.2.2 Processing of n Jobs, and 2 Machines, with the Same Processing Order Through the Two Machines for all the n Jobs

5.2.3 Processing of n Jobs, and 3 Machines, with the Same Processing Order Through the Three Machines for all the n Jobs

5.2.4 Processing of n Jobs, and m Machines, with the Same Processing Order Through all the m Machines for all the n Jobs

5.2.5 Processing of 2 Jobs, through m Machines, and the Processing Order of the 2 Jobs through all the m Machines being not Necessarily the Same

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 6 TRANSPORTATION PROBLEM**

6.1 Mathematical Formulation of the Problem

6.2 To find the Basic Feasible Solution (BFS)

6.2.1 North-West Corner Method

6.2.2 Least Cost Entry Method

6.2.3 Unit Cost Penalty Method or Vogel’s Approximation Method (VAM)

6.3 Unbalanced Transportation Problem

6.4 Degeneracy in Transportation Problems

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 7 REPLACEMENT**

7.1 Replacement of Capital Cost Items

7.2 Replacement of Capital Cost Items when Money’s Worth (over time) is Considered

7.3 Replacement of Low-Cost Items in Bulk Quantities (Group Replacement)

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 8 INVENTORY CONTROL**

8.1 Objective of an Inventory Problem

8.1.1 Controllable Variables

8.1.2 Uncontrollable Variables

8.1.2.1 Cost variables

8.1.2.2 Other variables

8.2 Need of Inventory

8.3 Inventory Models

8.3.1 Deterministic Models

8.3.2 Probabilistic Models

8.4 MODEL I: Economic Lot Size Model with Uniform Rate of Demand, Infinite Production Rate and Having No Shortages

8.4.1 Description of the Model

8.5 MODEL II: Economic Lot Size Model with Different Rates of Demand in Different Production Cycles, Infinite Production Rate and having No Shortages

8.6 MODEL III: Economic Lot Size Model with Uniform Rate of Demand, Finite Rate of Replenishment and Having no Shortages

8.7 MODEL IV: Deterministic Models with Shortages and Backlogging (Fixed Time Model) with Uniform Rate of Demand ‘Q’ to be fulfilled in Constant Time t–Infinite Rate of Production and having Shortages which are to be Fulfilled

8.8 MODEL V: ELS Model with Uniform Rate of Demand, Infinite Rate of Production having Shortages which are to be Fulfilled (Varying Time Model)

8.9 MODEL VI: ELS Model with Uniform Rate of Demand, Finite Rate of Production having Shortages which are to be Fulfilled

8.10 MODEL VII: Single Period Model with Discontinuous or Instantaneous Demand and No Set-up Cost Model

8.11 MODEL VIII: Single Period Model with Uniform Rate of Demand (No Set-up Cost Model)

8.12 MODEL IX: General Single Period Model of Profit Maximization with Time-Independent Cost

8.13 MODEL X: Purchase Inventory Model with One Price Break

8.14 MODEL XI: Purchase–Inventory Model with Two Price Breaks

8.15 Purchase Inventory Model with Multiple Price Breaks

8.15.1 To find the Optimal Order Quantity q which Minimizes the Total Cost

8.16 ABC Analysis

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 9 QUEUING MODELS**

9.1 Objective of Queuing Models

9.2 Service Mechanism

9.3 Operating Characteristics of a Queuing System

9.4 Queing Theory

9.4.1 Important Definitions

9.4.2 State of the System

9.4.3 Poisson Process

9.4.4 Poisson Arrivals

9.4.5 Inter-Arrival Time/Service Time Distributions (Used in Queuing Theory)

9.4.6 Erlang Distribution

9.5 Queing Theory

9.5.1 Terminology of Queing Theory

9.5.2 Objectives of Queuing Theory

9.5.3 Basic Elements of Queuing System

9.6 Queing Models

9.6.1 Poisson Arrivals and Exponential Service Times

9.7 Basic Equations of Queuing System

9.8 Classification of Queing Models

9.9 MODEL I: {(M/M/1):(∞/FCFS)}

9.9.1 Characteristics of Model 1

9.10 MODEL II: {(M/M/1):(N/FCFS)}-(Finite Queue Model)

9.10.1 Characteristics of Model II

9.11 MODEL III: {(M/M/S):(∞/FCFS)}

9.11.1 System of Differential Difference Equations

9.11.2 Solution of Equations

9.11.3 Characteristics of Model III

9.12 MODEL IV: {(M/M/S):(N/FCFS)}

9.12.1 Characteristics of Model IV

9.13 MODEL V: {(M/M/S):(M/GD)}

9.13.1 Characteristics of the Model for S = 1

9.13.2 Characteristics of the Model for S > 1

9.14 MODEL VI: {(M/Ek/1):(∞/FCFS)}

9.14.1 Characteristics of the Model

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 10 THEORY OF GAMES OR COMPETITIVE STRATEGIES**

10.1 A Simple Illustration to Understand the Theory of Games

10.2 Features of Competitive Games

10.2.1 Zero-Sum Game

10.2.2 Two-Person Zero-Sum Game (Rectangular Game)

10.2.3 Pay-Off Matrix (Gain Matrix)

10.2.4 Strategy of a Player

10.2.5 Difference between the Course of Action and Strategy of a Player

10.2.6 Pure Strategy

10.2.7 Mixed Strategy

10.2.8 Value of the Game (n)

10.3 Maximin or Minimax Criterion of Optimality

10.4 Saddle Point

10.5 Game Theory—Problems

10.5.1 Problems with Saddle Point

10.6 Mixed Strategies

10.7 Dominance Property: Rules of Dominance

10.8 Solution of Problems without Saddle Point

10.9 Problems without Saddle Point—Solution using Dominance

10.10 Solution of Problems by Graphical Method

10.11 Solution Methodology for Problems without Saddle Point, without Dominance, and which can NOT be reduced to either (2 × n) size or (m × 2) size

10.12 Solution of Theory of Games Problems without saddle point 637

10.13 Solution of Rectangular Games without Saddle Point by Approximate Method (Iterative Method)

10.14 Algebraic Method

10.15 Solution of Game Theory Problem By Converting into L.P. Problem

10.16 Game Theory—Additional Problems

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 11 DECISION ANALYSIS**

11.1 Definition of Decision Analysis

11.2 Steps in Decision-Making

11.3 Different Environments in which Decisions may be Made

11.3.1 Decision-Making Under Conditions of Certainty

11.3.2 Decision-Making Under Conditions of Uncertainty

11.3.3 Decision-Making Under Conditions of Risk

11.4 Criteria for Decision-Making Under Uncertainty

11.4.1 Maximax Criterion

11.4.2 Maximin Criterion

11.4.3 Minimax Criterion

11.4.4 Minimax Regret Criterion

11.4.5 The Criterion of Realism

11.5 Decision-Making Under Conditions of Risk

11.5.1 Solution Methods for Decision-Making Under Risk Environment

11.5.1.1 Expected value criterion

11.5.1.2 Expected value combined with variance criterion

11.6 Decision-Making Under Uncertainty

11.6.1 Laplace Criterion

11.6.2 Maximin Criterion

11.6.3 Minimax Criterion

11.6.4 Savage Minimax Regret Criterion

11.6.5 Hurwicz Criterion

11.7 Decision-Making under Risk: [Expected Monetary Value (EMV) Criterion]

11.8 Decision Tree

11.8.1 Symbols Used in the Decision Tree Approach

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 12 SIMULATION**

12.1 Usefulness of Simulation

12.1.1 Analytical Complexity

12.1.2 Complex Interdependence

12.1.3 Effect of Change

12.1.4 Time Compression

12.1.5 Physical Simulation

12.2 Significant Parameters of Simulation

12.2.1 State Variables

12.2.2 Event

12.2.3 Static and Dynamic Simulation Models

12.3 Monte Carlo Simulation Technique

12.4 Normal Distribution

12.4.1 Characteristics of Normal Distribution

12.4.2 How to Calculate the Variance from the Available Data?

12.4.3 How to Measure the Area Under the Normal Curve?

12.5 Simulation Languages

12.6 Types of Simulation

12.7 Advantages of Simulation

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 13 NETWORK ANALYSIS (CPM AND PERT)**

13.1 Critical Path Method

13.2 Program Evaluation and Review Technique

13.3 Basic Concepts of Network Analysis

13.4 Types of Activities

13.5 Drawing a Network diagram

13.6 Rules for Numbering of the Events

13.6.1 D.R. Fulkerson’s Rule (for numbering of the events)

13.7 Hints for drawing the network diagrams

13.7.1 General Hints

13.7.2 Avoiding the Loop Network

13.7.3 Dummy Activity

13.8 Critical Path Method

13.8.1 Important Terminology in CPM

13.8.2 Float

13.8.3 Time–Cost Trade-off

13.8.4 Project Cost Analysis

13.8.5 Limitations of CPM

13.9 Program Evaluation and Review Technique (PERT)

13.9.1 Calculation of the Probabilities

13.10 Comparison between CPM and PERT

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

**CHAPTER 14 DYNAMIC PROGRAMMING**

14.1 Some Important Terms

14.2 Basic Features (Characteristics) of Dynamic Programming Problems 14.3 Model-I: Shortest Route Problem

14.4 Model-II: Optimal Sub-Division Problems

14.5 Bellman’s Principle of Optimality

14.6 Model-III: Solution of L.P. Problems by Dynamic Programming

14.7 Model-IV: Cargo Loading Problems

14.8 Model-V: Optimization of Study Days for Examination Preparation

14.9 Capital Budgeting Problem

14.10 Reliability Problems

14.11 Warehouse problem

14.12 Product Allocation Problem

14.13 Salesmen Allocation Problem

14.14 Advertising Media Problem

Summary

Review Questions

Objective Questions

Descriptive Questions

Exercise Problems

Appendices

Index

## About The Author

**Dr. A. Chandra Sekhara Kumar** is a Professor of Mechanical Engineering who had retired as Vice Principal from Jawaharlal Nehru Technological University College of Engineering, Hyderabad. He has more than 40 years of teaching experience in B.I.T.S., Pilani, Higher College of Technology, Muscat, Sultanate of Oman and J.N.T.U.H., Hyderabad put together. He had taught the subject of Theory of Machines for many batches of students of B.Tech., during his long service in J.N.T.U.H., Hyderabad. Besides, he is an active Researcher in the field of Production Engineering, and so far ten research scholars obtained Ph.D under his guidance, and another five research scholars are currently working under his supervision. He has nearly 75 research publications to his credit in various reputed International and National Journals and Conferences.

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