OPERATIONS RESEARCH

M. Sreenivasa Reddy

M.Tech, (Ph. D) MISTE, MIE

CONTENTS

Chapter – 1 Introduction 1

1.1 Concept of Operations Research 1

1.2 Definitions of Operations Research 2

1.3 Evolution of Operations Research (Historical Development) 2

1.4 Scope of Operations Research (Applications) 3

1.5 Approach of Operations Research (Phases of OR) 4

1.6 Limitations of OR Models 5

1.7 Characteristics of OR (Features of OR) 5

1.8 Linear Programming 5

1.9 Steps in Formulation of a Linear Programming Problem (LPP) 6

1.10 General (Mathematical) Formulation of LPP 6

Worked Examples 7

1.11 Graphical Method 13

1.12 Various Cases in Graphical Method 20

Questions 24

Problems 24

Chapter – 2 Linear Programming Problem – Simplex Method 27

2.1 Introduction 27

2.2 Basic Terms / Definitions 27

2.3 Standard Form of a LP Problem (Characteristics of LPP) 28

2.3.1 Steps in Simplex Method 28

2.3.2 Computational Procedure (Working Procedure) of Simplex Method 29

Worked Examples 32

2.4 Artificial Variables 50

2.5 Big - M Method (Penaulty Method) 50

2.6 Two Phase Method 56

2.7 Dual Simplex Method 64

2.7.1 Procedure of Dual Simplex Method 64

2.8 Degeneracy in LPP 71

2.8.1 Procedure to Resolve Degeneracy 71

2.9 Concept of Duality 73

Questions 74

Problems 75

Chapter – 3 Transportation Problem 78

3.1 Introduction 78

3.2 Formulation of a Transformation Problem 78

3.3 Basic Feasible Solution 79

3.4 Steps in solving a Transportation Problems 79

Worked Examples 80

3.5 Optimality Check 84

3.6 Variations in Transportation Problem 96

3.7 Assignment Problems 110

3.8 Steps in Solving an Assignment Problem (Hungarian Method) 110

3.9 Differences Between Transportation Problem and Assignment Problem 129

3.10 The Traveling Sales Man Problem (Routing Problem) 129

Questions 134

Problems 134

Chapter – 4 Sequencing Problems 140

4.1 Introduction 140

4.2 Terms Commonly Used 140

4.3 Assumptions Made in Sequencing 141

4.4 Types of Sequencing Problems 142

4.4.1. Sequencing n Jobs Through Two Machines 142

Worked Examples 142

4.4.2. n – Jobs and Three Machines 150

4.4.3. ‘n’ Jobs and ‘m’ Machines 155

4.4.4. Processing 2 – Jobs Through m Machines 161

Questions 166

Problems 166

Chapter – 5 Queuing Theory (Waiting Lines) 170

5.1 Introduction 170

5.2 Terminology 170

5.3 Queuing System Characteristics 171

5.4 Notations for Representing Queuing Models 173

5.5 Classification of Queuing Models 173

5.6. Operating Characteristics of a Queueing System 177

Worked examples 179

Questions 194

Problems 194

Chapter – 6 PERT – CPM Techniques 197

6.1 Introduction 197

6.2 Basic Steps in PERT / CPM (Project Scheduling) 197

6.3 Applications and Objectives of PERT/CPM 198

6.4 Terminology used in PERT/CPM 198

6.5 Network Construction 200

6.6 Fulkerson’s I – J Rule (For labeling the network diagram) 200

Worked Examples 201

6.7 Important Concepts 203

6.8 Project Evaluation and Review Techniques (PERT) 211

6.9 Related Statistical Terms 212

6.10 PERT – Procedure / Steps 213

6.11 Differences Between PERT and CPM 221

6.12 Crashing of Networks 222

6.13 Terms Related to Project Cost Analysis 222

6.14 Steps for Crashing 223

Questions 229

Problems 230

Chapter – 7 Game Theory 234

7.1 Introduction 234

7.2 Basic Terms used in Game Theory 234

7.3 Formulation of a game 236

7.4 Properties of Game 236

Worked Examples 236

7.5 Maxi min Principle 237

7.6 Mini max Principle 238

7.7 Procedure to Determine Saddle Point 238

7.8 Solution of a Game 239

7.9 Games having no Saddle Point 245

7.10 Dominance rule 246

7.11 Graphical Method 257

Questions 267

Problems 267

Chapter – 8 Integer Programming 270

8.1 Introduction 270

8.2 Applications of Integer Programming 270

8.3 Formulation of Integer Programming 271

8.4 Methods of Integer Programming 271

8.5 Cutting Plane Algorithm 271

8.6 Branch and Bound Method 273

8.7 Limitations of Integer Linear Programming 274

8.8 Zero One Algorithm 274

Worked Examples 275

Questions 287

Problems 287

Glossary 289

Answers to the problems 303

Index 311

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