QUANTITATIVE TECHNIQUES IN MANAGEMENT

QUANTITATIVE TECHNIQUES IN MANAGEMENT

Books by the Same Author Operations Research: Theory and Applications (5 th Ed) Operations Research: Problems and Solutions (3 rd Ed) Linear Programming: Theory and Applications Discrete Mathematics (3 rd Ed) Management of Systems

QUANTITATIVE TECHNIQUES IN MANAGEMENT Third Edition J K Sharma Professor, Amity Bussines School Amity University Uttar Pradesh, Noida (An Imprint of Laxmi Publications Pvt. Ltd.) BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI PATNA NEW DELHI BOSTON, USA

QUANTITATIVE TECHNIQUES IN MANAGEMENT Copyright by the Author Typeset at ISBN Price: ` Limits of Liability/Disclaimer of Warranty: PUBLISHED IN INDIA BY & & Chennai & Cochin & & & & Kolkata & & (An Imprint of Laxmi Publications Pvt. Ltd.) & C Printed at: Branches

Preface to the Third Edition It gives me great pleasure and satisfaction to present the third edition of the book Quantitative Techniques in Management to the teachers and students of this subject. This edition continues to provide readers an understanding of problem-solving methods based on a careful discussion of model formulation, solution procedure and analysis. I hope this easy-to-understand approach would enable readers to develop the required skills and apply quantitative techniques to all kinds of decision-making problems. The text revision in this edition is extensive and in accordance with the objective of enhancing and strengthening the conceptual as well as practical knowledge of readers about various quantitative techniques. A large number of new businessoriented solved as well as practice problems have been added, thus creating a bank of problems that give a better representation of the various quantitative techniques. This edition has a completely new look and feel. I hope this revision will facilitate the teaching of quantitative techniques as well as enhance the learning experience for students. Following are some of the key changes: The text of almost each chapter has been reorganized and/or rewritten to make explanations more cogent through relevant and interesting examples. This will provide a more meaningful, easier and effective learning experience. Each chapter contains Preview and Learning Objectives to guide the students and help them focus their attention on understanding a specific topic under study. Most chapters contain Management Cases to help students understand various business situations and suggest solutions to managerial issues that are raised while using specific techniques. Each chapter contains Concept Quizzes to help students reinforce their understanding of the principles and applications of quantitative techniques. Explanations are well illustrated with numerous interesting and varied business-oriented examples. Conceptual Questions, Self Practice Problems with Hints and Answers are given in each chapter to enable students to learn at their own pace. Complete conformity to the latest trends of questions appearing in universities and professional examinations. References to questions set in examinations of various Indian universities have been updated. The book is intended to serve as a core textbook for students of BBA, BCA, MBA/PGDBM, BCom, MCom, CA, ICWA and those who need to understand the basic concepts of quantitative techniques in management and apply results directly to reallife business problems. It would also prove to be useful for those preparing for IAS, NET, ISI and other competitive examinations. Prof (Dr) J K Sharma

Preface to the First Edition The primary objective in writing this book is to provide the readers the insight into structures and processes that Quantitative Techniques can offer and the enormous practical utility of its various techniques. The aim is to explain the concepts and simultaneously to develop in readers an understanding of problem-solving methods based upon a careful discussion of model formulation, solution procedures and analysis. To this end, numerous solved businessoriented examples have been presented throughout the text. Unsolved Self Practice Problems with Hints and Answers, and Review Questions have been added in each chapter to strengthen the conceptual as well as practical knowledge of the reader. The book is designed to be self-contained and comprises of 20 chapters. The book is intend to serve as a core text primarily for students of BBA, BCA, BCom, MCom, MBA/PGDBM, CA, ICWA etc. who need to understand basic concepts of and apply results directly to real-life business problems. The book also suits the requirements for those preparing for IAS, NET, ISI and other competitive examinations. I hope that the presentation and sequence of chapters have made the text interesting and lucid. In writing this book I have benefitted immensely by referring to many books and publications. I express my gratitude to all such authors, publishers and institutions; many of them have been listed in the references. If anybody has been left out inadvertently, I seek their pardon. I express my sincere gratitude to my teachers Prof. Kanti Swarup and Dr S D Sharma for their blessings and inspiration. I wish to acknowledge my sincere thanks to my students, friends and colleagues, particularly to Prof M P Gupta and Prof A S Narag for their valuable suggestions and encouragement during the preparation of this text. I would like to thank the publishers for the efficient and thoroughly professional way in which the whole project was managed. In the end let me thank my wife and children for the unflagging support and encouragement they gave me while I worked on this book. Any suggestions to improve the book in contents or in style are always welcome and will be appreciated and acknowledged. J K Sharma

Acknowledgements I express my heartfelt gratitude to Founder President Dr. Ashok K Chauhan and Chancellor Mr. Atul K Chauhan, Amity University Uttar Pradesh, Noida for their inspiration, overwhelming support, and motivation. The support of Prof. B Shukla, Acting Vice-Chancellor, Amity University Uttar Pradesh, Noida; Prof. Sanjay Srivastava, Dean, Faculty of Management Studies and Director General, Amity Business School, Amity University Uttar Pradesh, Noida and Prof. Sanjeev Bansal, Director, Amity Business School, Amity University Uttar Pradesh, Noida was very reassuring and invaluable. I thank them from the core of my heart. In preparing the text of this book, I have benefitted immensely by referring many books and publications. I express my gratitude to those authors, publications, publishers and institutions, most of them have been listed in the references. I would also like to thank Wikipedia, (www.wikipedia.org as accessed on 6/5/09) from where I have taken quotes that I have placed at the beginning of each chapter. If anybody is left out inadvertently, I seek their pardon. I am thankful to my esteemed colleagues, and students who have contributed to this book through their valuable advice and feedback. I express my sincere thanks to Trinity Press and Editorial Staff for their excellent job during the publication of the book. Last but never ever the least I thank God Almighty and my family for being there whenever I need them. I hope that the book serves the purpose for its readers and that I will continue to get their support and suggestions. I retain the responsibility of errors of any kind in the book. Suggestions and comments to improve the book in content and in style are always welcome and will be appreciated and acknowledged. Email: jks_sharma@yahoo.com Prof (Dr) J K Sharma

Contents Preface to the Third Edition Preface to the First Edition Acknowledgements v vii ix Chapter 1 Quantitative Techniques: A Managerial Perspective 1 21 1.1 Quantitative Approach to Decision-making 2 1.2 Features of Quantitative Approach 3 1.3 Quantitative Approach to Problem-solving 4 Conceptual Questions A 5 1.4 Models and Modelling for Quantitative Analysis 5 Precautions in Model Building 5 Classification Based on Structure 6 Classification Based on Function (Or Purpose) 8 Classification Based on Time Reference 8 Classification Based on Degree of Certainty 8 Classification Based on Solution Method 9 1.5 Advantages of Model Building 9 1.6 Methods for Solving Quantitative Models 9 1.7 Quantitative Analysis Process 10 1.8 Advantages of Quantitative Analysis 12 1.9 Opportunities and Shortcomings of the Quantitative Approach 12 1.10 Features of Quantitative Analysis 13 1.11 Quantitative Models in Practice 13 1.12 Computer Software for Quantitative Analysis 14 Conceptual Questions B 15 Chapter Concepts Quiz 16 Case Study 17 Chapter 2 Linear Programming: Applications and Model Formulation 22 64 2.1 Introduction 23 2.2 Structure of Linear Programming Model 23 General Structure of an LP Model 23 Assumptions of an LP Model 24 2.3 Advantages of Using Linear Programming 24 2.4 Limitations of Linear Programming 24

xii Contents 2.5 Application Areas of Linear Programming 25 2.6 General Mathematical Model of Linear Programming Problem 26 2.7 Guidelines on Linear Programming Model Formulation 27 2.8 Examples of LP Model Formulation 27 Examples on Production 27 Examples on Marketing 38 Examples on Finance 40 Examples on Agriculture 46 Examples on Transportation 48 Examples on Personnel 50 Conceptual Questions 52 Self Practice Problems 53 Hints and Answers 58 Chapter Concepts Quiz 61 Case Study 62 Chapter 3 Linear Programming: The Graphical Method 65 95 3.1 Introduction 66 3.2 Important Definitions 66 3.3 Graphical Solution Methods of LP Problem 66 Extreme Point Solution Method 67 Examples on Maximization LP Problem 67 Examples on Minimization LP Problem 72 Examples on Mixed Constraints LP Problems 75 Iso-profit (Cost) Function Line Method 83 Comparison of Two Graphical Solution Methods 84 3.4 Special Cases in Linear Programming 84 Alternative (or Multiple) Optimal Solutions 84 Unbounded Solution 85 Infeasible Solution 87 Redundancy 89 Conceptual Questions 89 Self Practice Problems 89 Hints and Answers 92 Chapter Concepts Quiz 93 Case Study 94 Chapter 4 Linear Programming: The Simplex Method 96 139 4.1 Introduction 97 4.2 Standard Form of an LP Problem 97 4.3 Simplex Algorithm (Maximization Case) 99 4.4 Simplex Algorithm (Minimization Case) 108 Two-Phase Method 110 Big-M Method 115 Self Practice Problems A 123 Hints and Answers 126 4.5 Some Complications and their Resolution 127 Unrestricted Variables 127 Tie for Entering Basic Variable (Key Column) 130 Tie for Leaving Basic Variable (Key Row) Degeneracy 130

Contents xiii xiii 4.6 Types of Linear Programming Solutions 131 Alternative (Multiple) Optimal Solutions 132 Unbounded Solution 133 Infeasible Solution 134 Conceptual Questions 135 Self Practice Problems B 135 Hints and Answers 137 Chapter Concepts Quiz 137 Case Study 139 Chapter 5 Duality in Linear Programming 140 160 5.1 Introduction 141 5.2 Formulation of Dual Linear Programming Problem 141 Symmetrical Form 141 Economic Interpretation of Dual Variables 142 Economic Interpretation of Dual Constraints 143 Rules for Constructing the Dual from Primal 143 Self Practice Problems A 147 Hints and Answers 147 5.3 Standard Results on Duality 148 Principle of Complementary Slackness 148 5.4 Managerial Significance of Duality 148 5.5 Advantages of Duality 154 Conceptual Questions 154 Self Practice Problems B 154 Hints and Answers 156 Chapter Concepts Quiz 158 Case Study 160 Chapter 6 Sensitivity Analysis in Linear Programming 161 192 6.1 Introduction 162 6.2 Sensitivity Analysis 162 Change in Objective Function Coefficient (c j ) 162 Change in the Availability of Resources (b i ) 169 Changes in the Input-Out Coefficients (a ij s ) 176 Addition of a New Variable (Column) 180 Addition of a New Constraint (Row) 181 Conceptual Questions 188 Self Practice Problems 188 Hints and Answers 190 Chapter Concepts Quiz 191 Case Study 192 Chapter 7 Integer Linear Programming 193 226 7.1 Introduction 194 7.2 Types of Integer Programming Problems 194 7.3 Enumeration and Cutting Plane Solution Concept 195 7.4 Gomory s All Integer Cutting Plane Method 195 Method for Constructing Additional Constraint (Cut) 196 Steps of Gomory s All Integer Programming Algorithm 196

xiv Contents Self Practice Problems A 204 Hints and Answers 207 7.5 Gomory s Mixed-Integer Cutting Plane Method 207 Method for Constructing Additional Constraint (Cut) 208 Steps of Gomory s Mixed-Integer Programming Algorithm 209 7.6 Branch and Bound Method 213 7.7 Applications of Zero-one Integer Programming 220 Capital Budgeting Problem 220 Fixed Cost (or Charge) Problem 220 Plant Location Problem 221 Self Practice Problems B 222 Conceptual Questions 222 Hints and Answers 223 Chapter Concepts Quiz 224 Case Study 225 Chapter 8 Goal Programming 227 246 8.1 Introduction 228 8.2 Difference between LP and GP Approach 228 8.3 Concept of Goal Programming 228 Distinction Among Objectives, Goals and Constraints 229 8.4 Goal Programming Model Formulation 229 Single Goal with Multiple Subgoals 229 Equally Ranked Multiple Goals 230 Ranking and Weighting of Unequal Multiple Goals 231 General GP Model 232 Steps to Formulate GP Model 232 8.5 Graphical Solution Method for Goal Programming 232 8.6 Modified Simplex Method of Goal Programming 236 8.7 Alternative Simplex Method for Goal Programming 238 Conceptual Questions 241 Self Practice Problems 241 Chapter Concepts Quiz 243 Case Study 245 Chapter 9 Transportation Problem 247 298 9.1 Introduction 248 9.2 Mathematical Model of Transportation Problem 248 General Mathematical Model of Transportation Problem 249 9.3 The Transportation Algorithm 250 9.4 Methods of Finding Initial Solution 250 North-West Corner Method (NWCM) 250 Least Cost Method (LCM) 251 Vogel s Approximation Method (VAM) 253 Conceptual Questions A 256 Self Practice Problems A 256 Hints and Answers 256 9.5 Test for Optimality 257 Dual of Transportation Model 257 Economic Interpretation of u i s and v j s 258 Steps of MODI Method (Transportation Algorithm) 259 Close-Loop in Transportation Table and its Properties 260

Contents xv xv Conceptual Questions B 269 Self Practice Problems B 269 Hints and Answers 271 9.6 Variations in Transportation Problem 271 Unbalanced Supply and Demand 271 Degeneracy and its Resolution 274 Alternative Optimal Solutions 278 Prohibited Transportation Routes 281 9.7 Maximization Transportation Problem 285 9.8 Trans-shipment Problem 287 Conceptual Questions C 289 Self Practice Problems C 289 Hints and Answers 293 Chapter Concepts Quiz 295 Case Study 296 Chapter 10 Assignment Problem 299 327 10.1 Introduction 300 10.2 Mathematical Model of Assignment Problem 300 10.3 Solution Methods of Assignment Problem 301 Hungarian Method for Solving Assignment Problem 301 Conceptual Questions A 307 Self Practice Problems A 307 Hints and Answers 309 10.4 Variations of The Assignment Problem 309 Multiple Optimal Solutions 309 Maximization Case in Assignment Problem 309 Unbalanced Assignment Problem 312 Restrictions on Assignments 312 Conceptual Questions B 316 Self Practice Problems B 316 Hints and Answers 318 10.5 A Typical Assignment Problem 319 10.6 Travelling Salesman Problem 320 Self Practice Problems C 323 Hints and Answers 324 Chapter Concepts Quiz 324 Case Study 326 Chapter 11 Theory of Games 328 362 11.1 Introduction 329 11.2 Two-person Zero-sum Games 330 11.3 Pure Strategies (Minimax and Maximin Principles): Games With Saddle Point 332 Rules to Determine Saddle Point 332 Conceptual Questions A 334 Self Practice Problems A 335 Hints and Answers 336 11.4 Mixed Strategies: Game without Saddle Point 336 11.5 The Rules (Principles) of Dominance 337 11.6 Solution Methods for Games without Saddle Point 338