Operations research

304 Pages·2008·22.318 MB·English

by  P Sankara Iyer

OPERATIONS
RESEARCH
About the Author
P. Sankara Iyer is presently Retired Reader in Mathematics at Alagappa University, Karaikudi in Tamil
Nadu. After holding the position of a Lecturer in Mathematics at Alagappa College from 1966 to 1981,
he became Professor in 1981. He then joined Alagappa University in 1985 and retired as Reader in
Mathematics in the year 2001. He is M.Sc. (Mathematics), M.Phil. (Mathematics) and Ph.D. (Computer
Applications in Operations Research). He has done a Certificate course in Computer Programming and
a Certificate course in Evaluation Methodology and Examination by A.I.U. His teaching experience is
vast—B.Sc. (19 yrs), M.Sc. (19 yrs), M.C.A. (15 yrs) and M.Phil. (18 yrs). He has taught all subjects at
B.Sc. level; Operations Research, Differential Equations, Differential Geometry, Classical Mechanics,
Space Mechanics, Quantum Mechanics, Algebra, Statistics and Numerical Analysis at M.Sc. level;
Resource Management Techniques (OR) and Computer based Numerical Methods at M.C.A. level; and
Algebra at M.Phil. level. His study material for M.Sc. Mathematics (Operations Research and Differential
Equations), M.Sc. Industrial Mathematics (Operations Research)—2 books, M.C.A. (Resource
Management Techniques) and M.B.A. (Operations Research) of Alagappa University book are very
popular among the students. He is a member of (i) Operations Research Society of India, (ii) Ramanujan
Mathematical Society and (iii) Forum for Advancement of Science and Human Development.
Prof. Sankara Iyer’s primary area of research is Operations Research: Fractional Programming
(Computer Applications). He has published seven research papers and guided more than 50 students for
M.Phil. Research Project. He has attended twelve conferences and seminars in Algebra, Operations
Research, Analysis Statistics and Stochastic Processes. He has been the Cultural Coordinator of Alagappa
University for 15 years, Director of Social Service in Alagappa College for 10 Years, Director of Fine
Arts in Alagappa College for 8 Years and Secretary of Staff Club for 6 Years. He is musically inclined
and has composed and tuned nearly 200 songs and given public performances. He has written and
directed social dramas (besides acting as the main protagonist), promoting national integration, communal
harmony and eradication of dowry. Tamil Isai Sangam, Karaikudi conferred the title ‘MUTHAMIZH
SELVAR’ (year 2001), in appreciation of his service in the field of literature, music, and drama. He
gives a musical discourse every year on the life of Saint Thyagaraja during Thyagaraja Aradhana Festival
at Tiruvaiyaru in Thanjavur District, Tamil Nadu.
OPERATIONS
RESEARCH
P. Sankara Iyer
Reader in Mathematics (Retd)
Alagappa University
Karaikudi, Tamil Nadu
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Foreword
I had the golden opportunity of being a student of Prof. P. Sankara Iyer, the author of this book. He is a
model teacher and we (all his students) consider him as our great Guru and Guide. I feel proud to write
this foreword for a book written by my beloved teacher.
Mathematics is the source of all knowledge. It is a vast ocean and the author of this book has dived
deep into a particular branch of this subject, namely Operations Research.
The contents of this book have been arranged on the principle of gradation. The chapters are in a
logical sequence: Introduction, Linear Programming, Integer Programming, Transportation, Assignment,
Decision Theory, Games Theory, Dynamic Programming, Sequencing, Queueing, Inventory, Replacement,
PERT and CPM, and Simulation.
There are two distinctive features that make this book stand apart. The first one is that the examples
and exercises provided in each chapter are indeed a boon to the students and they can assess their
understanding of each chapter clearly. Another unique feature is that the entire book has been based on
a self-taught new method, culminating from Prof. Sankara Iyer’s rich teaching experience of 35 years.
Also, while structuring this book, he has kept both the instructor and the student in his purview, and
hence it will prove to be a valuable resource for them.
The book covers the syllabi of almost all Indian universities. Therefore not only students of
Mathematics but also students of M.C.A., M.B.A., C.A. and I.C.W.A. will benefit from this book.
I earnestly wish that the student community and faculty members derive maximum benefit from this
magnificent creation.
Dr V. VEMBAIYAN
Principal,
Alagappa Govt. Arts College,
Karaikudi, Tamil Nadu.

Preface
In recent years, Operations Research—the mathematical analysis of a process, used in making decisions—
has become a very popular branch of mathematics studied by undergraduate and postgraduate students.
This interdisciplinary branch of Applied Mathematics has wide applications in various types of real life
situations. A large number of research papers have been published on this subject and a great deal of
research is being carried out.
Purpose
The primary objective of writing this book is to provide the students with a basic knowledge of the
processes involved in Operations Research and the enormous practical applications of the techniques to
daily life problems. Presently, there are limited good books in the market, with even fewer offering well
illustrated problems and also, the style of presentation in foreign books is not very helpful to our Indian
students—this generates a need for a single book that covers all the topics of this subject in order to
cater to the requirement of students in various universities. This book fulfills the purpose.
Users
This book is primarily meant for the course on Operations Research to be studied by the engineering
students of all branches. It offers coverage of different topics pertaining to the syllabus prescribed for
the students of Mathematics and Statistics (UG and PG), M.C.A., M.B.A., M.Com., B.E. and B.Tech. of
Indian Universities and also, C.A. and I.C.W.A. This book is a fine reference for competitive (UG and
PG levels) and I.A.S. examinations.
Highlights
What makes this book unique is its lucid presentation of the subject aided by plenty of solved and
unsolved problems. Brief yet adequate theory clearly explained in simple language supported by worked
out examples (with step by step solution) and exercise problems facilitate quick learning by the students.
The chapters have been structured in a planned manner so that the students can study the subject
independently.
It is a comprehensive text which helps students in understanding problem solving methods based on
model formulation, solution procedure and analysis. The salient features of the book are
(i) Provides sound knowledge of the OR techniques in managerial decision making.
(ii) Incorporates important chapters like Sequencing Problems, Replacement Models, Project
Scheduling, Artificial Variables and Bounded Variable Techniques.
viii Preface
(iii) Numerous solved examples give an overall view of the applications of OR techniques. Solved
examples from various university question papers are an effective tool for examinations.
(iv) To clarify concepts, figures and tables are used exhaustively.
(v) To carry out self evaluation, students can solve multitude of exercises that are provided in each
chapter along with their answers.
The Sigma Format
This book is based on the Sigma Format that offers a clear understanding of fundamental theory supported
with exhaustive examples, solved using a step-by-step approach. To get a better grasp on the subject,
lots of exercises along with answers are provided in the chapters. This well structured format has proven
to be a handy resource material for students as well as teachers.
Chapters Organisation
In the first chapter, the importance of the study of OR and its various applications have been discussed.
Chapter 2 describes the method of mathematical formulation of a real life problem (in the form of a
Linear Programming Problem) and the graphical method of solving an L.P.P having two decision variables.
Simplex Method, an important method of solving a general L.P.P has been discussed in detail in chapter
3. Various modifications of simplex method such as Big M method, Two Phase Method, Dual Simplex
Method, Principle of Duality, and Revised Simplex Method have been covered in the subsequent chapters
4, 5, 6 and 7. When the variables are restricted to assume values within certain given limits, we have to
apply Bounded Variable Technique described in chapter 8. Another restriction that the variables can
take only integer values, leads to Integer Programming which is given in chapter 9. Also, a study of the
effect on the optimal solution due to changes in the values of the parameters, which is called Sensitivity
Analysis, has been discussed in chapter 10.
Then follow chapters 11, 12, 13, 14, 15 and 16 dealing with applications of OR techniques namely
Transportation and Assignment Problems, Decision Theory, Game Theory, Dynamic Programming and
Sequencing of Jobs. Another set of applications have been provided in detail in chapters 17 (Queueing
theory), 18 (Inventory models), 19 (Replacement theory), and 20 (P.E.R.T - C.P.M). Simulation Techniques
of solving different types of problems have been taken up for discussion in the last chapter (chapter 21).
Acknowledgements
I would like to express my gratitude to all the authors who were a source of inspiration while writing this
book. I wish to express my indebtedness to my teachers and well wishers. I wish to acknowledge my
sincere thanks to late Dr M K Venkataraman for his inspiration and encouragement in this project. My
thanks are also due to my wife and children, and M.Phil. Scholar, R G N Nagarajan of Alagappa University
for their continuous help and cooperation in fulfilling the project. I express my deepest sense of gratitude
to the following reviewers, whose suggestions have helped me bring the text to its present form.
J S Oberoi S K Tiwari
Baba Banda Singh Bahadur Institute of National Institute of Technology, Jalandhar
Engineering,
C M Agarwal
Fatehgarh sahib, Punjab
Maulana Azad National Institute of Technology,
Bhopal
Preface ix
Rajiv Lochan S Srinivasan
The Institute of Chartered Financial Analysts of Crescent Engineering College, Chennai
India, Jaipur
S Nadarajan
S R Kapoor Dept. of Mechanical Engineering
Dept. of Electrical Engineering GCT, Coimbatore
Kota Engineering College, Kota
A Rajesh
Sinha Roy Dept. of Mechanical Engineering
Dept. of Mathematics Kumaraguru College of Technology
Future Institute of Technology, Kolkata Coimbatore
Shila Das A Ramakrishna Rao
Dept. of Statistics Adi Shankara Instiute of Engineering and
Dibrugarh University, Assam Technology,
Kalady
Chakradhar Dash
N M Institute of Technology K R Divakara Roy
Khandagiri, Bhubaneswar Dept. of Mechanical Engineering,
College of Engineering
Reena Pant
Andhra University, Visakhapatnam
Mumbai University, Mumbai
T. Chandrashekar
Arun Kumar,
Dept. of Mechanical Engineering,
M H Saboo Siddik College of Engineering,
M S Ramaiah Institute of Technology, Bengaluru
Mumbai
I am truly grateful to the publishers at McGraw-Hill Education India for coming forward to publish
this book and for their efficient management of the project. I am sure that this book will receive good
response from the teachers and the taught. Suggestions from the readers for the improvement of the
book are most welcome and will be duly acknowledged and incorporated.
P. Sankara Iyer
Email id: [email protected]