Operations Research

882 Pages·2015·96.41 MB·english

by  R K Gupta

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nPERATIONS
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(For Honours and Post-Graduate Students of Mathematics, Statistics,
Commerce, Management and Engineering of all Universities)
(As per
UGC
Syllabus
By
Dr. R.K. Gupta
Rtd. Principal & Head
Department of Mathematics
S.S.V. College, Hapur
(C.C.S. University, Meerut)
KRISHNA Prakashan Media (P) Ltd.
KRISHNA HOUSE, 11, Shivaji Road, Meerut-250 001 (U.P.), India
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pERATIONS RESEARCh
First Edition: 1984
Twentieth Revised and Enlarged Edition: 2005
Twenty Eighth Revised Edition: 2010
Thirty Fourth Edition : 2015
Name, style or any part of this book thereof may not be reproduced in any form or by any means without the written permissior
from the publishers and the author. Every effort has been made to avoid errors or omissions in this publication. In spite of this,
some errors might have crept in. Any mistake, error or discrepancy noted may be brought to our notice which shall be token core of
in the next edition. It is notified that neither the publisher nor the author or seller will be responsible for any damage or loss of
action to anyone, of any kind, in any manner, therefrom. For binding mistakes, misprints or for missing pages, etc. the publisher's
liability is limited to replacement within one month of purchase by similar edition. All expenses in this connection are to be borne
by the purchaser.
I.S.B.N. : 978-81-8283-721-8
Book Code No. : 241-34
Price : 495.00 Only
Published by : Satyendra Rastogi "Mitre"
for KRISHNA Prakashan Media (P) Ltd.
11, Shivaji Road, Meerut — 250 001 (U.P.) India
Phones: 91.121.2644766, 2642946, 4026111/12; Fax: [phone]
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E-mail: [email protected]
Chitf Editor : Sugam Rastogi
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Printed at : Vimal Offset Printers, Meerut.
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Preface to the Present Edition
In this edition, the book has been revised thoroughly according to the latest
syllabus proposed by UGC for all Indian Universities. The book covers the syllabi of
all Engineering and Professional institutes. We hope that this revised edition will be of
great help for whom the book is written. For further improvement, the author will
welcome all sorts of criticisms and comments from all readers.
—R. K. Gupta
Preface to the First Edition
The book entitled 'Operations Research', has been written to meet the
requirements of Honours and Post-Graduate students of mathematics, statistics,
commerce, management and engineering of all Universities on this subject. The book
covers almost the entire syllabi of various Universities.
Each topic in the book has been treated in an easy and lucid style without
sacrificing any rigor. The language is simple and easily understandable. The book
contains large number of important and interesting worked out examples. A set of
exercises is given at the end of each chapter so that the students may have enough
practice in the subject. Up-to-data references from question papers of various
Universities are also given in the book.
No claim to originality can be made but the treatment of the subject is in our own
style. Though I have taken great care in eliminating the misprints, but if there are still
any, I shall feel highly obliged to those who will take trouble of pointing them out.
Suggestions for the improvement of the book will be gratefully acknowledged.
The author is grateful to his publishers and printers for their full co-operation in
bringing out the book in the present nice form.
—R. K. Gupta
10, Ganga Nagar,
Railway Road,
Hapur
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Brief Contents
Dedication (iii)
Preface (iv)
Brief Contents (1)
Detailed Contents (vi-xvi)
S.N. Chapters Pages
1. Introduction to Operations Research 1-10
2. Mathematical Preliminaries I 1-40
3. Inventory Theory 41-114
4. Replacement Problems 115-160
5. Waiting Line or Queuing Theory 161-234
6. Allocation (General linear prog. problems) 235-276
7. Convex Sets and Their Properties 277-292
8. Simplex14-,thod 293-370
9. Duality in Linear Programming 371-398
10. Sensitivity Analysis 399-434
11. Parametric Linear Programming 435-456
12. Integer Programming 457-488
13. Assignment Problem 489-538
14. Transportation Problem 539-602
15. Sequencing (Including Travelling Salesman Problem) 603-630
16. Dynamic Programming 631-658
17. Game Theory (Competitive Strategies) 659-724
18. Goal Programming 725-750
19. Network Analysis (PERT/CPM) 751-800
20. Information Theory 801-820
21. Non-Linear Programming 821-856
Index 857-862
•

I
•
Detailed Contents
S.No. Chapters Pages
1. Introduction to Operations Research 1-10
1.1 Introduction (The Origin and the Development of OR)
1.2 Nature and Definition of OR 1
1.3 Objective of OR 2
1.4 Phases of OR Method 2
1.5 Areas of Applications (Scope) of OR 3
1.6 Operations Research and Decision-Making
1.7 Scientific Method in OR 5
1.8 Characteristics of Operations Research 5
1.9 Modeling in OR 6
1.10 Types of Models 7
1.1 1 General Methods of Solution for OR Models 8
• Exercise on Chapter 1 10
2. Mathematical Preliminaries - ID
2.1 Introduction 11
I. Elementary Probability Theory 1 I
2.2 Sample Space 11
2.3 Events 11
2.4 Algebra of Events 12
2.5 Classical Definition of Probability 12
2.6 Odds in Favour and Odds Against 13
2.7 The Statistkal (or Empirical) Definition of Probability 13
2.8 Axiomatic Definition of Probability 13
2.9 Natural Assignment of Probabilities 14
2.10 Theorem of Total Probability or Additional Theorem of Probability 14
2.11 Compound Events 14
2.12 Independent and Dependent Events 15
2.13 Conditional Probability 15
2.14 Multiplication Theorem of Probability 15
2.15 I&ndom Variable ' 16
2.16 Discrete Probability Distributions 16
2.17 Expectation of a Random Variable 16
2.18 Special Discrete Probability Distributions 20
2.19 Continuous Probability Distributions 22
2.20 Special Continuous Probability Distributions 22
2. Matrices and Determinants 28
2.21 Definitions 28
2.22 Operations of Matrix Addition and Multiplication 28
2.23 Sub-Matrix 29
2.24 Minor of Order k 29
2.25 Determinant 30
2.26 Important Properties of Determinants 30
2.27 Minors 31
2.28 Cofactors 31
2.29 Rank of a Matrix 31
2.30 Adjoint of a Matrix 31
2.31 Singular and Non-Singular Matrices 32
2.32 Inverse of a Matrix 32
3. Vectors and Vector Spaces 32
2.33 Definitions 32
2.34 Euclidean Space 34
2.35 Linear Dependence and Independence of Vectors 34
2.36 Linear Combination (L.C.) of Vectors 34
2.37 Spanning Set 34
2.38 Basis Set 34
2.39 Some Useful Theorems of Linear Algebra 35
4. Simultaneous Linear Equations 35
2.40 Simultaneous Linear Equations 35
5. Finite Difference 36
2.41 First Difference off (x) 36
2.42 Second Difference of f(x) 37
2.43 Conditions for a Maximum or Minimum of f(x) 37
6. Differentiation of Integrals 37
7. Generating Functions 38
¤ Exercise on Chapter 2 40
(WO
. Inventory Theory 41-114
3.1 Inventory 41
3.2 Variables in Inventory Problems 41
3.3 Need of Inventory 43
3.4 Inventory Problems 43
3.5 Advantages and Disadvantages of Inventory 43
3.6 Classification or Categories of Inventory Models 43
3.7 Some General Notations Used in Inventory Models 43
Deterministic Models 44
3.8 Economic Lot Size Models 44
3.9 Model I : Economic Lot Size Model with Uniform Rate of Demand
Infinite Production Rate and having no Shortages 44
3.10 Another form of Model I 46
3.11 Model II : Economic Lot-size Model with Different Rates of Demand
in Different Production Cycles, Infinite Production Rate and having
no Shortages 47
3.12 Model III : Economic Lot-size Model with Uniform Rate of Demand,
Finite Rate of Replenishment having no Shortages 48
Deterministic Models With Shortages 50
3.13 Model IV : Fixed Time Model 50
3.14 Model V : Economic Lot-size Model with Uniform Rate of Demand,
Infinite Rate of Production and Having Shortages which are
to be Fulfilled. 52
3.15 Model VI : Economic Lot-size Model with Uniform Rate of Demand,
Finite Rate of Production and having Shortages which are to be
Fulfilled 55
3.16 Multi Item, Deterministic Models with One Constant 59
3.17 Probabilistic Models 77
3.18 Model VII : Single Period Model with Discontinuous or
Instantaneous Demand and Time Independent Costs
(No Set up Cost Model) 77
3.19 Model VIII : Single Period Model with Uniform Demand.
(No Set up Cost Model) 80
3.20 Model IX : The General Single Period Model of Profit Maximization
with Time Independent Cost 85
3.21 Model X : Probabilistic Order Level System with Lead-Time 87
3.22 Purchase Inventory Models with Price Breaks 100
3.23 Model XI : Purchase Inventory Model 100
3.24 Model XII : Purchase Inventory Model with One Price Break 101
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3.25 Model XIII : Purchase Inventory MIode with Two Pr ice Breaks 103
3.26 Model XIV : Purchase Inventory Model with Multiple Price Breaks 104
• Exercise on Chapter 3 112
4. Replacement Problems 1 1 5- 1 60
4.1 Introduction 115
4.2 Replacement of Major or Capital Item (Equipment) that
Deteriorates with Time 115
4.3 To find the Best Replacement Age (Time) of a Machine when
(i) Its Maintenance Cost is given by a Function Increasing with Time
(ii) Its Scrap Value is Constant and
(iii) The Money Value is not Considered 115
4.4 Few important Terms 124
4.5 To Determine the Best Replacement Age of Items whose
Maintenance Costs Increase with Time and the Value of Money
also Changes with Time 125
4.6 A Discounted Cost P(n) is Invested by taking Loan at the Interest
Rate r; and the Loan is Repaid by Fixed Annual Payments say x,
throughout the Life of the Machine. To Find the Minimum Value
of x for Optimum Period n at which to Replace the Machine 128
4.7 Replacement of Items in Anticipation of Complete Failure the
Probability of which increases with Time 137
4.8 To determine the Interval of Optimum Replacement 138
4.9 Problems in Mortality 139
4.10 Staffing Problem 142
4.11 Mortality Tables 142
• Exercise on Chapter 4 153
5. Waiting Line or Queuing Theory 161-234
5.1 Introduction 161
5.2 Basic Queuing Process (system) and Its Characteristics 161
5.3 Customers Behaviour in a Queue 162
5.4 Important Definitions in Queuing Problem 163
5.5 The State of the System 163
5.6 Poisson Process 164
5.7 Poisson Arrivals 165
5.8 Theorem 166