PROBLEMS INOPERATIONS RESEARCH

(Principles and Solutions)For Engineering, Computer Science, Commerce, Management,

Economics, Statistics, Mathematics, C.A., I.C.W.A., C.S.Also Useful for I.A.S. and Other Competitive Examinations

PREM KUMAR GUPTAB.Sc. Mech. Engg. (Hons.), M.Sc. Mech. Engg. (Distinction), M.I.E. (India)

Formerly Assistant ProfessorP.E.C. Institute of Engineering & Technology (Deemed University), Chandigarh

Dr. D.S. HIRAB.Sc. Mech. Engg., M.Sc. Mech. Engg., Ph.D. (Roorkee)

Director GeneralSwami Vivekanand Group of Institutes, Banur, Distt. Patiala, Punjab

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© 1991, Prem Kumar Gupta & Dr. D.S. HiraAll rights reserved. No part of this publication may be reproduced or copied in any material form (including photo copying or storingit in any medium in form of graphics, electronic or mechanical means and whether or not transient or incidental to some otheruse of this publication) without written permission of the copyright owner. Any breach of this will entail legal action and prosecutionwithout further notice.Jurisdiction : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts, tribunals and forumsof New Delhi, India only.First Edition 1991Subsequent Editions and Reprints : 1993, 95, 98, 99, 2000, 2001, 2002, 2003, 2005, 2007, 2008, 2009, 2010, 2012(Twice), 2013Fourth Edition : 2015

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iii

PREFACE TO THE FOURTH EDITIONIt is our great pleasure to present to the readers the fourth thoroughly revised edition of the

book after a number of reprints. Every effort has been made to incoporate the valuable suggestionsreceived from a number of readers. All errors and omissions have been rectified. A number oftypical solved examples have been added. A large number of exercises from the latest examinationpapers have been included and the book now covers questions up to year 2013 examinations. Thebook now contains 644 solved examples, 1695 exercises and 257 illustrations.

We are indebted to the Publishers S.Chand & Company Pvt. Ltd. for their sincere efforts,unfailing courtesy and cooperation in bringing out the book in this elegant form.

Suggestions for further improvement of the book will be highly appreciated and thankfullyacknowledged.

Prem Kumar GuptaDr. D.S. Hira

iv

PREFACE TO THE FIRST EDITION

The authors have been feeling the need of a book on Operations Research that wouldcontain a large number of problems with complete solutions. The overwhelming response to ourbook ‘Operations Research’ has prompted us to write and place in the hands of the readers thepresent volume. The book has been designed and tailored to meet the long standing requirementsof those appearing in Engineering, C.A., I.C.W.A., I.C.M.A., I.A.S., M.B.A., M.M.S., M.Com.,M. Math. & Stat., B. Com. and B. Stat. examinations.

Efforts have been made to simplify the technical material without distorting it. The bookdoes not require a high level of mathematical knowledge on the part of the reader. An elementaryknowledge of integral and differential calculus and matrix algebra is all that is required tounderstand the subject.

While the main emphasis has been in each chapter (with the exception of the first whichdeals with ‘basics of OR’) on problems and their solutions, brief theory and related principleshave also been provided to enable the readers to understand the solution procedures. Each chaptercontains a large number of important and interesting problems taken from a variety of fields.Almost each problem presents a new idea. Unsolved exercises (with answers) at the end of eachchapter are provided to test the reader’s understanding of the subject matter. The book containsaround 480 completely solved examples and 490 unsolved exercises.

Every effort has been made to present the material in an easy, clear, lucid and systematicmanner. Though all efforts have been made to make the book reasonably exhaustive andcomprehensive, there still may be ways in which the presentation can be further improved.Valuable suggestions for further improvement of the book will be gratefully accepted.

AUTHORS

v

CONTENTS

Chapters Pages1. BASICS OF OPERATIONS RESEARCH 1–37

1.1 Development of Operations Research 11.2 Definition of Operations Research 41.3 Characteristics of Operations Research 51.4 Scientific Method in Operations Research 71.5 Necessity of Operations Research in Industry 81.6 Scope of Operations Research 91.7 Operations Research and Decision-Making 101.8 Scope of OR in Management 111.9 Scope of OR in Financial Management 12

1.10 Applications of Various OR Techniques 131.11 Objectives of Operations Research 131.12 Phases of OR 141.13 Models in OR 181.14 Classification Schemes of Models 181.15 Characteristics of a Good Model 211.16 Advantages of a Model 211.17 Limitations of a Model 211.18 Constructing the Model 211.19 Approximations (Simplifications) in OR Models 231.20 Types of Mathematical Models 241.21 Role of Computers in Operations Research 311.22 Difficulties in Operations Research 321.23 Limitations of Operations Research 33

Exercises (1 to 55) 33

2. LINEAR PROGRAMMING 38–2272.1 Introduction 382.2 Formulation of Linear Programming Problems 38

Exercises 2.1 (1 to 68) 792.3 Graphical Method of Solution 1042.4 Some Exceptional Cases 117

Exercises 2.2 (1 to 71) 1232.5 The General Linear Programming Problem 1352.6 Canonical and Standard Forms of Linear Programming Problem 1362.7 Theory of Simplex Method 1412.8 Some Important Definitions 1432.9 Analytical Method or Trial and Error Method 144

Exercises 2.3 (1 to 18) 1502.10 The Simplex Method (Technique or Algorithm) 1522.11 Artificial Variables Techniques 165

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2.11-1 The Big M-Method 1652.11-2 The Two-Phase Method 175

2.12 Special Cases in the Simplex Method Application 1862.13 Solution of Simultaneous Equations by Simplex Method 2052.14 Some Additional Points 208

Exercises 2.4 (1 to 117) 208

3. THE TRANSPORTATION MODEL 228–3343.1 Introduction to the Model 2283.2 Definition of the Transportation Model 2283.3 Matrix Terminology 2303.4 Formulation and Solution of Transportation Models 2303.5 Variants in Transportation Problems 2643.6 Additional Problems 2813.7 Least-Time Transportation Problems 2903.8 Post Optimality Analysis in Transportation 2923.9 The Transshipment Problem 299

3.10 Dual of the Transportation Problem 301Exercises (1 to 113) 304

4. THE ASSIGNMENT MODEL 335–4194.1 Definition of the Assignment Model 3354.2 Mathematical Representation of the Assignment Model 3354.3 Comparison with the Transportation Model 3364.4 The Hungarian Method for Solution of the Assignment Problems 3364.5 Formulation and Solution of the Assignment Models 3374.6 Variations of the Assignment Problem 3474.7 Additional Problems 363

Exercises 4.1 (1 to 81) 3724.8 The Travelling Salesman Problem (Shortest Cyclic Route Models) 397

Exercises 4.2 (1 to 20) 414

5. SEQUENCING MODELS AND RELATED PROBLEMS 420–4665.1 Sequencing Problems 4205.2 Assumptions in Sequencing Problems 4205.3 Processing n Jobs through one Machine 4215.4 Processing n Jobs through two Machines 4265.5 Processing n Jobs through three Machines 4325.6 Processing two Jobs through m Machines 4375.7 Processing n Jobs through m Machines 4435.8 Problems related to Sequencing (Routing Problems in Networks) 4485.9 Minimal Path Problem (Shortest Acyclic Route Models) 448

Exercises (1 to 75) 451

6. ADVANCED TOPICS IN LINEAR PROGRAMMING 467–6336.1 Duality in Linear Programming 467

Exercises 6.1 (1 to 67) 4886.2 The Dual Simplex Method 500

Exercises 6.2 (1 to 15) 508

vii

6.3 The Revised Simplex Method 510Exercises 6.3 (1 to 14) 526

6.4 The Bounded Variables Problem 528Exercises 6.4 (1 to 9) 532

6.5 The Decomposition Method 5336.6 Sensitivity Analysis 534

Exercises 6.6 (1 to 49) 5566.7 Parametric Linear Programming 566

Exercises 6.7 (1 to 20) 5776.8 Goal Programming 580

Exercises 6.8 (1 to 21) 5906.9 Linear Fractional Programming 595

Exercises 6.9 (1 to 4) 5966.10 Integer Programming 596

Exercises 6.10 (1 to 48) 628

7. DYNAMIC PROGRAMMING 634–6787.1 Introduction 6347.2 Optimal Subdivision Problem 6547.3 System Reliability 6667.4 Solution of L.P.P. by Dynamic Programming 668

Exercises (1 to 39) 671

8. DECISION THEORY, GAMES 679–8158.1 Decision Theory 679

8.1-1 Steps in Decision Theory Approach 6798.1-2 Decision-Making Environments 6798.1-3 Decision-Making under Conditions of Uncertainty 6808.1-4 Decision-Making under Conditions of Risk 6848.1-5 Use of Incremental (Marginal) Analysis 6908.1-6 Expected Value Criterion for Continuously Distributed Random Variables 6928.1-7 Additional Examples 692

Exercises 8.1 (1 to 55) 7048.2 Decision Trees 717

Exercises 8.2 (1 to 14) 7388.3 Utility Theory 742

Exercises 8.3 (1 to 7) 7478.4 The Theory of Games 748

8.4-1 Characteristics of Games 7498.4-2 Definitions 7498.4-3 Rule 1. Look for a Pure Strategy (Saddle Point) 7528.4-4 Rule 2. Reduce Game by Dominance 7568.4-5 Rule 3. Solve for a Mixed Strategy 7578.4-6 Mixed Strategies (2 × 2 Games) 7588.4-7 Mixed Strategies (2 × n Games or m × 2 Games) 7698.4-8 Mixed Strategies (3 × 3 or Higher Games) 7808.4-9 n-Person Zero-Sum Games 794

8.4-10 Bidding Problems 796Exercises 8.4 (1 to 98) 798

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9. INVESTMENT ANALYSIS AND ANNUITIES 816–8459.1 Introduction to Investment Analysis 8169.2 Methods of Investment Analysis 816

9.2-1 Break-Even Analysis 8169.2-2 Payback Period Method 8229.2-3 Average (Accounting) Rate of Return Method 8249.2-4 Time-Adjusted or Discounted Cash Flow (DCF) Methods 8249.2-5 Net Present Value (NPV) Method 8259.2-6 Internal Rate of Return (IRR) Method 8269.2-7 Discounted Payback Period Method 827

9.3 Probabilistic Models 8289.3-1 Risk Adjusted Discount Rate 8289.3-2 Certainty-Equivalent Approach 8299.3-3 Expected Monetary Value (EMV) 8309.3-4 Hillier and Hertz’s Models 830

Exercises 9.1 (1 to 24) 8359.4 Annuity 839

Exercises 9.2 (1 to 12) 844

10. QUEUING MODELS 846–90710.1 Introduction 84610.2 Elements of a Queuing System 84710.3 Operating Characteristics of a Queuing System 84810.4 Waiting Time and Idle Time Costs 84910.5 Transient and Steady States of the System 85010.6 Kendall’s Notation for Representing Queuing Models 85010.7 Models for Arrival and Service Times 851

Exercises 10.1 (1 to 3) 85210.8 Model 1. Single-Channel Poisson Arrivals with Exponential Service

Times, Infinite-Population Model (M/M/1): (FCFS/∞/∞) 85210.9 An Explanatory Note on the Queuing Formulae 854

Exercises 10.2 (1 to 56) 87210.10 Model II. Generalisation of Model (M/M/1): (FCFS/∞/∞) 881

(Birth-Death Process)10.11 Model III. Single-Channel Poisson Arrivals, Exponential Service,

Infinite-Population, Service in Random Order Model 883(M/M/1): (SIRO/∞/∞)

10.12 Model IV. (M/M/1): (FCFS/N/∞) Finite Queue Length Model 884Exercises 10.3 (1 to 8) 886

10.13 Model V. Single-Channel, Finite-Population Model with 887Poisson Arrivals and Exponential Service Times (LimitedSource Model) (M/M/1): (FCFS/n/M)Exercises 10.4 (1 to 2) 888

10.14 Model VI. Multi-Channel Queuing Theory Model (M/M/C): (FCFS/∞/∞) 888Exercises 10.5 (1 to 11) 898

10.15 Erlang Family Distribution 900

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10.16 Model VII. (M / Ek /1) : (FCFS ∞ /∞ ) [Multi-Phase Service Model] 900Exercises 10.6 (1 to 12) 903

10.17 Deterministic Models 90510.18 Model XI. D/D/1 90510.19 Model XII. M/D/1 906

11. REPLACEMENT MODELS 908–94811.1 Introduction 90811.2 Replacement of Items Whose Maintenance and Repair 908

Costs Increase with Time, Ignoring Changes in the Value of MoneyDuring the PeriodExercises 11.1 (1 to 21) 916

11.2.1 Replacement of Items Whose Maintenance Costs 920Increase with Time and Value of Money also Changes with TimeExercises 11.2 (1 to 15) 927

11.3 Replacement of Items that Fail Suddenly 93011.4 Group Replacement Policy 930

Exercises 11.3 (1 to 23) 93611.5 Mortality and Staffing Problems 94111.6 Miscellaneous Replacement Problems 94311.7 Renewal Theory 946

Exercises 11.4 (1 to 4) 947

12. INVENTORY MODELS 949–102312.1 Necessity for Maintaining Inventory 94912.2 Inventory Costs 94912.3 Inventory Control Problem 95012.4 Classification of Fixed Order Quantity Inventory Models 95112.5 Inventory Models with Deterministic Demand 951

12.5-1 Model 1(a). Classical EOQ Model (Demand Rate Uniform, 951Replenishment Rate Infinite)Exercises 12.1 (1 to 43) 964

12.5-2 Model 1(b). (Demand Rate Non-uniform, Replenishment Rate Infinite) 96812.5-3 Model 1(c). (Demand Rate Uniform, Replenishment or Production

Rate Finite) 96912.5-4 Model 2(a). (Demand Rate Uniform, Replenishment Rate Infinite,

Shortages Allowed) 97112.5-5 Model 2(b). (Demand Rate Uniform, Production Rate Finite,

Shortages Allowed) 972Exercises 12.2 (1 to 16) 975

12.6 Inventory Models with Probabilistic Demand 978Exercises 12.3 (1 to 13) 986

12.7 Inventory Models with Price Breaks 988Exercises 12.4 (1 to 25) 992

12.8 Multi-Item Deterministic Model 995Exercises 12.5 (1 to 4) 1001

12.9 Forecasting of Demand 100212.10 Forecasting Methods 1002

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12.11 When to Order 100512.12 Selective Inventory Management Techniques 1013

Exercises 12.6 (1 to 34) 1019

13. SIMULATION 1024–107413.1 Introduction 102413.2 What is Simulation ? 102413.3 Monte Carlo Simulation 102513.4 Generation of Random Numbers 1053

Exercises (1 to 50) 1064

14. NETWORK ANALYSIS IN PROJECT PLANNING (PERT AND CPM)1075–1184

14.1 Introduction 107514.2 Phases of Project Management 107514.3 Work Breakdown Structure (W.B.S.) 107614.4 Basic Tools and Techniques of Project Management 107614.5 Network Logic (Network or Arrow Diagram) 107714.6 Numbering the Events (Fulkerson’s Rule) 107914.7 Activity on Node Diagram 1084

Exercises 14.1 (1 to 17) 108614.8 Critical Path Method 1089

Exercises 14.2 (1 to 42) 109814.9 Programme Evaluation and Review Technique (PERT) 1108

Exercises 14.3 (1 to 38) 112214.10 Cost Analysis and Crashing the Network 1132

Exercises 14.4 (1 to 36) 114914.11 Resource Scheduling 116114.12 Updating 1175

Exercises 14.5 (1 to 14) 117814.13 Applications of Network Techniques 118214.14 Distinctions Between PERT and CPM 118314.15 Linear Programming Formulation 1183

Exercises 14.6 (1 to 5) 1184

15. STATISTICAL QUALITY CONTROL 1185–121815.1 Definition 118515.2 Causes of Variation in Quality 118515.3 Techniques of SQC 118615.4 Control Charts 118615.5 Control Charts for Variables 118715.6 Control Charts for Attributes 119815.7 Product Control 121115.8 Acceptance Sampling 121115.9 Single Sampling Plan 1212

15.10 Double Sampling Plan 121315.11 Multiple or Sequential Sampling Plan 1213

Exercises (1 to 41) 1214

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16. NON-LINEAR PROGRAMMING 1219–129016.1 Introduction 121916.2 Problem Formulation Examples 121916.3 Local and Global Optimum 122216.4 Concave and Convex Functions 122316.5 Constrained Extremal Problems 123216.6 Constrained Extremal Problem with more than one Equality Constraint 124116.7 Non-Linear Programming Problem with Inequality Constraints 124516.8 Non-Linear Programming Problem with more than One Inequality

Constraint 124916.9 The Graphical Method 1254

16.10 One-Variable Unconstrained Optimization 126216.11 One-Dimensional Search Procedure 126316.12 Multi-Variable Unconstrained Optimization 126716.13 The Gradient Search Procedure 126716.14 Quadratic Programming Problem 127816.15 Wolf’s Modified Simplex Method 1278

Exercises (1 to 50) 1284Table C-1 1291Table C-2 1292–1293

xii

This chapter provides an overall view of the subject of operations research. It covers somegeneral ideas on the subject, thus providing a perspective. The remaining chapters deal withspecific ideas and specific methods of solving OR problems.

1.1 DEVELOPMENT OF OPERATIONS RESEARCH

(i) Pre-World War II: No science has ever been born on a specific day. Operations researchis no exception. Its roots are as old as science and society. Though the roots of OR extend to evenearly 1800s, it was in 1885 when Ferderick W. Taylor emphasised the application of scientificanalysis to methods of production, that the real start took place. Taylor conducted experiments inconnection with a simple shovel. His aim was to find that weight load of ore moved by shovelwhich would result in maximum of ore moved with minimum of fatigue. After many experimentswith varying weights, he obtained the optimum weight load, which though much lighter than thatcommonly used, provided maximum movement of ore during a day.

Another man of early scientific management era was Henry L. Gantt. Most job-schedulingmethods at that time were rather haphazard. A job, for instance, may be processed on a machinewithout trouble but then wait for days for acceptance by the next machine. Gantt mappedeach job from machine to machine, minimizing every delay. Now, with the Gantt procedureit is possible to plan machine loadings months in advance and still quote delivery datesaccurately.

The well-known economic lot size model is attributed to F.W. Harris, who published hiswork on the area of inventory control in 1915.

In 1917, A.K. Erlang, a Danish mathematician, published his work on the problem ofcongestion of telephone traffic. The difficulty was that during busy periods, telephone operatorswere unable to handle the calls the moment they were made, resulting in delayed calls. A fewyears after its appearance, his work was accepted by the British Post Office as the basis forcalculating circuit facilities. The formulae he developed on waiting time are of fundamentalimportance to the theory of telephone traffic.

During the 1930s, H.C. Levinson, an American astronomer, applied scientific analysis to theproblems of merchandising. His work included scientific study of customers’ buying habits,response to advertising and relation of environment to the type of article sold.

However, it was the First Industrial Revolution which contributed mainly towards thedevelopment of OR. Before this revolution, most of the industries were small scale, employingonly a handful of men. The advent of machine tools — the replacement of man by machine as asource of power and improved means of transportation and communication resulted in fastflourishing industry. It became increasingly difficult for a single man to perform all the managerialfunctions (or planning, sale, purchase, production, etc.). Consequently, a division of managementdevelopment etc., began to appear. With further industrial growth, further subdivisions of

1

BASICS OF OPERATIONS

RESEARCH

CHAPTER

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1

❚ ❙ ❘ 2 ❘ ❙ ❚ Problems in Operations Research

management functions took place. For example, production department was sub-divided intosections like maintenance, quality control, procurement, production planning, etc.

This industrial development, brought with it, a new type of problems called executive-typeproblems. These problems are a direct consequence of functional division of labour in anorganization. In an organization, each functional unit (department or section) performs a part ofthe whole job and for its successful working, develops its own objectives. These objectives,though in the best interest of the individual department, may not be in the best interest of theorganization as a whole. In fact, these objectives of individual departments may be inconsistentand clashing with each other. Consider, for example, the attitudes of the various departments ofa business organization towards the inventory policy. The production department wants to havemaximum production, associated with the lowest possible cost. This can be achieved by producingonly one item continuously. Thus it is interested in long, uninterrupted production runs, becausesuch runs minimise set-up and clean-up costs. Thus it prefers to have a large inventory inrelatively few product lines.

The marketing department also wants a large but diverse inventory so that a customer maybe provided immediate delivery over a wide variety of products. It would also like to have aflexible production policy so as to meet special demands at a short notice.

The finance department wants to minimize inventory so as to minimize the unproductivecapital investments ‘tied up’ in it. Funds could be used elesewhere for better returns. It alsobelieves that inventories should rise and fall with rise and fall in company’s sales.

The personnel department wants to hire good labour and to retain it. This is possible onlywhen goods are produced continuously for inventory during slack periods also. In other words, itis interested in maintaining a constant production level resulting in large inventory.

To set an inventory policy which serves the interest of the organization as a whole and notthat of any individual department is an executive-type problem, which can be satisfactorily solvedby the application of OR techniques. The decision which is in the best interest of the organizationas a whole is called Optimal (optimum or global optimum) decision and the one in the best interestof an individual department is called sub-optimal decision.

(ii) World War II: During World War II, the military management in England called on ateam of scientists to study the strategic and tactical problems of air and land defence. Theobjective was to find out the most effective allocation of limited military resources to the variousmilitary operations and to the activities within each operation. The application included theeffective use of newly invented radar, allocation of British Air Force Planes to missions and thedetermination of best patterns for searching submarines. This group of scientists formed the firstOR team.

The name operations research (or operational research) was apparently coined in 1940because the team was carrying out research on (military) operations. The encouraging results ofthese efforts led to the formation of more such teams in British Armed Services and the use ofsuch scientific teams soon spread to Western Allies — the United States, Canada and France. Thusthough this science of operations research originated in England, the United States soon took thelead. In United States these OR teams helped in developing strategies for mining operations,inventing new flight patterns and planning of sea mines.

(iii) Post-World War II: Immediately after the war, the success of military teams attractedthe attention of industrial managers who were seeking solutions to their problems. Industrialoperations research in U.K. and U.S.A. developed along different lines. In U.K. the criticaleconomic situation required drastic increase in production efficiency and creation of new markets.Nationalisation of a few key industries further increased the potential field for OR. ConsequentlyOR soon spread from military to government, industrial, social and economic planning.

Problems In Operation Research(Principles And Solution)

Publisher : SChand Publications ISBN : 9788121909686Author : Prem KumarGupta, D. S. Hira

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