OPERATIONSRESEARCH

Principles and Practice

PRADEEP PRABHAKAR PAIAssociate Professor

Chetana's Institute of Management and ResearchMumbai

OXFORDUNIVERSITY PRESS

ContentsFeatures of the Book iv

Preface vi

Brief Contents x

1. Operations Research: An Introduction 1Introduction /Origin of Operations Research 3

Historical Standpoint 3

Methodology of Operations Research 4

Conclusion 5Case Study: Janmarg Overview 6

2. Assignment Problem 8Introduction 8

Schematic Introduction to AssignmentProblems 9

Understanding the Logic orthe 'Why Part 9

Procedure to be Followed 10Problems Involving Blocked !

Allocations 13Problem of Imbalance 13Hungarian Assignment Method 15Alternate Optima 15Maximization Problem 16Travelling Salesman Problem 18Avoiding Common Mistakes while

Solving Assignment Problems 22Conclusion 22Solved Problems 22Annexure 1: Solution to Assignment

Problems Using Microsoft Excel Solver 32

3. Transportation ProblemIntroduction 39Solving Transportation Problems 40Complications to Basic Transportation

Problem 46

39

When Initial Feasible Solution Is NotOptimal 47

Unbalanced TransportationProblem 49

Multiple Alternate Solutions/AlternateOptimal Solution 53

Degeneracy in TransportationProblems 53

Maximization Problem 56Transshipment Problem 59Inventory Control Problems 62

Solved Problems 63Avoiding Common Mistakes while

Solving Transportation/Transshipment Problems 81

Annexure 2: Solution to TransportationProblems Using Microsoft Excel Solver 92

4. Linear Programming Problem 101Introduction 101Basic Assumptions 102

Formulation of LPP 103Production Allocation Problem 104Agriculturist's Yield Maximization

Problem 105Problems where Profit or Cost

Needs to be Calculated 106Blending Problem 107Production Scheduling Problems 108

General Statement of LPP " 111

Graphical Solution to LPP 112Procedure 112Limitations 115Complications in LPP and Their

Effects on Graphical Solution 116

XII DETAILED CONTENTS

Simplex Method for Solving LPP 118Requirements for Application

of Simplex Algorithm 119Elements in the Simplex Table 121Simplex Table Structure 122Readings from the Simplex Table 122Steps to be Performed in Iterating

Towards the Optimal Solution 123Using the Optimal Table for

'What-if' Analysis 127Minimization Problems 131

Big M Method 131Two-phase Method 135

Complications Encountered whileUsing Simplex Method 138

Unrestricted Variables 139Operational Difficulties 140Multiple Optimal Solutions 142Infeasible Solution 143Unbounded Solution 144Degeneracy 145

Cycling 146

Sensitivity Analysis 146Changes in Objective Coefficient (c) 148Introducing a New Product 149Changes in b. Values or RHS

Ranging 150Changes in Technology Coefficient

(*,) 151Deletion of Decision Variable 152Deletion of Constraint 153Sensitivity Analysis for Minimization

Problems 153Primal and Dual 155

Obtaining Dual from Primal 155Symmetrical Relationship Between« Primal and Dual 156Economic Interpretation of Dual , 163

Specially Structured LPPs 763Transportation Problem 163Assignment Problem 164

Avoiding Common Mistakeswhile Solving LPPs 765

Conclusion 765

Solved Problems 766'Annexure 3: Solution to LPPs Using

Microsoft Excel Solver 187

5. Extension of Linear ProgrammingProblem 195Introduction 7.95Parametric Linear Programming 7.95

Parametric Cost (or Profit) Problem 7.96'Parametric RHS Constraints 200

Dual Simplex Method 205Application 206Stages 206

Goal Programming 210Essential Steps 270Models with a Single Goal 210Models with Multiple Goals 212Non-preemptive Goal Programming 213Preemptive Goal Programming 27 o'Solution by Graphical Method 27.9

Integer Programming Problems 221Pure and Mixed IPPs 222Gomory's Cutting Plane Method 222Zero-One Model of IPP 237Branch and Bound Method for

Solving IPP 235

Avoiding Common Mistakes whileSolving Extension of LPPs 242

Conclusion 243

Solved Problems 243

Annexure 4: Solution to IPPs UsingMicrosoft Excel Solver 271

6. Sequencing ModelsIntroduction 277

Sequencing Problem 278Gantt Chart 279

• Methods of Sequencing 279Johnson's Algorithm 280Johnson's Algorithm for a Three-machine

Problem 285Johnson's Algorithm for an M-machine

Problem 285

277

DETAILED CONTENTS xiii

Processing Two Jobs ThroughM Machines 288

Avoiding Common Mistakes whileSolving Sequencing Problems 2.97

Conclusion 291Solved Problems 291

7. Inventory Management 299Introduction 299

Classification of InventoryCategories 300

Independent Demand InventorySystems 301

Fixed Order or Q Systems 307Inventory Costs 307Basic EOQ Model for Retailers 303EOQ for Manufacturers 373Periodic Review System 375(s, S) System of Inventory

Management 376^Dependent Demand Inventory Systems 377

MRP System 317JIT System 317Inventory Classification Systems 318Analysis on the Basis of Consumption

Value 375HML Analysis 320 !

FSND Analysis 320VED Analysis 327SDE and GOLF Analyses 327S-OS Analysis 327XYZ Analysis 322

Avoiding Common Mistakes while SolvingInventory Management Problems 322

Conclusion 322Solved Problems 323

8. Dynamic ProgrammingIntroduction 335

Steps Involved in DynamicProgramming Problems 336"

Unique Characteristics ofDynamic Programming 337

Explanation of DynamicProgramming Problem 338

335

Formulation of Dynamic ProgrammingProblem 343

Deterministic and ProbabilisticDynamic Programming Problems 344

Solution of Linear ProgrammingProblems by DynamicProgramming 344

Avoiding Common Mistakes whileSolving Dynamic ProgrammingProblems 346

Conclusion 346

Solved Problems 347

9. Queuing Theory 363Introduction 363

Basis of Queuing Theory 364Models 365

Elements of a QueuingSystem \366

Kendall's Notation 368Operating Characteristics of a Queuing

System 369Waiting and Server Idle Time costs 370Classification of Queuing Models 377

Deterministic Queuing Model 377Probabilistic Queuing Model 373Mixed Queuing Model 383

Avoiding Common Mistakes When .Solving Queuing Problems 383

Simulation 383Conclusion 384Solved Problems 384Formulae at a Glance for Different

Queuing Models 392

10. Game TheoryIntroduction 399

Background 400Characteristics of Game Theory

Applications 400Methodology 401Steps Involved in Identifying the Saddle

Point ^03Rule of Dominance 405

399

XIV DETAILED CONTENTS

Mixed Strategies for 2 x 2 games 406Arithmetic Mean, Method of Odds, or Short-

cut Method 406Algebraic Method for Finding Optimum

Strategies 410

Mixed Strategies for 2 x « games or.m x. 2 games 477

Algebraic Method 411Graphical Method -#73Method of Sub-games 417

Mixed Strategies for 3 x 3 or LargerGames 419

Method of Matrices or OddmentsMethod 419

Method of Linear Programming 422Iterative Method of Approximate

Solution 430Avoiding Common Mistakes while

Solving Game Theory Problems 432Conclusion 433Solved Problems 433

11. Replacement Theory 454Introduction 454Replacement Policy for Items that

Deteriorate Over Time 455While Not Considering Salvage

Value 456Important Deductions 457- •While Considering Salvage Value 457

Replacement Policy When Time Valueof Cash Flows is Considered 461

Replacement of Items that FailSuddenly 465

Group Replacement 466Individual Replacement 466

/ Failure Tree 466Mortality and Staff Replacement

Problems 477Avoiding Common Mistakes while

Solving Replacement TheoryProblems 475

Conclusion 475Solved Problems 476

12. Network Analysis 493Introduction 493

Work Breakdown Structure 494

Project Management 495

Parameters for Success of a Project 495

Network Analysis 495

Critical Path Activities 496Constructing a Network 496Identifying the Critical Path 507Float Analysis 503Calculation Considerations " 505

Program Evaluation and ReviewTechnique Analysis 57 7

Crashing Analysis 577Activity on Node Analysis 537

Activity on Node Convention 532Activity on Node Network

Relationships 532Float Calculations in Activity on

Node Network 536Resource Scheduling 547Avoiding Common Mistakes while

Solving Network Analysis Problems 550Conclusion 550Solved Problems 557

13. Simulation 582Introduction 582

Background 583Monte Carlo Simulation 584Selecting Random Numbers for

Experimentation 586Simulation using Excel Spreadsheet 589Simulation of an Inventory System 591Simulation of a Queuing System 596Avoiding Common Mistakes while

Solving Simulation Problems 598Conclusion 598Solved Problems 599

14. Markov ChainsIntroduction 613

Uses of Markov Chains

613

674

DETAILED CONTENTS

Background 674Stochastic Process 675Gambler's Ruin Problem 616Share Price Fluctuation 676^

Markov Process 676^Finite States 677Recent Order Processes 677Constancy 677Uniform Periodicity of Time Periods 618

Absorbing Chains 618Possible Input and Output

Parameters in Markovian Analysis 619Transition Probabilities 619Initial Condition 67.9Steady State Probability 620Specific State Probability 622 .

Analysis of Absorbing Chains 625Avoiding Common Mistakes while

Solving Markov Chain Analysis.Problems 626 .

Conclusion 627Solved Problems 627

15. Forecasting 644

Introduction 644

Forecasting Models 646

Qualitative Forecasting Techniques 646Customer Surveys 646Sales Force Composite 647Expert Opinion 647Delphi Technique 647Past Sales Analogy 647

Forecasting using Time Series 648Simple Moving Average 648Weighted Moving Average 648

j, Simple Exponential Smoothing 648Double Exponential Smoothing 650Forecasting by Linear Regression 657

Forecasting for Causal Series 654Simple Regression Analysis 654Multiple Regression Analysis 655

Errors in Forecasting 657Forecast Control Limits 660Goodness of Fit 661

Avoiding Common Mistakes whileSolving Forecasting Problems , 663

Conclusion 663Solved Problems 663

16. Decision Theory 672Introduction 672Decision-making Under Conditions of

Uncertainty 673Developing the Payoff and Regret

Tables 674Decision Rules 675

Decision-making Under Conditionsof Risk 682

Maximum Likelihood principle 682Expected Value Criterion 683Expected Opportunity Loss

Criterion 684Expected Value of Perfect

Information 6S5Expected Monetary Value While Considering

Salvage Cost 690Marginal Analysis Method for Continuously

Distributed Random Variable • 693Posterior Analysis and Bayesian

Approach to Decision-making 695Decision Trees in Decision-making 659Utility Theory as Basis for

Decision-making 705Assumptions in Utility Theory 706Von Neumann and Morgenstern

Method of Measuring Utility 707Standard Gambling Technique for Measuring

Utility 708Logarithmic Utility Function for

Measuring Utility 708Avoiding Common Mistakes while

Solving Decision Analysis Problems 777Conclusion 772Solved Problems 772

17. Investment Analysis 730Introduction 730

Overview of Situations and Methods 737

Break-even Analysis 737

DETAILED CONTENTS

Assumptions in Break-even Analysis 732Calculation of Break-even Point 732Margin of Safety 735Sensitivity Analysis 735Volume-Profit Graph 736Break-even Analysis for

Multiproduct Situations 736Payback Period Method 738Average Rate of Return Method 739Net Present Value Method 740

Net Present Value with Annuities 742Equivalent Annual Annuity Method 744Internal Rate of Return Method 744

Extrapolation Method 745Comparison of NPV and IRR

Methods of Analysis 746Discounted Payback Period Method 748Profitability Index Method 749Benefit Cost Ratio and Net benefit

Cost Ratio Methods 749Common Time Horizon Method 750Probabilistic Situations 757Risk-adjusted Discount Rate Method 757Certainty Equivalent Approach 752Expected Monetary Value Method 754Hillier and Hertz's Model ' 755

Avoiding Common Mistakes whileSolving Investment AnalysisProblems 758

Conclusion 759

Solved Problems 759

18. Introduction to Non-linearProgramming Problems 771Introduction 777Examples of Non-linear Programming

Problems 773Product Mix Problem 773Transportation Problem 774Portfolio Selection of Securities 775

Concave and Convex Functions 778Graphical Illustration of Non-linear

Programming Problems 782

Types of Non-linear ProgrammingProblems 786

Unconstrained Optimization 786Linearly Constrained Optimization 789Quadratic Programming 790Convex Programming 794Separable Problem 794Non-convex Programming 795

Conclusion 795

Appendix: Statistical and Financial Tables

Bibliography 822

Index 824

'797