OPERATIONS RESEARCH. Principles and Practice PRADEEP PRABHAKAR PAI. Associate Professor Chetana's Institute of Management and Research Mumbai

Transcription

1 OPERATIONS RESEARCH Principles and Practice PRADEEP PRABHAKAR PAI Associate Professor Chetana's Institute of Management and Research Mumbai OXFORD UNIVERSITY PRESS

2 Contents Features of the Book Preface vi Brief Contents x iv 1. Operations Research: An Introduction 1 Introduction / Origin of Operations Research 3 Historical Standpoint 3 Methodology of Operations Research 4 Conclusion 5 Case Study: Janmarg Overview 6 2. Assignment Problem 8 Introduction 8 Schematic Introduction to Assignment Problems 9 Understanding the Logic or the 'Why Part 9 Procedure to be Followed 10 Problems Involving Blocked! Allocations 13 Problem of Imbalance 13 Hungarian Assignment Method 15 Alternate Optima 15 Maximization Problem 16 Travelling Salesman Problem 18 Solving Assignment Problems 22 Conclusion 22 Solved Problems 22 Annexure 1: Solution to Assignment Problems Using Microsoft Excel Solver Transportation Problem Introduction 39 Solving Transportation Problems 40 Complications to Basic Transportation Problem When Initial Feasible Solution Is Not Optimal 47 Unbalanced Transportation Problem 49 Multiple Alternate Solutions/Alternate Optimal Solution 53 Degeneracy in Transportation Problems 53 Maximization Problem 56 Transshipment Problem 59 Inventory Control Problems 62 Solved Problems 63 Solving Transportation/ Transshipment Problems 81 Annexure 2: Solution to Transportation Problems Using Microsoft Excel Solver Linear Programming Problem 101 Introduction 101 Basic Assumptions 102 Formulation of LPP 103 Production Allocation Problem 104 Agriculturist's Yield Maximization Problem 105 Problems where Profit or Cost Needs to be Calculated 106 Blending Problem 107 Production Scheduling Problems 108 General Statement of LPP " 111 Graphical Solution to LPP 112 Procedure 112 Limitations 115 Complications in LPP and Their Effects on Graphical Solution 116

3 XII DETAILED CONTENTS Simplex Method for Solving LPP 118 Requirements for Application of Simplex Algorithm 119 Elements in the Simplex Table 121 Simplex Table Structure 122 Readings from the Simplex Table 122 Steps to be Performed in Iterating Towards the Optimal Solution 123 Using the Optimal Table for 'What-if' Analysis 127 Minimization Problems 131 Big M Method 131 Two-phase Method 135 Complications Encountered while Using Simplex Method 138 Unrestricted Variables 139 Operational Difficulties 140 Multiple Optimal Solutions 142 Infeasible Solution 143 Unbounded Solution 144 Degeneracy 145 Cycling 146 Sensitivity Analysis 146 Changes in Objective Coefficient (c) 148 Introducing a New Product 149 Changes in b. Values or RHS Ranging 150 Changes in Technology Coefficient (*,) 151 Deletion of Decision Variable 152 Deletion of Constraint 153 Sensitivity Analysis for Minimization Problems 153 Primal and Dual 155 Obtaining Dual from Primal 155 Symmetrical Relationship Between «Primal and Dual 156 Economic Interpretation of Dual, 163 Specially Structured LPPs 763 Transportation Problem 163 Assignment Problem 164 Avoiding Common Mistakes while Solving LPPs 765 Conclusion 765 Solved Problems 766' Annexure 3: Solution to LPPs Using Microsoft Excel Solver Extension of Linear Programming Problem 195 Introduction 7.95 Parametric Linear Programming 7.95 Parametric Cost (or Profit) Problem 7.96' Parametric RHS Constraints 200 Dual Simplex Method 205 Application 206 Stages 206 Goal Programming 210 Essential Steps 270 Models with a Single Goal 210 Models with Multiple Goals 212 Non-preemptive Goal Programming 213 Preemptive Goal Programming 27 o' Solution by Graphical Method 27.9 Integer Programming Problems 221 Pure and Mixed IPPs 222 Gomory's Cutting Plane Method 222 Zero-One Model of IPP 237 Branch and Bound Method for Solving IPP 235 Solving Extension of LPPs 242 Conclusion 243 Solved Problems 243 Annexure 4: Solution to IPPs Using Microsoft Excel Solver Sequencing Models Introduction 277 Sequencing Problem 278 Gantt Chart 279 Methods of Sequencing 279 Johnson's Algorithm 280 Johnson's Algorithm for a Three-machine Problem 285 Johnson's Algorithm for an M-machine Problem

4 DETAILED CONTENTS xiii Processing Two Jobs Through M Machines 288 Solving Sequencing Problems 2.97 Conclusion 291 Solved Problems Inventory Management 299 Introduction 299 Classification of Inventory Categories 300 Independent Demand Inventory Systems 301 Fixed Order or Q Systems 307 Inventory Costs 307 Basic EOQ Model for Retailers 303 EOQ for Manufacturers 373 Periodic Review System 375 (s, S) System of Inventory Management 376^ Dependent Demand Inventory Systems 377 MRP System 317 JIT System 317 Inventory Classification Systems 318 Analysis on the Basis of Consumption Value 375 HML Analysis 320! FSND Analysis 320 VED Analysis 327 SDE and GOLF Analyses 327 S-OS Analysis 327 XYZ Analysis 322 Solving Inventory Management Problems 322 Conclusion 322 Solved Problems Dynamic Programming Introduction 335 Steps Involved in Dynamic Programming Problems 336" Unique Characteristics of Dynamic Programming 337 Explanation of Dynamic Programming Problem Formulation of Dynamic Programming Problem 343 Deterministic and Probabilistic Dynamic Programming Problems 344 Solution of Linear Programming Problems by Dynamic Programming 344 Solving Dynamic Programming Problems 346 Conclusion 346 Solved Problems Queuing Theory 363 Introduction 363 Basis of Queuing Theory 364 Models 365 Elements of a Queuing System \366 Kendall's Notation 368 Operating Characteristics of a Queuing System 369 Waiting and Server Idle Time costs 370 Classification of Queuing Models 377 Deterministic Queuing Model 377 Probabilistic Queuing Model 373 Mixed Queuing Model 383 Avoiding Common Mistakes When. Solving Queuing Problems 383 Simulation 383 Conclusion 384 Solved Problems 384 Formulae at a Glance for Different Queuing Models Game Theory Introduction 399 Background 400 Characteristics of Game Theory Applications 400 Methodology 401 Steps Involved in Identifying the Saddle Point ^03 Rule of Dominance

5 XIV DETAILED CONTENTS Mixed Strategies for 2 x 2 games 406 Arithmetic Mean, Method of Odds, or Shortcut Method 406 Algebraic Method for Finding Optimum Strategies 410 Mixed Strategies for 2x«games or. m x. 2 games 477 Algebraic Method 411 Graphical Method -#73 Method of Sub-games 417 Mixed Strategies for 3 x 3 or Larger Games 419 Method of Matrices or Oddments Method 419 Method of Linear Programming 422 Iterative Method of Approximate Solution 430 Solving Game Theory Problems 432 Conclusion 433 Solved Problems Replacement Theory 454 Introduction 454 Replacement Policy for Items that Deteriorate Over Time 455 While Not Considering Salvage Value 456 Important Deductions 457- While Considering Salvage Value 457 Replacement Policy When Time Value of Cash Flows is Considered 461 Replacement of Items that Fail Suddenly 465 Group Replacement 466 Individual Replacement 466 / Failure Tree 466 Mortality and Staff Replacement Problems 477 Solving Replacement Theory Problems 475 Conclusion 475 Solved Problems Network Analysis 493 Introduction 493 Work Breakdown Structure 494 Project Management 495 Parameters for Success of a Project 495 Network Analysis 495 Critical Path Activities 496 Constructing a Network 496 Identifying the Critical Path 507 Float Analysis 503 Calculation Considerations " 505 Program Evaluation and Review Technique Analysis 57 7 Crashing Analysis 577 Activity on Node Analysis 537 Activity on Node Convention 532 Activity on Node Network Relationships 532 Float Calculations in Activity on Node Network 536 Resource Scheduling 547 Solving Network Analysis Problems 550 Conclusion 550 Solved Problems Simulation 582 Introduction 582 Background 583 Monte Carlo Simulation 584 Selecting Random Numbers for Experimentation 586 Simulation using Excel Spreadsheet 589 Simulation of an Inventory System 591 Simulation of a Queuing System 596 Solving Simulation Problems 598 Conclusion 598 Solved Problems Markov Chains Introduction 613 Uses of Markov Chains

6 DETAILED CONTENTS Background 674 Stochastic Process 675 Gambler's Ruin Problem 616 Share Price Fluctuation 676^ Markov Process 676^ Finite States 677 Recent Order Processes 677 Constancy 677 Uniform Periodicity of Time Periods 618 Absorbing Chains 618 Possible Input and Output Parameters in Markovian Analysis 619 Transition Probabilities 619 Initial Condition 67.9 Steady State Probability 620 Specific State Probability 622. Analysis of Absorbing Chains 625 Solving Markov Chain Analysis. Problems 626. Conclusion 627 Solved Problems Forecasting 644 Introduction 644 Forecasting Models 646 Qualitative Forecasting Techniques 646 Customer Surveys 646 Sales Force Composite 647 Expert Opinion 647 Delphi Technique 647 Past Sales Analogy 647 Forecasting using Time Series 648 Simple Moving Average 648 Weighted Moving Average 648 j, Simple Exponential Smoothing 648 Double Exponential Smoothing 650 Forecasting by Linear Regression 657 Forecasting for Causal Series 654 Simple Regression Analysis 654 Multiple Regression Analysis 655 Errors in Forecasting 657 Forecast Control Limits 660 Goodness of Fit 661 Solving Forecasting Problems, 663 Conclusion 663 Solved Problems Decision Theory 672 Introduction 672 Decision-making Under Conditions of Uncertainty 673 Developing the Payoff and Regret Tables 674 Decision Rules 675 Decision-making Under Conditions of Risk 682 Maximum Likelihood principle 682 Expected Value Criterion 683 Expected Opportunity Loss Criterion 684 Expected Value of Perfect Information 6S5 Expected Monetary Value While Considering Salvage Cost 690 Marginal Analysis Method for Continuously Distributed Random Variable 693 Posterior Analysis and Bayesian Approach to Decision-making 695 Decision Trees in Decision-making 659 Utility Theory as Basis for Decision-making 705 Assumptions in Utility Theory 706 Von Neumann and Morgenstern Method of Measuring Utility 707 Standard Gambling Technique for Measuring Utility 708 Logarithmic Utility Function for Measuring Utility 708 Solving Decision Analysis Problems 777 Conclusion 772 Solved Problems Investment Analysis 730 Introduction 730 Overview of Situations and Methods 737 Break-even Analysis 737

7 DETAILED CONTENTS Assumptions in Break-even Analysis 732 Calculation of Break-even Point 732 Margin of Safety 735 Sensitivity Analysis 735 Volume-Profit Graph 736 Break-even Analysis for Multiproduct Situations 736 Payback Period Method 738 Average Rate of Return Method 739 Net Present Value Method 740 Net Present Value with Annuities 742 Equivalent Annual Annuity Method 744 Internal Rate of Return Method 744 Extrapolation Method 745 Comparison of NPV and IRR Methods of Analysis 746 Discounted Payback Period Method 748 Profitability Index Method 749 Benefit Cost Ratio and Net benefit Cost Ratio Methods 749 Common Time Horizon Method 750 Probabilistic Situations 757 Risk-adjusted Discount Rate Method 757 Certainty Equivalent Approach 752 Expected Monetary Value Method 754 Hillier and Hertz's Model ' 755 Solving Investment Analysis Problems 758 Conclusion 759 Solved Problems Introduction to Non-linear Programming Problems 771 Introduction 777 Examples of Non-linear Programming Problems 773 Product Mix Problem 773 Transportation Problem 774 Portfolio Selection of Securities 775 Concave and Convex Functions 778 Graphical Illustration of Non-linear Programming Problems 782 Types of Non-linear Programming Problems 786 Unconstrained Optimization 786 Linearly Constrained Optimization 789 Quadratic Programming 790 Convex Programming 794 Separable Problem 794 Non-convex Programming 795 Conclusion 795 Appendix: Statistical and Financial Tables Bibliography 822 Index 824 '797