Operations Research

1. Introduction to Operations Research ; 1.1 INTRODUCTION ; 1.2 HISTORICAL DEVELOPMENT ; 1.3 DEFINITIONS ; 1.4 MODELS ; 1.5 SCOPE AND APPLICATIONS ; 1.6 PHASES ; 2. Linear Programming Problem I-Formulation ; 2.1 INTRODUCTION ; 2.2 LINEAR PROGRAMMING PROBLEM ; 2.3 BASIC ASSUMPTIONS OF LINEAR PROGRAMMING PROBLEM ; 2.4 FORMULATION OF LINEAR PROGRAMMING MODEL ; 2.5 LIMITATIONS OF LINEAR PROGRAMMING PROBLEM ; 2.6 APPLICATIONS OF LINEAR PROGRAMMING PROBLEM IN BUSINESS AND INDUSTRIES ; 3. Linear Programming Problem II-Graphical Method ; 3.1 INTRODUCTION ; 3.2 SOME DEFINITIONS ; 3.3 SOME IMPORTANT THEOREMS ; 3.4 GRAPHICAL METHOD ; 3.4.1 Corner Point Method ; 3.4.2 Iso-profit Method or Isovalue Line Method ; 3.5 SPECIAL CASES IN GRAPHICAL METHOD ; 3.5.1 Alternate Optimal Solution ; 3.5.2 No Feasible Solution ; 3.5.3 Unbounded Solution Space but Bounded Optimal Solution ; 3.5.4 Unbounded Solution Space and Unbounded Solution ; 3.6 LIMITATIONS OF GRAPHICAL METHOD ; 4. Linear Programming Problem III-Simplex Method ; 4.1 INTRODUCTION ; 4.2 STANDARD FORM OF LINEAR PROGRAMMING PROBLEM ; 4.3 SOME IMPORTANT TERMINOLOGIES ; 4.4 SOME IMPORTANT RESOLUTIONS USED IN LPP FOR SIMPLEX METHOD ; 4.5 SIMPLEX METHOD ; 4.6 SIMPLEX TABLE ; 4.7 CRITERIA OF OPTIMALITY ; 4.8 COMPUTATIONAL OR ITERATIVE PROCEDURE FOR SOLVING LINEAR PROGRAMMING PROBLEM USING SIMPLEX METHOD ; 4.9 SPECIAL CASES IN SIMPLEX METHOD ; 4.9.1 Infeasibility ; 4.9.2 Unboundedness ; 4.9.3 Degeneracy ; 4.9.4 Alternate or More Than One Optimal Solution ; 4.9.5 Cycling ; 4.10 ARTIFICIAL VARIABLE TECHNIQUE FOR SOLVING LINEAR ; PROGRAMMING PROBLEMS ; 4.10.1 Big-M Method ; 4.10.2 Two-phase Method ; 4.10.3 Comparison between Big-M and Two-phase Methods ; 4.11 SOLVING SIMULTANEOUS LINEAR EQUATIONS USING SIMPLEX METHOD ; 4.12 FINDING INVERSE OF SQUARE MATRIX USING SIMPLEX METHOD ; 5. Linear Programming Problem IV-Revised Simplex Method ; 5.1 INTRODUCTION ; 5.2 REVISED SIMPLEX METHOD ; 5.3 COMPUTATIONAL PROCEDURE FOR SOLVING LPP BY REVISED SIMPLEX METHOD ; 6. Duality in Linear Programming ; 6.1 INTRODUCTION ; 6.2 SYMMETRIC FORM ; 6.3 DEFINITION OF DUAL OF LINEAR PROGRAMMING PROBLEM ; 6.4 PRIMAL-DUAL RELATIONSHIP ; 6.5 ECONOMIC INTERPRETATION OF DUALITY ; 6.6 IMPORTANT THEOREMS ; 6.7 DUAL SIMPLEX METHOD ; 6.7.1 Procedure for Solving a Linear Programming Problem ; 7. Post-optimality Analysis or Sensitivity Analysis ; 7.1 INTRODUCTION ; 7.2 CHANGES AFFECTING FEASIBILITY AND OPTIMALITY ; 7.3 GRAPHICAL SENSITIVITY ANALYSIS ; 7.4 CHANGES IN COST CJ IN OBJECTIVE FUNCTION ; 7.5 CHANGES IN BI'S AVAILABILITIES ; 7.6 ADDITION OF NEW VARIABLE ; 7.7 DELETION OF CONSTRAINTS ; 7.8 DELETION OF VARIABLES ; 7.9 ADDITION OF CONSTRAINTS ; 7.10 CHANGE IN AIJ'S ; 7.11 PARAMETRIC LINEAR PROGRAMMING ; 7.11.1 Parametric Changes in Cost Vector c ; 7.11.2 Parametric Changes in Requirement Vector b ; 7.12 DIFFERENCE BETWEEN SENSITIVITY ANALYSIS AND PARAMETRIC LINEAR PROGRAMMING ; 8. Transportation Problems ; 8.1 INTRODUCTION ; 8.2 FORMULATION OF TRANSPORTATION PROBLEM ; 8.3 DEVELOPMENT OF TRANSPORTATION ALGORITHM ; 8.4 SOLUTION OF TRANSPORTATION PROBLEM ; 8.4.1 North-west Corner Method ; 8.4.2 Least Cost Entry or Matrix Minima Method ; 8.4.3 Vogel's Approximation Method ; 8.5 TEST OF OPTIMALITY ; 8.5.1 MODI Method ; 8.5.2 Stepping Stone Method ; 8.6 DEGENERACY IN TRANSPORTATION PROBLEM ; 8.7 UNBALANCED TRANSPORTATION PROBLEM ; 8.8 TRANSSHIPMENT PROBLEM ; 9. Assignment Problems ; 9.1 INTRODUCTION ; 9.2 SOLVING ASSIGNMENT PROBLEMS USING HUNGARIAN METHOD ; 9.3 MINIMAL ASSIGNMENT PROBLEM ; 9.4 MAXIMAL ASSIGNMENT PROBLEM ; 9.5 UNBALANCED ASSIGNMENT PROBLEM ; 9.6 ASSIGNMENT PROBLEMS UNDER CERTAIN RESTRICTIONS ; 9.7 TRAVELLING SALESMAN PROBLEM ; 9.8 DIFFERENCE BETWEEN ASSIGNMENT AND TRANSPORTATION PROBLEMS ; 10. Sequencing ; 10.1 INTRODUCTION ; 10.2 ASSUMPTIONS, NOTATIONS, AND TERMINOLOGIES ; 10.2.1 Assumptions ; 10.2.2 Notations ; 10.2.3 Terminologies ; 10.3 JOHNSON'S ALGORITHM FOR PROCESSING N JOBS THROUGH TWO MACHINES ; 10.4 JOHNSON'S ALGORITHM FOR PROCESSING N JOBS THROUGH K MACHINES ; 10.5 PROCESSING TWO JOBS THROUGH K MACHINES ; 11. Project Scheduling ; 11.1 NTRODUCTION ; 11.2 PROJECT SCHEDULING ; 11.2.1 Planning ; 11.2.2 Scheduling ; 11.2.3 Controlling ; 11.3 NETWORK ; 11.3.1 Notations ; 11.3.2 Fulkerson's Rule for Numbering Events ; 11.4 CRITICAL PATH METHOD ; 11.5 PROGRAM EVALUATION AND REVIEW TECHNIQUE ; 11.6 OPTIMUM SCHEDULING BY CRITICAL PATH METHOD ; 11.7 TIME-COST OPTIMIZATION ALGORITHM ; 12. Dynamic Programming ; 12.1 INTRODUCTION ; 12.2 TERMINOLOGY USED IN DYNAMIC PROGRAMMING ; 12.3 MULTI-DECISION PROCESS ; 12.4 BELLMAN'S PRINCIPLE OF OPTIMALITY ; 12.5 CHARACTERISTICS OF DYNAMIC PROGRAMMING PROBLEMS ; 12.6 DYNAMIC PROGRAMMING ALGORITHM ; 12.7 DETERMINISTIC AND PROBABILISTIC DYNAMIC PROGRAMMING ; 12.8 MODELS OF DYNAMIC PROGRAMMING ; 12.8.1 Model I-Shortest Route Problem ; 12.8.2 Model II-Solving Dynamic Programming using Calculus Method ; 12.8.3 MODEL III ; 12.9 SOLVING LINEAR PROGRAMMING PROBLEMS USING DYNAMIC PROGRAMMING ; 12.10 APPLICATIONS OF DYNAMIC PROGRAMMING ; 13. Integer Programming ; 13.1 INTRODUCTION ; 13.2 MATHEMATICAL FORMULATION OF INTEGER PROGRAMMING ; PROBLEMS ; 13.3 TYPES OF INTEGER PROGRAMMING PROBLEMS ; 13.4 GOMORY'S CUTTING PLANE METHOD FOR AIPP ; 13.4.1 Algorithm for Gomory's Cutting Plane Method ; 13.5 GOMORY'S CUTTING PLANE METHOD FOR MIPP ; 13.6 DIFFERENCE BETWEEN GOMORY'S CUTTING PLANE METHOD FOR AIPP AND MIPP ; 13.7 BRANCH AND BOUND TECHNIQUE TO FIND SOLUTION OF IPP () ; 13.8 ZERO-ONE INTEGER PROGRAMMING PROBLEM ; 13.8.1 Format of Balas-Zero-One Additive Algorithm ; 13.8.2 Some Important Terms used in Balas Additive Algorithm ; 13.8.3 Solution Procedure of Zero-One IPP ; 14. Queuing Theory ; 14.1 INTRODUCTION ; 14.2 BASIC ELEMENTS OF QUEUING SYSTEMS ; 14.2.1 State of Systems ; 14.3 MARKOVIAN QUEUES ; 14.4 TERMINOLOGY AND NOTATIONS USED IN QUEUING SYSTEMS ; 14.5 SYMBOLIC REPRESENTATION OF QUEUING MODELS ; 14.6 PROBABILITY DISTRIBUTION OF N ARRIVALS IN TIME INTERVAL (T, T+ IN PURE BIRTH PROCESSES ; 14.7 DISTRIBUTION OF INTER-ARRIVALS TIME ; 14.8 DISTRIBUTION OF DEPARTURES IN PURE DEATH PROCESSES ; 14.9 BIRTH AND DEATH PROCESS ; 14.10 VARIOUS QUEUING MODELS WITH THEIR CHARACTERISTIC PROPERTIES ; 14.10.1 Model-I-(M/M/1):(FCFS/?) ; 14.10.2 Finite Storage Queue System with One Server (M/M/1):(FCFS/N) ; 14.10.3 S-Server case (M/M/S):(FCFS/?) ; 14.10.4 S-Server Case with Finite Accommodation Capacity (M/M/S):(FCFS/N) ; 14.11 ADVANTAGES OF QUEUING THEORY ; 15. Goal Programming ; 15.1 INTRODUCTION ; 15.2 FORMULATION OF GOAL PROGRAMMING ; 15.3 BASIC TERMINOLOGIES ; 15.4 SINGLE-GOAL MODELS ; 15.5 GP ALGORITHM OR MODIFIED SIMPLEX METHOD ; 15.6 MULTIPLE-GOAL MODELS ; 15.6.1 Multiple-goal Models with Equal or No Priorities ; 15.6.2 Multiple-goal Models with Priorities ; 15.6.3 Multiple-goal Models with Priorities and Weights ; 15.7 GRAPHICAL SOLUTION OF GOAL PROGRAMMING PROBLEMS ; 16. Game Theory ; 16.1 INTRODUCTION ; 16.2 CHARACTERISTICS OF GAMES ; 16.3 BASIC TERMINOLOGY USED IN GAME THEORY ; 16.4 LOWER AND UPPER VALUE OF GAME-'MINIMAX' PRINCIPLE WITH PURE STRATEGIES ; 16.5 PROCEDURE TO DETERMINE SADDLE POINT ; 16.6 MATRIX REDUCTION BY DOMINANCE PRINCIPLE ; 16.7 GAMES WITHOUT SADDLE POINT ; 16.7.1 2 x 2 Game Without Saddle Point ; 16.8 (3 x 3) GAMES WITH NO SADDLE POINT ; 16.9 GRAPHICAL METHOD FOR (2 X N) AND (M X 2) GAMES ; 16.9.1 Graphical Method for n x 2 Games ; 16.9.2 Graphical Method for m x 2 Games ; 16.10 METHOD OF SUBMATRICES OR SUBGAMES FOR (2 X N) OR (M X 2) GAMES WITH NO SADDLE POINT ; 16.11 TWO-PERSON ZERO-SUM GAME WITH MIXED STRATEGIES OR LINEAR PROGRAMMNING METHOD ; 16.12 LIMITATIONS OF GAME THEORY ; 17. Decision Theory (Analysis) ; 17.1 INTRODUCTION ; 17.2 DECISION MODELS ; 17.2.1 Decision Alternatives ; 17.2.2 States of Nature or Events ; 17.2.3 Payoff ; 17.3 DECISION-MAKING SITUATIONS ; 17.3.1 Decision-making Under Certainty ; 17.3.2 Decision-making Under Risk ; 17.3.3 Decision-making Under Uncertainty (Fuzzy Environment) ; 17.3.4 Posterior Probability and Bayesian Analysis ; 17.3.5 DECISION-MAKING UNDER CONFLICT (GAME THEORY) ; 18. Networking ; 18.1 INTRODUCTION ; 18.2 DEFINITIONS AND NOTATIONS USED IN NETWORKING ; 18.3 SHORTEST ROUTE PROBLEM ; 18.4 MINIMUM SPANNING TREE PROBLEM ; 18.5 MAXIMUM FLOW PROBLEMS ; 19. Replacement Models ; 19.1 INTRODUCTION ; 19.2 REPLACEMENT POLICY MODELS ; 19.3 REPLACEMENT POLICY WHEN THE VALUE OF MONEY DOES NOT CHANGE WITH TIME ; 19.4 REPLACEMENT POLICY WHEN THE VALUE OF MONEY CHANGES WITH TIME ; 19.5 PROCEDURE TO SELECT THE BETTER EQUIPMENT ; 19.6 REPLACEMENT OF EQUIPMENT THAT FAILS SUDDENLY ; 19.7 GROUP REPLACEMENT THEOREM ; 20. Simulation ; 20.1 INTRODUCTION ; 20.2 BASIC TERMINOLOGIES ; 20.3 RANDOM NUMBERS AND PSEUDO-RANDOM NUMBERS ; 20.3.1 Mid-square Method or Technique of Generating Pseudo-random Numbers ; 20.3.2 Limitations of Mid-square Method ; 20.3.3 Multiplicative Congruential or Power Residual Technique ; 20.3.4 Mixed Congruential Method ; 20.4 MONTE CARLO SIMULATION ; 20.5 GENERATION OF RANDOM VARIATES ; 20.5.1 Continuous Random Variate X ; 20.5.2 Discrete Case ; 20.6 APPLICATIONS OF SIMULATION IN QUEUING MODELS ; 20.7 ADVANTAGES AND DISADVANTAGES OF SIMULATION ; 20.8 SIMULATION LANGUAGES ; 21. Inventory Models ; 21.1 INTRODUCTION ; 21.2 INVENTORY ; 21.3 SOME BASIC TERMINOLOGIES USED IN INVENTORY ; 21.4 INVENTORY CONTROL ; 21.5 INVENTORY COSTS ; 21.6 INVENTORY MANAGEMENT AND ITS BENEFITS ; 21.7 ECONOMIC ORDER QUANTITY ; 21.7.1 Deterministic Inventory Models With No Shortages ; 21.8 DETERMINISTIC INVENTORY MODELS WITH SHORTAGES ; 21.9 EOQ PROBLEM WITH PRICE BREAKS OR QUANTITY DISCOUNT ; 21.10 PROBABILISTIC INVENTORY MODELS ; 21.10.1 Single Period Problem without Set-up Cost and Uniform Demand ; 21.10.2 Single Period Problems without Set-up Cost and Instantaneous Demand ; 21.11 SOME IMPORTANT INVENTORY CONTROL TECHNIQUES ; 22. Classical Optimization Techniques ; 22.1 INTRODUCTION ; 22.2 UNCONSTRAINED OPTIMIZATION PROBLEMS ; 22.2.1 Single-variable Unconstrained Optimization Problems ; 22.2.2 Conditions for Local Maxima or Minima of Single-variable Function ; 22.2.3 Procedure to Find Extreme Points of Functions of Single Variables ; 22.3 MULTIVARIABLE OPTIMIZATION PROBLEMS ; 22.3.1 Working Rule to Find Extreme Points of Functions of Two Variables ; 22.3.2 Working Rule to Find Extreme Points of Functions of n Variables ; 22.4 MULTIVARIABLE CONSTRAINED OPTIMIZATION PROBLEMS WITH EQUALITY CONSTRAINTS ; 22.4.1 Direct Substitution Method ; 22.4.2 Lagrange Multipliers Method ; 22.5 MULTIVARIABLE CONSTRAINED OPTIMIZATION PROBLEMS WITH INEQUALITY CONSTRAINTS ; 23. Non-Linear Programming Problem 1-Search Techniques ; 23.1 INTRODUCTION ; 23.2 UNCONSTRAINED NON-LINEAR PROGRAMMING PROBLEM ; 23.3 DIRECT SEARCH METHODS ; 23.4 SEARCH TECHNIQUES OR ONE DIMENSION ; 23.4.2 Golden Section Method arch ; 23.4.3 Univariate Method ; 23.4.4 Pattern Search Methods ; 23.5 INDIRECT SEARCH METHODS ; 23.5.1 Steepest Descent or Cauchy's Method ; 23.6 CONSTRAINED NON-LINEAR PROGRAMMING PROBLEMS ; 23.7 DIRECT METHODS ; 23.7.1 Complex Method ; 23.7.2 Zoutendijk Method or Method of Feasible Direction ; 23.8 INDIRECT METHODS ; 23.8.1 Transform Techniques ; 23.8.2 Penalty Function Methods ; 23.9 ROSEN'S GRADIENT PROJECTION METHOD ; 24. Non-Linear Programming 2-Quadratic and Separable Programming ; 24.1 INTRODUCTION ; 24.2 KUHN-TUCKER CONDITIONS ; 24.3 QUADRATIC PROGRAMMING ; 24.3.1 Wolfe's Modified Simplex Method ; 24.3.2 Beale's Method ; 24.4 SEPARABLE PROGRAMMING ; Appendix A: Linear Algebra ; Appendix B: Matrices ; Appendix C: Calculus ; Appendix D: Probability ; Appendix E: Poisson Probability Distribution X Table ; Appendix F: Area under the Standard Normal Distribution Z ; Appendix G: Table of Random Nu