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