Operations Research and Optimization

  1. Linear Programming
    1. Fundamental Concepts
      1. Decision Variables
        1. Variable Definition
          1. Variable Types
            1. Variable Bounds
            2. Objective Function
              1. Maximization Problems
                1. Minimization Problems
                  1. Linear Objective Functions
                  2. Constraints
                    1. Equality Constraints
                      1. Inequality Constraints
                        1. Non-negativity Constraints
                          1. Constraint Types
                          2. Feasible Region
                            1. Feasible Solutions
                              1. Bounded Regions
                                1. Unbounded Regions
                                  1. Empty Feasible Regions
                                  2. Optimal Solutions
                                    1. Existence of Optimal Solutions
                                      1. Uniqueness of Optimal Solutions
                                        1. Multiple Optimal Solutions
                                          1. Corner Point Solutions
                                        2. Linear Programming Assumptions
                                          1. Proportionality Assumption
                                            1. Additivity Assumption
                                              1. Divisibility Assumption
                                                1. Certainty Assumption
                                                  1. Linearity Assumption
                                                    1. Implications of Violations
                                                    2. Model Formulation
                                                      1. Problem Translation Process
                                                        1. Resource Allocation Problems
                                                          1. Production Planning
                                                            1. Capital Budgeting
                                                              1. Workforce Allocation
                                                              2. Blending Problems
                                                                1. Diet Problems
                                                                  1. Product Mix Problems
                                                                    1. Chemical Blending
                                                                    2. Transportation Problems
                                                                      1. Supply and Demand Constraints
                                                                        1. Cost Minimization
                                                                          1. Balanced Transportation
                                                                          2. Assignment Problems
                                                                            1. One-to-One Assignments
                                                                              1. Cost Minimization
                                                                                1. Hungarian Method Setup
                                                                                2. Network Flow Problems
                                                                                  1. Node Balance Constraints
                                                                                    1. Arc Capacity Constraints
                                                                                  2. Graphical Solution Method
                                                                                    1. Two-Variable Problems
                                                                                      1. Plotting Constraints
                                                                                        1. Boundary Lines
                                                                                          1. Feasible Half-Planes
                                                                                          2. Identifying Feasible Region
                                                                                            1. Intersection of Half-Planes
                                                                                              1. Corner Points
                                                                                              2. Objective Function Lines
                                                                                                1. Iso-profit Lines
                                                                                                  1. Iso-cost Lines
                                                                                                  2. Locating Optimal Solutions
                                                                                                    1. Corner Point Theorem
                                                                                                      1. Optimal Point Identification
                                                                                                    2. The Simplex Method
                                                                                                      1. Standard Form Conversion
                                                                                                        1. Slack Variables
                                                                                                          1. Surplus Variables
                                                                                                            1. Artificial Variables
                                                                                                              1. Non-negativity Requirements
                                                                                                              2. Initial Basic Feasible Solution
                                                                                                                1. Basic Variables
                                                                                                                  1. Non-basic Variables
                                                                                                                    1. Basic Feasible Solutions
                                                                                                                    2. Simplex Tableau Construction
                                                                                                                      1. Tableau Format
                                                                                                                        1. Coefficient Matrix
                                                                                                                          1. Right-Hand Side Vector
                                                                                                                            1. Objective Row
                                                                                                                            2. Simplex Algorithm Steps
                                                                                                                              1. Optimality Test
                                                                                                                                1. Entering Variable Selection
                                                                                                                                  1. Leaving Variable Selection
                                                                                                                                    1. Pivot Operations
                                                                                                                                      1. Tableau Updates
                                                                                                                                      2. Special Cases in Simplex
                                                                                                                                        1. Unbounded Solutions
                                                                                                                                          1. Infeasible Solutions
                                                                                                                                            1. Alternative Optimal Solutions
                                                                                                                                              1. Degeneracy
                                                                                                                                                1. Cycling Prevention
                                                                                                                                                2. Two-Phase Method
                                                                                                                                                  1. Phase I Objective
                                                                                                                                                    1. Artificial Variable Elimination
                                                                                                                                                      1. Phase II Optimization
                                                                                                                                                        1. Transition Between Phases
                                                                                                                                                        2. Big M Method
                                                                                                                                                          1. Large Penalty Parameter
                                                                                                                                                            1. Artificial Variable Treatment
                                                                                                                                                              1. Solution Interpretation
                                                                                                                                                            2. Duality Theory
                                                                                                                                                              1. Primal Problem Formulation
                                                                                                                                                                1. Dual Problem Construction
                                                                                                                                                                  1. Dual Variables
                                                                                                                                                                    1. Dual Constraints
                                                                                                                                                                      1. Dual Objective Function
                                                                                                                                                                      2. Primal-Dual Relationships
                                                                                                                                                                        1. Weak Duality Theorem
                                                                                                                                                                          1. Strong Duality Theorem
                                                                                                                                                                            1. Complementary Slackness
                                                                                                                                                                            2. Economic Interpretation
                                                                                                                                                                              1. Shadow Prices
                                                                                                                                                                                1. Marginal Values
                                                                                                                                                                                  1. Resource Valuation
                                                                                                                                                                                  2. Dual Simplex Method
                                                                                                                                                                                    1. Dual Feasibility
                                                                                                                                                                                      1. Primal Infeasibility
                                                                                                                                                                                        1. Algorithm Steps
                                                                                                                                                                                      2. Sensitivity Analysis
                                                                                                                                                                                        1. Objective Function Coefficient Changes
                                                                                                                                                                                          1. Allowable Ranges
                                                                                                                                                                                            1. Optimality Preservation
                                                                                                                                                                                            2. Right-Hand Side Changes
                                                                                                                                                                                              1. Shadow Price Interpretation
                                                                                                                                                                                                1. Feasibility Ranges
                                                                                                                                                                                                  1. Dual Price Validity
                                                                                                                                                                                                  2. Constraint Coefficient Changes
                                                                                                                                                                                                    1. Structural Changes
                                                                                                                                                                                                      1. Feasibility Impact
                                                                                                                                                                                                      2. Adding New Variables
                                                                                                                                                                                                        1. Reduced Cost Analysis
                                                                                                                                                                                                          1. Profitability Assessment
                                                                                                                                                                                                          2. Adding New Constraints
                                                                                                                                                                                                            1. Constraint Violation Check
                                                                                                                                                                                                              1. Dual Variable Introduction
                                                                                                                                                                                                              2. Parametric Programming
                                                                                                                                                                                                                1. Systematic Parameter Changes
                                                                                                                                                                                                                  1. Solution Path Analysis

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                                                                                                                                                                                                              2. Mathematical Foundations for Optimization

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                                                                                                                                                                                                              4. Network Optimization