Operations Research and Optimization

  1. Network Optimization
    1. Graph Theory Foundations
      1. Basic Definitions
        1. Nodes and Vertices
          1. Arcs and Edges
            1. Directed Graphs
              1. Undirected Graphs
              2. Graph Properties
                1. Paths and Walks
                  1. Cycles and Circuits
                    1. Trees and Forests
                      1. Connected Components
                      2. Network Representation
                        1. Adjacency Matrices
                          1. Adjacency Lists
                            1. Incidence Matrices
                              1. Arc-Node Formulations
                            2. Shortest Path Problems
                              1. Problem Formulation
                                1. Single Source Shortest Path
                                  1. All Pairs Shortest Path
                                    1. Path Length Definitions
                                    2. Dijkstra's Algorithm
                                      1. Algorithm Description
                                        1. Implementation Steps
                                          1. Complexity Analysis
                                            1. Non-negative Weight Requirement
                                            2. Bellman-Ford Algorithm
                                              1. Negative Weight Handling
                                                1. Algorithm Steps
                                                  1. Negative Cycle Detection
                                                    1. Complexity Analysis
                                                    2. Floyd-Warshall Algorithm
                                                      1. All Pairs Implementation
                                                        1. Dynamic Programming Approach
                                                          1. Algorithm Steps
                                                        2. Minimum Spanning Tree Problems
                                                          1. Problem Definition
                                                            1. Spanning Tree Properties
                                                              1. Minimum Weight Criteria
                                                              2. Prim's Algorithm
                                                                1. Greedy Approach
                                                                  1. Algorithm Steps
                                                                    1. Implementation Details
                                                                      1. Complexity Analysis
                                                                      2. Kruskal's Algorithm
                                                                        1. Edge-Based Approach
                                                                          1. Union-Find Data Structure
                                                                            1. Algorithm Steps
                                                                              1. Complexity Analysis
                                                                            2. Maximum Flow Problems
                                                                              1. Problem Formulation
                                                                                1. Source and Sink Nodes
                                                                                  1. Arc Capacities
                                                                                    1. Flow Conservation
                                                                                    2. Ford-Fulkerson Method
                                                                                      1. Augmenting Path Concept
                                                                                        1. Residual Networks
                                                                                          1. Algorithm Framework
                                                                                          2. Edmonds-Karp Algorithm
                                                                                            1. Breadth-First Search Implementation
                                                                                              1. Complexity Improvement
                                                                                              2. Max-Flow Min-Cut Theorem
                                                                                                1. Cut Definition
                                                                                                  1. Theorem Statement
                                                                                                    1. Proof Concepts
                                                                                                  2. Minimum Cost Flow Problems
                                                                                                    1. Problem Formulation
                                                                                                      1. Supply and Demand Nodes
                                                                                                        1. Arc Costs and Capacities
                                                                                                          1. Flow Balance Constraints
                                                                                                          2. Network Simplex Method
                                                                                                            1. Spanning Tree Solutions
                                                                                                              1. Entering and Leaving Arcs
                                                                                                                1. Optimality Conditions
                                                                                                                2. Cycle-Canceling Algorithms
                                                                                                                  1. Negative Cycle Detection
                                                                                                                    1. Flow Augmentation
                                                                                                                  2. Transportation Problems
                                                                                                                    1. Problem Formulation
                                                                                                                      1. Supply Constraints
                                                                                                                        1. Demand Constraints
                                                                                                                          1. Transportation Costs
                                                                                                                          2. Initial Solution Methods
                                                                                                                            1. Northwest Corner Rule
                                                                                                                              1. Minimum Cost Method
                                                                                                                                1. Vogel's Approximation Method
                                                                                                                                2. Optimality Testing
                                                                                                                                  1. MODI Method
                                                                                                                                    1. Dual Variables
                                                                                                                                      1. Opportunity Costs
                                                                                                                                      2. Solution Improvement
                                                                                                                                        1. Stepping Stone Method
                                                                                                                                          1. Loop Construction
                                                                                                                                            1. Flow Adjustments
                                                                                                                                          2. Assignment Problems
                                                                                                                                            1. Problem Formulation
                                                                                                                                              1. One-to-One Assignments
                                                                                                                                                1. Cost Matrix
                                                                                                                                                  1. Binary Variables
                                                                                                                                                  2. Hungarian Method
                                                                                                                                                    1. Row and Column Reductions
                                                                                                                                                      1. Line Covering
                                                                                                                                                        1. Assignment Identification
                                                                                                                                                          1. Algorithm Steps
                                                                                                                                                          2. Variations
                                                                                                                                                            1. Maximization Problems
                                                                                                                                                              1. Unbalanced Assignments
                                                                                                                                                                1. Prohibited Assignments
                                                                                                                                                              2. Project Scheduling
                                                                                                                                                                1. Project Network Representation
                                                                                                                                                                  1. Activity-on-Node Networks
                                                                                                                                                                    1. Activity-on-Arrow Networks
                                                                                                                                                                      1. Precedence Relationships
                                                                                                                                                                      2. Critical Path Method
                                                                                                                                                                        1. Forward Pass Calculations
                                                                                                                                                                          1. Backward Pass Calculations
                                                                                                                                                                            1. Total Float and Free Float
                                                                                                                                                                              1. Critical Path Identification
                                                                                                                                                                              2. Program Evaluation and Review Technique
                                                                                                                                                                                1. Three-Time Estimates
                                                                                                                                                                                  1. Expected Activity Times
                                                                                                                                                                                    1. Project Duration Distribution
                                                                                                                                                                                      1. Probability Calculations

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                                                                                                                                                                                  5. Integer Programming