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“Concept of Atomic Physics” authored by S. P. Kuila. (NCBA)
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1. Mathematical Preliminaries & Convex Sets
- Vector Spaces & Matrices: Review of simultaneous linear equations, basis vectors, and linear independence.
- Convex Sets: Hyperplanes, open and closed half-spaces, convex combinations, and extreme points.
- Theorems: Separation and supporting hyperplanes, and the foundational property that optimal LPP solutions occur at the extreme points of a convex feasible region.
2. Formulation of Linear Programming Problems (LPP)
- Real-World Modeling: Translating practical problems into mathematical expressions.
- Components: Identifying decision variables, maximizing/minimizing the objective function, and writing linear constraints.
- Applications: Product mix optimization, diet problems, blending/manufacturing processes, and media scheduling.
3. Graphical Method
- Two-Variable Systems: Visualizing linear constraints on a 2D Cartesian plane.
- Feasible Region: Identifying bounded, unbounded, or empty solution spaces.
- Corner-Point Evaluation: Finding the optimal vertex or demonstrating multiple alternative optima, infeasibility, and degeneracy.
4. The Simplex Method (Algorithmic Approach)
- Standard Form: Conversion of inequalities to equalities using slack, surplus, and artificial variables.
- Simplex Tableau: Setting up initial basic feasible solutions (BFS) and updating iterations.
- Optimality Criteria: Calculating net evaluations (\(z_j – c_j\)) to determine entering and leaving variables.
- Advanced Variations:
- Charnes’ Big-M Method: Penalizing artificial variables.
- Two-Phase Method: Phase I eliminates artificial elements; Phase II optimizes the original objective.
5. Duality in Linear Programming
- Symmetry: Formulating the “Dual” problem directly from the original “Primal” problem.
- Duality Theorems: Weak duality, strong duality, and complementary slackness conditions.
- Economic Interpretation: Understanding shadow prices, resource values, and opportunity costs.
- Dual Simplex Method: Solving problems that start with an optimal but infeasible condition.
6. Transportation & Assignment Problems
- Transportation Model: Allocating goods from sources to destinations while minimizing shipping costs.
- Initial BFS: Northwest Corner Rule, Least Cost Method, and Vogel’s Approximation Method (VAM).
- Optimality: Modified Distribution (MODI) method and the Stepping-Stone algorithm.
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