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Contents Preface Introduction 1. Linear Programming Problems Introduction Mathematical formulation of Practical Linear Programming Problems (LPP) Standardisation of LPP by Introducing Slack and Surplus Variables Graphical Method of Solving a Linear Programming Problem (LPP Examples 1.1 to 1.7 Nature of Solutions of LPP Exercises (1 –15) Preliminaries Vector Space Row Space and Column Space Rank of a Matrix Row Rank of a Matrix Column Rank of a Matrix Linear Independence Linear Dependence Useful Results of Linear Algebra Results I to III Basis Properties Basic Solution Examples 1.8 to 1.9 Important Definitions Hyperplane Hypersphere e-neighbourhood Interior Point Boundary Point8 Open Set Closed Set Bounded Set Convex Set Convex Combination of Finite Number of Points Convex Polyhedron Simplex Convex Hull of a Set A Theorems on Covex Set, Feasible Region, Basic Feasible Solution and Convex Hull Theorems 1.1 to 1.2 Lemmas I and II Theorem 1.3 Examples 1.10 to 1.12 Convex Hull of Finite Set Theorem 1.4 Examples 1.13 to 1.16 (Convex and Non-convex Sets) Euclidean Plane and Feasible Region of lpp Extreme Point Theorem 1.5 Half Spaces4 Supporting hyperplane Optimal hyperplane Theorem 1.6 Analytical Results Results I to IV Optimal Solution and Extreme Points Theorem 1.7 Optimality Criterion Some Notations Theorem 1.8 Corollary Unbounded Solution Theorem 1.9 Corollary Improving a Basic Feasible Solution (BFS) Theorem 1.10 Corollary Alternate Optima Theorem 1.11 Non-basic Alternate Optimal Solution Examples 1.17 to 1.18 Miscellaneous Exercises (1-10) SECTION 2—Solution Techniques 2. Simplex Algorithm Pre-requisites Formation of a New Table by Changing the Basis Simplex Algorithm Application to Problems with Slack Variables Only Example 2.1 Charnes’ M-Technique for Problems with Surplus and Artificial Variables Application of Simplex Method to Problems with Surplus and Artificial Variables Examples 2.2 to 2.10 Two-Phase method
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