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The exhaustive list of topics in Linear programming in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Linear models such as; Product mix problem, Nutrition Problem,a BlendingProblem, Formulation of these problems as Linear Programming problems (LLP). Axioms of linearity, General form of LPP, Slack and Surplus Variables. Standard Form of LPP.
  • Basic concepts of rank of a matrix, Solution of a system of linear equations, Examples. Basic feasible solution (bf s), degenerate and non-degenrate, examples of basic solutions which are not feasible. Upper bound on the number of bf s. Upper bound on the absolute value of the basic variables.
  • Existence of bf s, Moving from one bfs to another and improving the value of the objective function. Optimality Criteria. Optimal solution is a bfs. Simplex algorithm through a simple example.
  • Simplex algorithm - geometrically interpretation. Definition of an affine space, Polyhedron P, faces of a polyhedron – facets, edges and vertices. Representation of a polyhedron in terms of extreme points and extreme rays.
  • A basic feasible solution is an extreme point of the corresponding Polyhedron. More about degeneracy.
  • Supporting hyperplane of a polyhedron. Characterisation of an optimal solution in terms of supporting hyperplane. Graphical illustrations.
  • Simplex Algorithm- Tableau format.
  • Simplex algorithm – Starting feasible solution, Artificial variables, Phase I and Phase II methods.
  • Bounded variables case; modification of the Simplex algorithm.
  • Revised Simplex algorithm. Define the Dual problem and its various forms. Fundamental Theorem of Duality. Farka’s theorem. Complementary Slackness theorem.
  • Dual Simplex algorithm; Motivation , theory and a numerical example.
  • Primal Dual algorithm: Motivation , theory and a numerical example.
  • Sensitivity Analysis of the objective function coefficient, right hand side components and elements of the matrix A.
  • Adding of constraints and activities. A comprehensive numerical example.
  • Parametric analysis.
  • Min-cost flow problem- formulation and derivation of special cases such as Transportation problem.
  • Assignment problem, Max-flow problem and the shortest path problem.
  • Integer bfs property of Transportation problem.
  • Simplified Simplex algorithm for Transportation problem.
  • Sensitivity Analysis and Bounded Variable case.
  • Formulation of Shortest Path Problem, Dijkstra’s algorithm.
  • More general shortest Path algorithms, Sensitivity analysis.
  • Applications of Max-flow problem.
  • Algorithms and Sensitivity Analysis.
  • Network Simplex Algorithm for Min – cost flow problem.
  • Project Planning Control with PERT / CPM, linear programming formulations.
  • Dynamic Programming: Principle of Optimality with proof. Discrete and continuous problems.
  • Backward and forward formulations. Probabilistic cases.
  • Game theory. Two-person Zero-sum game. Pure and mixed strategies with examples.
  • Saddlepoint and graphical solutions.
  • Linear programming iterative solution method.
  • Computational complexity of Simplex algorithm. To show through an example that the Simplex algorithm can go through all the extreme points before reaching the optimal extreme point solution.
  • Ellipsoid algorithm- basic concepts and its applications.
  • Basic idea behind Karmarkar’s algorithm and its applications.