NPTEL Syllabus Linear programming and Extensions - Video course COURSE OUTLINE The objective of this course is to introduce those real life problems which can be formulated as Linear Programming Problems ( LPP ). The course will be taught as a first course in optimization, hence all the concepts will be properly motivated and explained with examples. Following will be discussed in particular: Simplex Algorithm Duality theory and its ramifications Basic ideas of the Ellipsoid algorithm and Karmarkar’s algorithm Special cases of LPP such as Transportation, Assignment and Network – flow Dynamic programming and PERT/CPM algorithms COURSE DETAIL Lectures Topic/s 1 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. 2 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. 3 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. 4 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. 5 A basic feasible solution is an extreme point of the corresponding Polyhedron. More about degeneracy. 6 Supporting hyperplane of a polyhedron. Characterisation of an optimal solution in terms of supporting hyperplane. Graphical illustrations. 7 Simplex Algorithm- Tableau format. 8 Simplex algorithm – Starting feasible solution, Artificial variables, Phase I and Phase II methods. 9 Bounded variables case; modification of the Simplex algorithm. NPTEL http://nptel.iitm.ac.in Mathematics Pre-requisites: Undergraduate Linear Algebra. Additional Reading: C.H. Papadimitrou and Ken Steiglitz, Combinatorial Optimisation Algorithms and Complexity, (Second edition) Dover, 1998. Coordinators: Prof. Prabha Sharma Department of Mathematics and Statistics IIT Kanpur