An experiment with simplex method for solving linear programming problems

Abstract

Linear programming (LP) is the most popular among optimization techniques used in production, industry, planning and in other areas. Simplex method for solving LP problems has been in use for more than 70 years. Despite the fact that pathological examples can be created in which simplex algorithm requires exponential number of iterations, in practice it has been found very e cient. In this research, we aim to devise an improvised algorithm for the simplex method so as to reach optimality in fewer iterations. In a feasible region with di erentiable surface, optimal solution will correspond to a point in which gradient of the objective function will coincide with the normal of the surface. LP theory asserts that if an LP has an optimal solution there is one at a vertex of the polyhedron. Unfortunately vertices of the polyhedron are in the intersection of n or more hyperplanes and therefore are not di erentiable. Moreover, LP theory asserts that there is an optimal vertex at which gradient of the objective function will lie in the cone determined by normals of hyperplanes intersection of which is the vertex itself. Unfortunately nding the right combination of the hyperplanes is a combinatorial problem. Therefore, we may think of starting simplex iterations from a point where gradient of the objective function makes minimum angles with the normals of hyperplanes determining the point. It may be noted here that such a point may well be beyond feasible region, and we may need iterations to reach feasibility. We would like to carry out simulation of this algorithm and compare its performance with the existing simplex algorithm. Furthermore, we have noted two other issues of LP problem. Firstly, we show that revenue maximizing and pro t maximizing LP problems are equivalent. Secondly, LP duality is robust in the sense that even if non-negativity constraints are included into the main constraints duality results hold.