2 - The Simplex Method

Abstract

This chapter describes an elementary version of the method that can be used to solve a linear programming problem systematically. A linear programming problem in canonical form can be solved by finding all the basic solutions, discarding those that are not feasible and finding an optimal solution among the remaining. This method determines the extreme points in the set of feasible solutions in a particular order that allows one to find an optimal solution in a small number of trials. The simplex algorithm consists of two steps: (1) a way of finding out whether a given basic feasible solution is an optimal solution and (2) a way of obtaining an adjacent basic feasible solution with the same or larger value for the objective function. In actual use, the simplex method does not examine every basic feasible solution; it checks only a relatively small number of them. After determining the entering and departing variables, a new tableau must be obtained, showing the new basic variables and the new basic feasible solution. Computer programs designed for large linear programming problems provide several options for dealing with degeneracy and cycling. One option is to ignore degeneracy and to assume that cycling will not occur. Another option is to use Bland's Rule for choosing entering and departing variables to avoid cycling.