The Theory of Linear Programming

Abstract

We noted in Chapter 3 that the primal feasible region Kp has a finite number of extreme points. Since each such point has associated with it a basic feasible solution (unique or otherwise), it follows that there exists a finite number of basic feasible solutions. Hence an optimal solution to the primal linear programming problem will be contained within the set of basic feasible solutions to AX = b. How many elements does this set possess? Since a basic feasible solution has at most m of n variables different from zero, an upper bound to the number of basic feasible solutions is

$$ \left( {\tfrac{n}{m}} \right) = \frac{{n!}}{{m!(n - m)!}}, $$

i.e., we are interested in the total number of ways in which m basic variables can be selected (without regard to their order within the vector of primal basic variables X_B) from a group of n variables.