Simplex Method Review

Canonical Form A is m x n Theorem 7.5: If an LP has an optimal solution, then at least one such solution exists at a basic feasible solution (BFS). For LP in canonical form a BFS is a solution for which there are m basic variables and n-m nonbasic variables Columns of A associated with basic variables (denoted by B) are linearly independent, x N = 0, x B uniquely solve B x B =b and x B  0.

Matrix Representation

Initial Solutions

Feasible Directions

Determining Improving Directions

More Than One Improving Direction We will use a greedy rule for selecting our improving simplex direction. For a maximization problem, we choose the simplex direction whose reduced cost is most positive. For a minimization problem, we choose the simplex direction whose reduced cost is most negative. Rule is called the Dantzig rule after George Dantzig, the founder of the simplex method.

No Improving Direction Optimality Condition for an LP (Exercise 8.26): If the simplex method does not identify a simplex direction that is improving, the current solution is a global optimal solution to the LP.

Determining Maximum Step Size

Updating the Basis The nonbasic variable corresponding to the chosen simplex direction enters the basis and becomes basic. Any one of the (possibly several) basic variables that define the maximum step size will leave the basis and become nonbasic.

Basic Simplex Method