Simplex Method Review Canonical Form A is m

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