Dual Simplex Method

Dual Simplex Method

Introduction

Primal Problem

Maximize $Z=3x_1+5x_2$
subject to
$$\begin{array}{rl}{x}_1& \le 4\\ 2x_2& \le 12\\ 3x_1+2x_2& \le 18\end{array}$$
and
$$x_1\ge 0,x_2\ge 0$$

Graphical Analysis

Solve primal problem using simplex method

Dual Problem

$$\text{ Maximize }Z=-4y_1-12y_2-18y_3$$
subject to
$$\begin{array}{r}{y}_1+3y_3\ge 3\\ 2y_2+2y_3\ge 5\end{array}$$
and
$$y_1\ge 0,y_2\ge 0,y_3\ge 0$$

Solve dual problem using dual simplex method

Compare simplex method and dual simplex method

1. The simplex method deals directly with basic solutions in the primal problem that are primal feasible($B^{-1}b\ge 0$) but not dual feasible. It then moves toward an optimal solution by striving to achieve dual feasibility (optimality test,$C-C_B{B}^{-1}A\le 0$) providing primal feasible(minimum ratio,$B^{-1}b\ge 0$).
2. The dual simplex method deals with basic solutions in the primal problem that are dual feasible($C-C_B{B}^{-1}A\le 0$) but not primal feasible. It then moves toward an optimal solution by striving to achieve primal feasibility (optimality test, $B^{-1}b\ge 0$) providing dual feasible(minimum ratio,$C-C_B{B}^{-1}A\le 0$).
3. According to the strong duality, both primal problem and dual problem approaches optimal solution when both are feasible.

Initial and later implex tableau in matrix form

Corresponding primal-dual forms