# Big M and Simplex methods

• Nov 10th 2007, 06:22 AM
khrst4
Big M and Simplex methods
Consider the following problem.

Minimize Z = 3x1 + 2x2,
subject to
2x1 + x2 ≥ 10
-3x1 + 2x2 ≤ 6
x1 + x2 ≥ 6
and
x1 ≥ 0, x2 ≥ 0.

(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify the initial entering basic variable and the leaving basic variable.

(b) Work through the simplex method step by step to solve the problem.
• Nov 10th 2007, 10:12 AM
Soroban
Hello, khrst4!

Quote:

Minimize: . $Z \:= \:3x + 2y$

subject to: . $\begin{Bmatrix}2x + y & \geq & 10 \\
-3x + 2y & \leq & 6 \\ x + y & \geq & 6 \end{Bmatrix}$
. and: . $\begin{Bmatrix}x & \geq & 0 \\ y & \geq & 0 \end{Bmatrix}$

(a) Using the Big M method, construct the complete first simplex tableau for the simplex method
and identify the corresponding initial (artificial) BF solution.
Also identify the initial entering basic variable and the leaving basic variable.

(b) Work through the simplex method step by step to solve the problem.

I haven't use the Simplex Method in decades.
. . But I can solve it with traditional graphic methods.

From $x \geq 0,\;y \geq 0$, we are in Quadrant 1.

Graph the line: . $2x + y \:=\:10$
It has intercepts: $(5,0),\:(0,10)$
Draw the line and shade the region above the line.

Graph the line: . $-3x + 2y \:=\:6$
It has intercepts: $(-2,0),\;(0,3)$
Draw the line and shade the region below the line.

Graph the line: . $x + y \:=\:6$
It has intercepts: $(6,0),\;(0,6)$
Draw the line and shade the region above the line.
Code:

                      |                   10 *                       |*                       | *                       |  *                       |    *                       |    *                       |      *                       |      *                     6 *        *              *                       | *      *        *::::                       |  *      *    *::::::::                       |    *    o::::::::::::                       |      *    *:::::::::::                       |  *    *  *::::::::::                     3 *          *  *:::::::::                   *  |            * *::::::::               *      |              **:::::::           *          |                o::::::       *              |                  **:::: - * - - - - - - - - - + - - - - - - - - - * o - -  -2                  |                  5 6

The shaded region has vertices: . $(6,0),\;(4,2),\;(2,6)$

Test those in the $Z$-function for minimum $Z.$