Chapter 6.7, Problem 1P

Explanation of Solution

a.

Primal of Giapetto:

Maximize, $z=3x_1+2x_2$

Subject to the constraints,

$\begin{array}{c}2x_1+x_2\le 100\\ x_1+x_2\le 80\\ x_1\le 40\end{array}$

Dual of Giapetto:

Minimize, ${\text{z}}_{\text{y}}{}_{}$

Explanation of Solution

b.

Optimal dual solution:

From the given optiml table of the Giapetto problem, the optimal solution of the optimal dual solution will be,

Minimize,

Explanation of Solution

c.

Verify Dual theorem holds:

The user has to solve dual to verify the primal solution with the dual solution.

After adding the surplus and artificial variables;

Minimize, ${\text{z}}_{\text{y}}{\text{=100y}}_{\text{1}}{\text{+80y}}_{\text{2}}{\text{+40y}}_{\text{3}}{\text{+0(S}}_{\text{1}}{\text{+S}}_{\text{2}}{\text{)+M(A}}_{\text{1}}{\text{+A}}_{\text{2}}\text{)}$

Subjected to,

$\begin{array}{l}{\text{y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{+y}}_{\text{3}}{\text{-S}}_{\text{1}}{\text{+A}}_{\text{1}}\text{=3}\\ {\text{y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{-S}}_{\text{2}}{\text{+A}}_{\text{2}}\text{=2}\end{array}$

And ${\text{y}}_{\text{1}}{\text{,y}}_{\text{2}}{\text{,y}}_{\text{3}}{\text{,S}}_{\text{1}}{\text{,S}}_{\text{2}}{\text{,A}}_{\text{1}}{\text{,A}}_{\text{2}}\ge \text{0}$

Iteration-1		$C_j$	100	80	40	0	0	M	M
B	$C_B$	$X_B$	$y_1$	$y_2$	$y_3$	$S_1$	$S_2$	${\text{A}}_{\text{1}}$	${\text{A}}_{\text{2}}$	Min Ratio
${\text{A}}_{\text{1}}$	0	2	-1	0	-1	1	0	0	0	$\text{1}$