Chapter 6.7, Problem 5P

Explanation of Solution

Using the Duel Theorem for proving that the computation must be incorrect:

Given LP:

Maximize, $Z=4x_1+x_2$

Subject to the constraints,

$\begin{array}{l}3x_1+2x_2\le 6\\ 6x_1+3x_2\le 10\end{array}$

And $x_1,x_2\ge 0$

$Z+2x_2+S_2=20/3$

Dual Theorem:

• In the dual theorem, the ${\text{i}}^{\text{th}}$ dual constraint corresponds to the primal variable ${\text{x}}_{\text{i}}$.
• For example,
  • The first dual constraint corresponds to $x_1$ because each number comes from the $x_1$ column of the primal.
  • The second dual constraint corresponds to $x_2$.
• In the same fashion, the ${\text{y}}_{\text{i}}$ is associated with the ${\text{i}}^{\text{th}}$ primal constraint.
• For example,
  • ${\text{y}}_{\text{1}}$ is associated with the first primal constraint as each coefficient of ${\text{y}}_{\text{1}}$ in the dual comes from the corresponding constraint.
  • $y_2$ is associated with the second primal constraint.

The table is given below,

Max Z
Min W
		$x_1\ge 0$	$x_2\ge 0$
${\text{y}}_1\ge 0$	${\text{y}}_1$	3	2	$\le 6$
${\text{y}}_2\ge 0$	${}_{}$