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Solve the fol...

Question

Solve the following Linear Programming Problem graphically:
Minimize $Z = - 3 x + 4 y$
Subject to: $x + 2 y \leq 8$ ,
$3 x + 2 y \leq 12, x \geq 0, y \geq 0$

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Solution

Minimize $Z = - 3 x + 4 y$
Subject to
$x + 2 y \leq 8$ , $3 x + 2 y \leq 12, x \geq 0, y \geq 0$

$x + 2 y = 8$
$\begin{matrix} x & 0 & 8 y & 4 & 0 \end{matrix}$

$3 x + 2 y = 12$
$\begin{matrix} x & 0 & 4 y & 6 & 0 \end{matrix}$
$\begin{matrix} Corner points & Value of Z = - 3 x + 4 y (0, 4) & 16 (2, 3) & 6 (4, 0) & - 12 (0, 0) & 0 \end{matrix}$

Hence, $Z = - 12$ is minimum at $(4, 0)$ .

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