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  4. Minimize Z = 20x1 + 9x2, subject to x1 ge 0, x2 ge 0, 2x1 + x2 ge 36, 6x1 + x2 ge 60.

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Q. Minimize $ Z = 20x_1 + 9x_2$, subject to $x_1 \ge 0$, $x_2 \ge 0$, $2x_1 + x_2 \ge 36$, $6x_1 + x_2 \ge 60$.

Linear Programming

Solution:

We have, Minimize $ Z = 20x_1 + 9x_2$,
subject to $2x_1 + x_2 \ge 36, 6x_1 + x_2 \ge 60$, $x_1 \ge 0$, $x_2 \ge 0$
Let $l_1 : 2x_1 + x_2 = 36, l_2 : 6x_1 + x_2 = 60$
$l_3 : x_1 = 0$ and $l_4 : x_2 = 0$
image
For B : Solving $l_1$ and $l_2$, we get $B(6,24)$
Shaded portion is the feasible region, where
$A(18, 0), B(6, 24), C(0, 60)$
Now minimize $Z = 20x_1 + 9x_2$
$Z$ at $A(18,0) = 20(18) + 9(0) = 360$
$Z$ at $B(6, 24) = 20(6) + 9(24) = 336$
$Z$ at $C(0, 60) = 20(0) + 9(60) = 540$
Thus, $Z$ is minimized at $B(6, 24)$ and its minimum value is $336$.