Question

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

• A. $360$ at $(18,0)$
• B. $336$ at $(6,24)$
• C. $540$ at $(0,60)$
• D. $0$ at $(0,0)$

Answer 1

Answer: (B)

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 $l_4:x_2=0$

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 $336$.