Question

Determine graphically the minimum value of the objective function.
$Z=-50x+20y$
Subject to constraints
$2x-y\ge -5$
$3x+y\ge 3$
$2x-3y\le 12$
$x\ge 0,y\ge 0$.

Answer 1

Given objective function is $Z=-50x+20y$

We have to minimize Z on given constraints

$2x-y\ge -5$

$3x+y\ge 3$

$2x-3y\le 12$

$x\ge 0,y\ge 0$

After plotting all the constraints we get the common region (Feasible region) as shown in the image.

There are four corner points $(0,5),(0,3),(1,0)and(6,0)$

Now, at corner points value of Z are as follows :

Corner Points	Value of $Z=-50x+20y$
(0,5)	100
(0,3)	60
(1,0)	-50
(6,0)	-300 (minimum)

Since common region is unbounded. So, value of Z may be minimum at $(6,0)$ and minimum value may be -300.

Now to check if this minimum is correct or not, we have to draw region $-50x+20y\le -300$

Since, there are some common region with feasible region(See image). So, -300 will not be minimum value of Z.

Hence, Z has no minimum value.