Question

Show that the maximum of $Z$ occurs at more than two points.
Maximise $Z=-x+2y$, subject to the constraints:
$x\ge 3,x+y\ge 5,x+2y\ge 6,y\ge 0$.

Answer 1

Objective function : $Z=-x+2y$

We have to minimize and maximize Z on constraints

$x\ge 3$

$x+y\ge 5$

$x+2y\ge 6$

$y\ge 0$

After plotting all the constraints on coordinate plane, we get the feasible region as shown in the image.

Corner points	Value of $Z=-x+2y$
(3,2)	1 (Maximum)
(4,1)	-2
(6,0)	-6 (Minimum)

Now, since region is unbounded. Hence we need to confirm that minimum and maximum value obtained through corner points is true or not.

Now, plot the region $Z\le -6\Rightarrow -x+2y\le -6$. Since, this region has common region with feasible region. Hence, there are many points where value of Z will be less than -6. So, minimum of Z not exists.

(Example :- check the value of Z at $(8,0.5)$ )

Similarly, plot the region $Z\ge 1\Rightarrow -x+2y\ge 1$. Since, this region has common region with feasible region. Hence there are many points where value of Z will be greater than 1. So, maximum of Z doesn't exists.

(Example :- check the value of Z at $(4,4)$ )