Question

The minimum and maximum values of $Z$ for the problem,
minimise and maximise $Z=3 x+9 y$...(i)
subject to the constraints
$x+3 y \leq 60$...(ii)
$x+y \geq 10$...(iii)
$x \leq y $...(iv)
$x \geq 0, y \geq 0$...(v)
are respectively

• A. 60 and 180
• B. 180 and 60
• C. 50 and 190
• D. 190 and 50

Answer 1

Answer: (A)

Given that,
Minimise and Maximise $Z=3 x+9 y$...(i)
Subject to the constraints are
$x+3 y \leq 60$...(ii)
$x+y \geq 10$...(iii)
$x \leq y $...(iv)
$x \geq 0, y \geq 0$...(v)
First of all, let us graph the feasible region of the system of linear inequalities (ii) to $(v)$. The feasible region $A B C D$ is shown in the figure. Note that the region is bounded. The coordinates of the corner points $A, B, C$ and $D$ are $(0,10)$, $(5,5),(15,15)$ and $(0,20)$, respectively.

We, now find the minimum and maximum value of $Z$. From the table, we find that the minimum value of $Z$ is 60 at the point $B(5,5)$ of the feasible region.
The maximum value of $Z$ on the feasible region occurs at the two corner points $C(15,15)$ and $D(0,20)$ and it is 180 in each case.
Remark Observe that in the above example, the problem has multiple optimal solutions at the corner points $C$ and $D$, i.e., the both points produce same maximum value 180 . In such cases, you can see that every point on the line segment $C D$ joining the two corner points $C$ and $D$ also give the same maximum value. Same is also true in the case, if the two points produce same minimum value.