Q.
Assertion (A) The maximum value of
$Z=11 x+7 y$
subject to the constraints
$2 x+y \leq 6$
$x \leq 2$
$x \geq 0, y \geq 0$
occurs at the corner point $(0,6)$.
Reason (R) If the feasible region of the given LPP is bounded, then the maximum and minimum value of the objective function occurs at corner points.
Linear Programming
Solution:
The given LPP is
Maximise $Z=11 x+7 y$
Subject to the constraints are
$2 x+y \leq 6$
$x \leq 2 $
$x \geq 0, y \geq 0$
The corresponding graph of the above LPP is
from the above graph, we see that the shaded region is the feasible region $O A B C$ which is bounded.
$\therefore$ The maximum value of the objective function $Z$ occurs at the corner points.
The corner points are $O(0,0), A(0,6), B(2,2), C(2,0)$.
The values of $Z$ at these corner points are given by
Corner point
Corresponding value of $ z-11 x+7 y $
$(0,0)$
0
$(0,6)$
$42 \leftarrow$ Maximum
$(2,2)$
36
$(2,0)$
22
Thus, the maximum valuo of $Z$ is 42 which occurs at the point $(0,6)$.
| Corner point | Corresponding value of $ z-11 x+7 y $ |
|---|---|
| $(0,0)$ | 0 |
| $(0,6)$ | $42 \leftarrow$ Maximum |
| $(2,2)$ | 36 |
| $(2,0)$ | 22 |