Question

Solve the following linear programming problem graphically:
Maximise $Z=4x+y...\left(1\right)$
subject to the constraints: $x+y\le 50...\left(2\right)$
$3x+y\le 90...\left(3\right)$
$x\ge 0,y\ge 0...\left(4\right)$.

Answer 1

The shaded region in fig. is the feasible region determined by the system of constraints $\left(2\right)$ to $\left(4\right)$. We observe that the feasible region $OABC$ is bounded. So,we now use Corner Point Method to determine the maximum value of $Z$.
The coordinates of the corner points $O,A,B$ and $C$ are $(0,0),(30,0),(20,30)$ and $(0,50)$ respectively. Now we evaluate $Z$ at each corner point.

Corner Point	Corresponding value of $Z$
$(0,0)$	$0$
$(30,0)$	$120\leftarrow Maximum$
$(20,30)$	$110$
$(0,50)$	$50$

Hence, maximum value of $Z$ is $120$ at the point $(30,0)$.