Solve the following linear programming problem by graphical metho… - Vidyadip

Question

Solve the following linear programming problem by graphical method.Under the constraints, maximise $Z=60 x+40 y$.
$
\begin{aligned}
x+2 y & \leq 12 \\
2 x+y & \leq 12 \\
x+\frac{5}{4} y & \geq 5 ; x \geq 0, y \geq 0
\end{aligned}
$

✓

Answer

The given constraints are as follows :
$
\begin{aligned}
x+2 y & \leq 12\quad \quad \ldots \ldots(1) \\
2 x+y & \leq 12\quad \quad \ldots \ldots(2) \\
x+\frac{5}{4} y & \geq 5 \quad \quad \ldots \ldots(3)\\
x \geq 0, y & \geq 0\quad \quad \ldots \ldots(4)
\end{aligned}
$
We draw the graph of the constraints (1) to (4). As shown with figure, the feasible region is ABCDE (shaded) which has been determined by the constraints (1) to (4). On observation we find that the feasible region is bounded.

The coordinates of the corner points $A , B , C , D$ and E and respectively $(5,0),(6,0),(4,4),(0,6)$ and $(0,4)$.

Corner Point	Value of Z = 60x + 40y
(5,0)	300
(6,0)	360
(4, 4)	400 ← Maximum
(0,6)	240
(0,4)	1600

We see that at the point (4, 4), the value of Z is maximum.

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