Question

A company uses three machines to manufacture two types of shirts, half sleeves and full sleeves. The number of hours required per week on machine $M_{1}, M_{2}$ and $M_{3}$ for one shirt of each type is given in the following table:

	$M_{1}$	$M_{2}$	$M_{3}$
Half sleeves	$1$	$2$	$\frac{8}{5}$
Full sleeves	$2$	$1$	$\frac{8}{5}$

None of the machines can be in operation for more than $40$ hours per week. The profit on each half sleeve shirt is $₹ 1$ and the profit on each full sleeve shirt is $₹ 1.50$ . How many of each type of shirts shoud be made per week to maximise the company's profit?

Accepted Answer

Let the number of half sleeves shirts produced $= x$

Let the number of full sleeves shirts produced $= y$

Maximise $Z = x + 1.5 y$

Subject to:

$x + 2 y \leq 40$

$2 x + y \leq 40$

$\frac{8}{5} x + \frac{8}{5} y \leq 40 \Rightarrow x + y \leq 25$

$x \geq 0, y \geq 0$

Now, we draw the three lines

$x + 2 y = 40$

$x$ $0$
$40$

$y$ $20$ $0$

$2 x + y = 40$

$x$ $0$ $20$

$y$ $M_{3}$ $0$

$x + y = 25$

$x$ $0$ $25$

$y$ $25$ $40$

Now, for $Z = x + 1.5 y$

At $A (0, 20)$ , $Z = 0 + 1.5 (20) = 30$

At $B (10, 15), Z = 10 + 1.5 (15) = 32.5$

At $C (15, 10), Z = 15 + 1.5 (10) = 30$

At $D (20, 0), Z = 20 + 1.5 (0) = 20$

Thus, the maximum value is at $B (10, 15) .$

Hence, the company should make $10$ half sleeves shirt and $15$ full sleeves shirts to maximize profit.

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