Question

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ required $5$ minutes each for cutting and $10$ minutes each for assembling. Souvenirs of type $B$ required $8$ minutes each for cutting and $8$ minutes each for assembling. There are $3$ hours $20$ minutes available for cutting and $4$ hours of assembling. The profit is Rs.$5$ each for type $A$ and $Rs.$6$eachfortype$B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

Answer 1

Let the company manufacture $x$ souverins of type $A$ and $y$ souvenirs of type $B$.
Therefore,
$x\ge 0$ and $y\ge 0$
The given information can be compiled in a table as follows.

	Type A	Type B	Availability
Cutting (min)	$5$	$8$	$3\times 60+20=200$
Assembling (min)	$10$	$8$	$4\times 60=240$

The profit on type $A$ souvenirs is Rs.$5$ and on type $B$ souvenirs is Rs.$6$.Therefore, the constraints are
$5x+8y\le 200$
$10x+8y\le 240$ i.e., $5x+4y\le 120$
Total profit, $Z=5x+6y$
The mathematical formulation of the given problem is
Maximise $Z=5x+6y.........\left(1\right)$
subject to the constraints
$5x+8y\le \mathrm{200........}\left(2\right)$
$5x+4y\le \mathrm{120........}\left(3\right)$
$x,y\ge \mathrm{0........}\left(4\right)$
The feasible region determined by the system of constraints is as shown.
The corner points are $A(24,0),B(8,20)$ and $C(0,25)$
The values of $Z$ at these corner points are as follows.

Corner point	$Z=5x+6y$
$A(24,0)$	$120$
$B(8,20)$	$160$	$\to$ Maximum
$C(0,25)$	$150$

The maximum value of $Z$ is $200$ at $(8,20)$.
Thus, $8$ souvenirs of type $A$ and $20$ souvenirs of type $B$ should be produced each day to get the maximum profit of Rs.$160$