Question

A toy company manufactures two types of dolls, $A$ and $B$. Market tests and available resources have indicated that the combined production level should not exceed $1200$ dolls per week and the demand for dolls of type $B$ is at most half of that for dolls of type $A$. Further, the production level of dolls of type $A$ can exceed three times the production of dolls of other types by at most $600$ units. If the company makes a profit of $Rs.12$ and $Rs.16$ per doll respectively on dolls $A$ and $B$, how many of each should be produced weekly in order to maximize the profit?

Answer 1

Let's assume that the number of dolls of type A is $X$

and number of dolls of type B be $Y$

Since combined production level should not exceed 1200 dolls

$\therefore X+Y\le 1200...\left(1\right)$

Since production levels of dolls of type A exceeds 3 times the production of type B by at most 600 units

$\therefore X-3Y\le 600...\left(2\right)$

Also, the demands of dolls of type B is at most half of that for dolls of type A

$\therefore Y\le \frac{X}{2}$

$\Rightarrow 2Y-X\le 0...\left(3\right)$

Since the count of an object can't be negative.

So, $X\ge 0,Y\ge 0...\left(4\right)$

Now, profit on type A dolls $=12Rs$

and profit on type B dolls $=16Rs$

So, total profit $Z=12X+16Y$

We have to maximize the total profit (Z) of the manufacturers.

After plotting all the constraints given by equation (1), (2), (3) and (4) we get the feasible region as shown in the image.

Corner points	Value of $Z=12X+16Y$
A (800,400)	16000 (maximum)
B (1050,150)	15000
C (600,0)	7200
O (0,0)	0

So, in order to maximize the profit, the company should produce $800$ type A dolls and $400$ type B dolls