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

• A. $(A,B)\equiv (400,800)$
• B. $(A,B)\equiv (800,400)$
• C. $(A,B)\equiv (600,600)$
• D. $(A,B)\equiv (0,1200)$

Answer 1

The correct option is B $(A,B)\equiv (800,400)$

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 $\therefore X+Y\le 1200\cdots \cdots \left(i\right)$
Since production levels of dolls of type $A$ exceeds $3$ times the production of type $B$ by at most $600$ units
$\therefore X-3Y\le 600\cdots \left(ii\right)$
Also, the demands of dolls of type $B$ is at most half of that for dolls of type $A$
$\therefore Y\le \frac{X}{2}$ $\Rightarrow 2Y-X\le 0\cdots \left(iii\right)$
Since the count of an object can't be negative.
So, $X\ge 0,Y\ge 0\cdots \left(iv\right)$
Now, profit on type $A$ dolls $=\text{Rs}12$ and profit on type $B$ dolls $=\text{Rs}16$
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 $\left(i\right),\left(ii\right),\left(iii\right)$ and $\left(iv\right)$

Corner points	$Z=12X+16Y$
$A(800,400)$	$16000$ (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