$A$ manufacturer of electronic circuits has a stock of $200$ resistors,$120$ transistors and $150$ capacitors and is required to produce two types of circuits $A$ and $B$. Type $A$ requires $20$ resistors,$10$ transistors and $10$ capacitors. Type $B$ requires $10$ resistors,$20$ transistors and $30$ capacitors. If the profit on type $A$ circuit is $Rs. 50$ and that on type $B$ circuit is $Rs. 60$,formulate this problem as a $LPP$ so that the manufacturer can maximize his profit.

Let the manufacturer produce $x$ units of type $A$ circuits and $y$ units of type $B$ circuits.
From the given information,we have the following corresponding constraint table:

Component	Type $A$ $(x)$	Type $B$ $(y)$	Maximum Stock
Resistors	$20$	$10$	$200$
Transistors	$10$	$20$	$120$
Capacitors	$10$	$30$	$150$
Profit	$Rs. 50$	$Rs. 60$	-

Thus,the objective function for profit is $Z = 50x + 60y$.
Now,we have the following mathematical model for the given problem:
Maximize $Z = 50x + 60y$
Subject to the constraints:
$20x + 10y \leq 200$ (Resistors constraint) $\Rightarrow 2x + y \leq 20$
$10x + 20y \leq 120$ (Transistors constraint) $\Rightarrow x + 2y \leq 12$
$10x + 30y \leq 150$ (Capacitors constraint) $\Rightarrow x + 3y \leq 15$
$x \geq 0, y \geq 0$ (Non-negativity constraints)
So,the $LPP$ is to maximize $Z = 50x + 60y$ subject to $2x + y \leq 20, x + 2y \leq 12, x + 3y \leq 15, x \geq 0, y \geq 0$.