Question

(Manufacturing problem) A manufacturing company makes two models $A$ and $B$ of a product. Each piece of Model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of Model $B$ requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing, the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of $Rs.8000$ on each piece of model $A$ and $Rs.12000$ on each piece of Model $B$. How many pieces of Model $A$ and Model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Answer 1

Suppose $x$ is the number of pieces of Model $A$ and $y$ is the number of pieces of Model $B$. Then
Total profit (in Rs) $=8000x+12000y$
Let $Z=8000x+12000y$
We now have the following mathematical model for the given problem.

Maximise $Z=8000x+12000y...\left(1\right)$

subject to the constraints :
$9x+12y\le 180$ (Fabricating constraint)
i.e. $3x+4y\le 60...\left(2\right)$
$x+3y\le 30$ (Finishing constraint) $...\left(3\right)$
$x\ge 0,y\ge 0$ (non-negative constraint) $...\left(4\right)$

The feasible region (shaded) $OABC$ determined by the linear inequalities $\left(2\right)$ to $\left(4\right)$ is shown in Fig. Note that the feasible region is bounded.
Let us evaluate the objective function $Z$ at each corner point as shown below:

Corner Point	$Z=8000x+12000y$
$0(0,0)$	$0$
$A(20,0)$	$160000$
$B(12,6)$	$168000\leftarrow Maximum$
$C(0,10)$	$120000$

We find that maximum value of $Z$ is $1,68,000$ at $B(12,6)$. Hence, the company should produce $12$ pieces of Model $A$ and $6$ pieces of Model $B$ to realise maximum profit and maximum profit then will be $Rs.1,68,000$.