Question

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes $2$ hours on grinding/cutting machine and $3$ hours on the sprayer to manufacture a pedestal lamp. It takes $1$ hour on the grinding/cutting machine and $2$ hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at least for at the most $20$ hours and the grinding/cutting machine for at the most $12$ hours. The profit from the sale of a lamp is Rs.$5$ and that from a shade is Rs. $3$ . Assuming that the manufacturer can sell all the lamps and shades that the produces , how should he schedule his daily production in order to maximise his profit?

Answer 1

Let the cottage industry manufacture $x$ pedestal lamps and $y$ wooden shades. Therefore, $x\ge 0$ and $y\ge 0$
The given information can be compiled in a table as follows.

	Lamps	Shades	Availability
Grinding/Cutting Machine (h)	$2$	$1$	$12$
Sprayer	$3$	$2$	$20$

The profit on a lamp is Rs.$5$ and on the shades is Rs.$3$. Therefore, the constraints are
$2x+y\le 12$
$3x+2y\le 20$
Total profit $Z=5x+3y$
The mathematical formulation of the given problem is
Maximise $Z=5x+3y.........\left(1\right)$
subject to the constraints
$2x+y\le \mathrm{12.......}\left(2\right)$
$3x+2y\le \mathrm{20.....}\left(3\right)$
$x,y\ge \mathrm{0........}\left(4\right)$
The feasible region determined by the system of constraints is as shown.
The corner points are $A(6,0),B(4,4)$ and $C(0,10)$
The values of $Z$ at these corner points are as follows

Corner point	$Z=5x+3y$
$A(6,0)$	$30$
$B(4,4)$	$32$	$\to$ Maximum
$C(0,10)$	$30$

The maximum value of $Z$ is $32$ at $(4,4)$
Thus, the manufacturer should produce $4$ pedestal lamps and $4$ wooden shades to maximise his profits.