Question

Use the graphical method to solve each of the following LP problems.
LESCO Engineering produces chairs and tables. Each table takes four hours of labour from the carpentry department and two hours of labour from the finishing department. Each chair requires three hours of carpentry and one hour of finishing.

During the current week, $240$ hours of carpentry time are available and $100$ hours of finishing time. Each table produced gives a profit of $E70$ and each chair a profit of $E50$. How many chairs and tables should be made in order to maximize profit?

Answer 1

Department	Tables	Chairs	Available Hours
Carpentry	4	3	240
Painting	2	1	100
Profit	7	5

Let $x_1$=number of tables to be produced

$x_2$=number of chairs to be produced

Maximize $P=7x_1+5x_2$ (objective Function)

${4x}_1+3x_2\le 2402x_1+x_2\le 100x_1\ge 0,x_2\ge 0$

To find the optional solution to this LP using a graphical method we first identify the region of feasible solutions

The corner point are (0,0) ,(50,0) ,(30,40) and (0,80)

Testing these corner points on$7x_1+5x_2$

Corner Point	Profit $(P=7x_1+5x_2)$
(0,0)	0
(50,0)	350
(30,40)	410 $\to$ Maximum
(0,80)	400

At the point (30,40) produces the highest profit i.e Rs.410

$(x,y)=(30,40)$ $P=410Rs$