Question

Use the graphical method to solve each of the following LP problems.
A factory manufactures two products, each requiring the use of three machines. The first machine can be used at most $70$ hours; the second machine at most $40$ hours; and the third machine at most $90$ hours. The first product requires $2$ hours on Machine 1, $1$ hour on Machine 2, and $1$ hour on Machine 3; the second product requires $1$ hour each on machine 1 and 2 and $3$ hours on Machine 3. If the profit in $E40$ per unit for the first product and $E60$ per unit for the second product, how many units of each product should be manufactured to maximize profit?

Answer 1

Given

Machine	Product A	Product B	Time
1	2	1	70
2	1	1	40
3	1	3	90
Profit	40	60

The constraints for given data

For Machine:1 $2x+y\le$ 70

2: $x+y\le$ 40

3: $x+3y\le$ 90

$x\ge$0 ,$y\ge$0

To maximize:Total profit($z$)= $40x+60y$

The bounded area is OEGB

For $z=40x+60y$

Point$(x,y)$	z value$(40x+60y)$
O(0,0)	0
E(0,30)	1800
G(20,30)	2600 $\to$ maximum
B(0,35)	2100

So at value G(20,30) the profit is maximum and the maximum value is 2600