Question

A manufacture of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients available to make 20000 bottles of A and 40000 bottles of B but there are only 45000 bottles into which either of the medicines can be put. Furthermore, it takes 3 hours to prepare enough material to fill 1000 bottles of A and it takes 1 hour to prepare enough material to fill 1000 bottles of B, and there are 66 hours available for this operation. The profit is ₹8 per bottle for A and ₹7 per bottle for B. 

How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

Answer 1

Let x and y be number of bottles of medicines A and B be prepared. 

∴According to the question, 

x + y ≤ 45000 , 3x + y ≤ 66000, x ≤ 20000, y ≤ 40000, x ≥ 0, y ≥ 0 

Maximize Z = 8x + 7y 

The feasible region determined by x + y ≤ 45000 , 3x + y ≤ 66000, x ≤ 20000, y ≤ 40000, x ≥ 0, y ≥ 0 is given by

The corner points of feasible region are A(0,0) , B(0,40000) ,C(5000,40000),D(10500,34500),E(20000,6000),F(20000,0).

The value of Z at corner points are

Corner Point	Z = 8x + 7y
A(0, 0)	0
B(0, 40000)	280000
C(5000, 40000)	320000
D(10500, 34500)	325500	Maximum
E(20000, 6000)	202000
F(20000, 0)	160000

The maximum value of Z is 325500 at point (10500,34500). 

Hence, the manufacturer should produce 10500 bottles of medicine A and 34500 bottles of medicine B.