A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials 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. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hours 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 manufacturer schedule his production in order to maximize his profit?
Let production of each bottle of A and B are x and y respectively.
Since profits on each bottle of A and B are Rs 8 and Rs 7 per bottle respectively. So, profit on x bottles of A and y bottles of of B are 8x and 7y respectively. Let Z be total profit on bottles so,
Z = 8x + 7y
Since, it takes 3 hours and 1 hour to prepare enough material to fill 1000 bottles of Type A and Type B respectively, so x bottles of A and y bottles of B are preparing is 
hours and 
hours respectively, bout only 66 hours are available, so,
3x + y 
66000
Since raw materials available to make 2000 bottles of A and 4000 bottles of B but there are 45000 bottles in which either of these medicines can be put so,
x 
20000
y 
40000
x + y 
45000
x,y 
0. [ Since production of bottles can not be negative]
Hence mathematical formulation of the given LPP is,
Max Z = 8x + 7y
Subject to constraints,
3x + y 
66000
x 
20000
y 
40000
x + y 
45000
x,y 
0
Region 3x + y
66000: line 3x + y = 66000 meets the axes at A(22000,0), B(0,66000) respectively.
Region containing origin represents 3x + y 
10000 as (0,0) satisfy 3x + y 
66000
Region x + y 
45000: line x + y = 45000 meets the axes at C(45000,0), D(0,45000) respectively.
Region towards the origin will satisfy the inequation as (0,00 satisfies the inequation
Region represented by x 
20000,
x = 20000 is the line passes through (20000, 0) and is parallel to the Y - axis. The region towards the origin will satisfy the inequation.
Region represented by y 
40000,
y = 40000 is the line passes through (0,40000) and is parallel to the X - axis. The region towards the origin will satisfy the inequation.
Region x,y 
0: it represents first quadrant.
The corner points are O(0,0), B(0,40000), G(10500,34500), H(20000,6000), A(20000,0).
The values of Z at these corner points are,
The maximum value of Z is 325500 which is attained at G(10500, 34500).
Thus the maximum profit is Rs 325500 obtained when 10500 bottles of A and 34500 bottles of B are manufactured.