A manufacturer makes two products A and B. Product A sells at 200 each and takes 1/2 hour to make. Product A sells at ₹ 300 each and takes 1 hours to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is ₹ 20 and on product B is Rs 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.
Let x units of product A and y units of product B were manufactured.
Number of units cannot be negative.
Therefore, x,y 
0.
According to question, the given information can be tabulated as:
Also, the availability of time is 40 hours and the revenue should be atleast Rs 10000.
Further, it is given that there is a permanent order for 14 units of Product A and 16 units of product B.
Therefore, the constraints are,
200x + 300y 
10000,
0.5x + y 
40
x 
14
y 
16.
If the profit on each of product A is Rs 20 and on product B is Rs 30. Therefore, profit gained on x units of product A and y units of product B is Rs 20x and Rs 30y respectively.
Total profit = 20x + 30y which is to be maximized.
Thus, the mathematical formulation of the given LPP is,
Max Z = 20x + 30y
Subject to constraints,
200x + 300y 
10000,
0.5x + y 
40
x 
14
y 
16
x,y 
0.
Region 200x + 300y
10000: line 200x + 300y = 10000 meets the axes at A(50,0), B(0,
) respectively.
Region not containing origin represents 200x + 300y
10000 as (0,0) does not satisfy 200x + 300y
10000.
Region 0.5x + y 
40: line 0.5x + y = 40 meets the axes at C(80,0), D(0,40) respectively.
Region containing origin represents 0.5x + y 
40 as (0,0) satisfies 0.5x + y 
40.
Region represented by x 
14,
x = 14 is the line passes through (14,0) and is parallel to the Y - axis. The region to the right of the line x = 14 will satisfy the inequation.
Region represented by y 
16,
y = 14 is the line passes through (16,0) and is parallel to the X - axis. The region to the right of the line y = 14 will satisfy the inequation.
Region x,y 
0: it represents first quadrant.
The corner points of the feasible region are E(26,16), F(48,16), G(14,33), H(14,24).
The values of Z at these corner points are as follow:
The maximum value of Z is Rs 1440 which is attained at F(48,16).
Thus, the maximum profit is Rs 1440 obtained when 48 units of product A and 16 units of product B are manufactured.