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, \(\frac{100}{3}\)) 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.
