A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize the profit? Formulate this as a linear programming problem and solve it graphically.
Let the dealer buy x fans and y sewing machines. The LPP is to maximize profit Z = 22x + 18y
Subject to constraints:
⇒ x + y ≤ 20
⇒ 360x + 240y ≤ 5760
⇒ 3x + 2y ≤ 48 and x ≥ 0, y ≥ 0
First we draw the lines AB and CD whose equations are x + y = 20
And 3x + 2y = 48
The feasible region is OBPCO which is shaded in the above figure.
P is the point of intersection of the lines x + y = 20 and 3x + 2y = 48.
Solving these equations, we get the point P (8, 12).
The vertices of the feasible region are O (0, 0), B (0, 20), P (8, 12), C (16, 0)
The value of objective function Z = 22x + 18y at these vertices are as follows:
∴ The maximum profit is Rs. 392 when 8 fans and 12 sewing machines are purchased.
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