Suppose the dealer buys x fans and y sewing machines. Since the dealer has space for at most 20 items. Therefore,
x + y ≤ 20
A fan costs Rs 360 and a sewing machine costs Rs 240. Therefore, total cost of x fans and y sewing machines is Rs (360x + 240y). But the dealer has only Rs 5760 to invest. Therefore,
360x + 240y ≤ 5760
Since the dealer can sell all the items that he can buy and the profit on a fan is of Rs 22 and on a sewing machine the profit is of Rs 18. Therefore, total profit on selling x fans and y sewing machines is of Rs (22x + 18y).
Let Z denote the total profit. Then, Z = 22x + 18y.
Clearly, x, y ≥ 0.
Thus, the mathematical formulation of the given problem is
Maximize Z = 22x + 18y
Subject to
x + y ≤ 20
360x + 240y ≤ 5760
and, x ≥ 0, y ≥ 0
To solve this LPP graphically, we first covert the inequations into equations and draw the corresponding lines. The feasible region of the LPP is shaded in fig. The corner points of the feasible region OA2 PB1 areO (0, 0), A2 (16, 0), P (8, 12) and B1 (0, 20).
These points have been obtained by solving the corresponding intersecting lines, simultaneously.
The values of the objective function Z at corner-points of the feasible region are given in the following table.
Point (x, y)
|
Value of the objective function
Z = 22x + 18y
|
O (0, 0)
|
Z = 22 × 0 + 18 × 0 = 0
|
A2 (16, 0)
|
Z = 22 × 16 + 18 × 0 = 352
|
P (8, 12)
|
Z = 22 × 8 + 18 × 12 = 392
|
B1 (0, 20)
|
Z = 22 × 0 + 20 × 18 = 360
|
Clearly, Z is maximum at x = 8 and y = 12. The maximum value of Z is 392.
Hence, the dealer should purchase 8 fans and 12 sewing machines to obtain the maximum profit under given conditions.