Question
A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5760.00 to invest and has space for at most 20 items. A fan costs him Rs 360.00 and a sewing machine Rs 240.00. His expectation is that he can sell a fan at a profit of Rs 22.00 and a sewing machine at a profit of Rs 18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Translate this problem mathematically and then solve it.
Solution
x = 8 and y = 12 & maximum value of Z is 392.
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 OA_{2} PB_{1} areO (0, 0), A_{2} (16, 0), P (8, 12) and B_{1} (0, 20).
These points have been obtained by solving the corresponding intersecting lines, simultaneously.
The values of the objective function Z at cornerpoints 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 
A_{2} (16, 0) 
Z = 22 × 16 + 18 × 0 = 352 
P (8, 12) 
Z = 22 × 8 + 18 × 12 = 392 
B_{1} (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.
SIMILAR QUESTIONS
Solve the following linear programming problem graphically:
Maximize Z = 50x + 15y
Subject to
5x + y ≤ 100
x + y ≤ 60
x, y ≥ 0.
Solve the following LPP graphically:
Maximize Z = 5x + 7y
Subject to
x + y ≤ 4
3x + 8y ≤ 24
10x + 7y ≤ 35
x, y ≥ 0
Solve the following LPP graphically:
Minimize Z = 3x + 5y
Subject to
– 2x + y ≤ 4
x + y ≥ 3
x – 2y ≤ 2
x, y ≥ 0
A house wife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A,12 units of vitamin B and 8 units of vitamin C.
The vitamin contents of one kg of food is given below:

Vitamin A 
Vitamin B 
Vitamin C 
Food X: 
1 
2 
3 
Food Y: 
2 
2 
1 
One kg of food X costs Rs 6 and one kg of food Y costs Rs 10. Find the least cost of the mixture which will produce the diet.
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A farm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. In view of the need to ensure certain nutrient constituents (call them X, Y and Z). it is necessary to buy two additional products, say A and B. One unit of product A contains 36 units of X, 3 units of Y, and 20 units of Z. One unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The minimum requirement of X, Y and Z is 108 units, 36 units and 100 units respectively. Product A costs Rs 20 per unit and product B costs Rs 40 per unit. Formulate the above as a linear programming problem to minimize the total cost, and solve the problem by using graphical method.