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प्रश्न
A dealer deals in two products X and Y. He has ₹ 1,00,000/- to invest and space to store 80 pieces. Product X costs ₹ 2500/- and product Y costs ₹ 1000/- per unit. He can sell the items X and Y at respective profits of ₹ 300 and ₹ 90. Construct the LPP and find the number of units of each product to be purchased to maximize its profit
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उत्तर
Let the dealer purchase x units of X and y units of Y.
The dealer can sell items X and Y at respective profits of ₹ 300 and ₹ 90
∴ Total profit = Z = 300x + 90y
∴ Given problem can be formulated as follows:
Maximize Z = 300x + 90y
Subject to x + y ≤ 80
2500x + 1000y ≤ 100000, x ≥ 0, y ≥ 0
To draw the feasible region, construct table as follows:
| Inequality | x + y ≤ 80 | 2500x + 1000y ≤ 100000 |
| Corresponding equation (of line) | x + y = 80 | 2500x + 1000y = 100000 |
| Intersection of line with X-axis | (80, 0) | (40, 0) |
| Intersection of line with Y-axis | (0, 80) | (0, 100) |
| Region | Origin side | Origin side |
Shaded portion OABC is the feasible region, whose vertices are O(0, 0), A(40, 0), B and C(0, 80).
B is the point of intersection of the lines 2500x + 1000y = 100000 and x + y = 80.
Solving the above equations, we get
x = `40/3`, y = `200/3`
∴ B ≡ `(40/3, 200/3)`
Here, the objective function is
Z = 300x + 90y
∴ Z at O(0, 0) = 300(0) + 90(0) = 0
Z at A(40, 0) = 300 (40) + 90(0)
= 12000
Z at B`(40/3, 200/3) = 300(40/3) + 90(200/3)`
= 4000 + 6000
= 10000
Z at C(0, 80) = 300 (0) + 90(80)
= 7200
∴ Z has maximum value 12000 at x = 40 and y = 0.
∴ The dealer should purchase 40 units of X and 0 units of Y to maximize its profit.
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