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प्रश्न
A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 on magazines A and B per copy. These are processed on three machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II and 2 hours on Machine III. Magazine B requires 3 hours on Machine I, 2 hours on Machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, 60 hours per week respectively. Formulate the Linear programming problem to maximize the profit.
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उत्तर
Let the company print x magazines of type A and y magazines of type B.
The profit on each copy of A and B is ₹ 10 and ₹ 15 respectively.
∴ Total profit = ₹ (10x + 15y)
We construct a table with the constraints of machines I, II, III as follows.
| Machine\Magazine | A (x) |
B (y) |
Available Time per week |
| I | 2 | 3 | 36 |
| II | 5 | 2 | 50 |
| III | 2 | 6 | 60 |
From the table, total time required for machines I, II, III are (2x + 3y) hours, (5x + 2y) hours and (2x + 6y) hours respectively.
∴ The constraints are:
2x + 3y ≤ 36,
5x + 2y ≤ 50,
2x + 6y ≤ 60
Since x, y cannot be negative, we have x ≥ 0, y ≥ 0
∴ Given problem can be formulated as,
Maximize Z = 10x + 15y
Subject to 2x + 3y ≤ 36, 5x + 2y ≤ 50, 2x + 6y ≤ 60, x ≥ 0, y ≥ 0.
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