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प्रश्न
A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 in 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, and 60 hours per week respectively. Formulate the LPP to determine weekly production of magazines A and B, so that the total profit is maximum.
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उत्तर
Let the company prints x magazine of type A and y magazine of type B.
Profit on sale of magazine A is ₹ 10 per copy and magazine B is ₹ 15 per copy. Therefore, the total earning z of the company is z = ₹(10x + 15y).
This is a linear function that is to be maximized. Hence, it is an objective function.
The constraints are as per the following table:
| Magazine type → | Time required per unit | Available time per week (in hours) | |
| Machine type ↓ | Magazine A (x) | Magazine B (y) | |
| Machine I | 2 | 3 | 36 |
| Machine II | 5 | 2 | 50 |
| Machine III | 2 | 6 | 60 |
From the table, the total time required for Machine I is (2x + 3y) hours, for Machine II is (5x + 2y) hours and Machine III is (2x + 6y) hours.
The machines I, II, III are available for 36, 50, and 60 hours per week. Therefore, the constraints are 2x + 3y ≤ 36, 5x + 2y ≤ 50, 2x + 6y ≤ 60.
Since x and y cannot be negative. We have, x ≥ 0, y ≥ 0. Hence, the given LPP 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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