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प्रश्न
A doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carbohydrates and proteins are 18, 28, 14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is ₹ 4.5 per unit and that of food B is ₹ 3.5 per unit. Form the LPP, so that the sick person’s diet meets the requirements at a minimum cost.
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उत्तर
Let x units of food A and y units of food B be prescribed in the weekly diet of a sick person.
The price for food A is ₹ 4.5 per unit and for food B is ₹ 3.5 per unit.
∴ Total cost is z = ₹ (4.5x + 3.5y)
We construct a table with constraints of fats, carbohydrates and proteins as follows:
| Nutrients\Foods | A (x) |
B (y) |
Minimum requirement |
| Fats | 4 | 6 | 18 |
| Carbohydrates | 14 | 12 | 28 |
| Protein | 8 | 8 | 14 |
From the table, diet of sick person must include (4x + 6y) units of fats, (14x + 12y) units of carbohydrates and (8x + 8y) units of proteins
∴ The constraints are
4x + 6y ≥ 18,
14x + 12y ≥ 28,
8x + 8y ≥ 14.
Since x and y cannot be negative, we have x ≥ 0, y ≥ 0
∴ Given problem can be formulated as follows:
Minimize z = 4.5x + 3.5y
Subject to 4x + 6y ≥ 18, 14x + 12y ≥ 28, 8x + 8y ≥ 14, x ≥ 0, y ≥ 0.
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