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प्रश्न
Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F1 and F2. F1 provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F2 provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F1 costs ₹ 3 and F2 costs ₹ 2 per 10 grams. How many grams of each F1 and F2 should buy every day to keep her food bill minimum
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उत्तर
Let food F1 be x grams and food F2 be y grams.
Since x and y cannot be negative, x ≥ 0, y ≥ 0.
F1 costs ₹ 3 and F2 costs ₹ 2 per 10 grams.
∴ Total cost = Z = 3x + 2y
We construct a table with constraints of vitamins A, B and C as follows:
| Vitamins/Food | F1 | F2 | Minimum requirement |
| A | 7 | 2 | 30 |
| B | 5 | 4 | 20 |
| C | 2 | 8 | 16 |
From the table, the constraints are
7x + 2y ≥ 30
5x + 4y ≥ 20
2x + 8y ≥ 16
∴ Given problem can be formulated as follows:
Minimize Z = 3x + 2y
Subject to 7x + 2y ≥ 30
5x + 4y ≥ 20,
2x + 8y ≥ 16, x ≥ 0, y ≥ 0
To draw the feasible region, construct table as follows:
| Inequality | 7x + 2y ≥ 30 | 5x + 4y ≥ 20 | 2x + 8y ≥ 16 |
| Corresponding equation (of line) | 7x + 2y = 30 | 5x + 4y = 20 | 2x + 8y = 16 |
| Intersection of line with X-axis | `(30/7, 0)` | (4, 0) | (8, 0) |
| Intersection of line with Y-axis | (0, 15) | (0, 5) | (0, 2) |
| Region | Non-Origin side | Non-Origin side | Non-Origin side |
Shaded portion XABCY is the feasible region, whose vertices are A(8, 0), B and C(0, 15),
B is the point of intersection of the lines 7x + 2y = 30 and 2x + 8y = 16
Solving the above equations, we get
x = 4, y = 1
∴ B ≡ (4, 1)
Here, the objective function is
Z = 3x + 2y
Z at A (8, 0) = 3(8) + 2(0) = 24
Z at B (4, 1) = 3(4) + 2(1)
= 12 + 2
= 14
Z at C (0, 15) = 3(0) + 2(15)
= 30
∴ Z has minimum value 14 at x = 4 and y = 1.
∴ Smita should buy 4 grams of food F1 and 1 gram of food F2 every day to keep her food bill minimum.
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