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Question
In a cattle breeding firm, it is prescribed that the food ration for one animal must contain 14, 22, and 1 unit of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit weight of these two contains the following amounts of these three nutrients:
| Nutrient\Fodder | Fodder 1 | Fodder2 |
| Nutrient A | 2 | 1 |
| Nutrient B | 2 | 3 |
| Nutrient C | 1 | 1 |
The cost of fodder 1 is ₹ 3 per unit and that of fodder ₹ 2 per unit. Formulate the L.P.P. to minimize the cost.
Solution
Let x units of fodder 1 and y units of fodder 2 be included in the food ration of an animal.
The cost of fodder 1 is ₹ 3 per unit and that of fodder 2 is ₹ 2 per unit.
∴ Total cost = ₹ (3x + 2y)
The minimum requirement of nutrients A, B, C for an animal are 14, 22 and 1 unit respectively.
We construct the given table with the minimum requirement column as follows:
| Nutrient\Fodder | Fodder 1 (x) |
Fodder 2 (y) |
Minimum requirement |
| Nutrient A | 2 | 1 | 14 |
| Nutrient B | 2 | 3 | 22 |
| Nutrient C | 1 | 1 | 1 |
From the table, the food ration of an animal must contain (2x + y) units of nutrient A, (2x + 3y) units of B and (x + y) units of C.
∴ The constraints are :
2x + y ≥ 14
2x + 3y ≥ 22
x + y ≥ 1
Since x and y cannot be negative, we have x ≥ 0, y ≥ 0
∴ The given problem can be formulated as follows:
Minimize Z = 3x + 2y
Subject to 2x + y ≥ 14,
2x + 3y ≥ 22,
x + y ≥ 1,
x ≥ 0,
y ≥ 0.
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