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Question
In a cattle breading firm, it is prescribed that the food ration for one animal must contain 14. 22 and 1 units of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit of these two contains the following amounts of these three nutrients:
| Fodder → | Fodder 1 | Fodder 2 |
| Nutrient ↓ | ||
| Nutrients A | 2 | 1 |
| Nutrients B | 2 | 3 |
| Nutrients C | 1 | 1 |
The cost of fodder 1 is ₹ 3 per unit and that of fodder 2 ₹ 2. Formulate the LPP to minimize the cost.
Solution
Let x units of fodder 1 and y units of fodder 2 be prescribed. The cost of fodder 1 is ₹ 3 per unit and cost of fodder 2 is ₹ 2 per unit.
∴ total cost is z = 3x + 2y
This is the linear function which is to be minimized. Hence it is the objective function. The constraints are as per the following table:
| Fodder → | Fodder 1 | Fodder 2 | Minimum requirements |
| Nutrient ↓ | |||
| Nutrients A | 2 | 1 | 14 |
| Nutrients B | 2 | 3 | 22 |
| Nutrients C | 1 | 1 | 1 |
From table fodder contains (2x + y) units of nutrients A,(2x + 3y) units of nutrients B and (x + y) units of nutrients C. The minimum requirements of these nutrients are 14 units, 22 units, and 1 unit respectively.
Therefore, the constraints are
2x + y ≥ 14, 2x + 3y ≥ 22, x + y ≥ 1
Since, number of units (i.e. x and y) cannot be negative, we have, x ≥ 0, y ≥ 0.
Hence, the given LPP can be formulated as
Minimize z = 3x + 2y, subject to
2x + y ≥ 14, 2x + 3y ≥ 22, x + y ≥ 1, x ≥ 0, y ≥ 0.
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