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Exercise 7.3 · Q31
Q.

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 :

NutrientFodder 1Fodder 2
Nutrients A21
Nutrients B23
Nutrients C11

The cost of fodder 1 is Rs.3 per unit and that of fodder 2 is Rs.2 per unit. Formulate the L.P.P. to minimize the cost.

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✓ Free question

Let xx = units of Fodder 1 and yy = units of Fodder 2 used per animal. Let xx = units of Fodder 1, yy = units of Fodder 2. The non-negativity constraints are x≥0,y≥0x\ge0, y\ge0 since a negative quantity produced has no meaning. The ration must supply at least 14 units of nutrient A, 22 of B and 1 of C; Fodder 1 costs Rs.3/unit and Fodder 2 costs Rs.2/unit, so all three nutrient rows become ≥\ge (minimum-requirement) constraints while the objective (cost) is minimized.

✓Final answer

The L.P.P. is — Minimize: Minimize z=3x+2y\text{Minimize } z = 3x + 2y subject to 2x+y≥14 (nutrient A), 2x+3y≥22 (nutrient B), x+y≥1 (nutrient C), x≥0, y≥02x+y\ge14 \text{ (nutrient A)},\ 2x+3y\ge22 \text{ (nutrient B)},\ x+y\ge1 \text{ (nutrient C)},\ x\ge0,\ y\ge0.

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