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 - Mathematics and Statistics

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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.

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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.

Concept: Linear Programming Problem (L.P.P.)
  Is there an error in this question or solution?
Chapter 6: Linear Programming - Exercise 6.1 [Page 98]

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