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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.

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