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.

Answer 1

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.