Question

The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs ₹ 20 per kg and sand costs of ₹ 6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the L.P.P. for the cost to be minimum.

Answer 1

Let the company use x₁ kg of cement and x₂ kg of sand to make concrete bricks.

Cement costs ₹ 20 per kg and sand costs ₹ 6 per kg.

∴ the total cost c = ₹ (20x₁ + 6x₂)

This is a linear function that is to be minimized.

Hence, it is an objective function.

Total weight of brick = (x₁ + x₂) kg

Since the weight of concrete brick has to be at least 5 kg, 

∴ x₁ + x₂ ≥ 5

Since concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand,

x₁ ≥ 4 and 0 ≤ x₂ ≤ 2

Hence, the given LPP can be formulated as:

Minimize c = 20x₁ + 6x₂, subject to

x₁ + x₂ ≥ 5, x₁ ≥ 4 , 0 ≤ x₂ ≤ 2.

Answer 2

Let the concrete brick contain x₁ kg of cement and x₂ kg of sand

Cement costs ₹ 20 per kg and sand costs ₹ 6 per kg.

∴ Total cost = ₹ (20x₁ + 6x₂)

Weight of a concrete brick has to be at least 5 kg.

∴ x₁ + x₂ ≥ 5

The brick should contain a minimum of 4 kg of cement.

∴ x₁ ≥ 4

The brick should contain not more than 2 kg of sand.

∴ x₂ ≤ 2

Since x₁ and x₂ cannot be negative, we have x₁ ≥ 0, x₂ ≥ 0

∴ Given problem can be formulated as follows:

Minimize Z = 20x₁ + 6x₂

Subject to x₁ + x₂ ≥ 5, x1 ≥ 4, x₂ ≤ 2, x₁ ≥ 0, x₂ ≥ 0.