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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.
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Solution 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.
Solution 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.
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