Question

A 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 INR 20 per kg and sand costs INR 6 per kg. Strength consideration dictate that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Formulate the L.P.P. for the cost to be minimum.

Hint:

When formulating an L.P.P., always follow three steps: 1. Identify the decision variables (what you can control, e.g., kg of cement/sand). 2. Write the objective function (what you want to maximize or minimize, e.g., cost). 3. List all the constraints (the rules and limitations).

Answer 1

Step 1: Define the variables.
Let \(x\) be the weight of cement in kg.
Let \(y\) be the weight of sand in kg.
Step 2: Formulate the objective function.
The objective is to minimize the cost of the brick. The cost function (\(Z\)) is: \[ \text{Minimize } Z = 20x + 6y \] Step 3: Formulate the constraints.
The constraints are based on the conditions given:
Total weight constraint: The total weight must be at least 5 kg. \[ x + y \geq 5 \] Cement constraint: The brick must contain a minimum of 4 kg of cement. \[ x \geq 4 \] Sand constraint: The brick must contain not more than 2 kg of sand. \[ y \leq 2 \] Non-negativity constraints: The amount of cement and sand cannot be negative. \[ x \geq 0, y \geq 0 \] The complete L.P.P. formulation is:
Minimize \(Z = 20x + 6y\)
Subject to the constraints: \[ x + y \geq 5 \] \[ x \geq 4 \] \[ y \leq 2 \] \[ x, y \geq 0 \]