Question 1

Formulating the Linear Programming Problem

Let's define the decision variables, objective function, and constraints for this problem.

1. Decision Variables:

• Let x represent the number of units of product A to be produced.
• Let y represent the number of units of product B to be produced.

2. Objective Function:

The objective is to minimize the total weekly cost of machines, men, and material. The cost per hour of machine is ₹50, per hour of men is ₹30, and per kg of material is ₹10. The total cost can be expressed as:

Minimize Z = (9x + 5y) * 50 + (7x + 3y) * 30 + (4x + 4y) * 10

Simplifying the objective function:

Minimize Z = (450 + 210 + 40)x + (250 + 90 + 40)y

Minimize Z = 700x + 380y

3. Constraints:

The production is limited by the available capacity of machines, men, and material.

• Machine Constraint: 9x + 5y ≤ 180 (Total machine hours used cannot exceed 180)
• Men Constraint: 7x + 3y ≤ 150 (Total men hours used cannot exceed 150)
• Material Constraint: 4x + 4y ≤ 200 (Total material used cannot exceed 200 kg)
• Non-Negativity Constraints: x ≥ 0, y ≥ 0 (Production quantities cannot be negative)

4. Complete Linear Programming Problem Formulation:

Minimize Z = 700x + 380y

Subject to:

• 9x + 5y ≤ 180
• 7x + 3y ≤ 150
• 4x + 4y ≤ 200
• x ≥ 0
• y ≥ 0

This LPP can now be solved using various methods like the graphical method, simplex method, or using software like Excel Solver to determine the optimal values of x and y that minimize the total cost while satisfying all the constraints.

Graphical Representation (Conceptual)

While not explicitly asked for, understanding the graphical representation helps visualize the feasible region. Each constraint represents a line on a graph. The feasible region is the area where all constraints are satisfied simultaneously. The optimal solution lies at one of the corner points of the feasible region.