Question

A company manufactures two models of voltage stabilizers viz., ordinary and auto-cut. All components of the stabilizers are purchased from outside sources, assembly and testing is carried out at the company’s own works. The assembly and testing time required for the two models are 0.8 hours each for ordinary and 1.20 hours each for auto-cut. Manufacturing capacity 720 hours at present is available per week. The market for the two models has been surveyed which suggests a maximum weekly sale of 600 units of ordinary and 400 units of auto-cut. Profit per unit for ordinary and auto-cut models has been estimated at ₹ 100 and ₹ 150 respectively. Formulate the linear programming problem.

Answer 1

(i) Variables: Let x₁ and x₂ denote the number of ordinary and auto-cut voltage stabilized.

(ii) Objective function:

Profit on x₁ units of ordinary stabilizers = 100x₁

Profit on x₂ units of auto-cut stabilized = 150x₂

Total profit = 100x₁ + 150x₂

Let Z = 100x₁ + 150x₂, which is the objective function.

Since the profit is to be maximized. We have to Maximize, Z = 100x₁ + 15x₂

(iii) Constraints: The assembling and testing time required for x₁ units of ordinary stabilizers = 0.8x₁ and for x₂ units of auto-cut stabilizers = 1.2x₂

Since the manufacturing capacity is 720 hours per week.

We get 0.8x₁ + 1.2x₂ ≤ 720

Maximum weekly sale of ordinary stabilizer is 600 i.e., x₁ ≤ 600

Maximum weekly sales of auto-cut stabilizer is 400 i.e., x₂ ≤ 400

(iv) Non-negative restrictions: Since the number of both the types of stabilizers is non-negative, we get x₁, x₂ ≥ 0.

Thus, the mathematical formulation of the LPP is, Maximize Z = 100x₂ + 150x₂

Subject to the constraints

0.8x₁ + 1.2x₂ ≤ 720, x₁ ≤ 600, x₂ ≤ 400, x₁, x₂ ≥ 0