Question

A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are ₹ 5 and ₹ 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.

Answer 1

(i) Variables: Let x₁ and x₂ denote the number of pens in type A and type B.

(ii) Objective function:

Profit on x₁ pens in type A = 5x₁

Profit on x₂ pens in type B is 3x₂

Total profit = 5x₁ + 3x₂

Let Z = 5x₁ + 3x₂, which is the objective function.

Since the B total profit is to be maximized, we have to maximize Z = 5x₁ + 3x₂

(iii) Constraints:

Raw materials required for each pen A is twice as that of pen B.

i.e., for pen A raw material required is 2x₁ and for B is x₂.

Raw material is sufficient only for 1000 pens per day

∴ 2x₁ + x₂ ≤ 1000

Pen A requires 400 clips per day

∴ x₁ ≤ 400

Pen B requires 700 clips per day

∴ x₂ ≤ 700

(iv) Non-negative restriction:

Since the number of pens is non-negative, we have x₁ > 0, x₂ > 0.

Thus, the mathematical formulation of the LPP is Maximize Z = 5x₁ + 3x₂

Subject to the constrains

2x₁ + x₂ ≤ 1000, x₁ ≤ 400, x₂ ≤ 700, x₁, x₂ ≥ 0