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 Rs 5 and Rs 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 denotes the number of pens in type A and type B.

(ii) Objective function: 

Profit on x₁ pens in type A is = 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 

Subj ect to the constrains 

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