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प्रश्न
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.
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उत्तर
(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
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