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Question
A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 on magazines A and B per copy. These are processed on three machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II and 2 hours on Machine III. Magazine B requires 3 hours on Machine I, 2 hours on Machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, 60 hours per week respectively. Formulate the Linear programming problem to maximize the profit.
Solution
Let the company print x magazines of type A and y magazines of type B.
The profit on each copy of A and B is ₹ 10 and ₹ 15 respectively.
∴ Total profit = ₹ (10x + 15y)
We construct a table with the constraints of machines I, II, III as follows.
| Machine\Magazine | A (x) |
B (y) |
Available Time per week |
| I | 2 | 3 | 36 |
| II | 5 | 2 | 50 |
| III | 2 | 6 | 60 |
From the table, total time required for machines I, II, III are (2x + 3y) hours, (5x + 2y) hours and (2x + 6y) hours respectively.
∴ The constraints are:
2x + 3y ≤ 36,
5x + 2y ≤ 50,
2x + 6y ≤ 60
Since x, y cannot be negative, we have x ≥ 0, y ≥ 0
∴ Given problem can be formulated as,
Maximize Z = 10x + 15y
Subject to 2x + 3y ≤ 36, 5x + 2y ≤ 50, 2x + 6y ≤ 60, x ≥ 0, y ≥ 0.
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