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Question
A manufacturer produces bulbs and tubes. Each of these must be processed through two machines M1 and M2. A package of bulbs requires 1 hour of work on Machine M1 and 3 hours of work on Machine M2. A package of tubes requires 2 hours on Machine M1 and 4 hours on Machine M2. He earns a profit of ₹ 13.5 per package of bulbs and ₹ 55 per package of tubes. Formulate the LPP to maximize the profit, if he operates the machine M1, for almost 10 hours a day and machine M2 for almost 12 hours a day.
Solution
Let the number of packages of bulbs produced by manufacturer be x and packages of tubes be y. The manufacturer earns a profit of ₹ 13.5 per package of bulbs and ₹ 55 per package of tubes.
Therefore, his total profit is p = ₹ (13.5x + 55y)
This is a linear function that is to be maximized. Hence, it is an objective function.
The constraints are as per the following table:
| Bulbs (x) | Tubes (y) | Available Time | |
| Machine M1 | 1 | 2 | 10 |
| Machine M2 | 3 | 4 | 12 |
From the table, the total time required for Machine M1 is (x + 2y) hours and for Machine M2 is (3x + 4y) hours. Given Machine M1 and M2 are available for at most 10 hours and 12 hours a day respectively.
Therefore, the constraints are x + 2y ≤ 10, 3x + 4y ≤ 12. Since, x and y cannot be negative, we have, x ≥ 0, y ≥ 0. Hence, the given LPP can be formulated as:
Maximize p = 13.5x + 55y, subject to
x + 2y ≤ 10, 3x + 4y ≤ 12, x ≥ 0, y ≥ 0
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