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.

Answer 1

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