Question

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs 60 per unit for the product A and Rs 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.

Answer 1

Let x units of product A and y units of product B be produced.
Then,
Since, it takes 5 hours to produce a unit of A and 3 hours to produce a unit of B.
Therefore, it will take 5x hours to produce x units of A and 3y hours to produce y units of B.
As, the total capacity is of 45000 man hours.

\[\Rightarrow 5x + 3y \leq 45000\]

Also,
The maximum number of units of A that can be sold is 7000 and that of B is 10,000 and number of units cannot be negative.
Thus, 

\[0 \leq x \leq 7000, 0 \leq y \leq 10000\]

Now,
Total profit =  \[60x + 40y\]

Here, we need to maximize profit
Thus, the objective function will be maximize

\[Z = 60x + 40y\]

​Hence, the required LPP is as follows:
Maximize Z = 60x + 40y
subject to  

\[5x + 3y \leq 45000\]
\[x \leq 7000\]
\[y \leq 10000\]
\[x, y \geq 0\]