#### 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.

#### Solution

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 5*x* hours to produce *x* units of A and 3*y* hours to produce *y* units of B.

As, the total capacity is of 45000 man hours.

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* = 60*x* + 40*y*

subject to

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

\[x \leq 7000\]

\[y \leq 10000\]

\[x, y \geq 0\]