A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?

It is being given that at least one of each must be produced.

#### Solution

Let the number of necklaces manufacture be *x*,

and the number of bracelets manufacture be *y*.

since the total number of items are at most 24

`x + y <= 24` .....(1)

Bracelets takes 1 hour to manufacture and necklaces takes half an hour to manufacture

x item takes x hour to manufacture and y items take y/2 hour to manufacture.

and maximum time available is 16 hours.

therefore

`x/2 + y <= 16` ....(2)

the profit on one necklace is Rs. 100 and the profit on one bracelet is Rs.300

Let the profit be Z. Now we wish to maximize the profit. So,

Max Z = 100*x* + 300*y* ...(3)

So, `x + y <= 24`

`x/2 + y <= 16`

Max Z = 100x + 300y is the required L.P.P.