# Formulation of Linear Programming Problem

A company produces two types of hats. Each hat of the first type requires twice as much labour time as does each hat of the second type . If all hats are of the second type only, the company can produce a total of 500 hats a day.


The market limits daily sales of the first and second types to 150 and 250 hats. Assume that profit per hat is $8 for type 1 and$ 5 for type 2. Determine number of hats of each type to produce maximim profit.

The formulation goes like:-

Let x be number of hats of 1st type to be produced. and y be no. of hats of 2nd type to be produced. The limiting of market daily sales gives the constraint x is less than or equal to 150 and y is less than or equal to 250.

what constraint should I write from the statement given in the question as " Each hat of the first time requires twice as much labour time as does each hat of the second type. If all hats are of the second type only the company can produce a total of 500 hats a day"? Is this 2X + Y is less than or equal to 500?

So it takes $\frac{1}{250}$day to make 1 hat of type 1 and $\frac{1}{500}$day to make 1 hat of type 2.
Hence it takes $\frac{x}{250}$day to make $x$ hats of type 1 and $\frac{y}{500}$day to make $y$ hats of type 2. $$\frac{x}{250}+\frac{y}{500}\le 1$$ or, $$2x+y \le 500$$ considering per day activities.
The objective function is maximum profit per day $$z=8x+5y$$