### Sample Problem

Suppose you are starting your own company selling chocolate covered strawberries. You decide to sell the milk chocolate covered strawberries at \$2.25/box and the white chocolate covered strawberries at \$2.50/box. Market tests and available resources, however, have given you the following constraints.

1.      The combined production level should not exceed 800 boxes per month.

2.      The demand for a box of white chocolate covered strawberries is no more than half the demand for a box of milk chocolate strawberries.

3.      The production level for milk chocolate strawberries should be less than or equal to 400 boxes plus two times the production level for white chocolate strawberries.

You, of course, want to maximize your profit.

a)What is the maximum monthly profit?  \$ (Round to nearest whole dollar)

b)How many boxes of each type of strawberry should be produced per month to yield maximum profit?

boxes of milk chocolate strawberries and ___ boxes of white chocolate strawberries

(Round to nearest whole box)

#### Solution

a)Objective function (for combined profit) is P=2.25M+2.50W where M is the number of boxes of milk chocolate strawberries and W is the number of boxes of white chocolate strawberries.

The three constraints translate into the following inequalities

M+W≤800

W≤1/2*M

M≤400+2W

We know that neither x nor y can be negative, so x≥0 and y≥0 are additional constraints.

The figure below shows the region determined by the constraints.

To find the maximum monthly profit, test the values of P at the vertices of the region

P=2.25M+2.50W

At (0, 0): P=2.25(0)+2.50(0)=0

At (1600/3, 800/3): P=2.25(1600/3)+2.50(800/3)=5600/31867

At (2000/3, 400/3): P=2.25(2000/3)+2.50(400/3)=5500/31833

At (400, 0): P=2.25(400)+2.50(0)=900

Maximum profit is \$1867

b)It occurs when the monthly production consists of 533 boxes of milk chocolate strawberries and 267 boxes of white chocolate strawberries.