how many packages of each blend should it make to maximize its profit for the week?

Question

For a given week, Joe's Coffee House has available 1792 ounces of A grade coffee and 1600 ounces of B grade coffee. These are blended into 1-pound packages as follows: an economy blend that contains 4 ounces of A grade coffee and 10 ounces of B grade coffee, and a superior blend that contains 8 ounces of A grade coffee and 2 ounces of B grade coffee. (The remainder of each blend is made of filler ingredients.) There is a $2 profit on each economy blend package sold and a $3 profit on each superior blend package sold. Assuming that the coffee house is able to sell as many blends as it makes, how many packages of each blend should it make to maximize its profit for the week?

Answer 1

This is an integer programming problem, but it can be solved by linear programming methods in this case.

The objective is to maximize 2x + 3y where x is the number of economy packages produced and y is the number of superior packages produced.

The constraints are:

x ≥ 0

y ≥ 0

4x + 8y ≤ 1792 (the limit on component A)

10x + 2y ≤ 1600 (the limit in component B)

The optimum value is one of the vertices on the boundaries. If you check all 4, you will find that the mix that maximizes profit is x = 128 and y = 160, resulting in a profit of $736.

Answer 2

Let x be the number of economy blend packages produced, and y be the number of superior blend packages produced.

Then, Profit = 2x+3y, since for each economy blend we earn $2 of profit, and for every superior blend - $3.

Each economy blend uses 4 ounces of A grade coffee, and each superior blend uses 8 ounces. The total amount of A grade coffee available is 1792 ounces. Hence 4x+8y=1792

Each economy blend uses 10 ounces of B grade coffee, and each superior blend uses 2 ounces. The total amount of B grade coffee available is 1600 ounces. Hence,10x+2y=1600

Solve the system of equations:

4x+8y=1792

10x+2y=1600

to get x=128 and y=160 which are the number of packages they need to sell to maximize the profit.

Then your profit in a week would be 2*128+3*160 = $736