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

Question

For a given week, Maya's Coffee House has available 1584 ounces of A grade coffee and 1440 ounces of B grade coffee. These are blended into 1 pound packages as follows: an economy blend that contains 3 ounces of A grade coffee and 6 ounces of B grade coffee, and a superior blend that contains 11 ounces of A grade coffee and 4 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 $4 profit on each superior blend package sold. Assuming that the store 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

Let us formulate the equations for the given data---

Say, we have x nos of packs of economy blend

                    y nos of  packs of superior blend.

 

3x+11y <=1584

 

6x+4y  <=1440

 

Making a graph of these two equations,we get the following three extreme points----

(240,0)   , (0,144)  ,  (176,96)

 

The objective function is

 

Max (2x +4y)

 

Let us now calculate value of objective function at these extreme points

 

For (240,0). Value of objective function =2*240=480

For (0,144)  value of objective function=4*144=576

For (176,96) value of objective function =2*176+4*96=736

 

So (176,96). Is the optimal point

That means x=176

                  y=96 

 

This is basically a linear programming problem.