Question

Optimized Cookie Production for a BCS Party. A friend was bringing small bags of cookies to sell at a fairly large BCS Championship Game Watch Party (there were no TCU fans present, however). Three kinds of cookies were sold: Stars (sold for $1 per bag), Circles (sold for $0.75 per bag), and Stars and Stripes (sold for $1.50 per bag). He was to bring the cookies to the Watch Party in three large boxes. (The boxes did not have to be full, but he could not bring more than three large boxes of cookies). By volume, it is a known fact that one of the large boxes can hold 100 bags of Stars, 120 bags of Circles, or 80 bags of Stars and Stripes (or a corresponding mix of cookies). HINT: Don’t concern yourself with what each box held; view this as an aggregate limit in the numbers of cookies. Previous parties had given him some hints on the demand for cookies – he knew that for the sake of variety, he needed to make at least 45 bags of each type of cookie. As he was planning his cookie composition, he also realized he was constrained by time in putting together the cookie bags. Circle cookies and Stars cookies took 1 minute per bag to finish; because Stars and Stripes had more icing, it took 2 minutes to finish each bag. He allocated 420 minutes (7 hours) to put the bags together. Can you determine how many of each of the three cookie types your friend should make to maximize sales (a surrogate for profit)?

Answer 1

This is a simple problem related to linear programming .

Let us try to solve the problem bit by bit .

But before that first things first .

We shall assume that the kid makes x, y and z bags of star, circles and star and stripes cookies respectively.

(let’s call cookie star as A ,circles as B and stripes as C for ease of representation)

We have to find this x ,y and z.

Now ,

it is said that A and B take 1 minutes and C takes 2 minutes for packaging with a maximum limit of 420 minutes allotted for it

For x bags of A we will need 1*x minutes .

For y bags of B , we will need 1*y minutes for packaging and

For z bags of C , we will need 2*z minutes for packaging

Overall ,total time =1*x+1*y+2*z

But the entire time has to be less than 420 minutes

Thus, 1*x+1*x+2*z <= 420    …. (i)

Now each pack should necessarily be greater than 45 bags

Thus ,

x>=45 , y>=45 , z>=45      … (ii)

Now to the selling price

it is said that A B and C sells at $ 1 , $ 0.75 and$1.50 per unit .

For x bags of A we will sell at $ 1*x .

For y bags of B , we will sell $ 0.75*y

For z bags of C , we sell $ 1.50*z

Overall ,total selling =1*x+0.75*y+1.50*z         … (iii)

This needs to be Maximized.

Now each box can contain at max of 100 bags of A ,120 bags of B or 80 Bags of C

Collectively all three bags can carry 100+120+80 bags at maximum =300 bags

But we have made collectively x+y+z bags overall.

This shouldn’t exceed 300 bags limits

Hence

x+y+z<=300     ….. (iv)

Now we are sorted .

Our objective function is

Maximize : 1*x+0.75*y+1.50*z

Under Constraints :

x+y+z<=300    

x>=45 ,

y>=45 ,

z>=45      and

1*x+1*x+2*z <= 420   

We now simply need to solve these equations .

We took help of excel solver to solve this and got the following results.

The optimal sales is equal to $ 348.75

The optimal number of bags of each type of cookie will be

x = 135
y = 45
z = 120