Linear Programming problem
 

Craig Browning bakes cookies for the elementary school cookie sale. His chocolate chip cookies sell for $1.00 a dozen, and his oatemal brownie cookies sell for $1.50 a dozen. He will bake up to 20 dozen chocolate chip cookies and up to 40 dozen oatmeal brownie cookies, but no more than 50 dozen cookies total. Also the number of oatmeal brownie cookies will be no more than three times the numer of chocolate chip cookies.

If C=chocolate chip cookies, B=Oatmeal brownie cookies, what are the constraints? How many of each kind should Craig make in order for the elementatry school to make the most money? How much money will this be?

MMMM.. Cookies...

...and homework problems. ;)

-SF

I could give you the answer right here and now, but i won't. Its really not very hard, and you do need to think about it.

If its not a homework question, then apologies.

For the others who might be reading, and wondering, you first need the objective function, which in this case, is an equation that describes the total amount of money to be made. This objective function will contain the variables C, and B.


Then there are the constraints, each of whom are equalities or inequalities containing the variables C and B.

The co-efficients in all cases for any of the variables can be any non-negative number, including zero. (you can make zero cookies but you cannot make negative numbers of cookies).

From the question, I am assuming that all the cookies that are baked will be sold. (or we could end up with the very interesting newsboy problem, and even more fun with 2-stage stochastic programming.... )

Speaking of assumptions, do cookies HAVE to be made in dozens? or can they be made in individual cookies? Are individual cookies divisible? If they are not, then we have a much harder integer programming question. Though i kind of suspect the answer you get from linear programming will turn out to be a nice integer number.

Well I suck at math, as evidenced by this Sophomore-level math attempt by a college student. I think I found four constraints:
c+b "<"/ 50

c '<'/' 20

b '<'/' 40

3c /> b

1.5b + c = P

you're pretty much there.

typically, you want all the variables on one side of the equality. You can have negative coefficients.

so it would be
-b + 3c >= 0

since its a 2 dimensional problem, you can solve this by graphing.
so you know how to set up the simplex method?

You don't need to graph it, just use subitution.

let B = 3C

so 3C + C = 50
4C = 50
C = 12.5

If C = 12.5 then B = 37.5

C x 1 = 12.5
B x 1.5 = 56.25

The Answer would be $68.75 (assumeing half dozens are alowed)
The Answer would be $68.50 (not alowing half dozens)