# Bad Boyz Breakfast: 1 product, 2 ingredients, min cost s.t. underlying quality

Pharr Owt's egg-and-oatmeal porridge has become so popular she has been asked to make it available in quantity to other institutions throughout the Southeast.   She wants to make it by the gallon, as cheaply as possible.

Figuring a gallon of oatmeal = 16 cups and a gallon of eggs = 80 eggs, she calculates that a gallon of eggs provides 80*5=400 units of fat, 80*0=0 units of fiber, and 80*5=400 units of protein.  A gallon of oatmeal provides 16*2=32 units of fat, 16*6=96 units of fiber, and 16*4=64 units of protein.

A gallon of "Bad Boyz Breakfast" porridge needs to serve 10 inmates, so it needs at least 10*8=80 units of fat, 11*8=80 units of fiber, and 11*12=120 units of protein.

A gallon of eggs costs 80*\$.10=\$8.00, a gallon of oatmeal costs 16*\$.05=\$0.80

## The question is:  What percent eggs and what percent oatmeal should go into Bad Boyz Breakfast?

Pharr's first attempt to set up the problem is as follows:

 min \$8*PctEgg + \$0.80*PctOat s.t. 400*PctEgg + 32*PctOat > 80 0*PctEgg + 96*PctOat > 80 400*PctEgg + 64*PctOat > 120
along with the usual nonnegativity constraints.

This gives a total cost of \$1.80 per ten servings, but the blend will not fit into a gallon container!   click here for spreadsheet.

We need to add another constraint to fill up the container:
min \$8*PctEgg+
\$0.80*PctOat

s.t. 400*PctEgg + 32*PctOat > 80

0*PctEgg + 96*PctOat > 80

400*PctEgg + 64*PctOat > 120

PctEgg + pctOat = 100%
Adding a constraint increases the cost of ten servings to \$2.00, but at least it fits into the  container.  click here for solution