Linear programming

Question

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 550 eggs and 900 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha. How many quarts of each flavor should you make in order to earn the largest profit?

Answer 1

Thomas,

Since the vanilla is most profitable, you make as much of it as possible subject to the constraints on the ingredients and to maximize the total profit. So

 

Let v = #quarts of creamy vanilla ice cream

     m = #quarts of continental mocha ice cream

     P = total profit of all the ice cream made

 

Ingredient constraints are:

     Eggs used    = 2v + 1m <= 550.  note that v cannot exceed 275 because that will use up all the eggs

     Cream used = 3v + 3m <= 900 cups,  therefore,  v + m <=300 quarts

 

Profit P = 3v + 2m  in dollars

 

Because of constraints, in general you solve it numerically using a linear programming algorithm that changes the v & m quantities to increase profits until it finds that combination that can no longer be improved.  We can simulate on paper by filling in a numerical table using the above P formula and V & m constraints.

 

v <=275          m<=(300 - v)*       P= 3v+2m                 * also subject to remaining eggs: m <=(550-2v)

-----------------------------------------------------------------------

200                  100                       $800

225                    75                       $825

245                    55                       $845

249                    51                       $849

250                    50                       $850

251                    48                       $849  

255                    40                       $845

275                      0                       $825

 

So v=250 and m=50 maximizes the ice cream profit at $850