Objective function and constraints

Question

To make one package of all beef hot dogs a manufacturer uses 1lb beef. To make regular hot dogs it uses 1/2 lb each of beef and pork. The profit on regular hot dogs is 30 cents. The print on all beef is 40 cents per pack. Of there are 200lb of beef and 150lb of pork available. How many packs each of all beef and regular hot dogs must the manufacturer make to maximise the profit? 

Answer 1

Hello, Alicia! Maximizing profit usually assumes that we utilize resources most efficiently - as wasted resources cost money & earn us no profit. According to this problem, we can make some combination of "all beef hot dogs" & regular hot dogs that are half beef & half pork. As both types use beef, let's take a closer look at the pork. We have 150 pounds of pork available, but we will throw any remainder away & earn no profit if we do not use all of it.

 

But this is a math problem, so let's run some numbers:

 

200 lb all beef: 200 x 0.40 = $80

 

(Remember that 100 lb of beef makes 200 lb of regular hot dogs)

 

150 lb all beef & 100 lb regular: (150 x 0.40) + (100 x 0.30) = $60 + $30 = $90 (and throw away 100 lb of pork)

100 lb all beef & 200 lb regular: (100 x 0.40) + (200 x 0.30) = $40 + $60 = $100 (and throw away 50 lb of pork)

 

Do you see a pattern? As we make fewer all beef hot dogs & blend the pork for regular hot dogs, we increase profit. What if we used all the pork & all the beef? We would sell fewer pounds of all beef hot dogs, but we would earn profit on all of the meat available - and the profit on regular hot dogs is not much less than the profit on all beef hot dogs:

 

300 lb regular & 50 lb all beef: (300 x 0.30) + (50 x 0.40) = $90 + $20 = $110

 

In this case, the most efficient use of the meat is also most profitable.