Question

Bill's Grill is a popular college restaurant that is famous for its hamburgers. The owner of the restaurant, Bill, mixes fresh ground beef and pork with a secret ingredient to make delicious quarter-pound hamburgers that are advertised as having no more than 25% fat. Bill can buy beef containing 80% meat and 20% fat at $0.85 per pound. He can buy pork containing 70% meat and 30% fat at $0.65 per pound. Bill wants to determine the minimum cost way to blend the beef and pork to make hamburgers that have no more than 25% fat.

Required:

1. What is objective function for the mathematical formulation, in words?

Answer 1

Step-by-step explanation:

Given that:

Bill can buy containing 80% meat and 20 % pound at $0.85 per pound

Also; He can buy pork containing 70% meat and 30% fat at $0.65 per pound.

Bill, mixes fresh ground beef and pork with a secret ingredient to make delicious quarter-pound hamburgers that are advertised as having no more than 25% fat.

From the information given:

The Objective is that Bill wants to determine the minimum cost way to blend the beef and pork to make hamburgers that have no more than 25% fat.

Also: Required is to find the objective function for the mathematical formulation, in words.

Assumptions:

Let assume that
`\mathbf{Z_1}` should be the percentage of the beef.

Let assume that
`\mathbf{Z_2}` should be the percentage of the beef.

The buying cost of beef is $0.85 and the buying cost of pork is $0.65

Hence; the Minimum Objective cost function for this model will be :

`\mathbf{Min:0.85Z_1 + 0.65Z_2}`

Also; from above:

We know that the fat in beef and pork is 20% and 30% respectively ( 0.2 and 0.3).

And Bill decided to make hamburgers that have fat not more than 25% (0.25)

Equally ; we can formulate a decision that the sum of the beef and pork should be less than or equal to 0.25

So:

`\mathbf{0.85Z_1 + 0.65Z_2 \leq 0.25}`

Since Bill is using both beef and pork for the production; there is need to add both entity together because He does not have to use either beef or pork alone;

So;

`\mathbf{Z_1+Z_2 =1}`

Of course , we know that the percentage of this aftermath result can't be zero. i.e it is definitely greater than 1.

So;
`\mathbf{Z_1,Z_2 > 1}`

Thus; the Objective function of this model is :

`\mathbf{Min:0.85Z_1 + 0.65Z_2}` which is subjected to
`\mathbf{0.85Z_1 + 0.65Z_2 \leq 0.25} \\ \\ \mathbf{Z_1+Z_2 =1} \\ \\ \mathbf{Z_1,Z_2 > 1}`