A meat packer wishes to determine the quantity of ingredients to use in making e

Question

A meat packer wishes to determine the quantity of ingredients to use in making each pound of sausage at a minimum cost. The available ingredients and their costs are provided in the following table:

Pound of ingredient

Cost ($)

Protein (oz.)

Fat (oz.)

Water (oz.)

Hog bellies

Tripe

Beef

Pork

Chicken

The meat packer is faced with some restrictions in doing this. Not more than 10% of the sausage weight can be composed of hog bellies and tripe, chicken cannot exceed 25% of the total content, and at least 30% of the sausage must consist of beef. In addition, a minimum of 3 ounces of protein must be present in each pound of sausage. Furthermore, each pound of sausage may contain a maximum of 4 ounces of fat and must contain a maximum of 8 ounces of water. Formulate the decision as a linear program.

Pound of ingredient

Cost ($)

Protein (oz.)

Fat (oz.)

Water (oz.)

Hog bellies

0.30 3 7 6

Tripe

0.20 5 4 7

Beef

0.70 4 2 10

Pork

0.60 3 4 9

Chicken

0.45 3 5 8

Explanation / Answer

Linear program is following:

Let H, T, B, P and C be the quantity (pounds) of respective ingredients to be used in making of a pound of sausage. The first letter of the respective ingredients represent the variable.

Min 0.3H+0.2T+0.7B+0.6P+0.45C

s.t.

H+T+B+P+C = 1

H+T <= 0.1

C <= 0.25

B >= 0.3

3H+5T+4B+3P+3C >= 3

7H+4T+2B+4P+5C <= 4

6H+7T+10B+9P+8C <= 8

H, T, B, P, C >= 0

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