Answer:
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 [tex]\mathbf{Z_1}[/tex] should be the percentage of the beef.
Let assume that [tex]\mathbf{Z_2}[/tex] 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 :
[tex]\mathbf{Min:0.85Z_1 + 0.65Z_2}[/tex]
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:
[tex]\mathbf{0.85Z_1 + 0.65Z_2 \leq 0.25}[/tex]
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;
[tex]\mathbf{Z_1+Z_2 =1}[/tex]
Of course , we know that the percentage of this aftermath result can't be zero. i.e it is definitely greater than 1.
So; [tex]\mathbf{Z_1,Z_2 > 1}[/tex]
Thus; the Objective function of this model is :
[tex]\mathbf{Min:0.85Z_1 + 0.65Z_2}[/tex] which is subjected to [tex]\mathbf{0.85Z_1 + 0.65Z_2 \leq 0.25} \\ \\ \mathbf{Z_1+Z_2 =1} \\ \\ \mathbf{Z_1,Z_2 > 1}[/tex]
