An alcohol manufacturer specializes in 3 different drinks, called Xena, Yolanda, and Zorgon. Each of these drinks has an ingredient unique to it, but these also share 3 ingredients in common.
1) Each keg of Xenas requires at least 3 gallons of special ingredient A, and must contain at least 3 gallons of special ingredient B, and must contain at least 6 gallons of special ingredient C. Plus each keg must also contain exactly 25 gallons in total of across all 3 of these special ingredients.
2) Each keg of Yolandas requires at least 4 gallons of special ingredient A, and must contain at least 3 gallons of special ingredient B, and must contain at least 2 gallons of special ingredient C. Plus each keg must also contain exactly 18 gallons in total of across all 3 of these special ingredients.
3) Each keg of Zorgons requires at least 8 gallons of special ingredient A, and must contain at least 9 gallons of special ingredient B, and must contain at least 8 gallons of special ingredient C. Plus each keg must also contain exactly 30 gallons in total of across all 3 of these special ingredients.
The mixture of these special ingredients cannot be dominated by any one particular ingredient. Specifically, in each keg, no one individual special ingredient can have more than double the amount of any other special ingredient.
The manufacturer must make 1 keg of each of the three drinks, but must also attempt to reduce the total cost of the special ingredients.
· 1 gallon of special ingredient A costs the manufacturer $18.
· 1 gallon of special ingredient B costs the manufacturer $20.
· 1 gallon of special ingredient C costs the manufacturer $15.
HINT: Note to solve this problem, you will need to have 9 variables (3 for Xena, 3 for Yolanda, and 3 for Zorgon; each of these 3 corresponding to the number of gallons of ingredient A, B, and C they each require).
Questions:
a) What is the optimal amount of each special ingredient for each drink?
b) What is the optimal cost of the special ingredients in total?