Real$-$Juice, a manufacturer of diet drinks, is planning to introduce a special drink that will magically burn away fat. The drink is a bit expensive, but Real$-$Juice guarantees that a person using this diet plan will lose up to $50$ pounds in just $3$ weeks. The drink is made up of four $$mystery$$ ingredients $($ingredients A$,$ B$,$ C$,$ and D$).$ The plan calls for a person to consume at least three $12-$ounce doses per day $($ie$.$ At least $36$ ounces per day$).$

Each of the four ingredients contains different levels of three chemical compounds $($chemicals X$,$ Y$,$ and Z$).$ Health regulations mandate that the dosage consumed per day should contain minimum prescribed levels of chemicals X and Y and should not exceed maximum prescribed levels for the third chemical, Z$.$

The composition of the four ingredients in terms of the chemical compounds $($units per ounce$)$ is shown in table $2,$ along with the unit cost prices of the ingredients. Real$-$Juice wants to find the optimal way to mix the ingredients to create the drink, at minimum cost per daily dose.

Ingredient A Ingredient B Ingredient C Ingredient D Requirements

Chemical X $34810$ At least $280$ units

Chemical Y $5366$ At least $200$ units

Chemical Z $10252040$ At most $1,050$ units

COST PER OUNCE $$0\mathrm{.}40$ $$0\mathrm{.}20$ $$0\mathrm{.}60$ $$0\mathrm{.}30$

REQUIRED:

a$.$ Formulate a linear programming model for this problem.

b$.$ Solve this model by using the computer.

c$.$ State the optimal solution to the problem.

d$.$ What constraints are non$-$binding?

e$.$ Interpret the non$-$zero shadow prices in your output constraint table.

f$.$ Real$-$Juice can decrease the minimum requirements for chemical X by $5$ units, provided that the maximum limit allowed for chemical Z is reduced to $1000$ units. Is this trade$-$off cost effective for Real$-$Juice to implement?