La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rose wine, and fruit juice. To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 20% and no more than 30% rose, and 20% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 10,000 gallons of white wine and 8000 gallons of rose wine can be purchased; there is no limit on the amount of fruit juice that can be ordered. The costs for the wine are $1.00 per gallon for the white and $1.50 per gallon for the rose; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all the wine cooler it can produce for $2.50 per gallon.
a. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Explain.
b. Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution they will realize from this blend.
c. If La Jolla could obtain additional amounts of the white wine, should they do so? If so, how much should they be willing to pay for each additional gallon and how many additional gallons would they want to purchase?
d. If La Jolla Beverage Products could obtain additional amounts of the rose wine, should they do so? If so, how much should they be willing to pay for each additional gallon and how many additional gallons would they want to purchase?
e. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this shadow price?
f. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this shadow price?