Problem 8-29 (Algorithmic)

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 exactly 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, 9000 gallons of white wine and 7500 gallons of rose wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1 per gallon for the white and $1.5 per gallon for the rose; the fruit juice can be purchased for $0.5 per gallon. La Jolla Beverage Products can sell all of the wine cooler it can produce for $3 per gallon.

1. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor.
2. 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 it will realize from this blend. If required, round your answers to one decimal place. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.

	Let W = gallons of white wine
	Let R = gallons of rose wine
	Let F = gallons of fruit wine

Max	fill in the blank 3W	+	fill in the blank 4R	+	fill in the blank 5F
s.t.
	fill in the blank 6W	+	fill in the blank 7R	+	fill in the blank 8F	?	fill in the blank 9	% white
	fill in the blank 10W	+	fill in the blank 11R	+	fill in the blank 12F	?	fill in the blank 13	% rose minimum
	fill in the blank 14W	+	fill in the blank 15R	+	fill in the blank 16F	?	fill in the blank 17	% rose maximum
	fill in the blank 18W	+	fill in the blank 19R	+	fill in the blank 20F	=	fill in the blank 21	% fruit juice
	fill in the blank 22W					?	fill in the blank 23	Available white
			fill in the blank 24R			?	fill in the blank 25	Available rose
W, R, F ? 0

Optimal Solution:
W	fill in the blank 26
R	fill in the blank 27
F	fill in the blank 28

1. Profit Contribution = fill in the blank 29
2. If La Jolla could obtain additional amounts of the white wine, should it do so? If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase? The input in the box below will not be graded, but may be reviewed and considered by your instructor.
3. If La Jolla Beverage Products could obtain additional amounts of the ros wine, should it do so? If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase? The input in the box below will not be graded, but may be reviewed and considered by your instructor.
4. 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? The input in the box below will not be graded, but may be reviewed and considered by your instructor.
5. 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? The input in the box below will not be graded, but may be reviewed and considered by your instructor.

YesNoNo

YesNoYes

Sunk costRelevant costRelevant cost

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 it will realize from this blend. If required, round your answers to one decimal place. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. Let W = gallons of white wine Let R = gallons of rose wine Let F = gallons of fruit wine Max 2 VW + 1.5 VR 2.5 VF s.t .5 VW + -.05 X R -.05 X F 12 45 X % white -.02 X W + 1 X R + X F X % rose minimum .030 X W + 1 X R + X F X % rose maximum 1 X W + X R + X F X % fruit juice 0 X W S X Available white 1 VR X Available rose W, R, F 20 Optimal Solution: W 9000 R 3600 X F 5400 X