Reference no: EM13230390

Smith Juice Co blends orange,apple,pineapple,grape,and cranberry juice to produce its beverage products. The products and their specifications are listed in the following table.

Product X: at least 40% is orange juice, no more than 15% is apple juice; at least 5% is pineapple juice, no grape juice is used, at least 25% is cranberry juice. The selling price per gallon is 3.05.

Product Y: no more than 25% is orange juice, at least 10% is apple juice, no pineapple juice is used, at least 25% is grape juice, no more than 5 % is cranberry juice. The selling price per gallon is $3.75

Product Z: no more than 10% is orange juice, no apple juice is used, at least 25% is pineapple juice; no more than 5 % is grape juice, at least 15 % is cranberry juice. The selling price per gallon is 3.15

The company has purchased and paid for 995,000 gallons of orange juice, 450,000 gallons of apple juice, 155,000 gallons of pineapple juice, 200,000 gallons of grape juice and 550,000 gallons of cranberry juice.

Set up and list the linear program that can maximize the revenue for this case.

Use Lingo to solve the program and report the maximized revenue for this blending problem.

How many gallons of product X,Y, and Z should be produced to maximize the revenue?

Report the actual mixing percentages of orange, apple , grape and cranberry juice in product Y?

Solution

Let the variables be

	Product X	Product Y	Product Z
Orange Juice	X1	X6	X11
Apple Juice	X2	X7	X12
Pineapple juice	X3	X8	X13
Grape Juice	X4	X9	X14
Cranberry Juice	X5	X10	X15

The objective function is

Maximize, Revenue Z = (X1+x2+X3+X4+X5)*$3.05 + (X6+X7+X8+X94+X10)*$3.75 (X11+X2+X13+X14+X15)*$3.15

The constraints are:

For product X

Orange Juice:                     X1 ≥ 0.4*(X1+X2+X3+X4+X5)

Apple Juice:                        X2 ≤ 0.15* (X1+X2+X3+X4+X5)

Pineapple Juice:               X3 ≥ 0.05*(X1+X2+X3+X4+X5)

Grape Juice:                       X4 = 0

Cranberry Juice:               X5 ≥ 0.25*(X1+X2+X3+X4+X5)

 

For product Y

Orange Juice:                     X6 ≤ 0.25*(X6+X7+X8+X9+X10)

Apple Juice:                        X7 ≥ 0.10* (X6+X7+X8+X9+X10)

Pineapple Juice:               X8 = 0

Grape Juice:                       X9 ≥ 0.25*(X11+X12+X13+X14+X15)

Cranberry Juice:               X10 ≤ 0.05*(X6+X7+X8+X9+X10)

 

For product Z

Orange Juice:                     X11 ≤ 0.10*(X11+X12+X13+X14+X15)

Apple Juice:                        X12 =0

Pineapple Juice:               X13 ≥ 0.25*(X11+X12+X13+X14+X15)

Grape Juice:                       X14 ≤0.05*(X11+X12+X13+X14+X15)

Cranberry Juice:               X15 ≥ 0.15*(X11+X12+X13+X14+X15)

 

Total Orange Juice

X1 + X6 + X11 ≤ 995000

 

Total Apple Juice

X2 + X7 + X12 ≤ 450000

 

Total Pineapple Juice

X3 + X8 + X13 ≤ 155000

 

Total Grape Juice

X4 + X9 + X14 ≤ 200000

 

Total Cranberry Juice

X5 + X10 + X15 ≤ 550000

The solution is as follows

	Product X	Product Y	Product Z
Orange Juice	756250	200000	38750
Apple Juice	57500	392500	0
Pineapple juice	58125	0	96875
Grape Juice	0	200000	0
Cranberry Juice	290625	7500	251875
Total	1162500	800000	387500