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
|