An organic apple processor produces apple juice in four locations in the United States with the following yearly capacities:
|
San Francisco
|
110,000
|
|
Chicago
|
50,000
|
|
San Antonio
|
40,000
|
|
New York
|
100,000
|
Total supply 300,000 gallons/year
It currently sells 300,000 gallons per year to the following regions of the United States:
|
East
|
150,000
|
|
South
|
25,000
|
|
West
|
75,000
|
|
North
|
50,000
|
Total demand 300,000 gallons/year
The company expects that the demand for the next year will increase by 30,000 gal- lons in the East, 5,000 gallons in the South, 10,000 gallons in the West, and 1,000 gal- lons in the North. Hence, total demand for the United States will increase from 300,000 to 346,000 gallons. In addition, the company has just negotiated a sale of 200,000 additional gallons to Japan. Thus, while the business currently can produce only 300,000 gallons, they face a demand of 546,000 gallons in the next year. The company has identified three options in meeting this new demand:
1. Purchase one new processing plant in Portland, Oregon, with a capacity of 300,000 gallons per year.
2. Purchase one new plant in Birmingham, Alabama, with a capacity of 150,000 gal- lons per year and another new plant in Syracuse, New York, with a capacity of 150,000 gallons.
3. Purchase a new plant in Seattle, Washington, with a capacity of 100,000 gallons, and expand their San Francisco plant from its current capacity of 110,000 to 310,000 gallons per year.
The total variable production cost (not including transportation costs) for each exist- ing facility and the three expansion options are listed below:
|
Plant
|
Variable Production Costs ($/gallon)
|
|
San Francisco (110,000 capacity)
|
5.00
|
|
Chicago
|
4.50
|
|
San Antonio
|
4.00
|
|
New York
|
5.50
|
|
Portland
|
5.00
|
|
Birmingham*
|
5.76
|
|
Syracuse*
|
6.00
|
|
Seattle*
|
6.20
|
|
San Francisco (310,000 capacity)*
|
6.30
|
|
*Includes expansion costs.
|
|
The transportation cost per gallon from each supply to each demand node is:
Demand Location
|
Plant
|
East
|
South West North (cents per gallon shipped)
|
Japan
|
Supply (1,000 gallons)
|
|
San Francisco
|
100
|
85
|
10
|
45
|
200
|
110
|
|
Chicago
|
50
|
50
|
55
|
5
|
300
|
50
|
|
San Antonio
|
75
|
5
|
80
|
75
|
400
|
40
|
|
New York
|
10
|
70
|
120
|
15
|
450
|
100
|
|
Portland
|
105
|
100
|
15
|
30
|
250
|
300
|
|
Birmingham
|
65
|
10
|
95
|
70
|
500
|
150
|
|
Syracuse
|
0
|
90
|
130
|
15
|
485
|
150
|
|
Seattle
|
110
|
110
|
25
|
25
|
220
|
100
|
|
San Francisco
|
100
|
85
|
10
|
45
|
200
|
310
|
|
New Demand
|
(1,000)
|
180
|
30
|
85
|
51
|
200
|
Assume that the firm's objective is to minimize the sum of total variable production costs and transportation costs. (Note that the transportation costs are in cents per gal- lon, while the variable production costs are in dollars per gallon.)
a. Formulate and solve an LP problem that involves the first option for expansion.
b. Formulate and solve an LP problem that involves the second option for expansion.
c. Formulate and solve an LP problem that involves the third option for expansion.
d. Based on the results of parts a, b, and c, which of the three options would you recommend?