Problem 1: East Coast Trucking:
East Coast Trucking provides service from Boston to Miami using regional offices located in Boston, New York, Philadelphia, Baltimore, Washington, Richmond, Raleigh, Florence, Savannah, Jacksonville, and Tampa. The number of miles between each of the regional offices is provided in the following table.
|
|
New York
|
Philadelphia
|
Baltimore
|
Washington
|
Richmond
|
Raleigh
|
Florence
|
Savannah
|
Jacksonville
|
Tampa
|
Miami
|
|
Boston
|
211
|
320
|
424
|
459
|
565
|
713
|
884
|
1056
|
1196
|
1399
|
1669
|
|
New York
|
|
109
|
213
|
248
|
354
|
502
|
673
|
845
|
985
|
1188
|
1458
|
|
Philadelphia
|
|
|
104
|
139
|
245
|
393
|
564
|
736
|
876
|
1079
|
1349
|
|
Baltimore
|
|
|
|
35
|
141
|
289
|
460
|
632
|
772
|
975
|
1245
|
|
Washington
|
|
|
|
|
106
|
254
|
425
|
597
|
737
|
940
|
1210
|
|
Richmond
|
|
|
|
|
|
148
|
319
|
491
|
631
|
834
|
1104
|
|
Raleigh
|
|
|
|
|
|
|
171
|
343
|
483
|
686
|
956
|
|
Florence
|
|
|
|
|
|
|
|
172
|
312
|
515
|
785
|
|
Savannah
|
|
|
|
|
|
|
|
|
140
|
343
|
613
|
|
Jacksonville
|
|
|
|
|
|
|
|
|
|
203
|
473
|
|
Tampa
|
|
|
|
|
|
|
|
|
|
|
270
|
The company's expansion plans involve constructing service facilities in some of the cities where a regional office is located. Each regional office must be within 400 miles of a service facility. For instance, if a service facility is constructed in Richmond, it can provide service to regional offices located in New York, Philadelphia, Baltimore, Washington, Richmond, Raleigh, and Florence. Management would like to determine the minimum number of service facilities needed and where they should be located.
a. Formulate and solve an integer linear programming model that can be used to determine the minimum number of service facilities needed and their location.
b. Suppose that each facility can only provide service to regional offices within 300 miles. Determine the minimum number of service facilities needed and their location.
Problem 2: Selecky Estates Winery
The Selecky Estates winery of Otter Creek, California (SE), produces three kinds of table wine - a blush, a white, and a red. The winery has 30,000 pounds of grapes available to produce wine this season. A cask of blush requires 360 pounds of grapes, a cask of white requires 375 pounds, and a cask of red requires 410 pounds. The winery has enough storage space in its aging room to store 67 casks of wine. The winery has 2,200 hours of production capacity, and it requires 14 hours to produce a cask of blush, 10 hours to produce a cask of white, and 18 hours for a cask of red. From prior years' sales, the winery knows it will sell at least twice as much blush as red and at least 1.5 times as much white as blush. The profit contribution for a cask of blush is $12,100, for a cask of white the contribution is $8,700, and for a cask of red the contribution is $10,500. Only full casks are sealed and stored in the aging room.
• Determine the number of full casks to produce of each variety.
Two neighboring wineries, Charles Buck winery (CB) and Otter Creek winery (OC) have made proposals to Selecky Estates with respect to possible collaborative efforts. Charles Buck has experienced a significant decrease in grape production this season and has offered space in its aging room (including casks) to Selecky for $6,500 per cask/space for the appropriate aging duration. Currently, Charles Buck has 50 cask/spaces available for such use. In addition, Charles Buck has offered to purchase grapes, in bulk, for $4.50 per pound from Selecky to minimize the underutilization of its own production processes. Otter Creek has experienced an opposite situation with a significantly larger volume of grape production this season than anticipated. Otter Creek has proposed to rent production capacity from Selecky Estates for $8.00 per hour for time needed to produce casks of Otter Creek wine if available. All three wineries have similar production characteristics and produce the same varieties.
• Develop a production plan and response to the proposals for Selecky Estates and explain the rationale used.
Problem 3: Ace Lumber and Building Supply
Ace Lumber and Building Supply in Andover, Maryland, has received the following order for standard 1x12 boards to be cut in three lengths:
|
Order for 1x12 Boards
|
|
Length
|
Quantity
|
|
7 ft
|
700
|
|
9 ft
|
1200
|
|
10 ft
|
300
|
The company maintains 1x12 boards in 25-foot standard-length in stock. Therefore, the 25-foot boards must be cut into the lengths necessary to meet the order requirements. The company wishes to complete this order with the most cost-effective and efficient use of lumber resources strategy possible. Requests for small quantities of 7-foot, 9-foot, and 10-foot boards occur frequently at Ace Lumber and, as a result of cutting 25-foot boards, these sizes are normally maintained in stock. None of these smaller board sizes are in current inventory.
a. Formulate a linear programming model that can be used to determine the optimal number of standard-length (25-foot) boards to cut in order to complete this order.
b. Determine the optimal solution using the Management Scientist software including the total number of boards used and explain the rationale used.