Linear Programming

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.