1. Mega-Mart, a discount store chain, is to build a new store in Rock Springs. The parcel of land the company has purchased is large enough to accommodate a store with 140,000 square feet of floor space. Based on marketing and demographic survey of the area and historical data from its other stores, Mega-Mart estimates its annual profit per square foot for each of the store's departments to be as shown in the table below.
Each department must have at least 15,000 square feet of floor space, and no department can have more than 20% of the total retail floor space. Men's, women's, and children's clothing plus housewares keep all of their stock on the retail floor; however, toys, electronics, and auto supplies keep some items (such as bicycles, televisions, and tires) in inventory. Thus, 10% of the total retail floor space devoted to these three departments must be set aside outside the retail area for stocking inventory. Mega-Mart wants to know the floor space that should be devoted to each department in order to maximize profit.
Formulate a linear programming model for this problem and solve the problem using the computer.
Data for Mega-Mart
|
Department
|
Profit Per Square Foot
|
|
Men's clothing
|
$4.25
|
|
Women's Clothing
|
$5.10
|
|
Children's Clothing
|
$4.50
|
|
Toys
|
$5.20
|
|
Housewares
|
$4.10
|
|
Electronics
|
$4.90
|
|
Auto Supplies
|
$3.80
|
2. Mountain Laurel Vineyards produces three kinds of wine - Mountain Blanc, Mountain Red, and Mountain Blush. The company has 17 tons of grapes available to produce wine this season. A cask of Blanc requires 0.21 tons of grapes, a cask of Red requires 0.24 ton, and a cask of Blush requires 0.18 ton. The vineyard has enough storage space in its aging room to store 80 casks of wine.
The vineyard has 2,500 hours of production capacity, and it requires 12 hours to produce a cask of Blanc, 14.5 hours to produce a cask of Red, and 16 hours to produce a cask of Blush. From past sales the vineyard knows that demand for the Blush will be no more than half of the sales of the other two wines combined. The profit for a cask of Blanc is $7,500, the profit for cask of Red is $8,200, and the profit for a cask of Blush is $10,500.
a. Formulate a linear programming model and solve using the computer.
b. If the vineyard could secure one additional unit of any of the resources used in the production of wine, which one should it select?
c. If the vineyard could obtain 0.5 ton more of grapes, 500 more hours of production capacity or enough storage capacity to store 4 more casks of wine, which should it choose?
3. The Roadnet Transport Company expanded its shipping capacity by purchasing 90 trailer trucks from a bankrupt competitor. The company subsequently located 30 of the purchased trucks at each of its shipping warehouses in Charlotte, Memphis, and Louisville. The company makes shipments from each one of these warehouses to terminals in St. Louis, Atlanta, and New York. Each truck is capable of making one shipment per week. The terminal managers have indicated their capacity for extra shipments. The manager in St. Louis can accommodate 40 additional trucks per week, the manager at Atlanta can accommodate 60 additional trucks, and the manager at New York can accommodate 50 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates:
|
|
Terminal
|
|
Warehouse
|
St. Louis
|
Atlanta
|
New York
|
|
Charlotte
|
$1,800
|
$2,100
|
$1,600
|
|
Memphis
|
1,000
|
700
|
900
|
|
Louisville
|
1,400
|
800
|
2,200
|
The company wants to know how many trucks to assign to each route (i.e., warehouse to terminal) to maximize profit. Formulate a total integer model for this problem and solve the model by using the computer
4. The Texas Consolidated Electronics Company is contemplating a research and development program encompassing eight research projects. The company is constrained from embarking on all projects by the number of available management scientists (40) and the budget available for R&D projects ($300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice versa). Following are the resource requirements and the estimated profit for each project. Formulate the integer programming model for this problem and solve it by using the computer.
|
Project
|
Expense ($1000s)
|
Management Scientists Required
|
Estimated Profit ($1,000,000s)
|
|
1
|
$60
|
7
|
$0.36
|
|
2
|
110
|
9
|
0.82
|
|
3
|
53
|
8
|
0.29
|
|
4
|
47
|
4
|
0.16
|
|
5
|
92
|
7
|
0.56
|
|
6
|
85
|
6
|
0.61
|
|
7
|
73
|
8
|
0.48
|
|
8
|
65
|
5
|
0.41
|