1. The Livewright Medical Supplies Company has a total of 12 salespeople it wants to assign to three regions - the South, the East, and the Midwest. A salesperson in the South earns $600 in profit per month of the company, a salesperson in the East earns $540, and a salesperson in the Midwest earns $375. The southern region can have a maximum assignment of 5 salespeople. The company has a total of $750 per day available for expenses for all 12 salespeople. A salesperson in the South has average expenses of $80 per day, a salesperson in the East has average expenses of $70 per day, and a salesperson in the Midwest has average daily expenses of $50. The company wants to determine the number of salespeople to assign to each region to maximize profit.
a. Formulate an integer programming model for this problem
b. Solve this model by using the computer.
2. Solve the following mixed integer linear programming model by using the computer:
Maximize Z = 5 x1 + 6 x2 + 4 x3
Subject to
5 x1 + 3 x2 + 6 x3 ≤ 20
x1 + 3 x2 ≤ 12
x1, x3 ≥ 0
x2 ≥ 0 and integer
3. 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.
Project
Expense ($1,000s)
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
Formulate the integer programming model for this problem and solve it using the computer.
4. During the war with Iraq in 1991, the Terraco Motor Company produced a lightweight, all-terrain vehicle code-named "J99-Terra" for the military. The company is now planning to sell the Terra to the public. It has five plants that manufacture the vehicle and four regional distribution centers. The company is unsure of public demand for the Terra, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table. The transportation costs are per thousand vehicles shipped; for example, the cost of shipping 1,000 vehicles from plant 1 to warehouse C is $32,000.
From Plant
Transportation Costs ($1000s)
to Warehouse
Annual Production Capacity
Annual Fixed Operating Costs
A
B
C
D
1
$56
$21
$32
$65
12,000
$2,100,000
2
18
46
7
35
18,000
850,000
3
12
71
41
52
14,000
1,800,000
4
30
24
61
28
10,000
1,100,000
5
45
50
26
31
16,000
900,000
Annual
Demand
6,000
14,000
8,000
10,000
Formulate and solve an integer programming model for this problem to assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plan to each warehouse to minimize total cost.