ADM2302 A, B, C, D, E
Fall 2015
Assignment # 3
Assignment #3
Transportation Problems, Integer and Binary Programming and Goal programming,
ADM2302 students are reminded that submitted assignments must be neat, readable, and well-organized.
Assignment marks will be adjusted for sloppiness, poor grammar and spelling, as well as for technical errors.
Please note that:
While working together is encouraged, plagiarism on assignments will not be accepted.
Each student must sign the individual statement of integrity to be included with the submission.
Each student must provide an individual original submission of completed Assignment #3.
Students should submit a PDF of their type-written (i.e., not handwritten) assignment via blackboard
learn by the due-date.
Assignments must be stapled or they will not be marked.
Problem1:
This problem is a combination of binary integer programming and transportation problem
A company has 6 sales centers in Ontario and has decided to open new depots to deliver its products from the
depots to the sales centers. There are two types of costs associated with the delivery: set-up costs (fixed
costs) are capital costs which may usually be written off over several years, and transportation costs which
depend on the distance covered. We assume that they have been put on some comparable basis, by taking
the costs over a year.
There are 6 sites available for the construction of new depots to deliver products to the sales centers.
The following table (Table 1) gives the transportation costs (in thousand dollars) of delivering the entire
demand of each sales center from a depot (not the unit costs). Certain deliveries that are impossible are
marked with the infinity symbol (∞).
Sales Centers
Depot
1
2
3
4
5
6
1
100
80
50
50
60
100
2
120
90
60
70
65
100
3
140
110
80
80
75
130
4
160
125
100
100
80
150
5
190
150
130
∞
∞
∞
6
200
180
150
∞
∞
∞
Table 1: Transportation costs for satisfying entire demand of each sales center
The construction costs (fixed cost) for each depot as well as the capacity of each depot are listed in Table 2.
Fall 2015
Page 1
ADM2302 A, B, C, D, E
Fall 2015
Assignment # 3
Depot
1
2
3
4
5
6
Cost(1000$)
3500
9000
10000
4000
3000
9000
Capacity(tons)
300
250
100
180
275
300
Table 2: Fixed costs and capacity limits of the depot locations
There are estimations for demand of each sales center which are shown in the following table.
Sales center
Demand (tons)
1
2
3
4
120
80
75
100
Table 3: Demand data
5
110
6
100
Considering that the demand of a sales center needs to be satisfied and a sales center may be delivered to
from several depots, which depots should be opened to minimize the total cost of construction and of
delivery, while satisfying all demands? Formulate algebraically the corresponding model for this problem, but
Do NOT solve the problem.
Problem2:
A car rental company has decided to provide roadside services for 6 zones in a large city. The company wants
to determine where (zone) to locate the roadside services. To reduce the company’s costs the manager wants
to locate a minimum number of roadside services and ensure that at least one service is within 15 minutes of
each zone.
The times (in minutes) required to drive between zones are:
To
From
1
2
3
4
5
6
1
0
10
20
30
20
20
2
10
0
25
35
20
10
3
20
25
0
15
30
20
4
30
35
15
0
15
25
5
20
20
30
15
0
14
6
20
10
20
25
14
0
Formulate an integer/binary programming model that will select the minimum number of roadside services
that the company will need to achieve its policy objective. Do NOT solve the problem.
Fall 2015
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ADM2302 A, B, C, D, E
Fall 2015
Assignment # 3
Problem3:
Capital budgeting
A firm has 6 projects that it would like to undertake over the next 5 years but because of budget limitations
not all can be selected. The total budget that the firm has considered to invest in the projects is $12,400,000.
The following table displays the expected revenue (NPV) of each project after 5 years and the required yearly
capital for each investment.
Table 1: Investment Details
Capital (in $000) required per year
Investment/
Project
1
2
3
4
5
6
Expected
NPV ($000)
$2700
$3330
$7010
$5770
$2900
$4870
Year 1
$ 975
$1200
$2500
$1550
$1400
$1900
Year 2
$ 350
$ 200
$1200
$1350
$ 350
$1900
Year 3
$ 200
$ 200
$ 850
$ 675
$87.5
$ 350
Year 4
$ 100
$ 200
$ 400
$ 337.5
$ 21.875
$ 350
Year 5
$
50
$ 200
$ 400
$168.75
$
0
$ 350
The capital available for the time period of each year is shown in the following table:
(The $12,400,000 in investment capital is spread over the 5 years)
Year 1
$ 5800
Capital (in $000) allocated per year
Year 2
Year 3
Year 4
$ 3500
$ 1300
$ 900
Year 5
$ 900
In addition, the firm must follow a few federal and state laws regarding these projects:
1.
2.
3.
Surplus capital funds in any year cannot be carried over from year to year.
If the firm decides to invest in the second investment/project, it must also invest in the fourth.
If the firm decides to invest in the first investment/project, it cannot invest in the third
investment/project.
Considering the budget limitations and the laws, which of the investments/projects should be chosen to
maximize potential NPV? Formulate the Integer Linear Programming model in algebraic form, and using
Excel’s Solver to find a solution. Provide a printout of your answer report (Attach the excel file to your
submission)
Fall 2015
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ADM2302 A, B, C, D, E
Fall 2015
Assignment # 3
Problem4:
Goal Programming
A large bookstore in Ottawa operates 7 days per week. This bookstore needs the following number of fulltime employees working each day of the week:
Day
Number of
employees
Sunday
47
Monday
22
Tuesday
28
Wednesday
35
Thursday
34
Friday
43
Saturday
53
Each employee must work 5 consecutive days each week and then have 2 days off. For example, any
employee who works Sunday through Thursday has Friday and Saturday off. This bookstore currently has a
total of 60 employees available to work. The director of the department has developed the following set of
prioritized goals for employee scheduling:
(1) The store wants to avoid hiring any additional employees. Weight=40
(2) The most important days for the department to be fully staffed are Saturday and Sunday. Weight=31
(3) The next most important day to be fully staffed is Friday. Weight=25
(4) The department would like to be fully staffed the remaining 4 days in the week. Weight=20
Formulate a goal programming model to determine the number of employees who should begin their 5-day
workweek each day of the week to achieve the department’s objectives. Do NOT solve the problem.
Fall 2015
Page 4
