Instructions:
1. For each question, your mathematical model and answers to each individual part
should be typed in Microsoft Word. For each question, state your model by clearly
defining your decision variables (with appropriate units), the objective function, and all
the relevant (including the non-negativity and binary if applicable) constraints.
2. For Question 1, 2 and 4, you only need to give the model formulation, no solution
required.
3. For Question 3, besides the model formulation, you also need to give the optimal
solution for each sub-question; you should solve for the optimal solution using Excel
solver; no need to submit the Excel work, just give the solutions in your Word submission.
Question #1.
Hardgrave Machine Company produces computer components at its factories in
Cincinnati, Kansas City, and Pittsburgh. These factories have not been able to keep up
with demand for orders at Hardgrave’s four warehouses in Detroit, Houston, New York,
and Los Angeles. As a result, the firm has decided to build a new factory to expand its
productive capacity. The two sites being considered are Seattle, Washington, and
Birmingham, Alabama. Both cities are attractive in terms of labour supply, municipal
services, and ease of factory financing.
Table 1 presents the production costs and monthly supplies at each of the three existing
factories, monthly demands at each of the four warehouses, and estimated production
costs at the two proposed factories. Transportation costs from each factory to each
warehouse are summarized in Table 2.
In addition to this information, Hardgrave estimates that the monthly fixed cost of
operating the proposed facility in Seattle would be $400,000. The Birmingham plant
would be somewhat cheaper, due to the lower cost of living at that location. Hardgrave
therefore estimates that the monthly fixed cost of operating the proposed facility in
Birmingham would be $325,000. Note that the fixed costs at existing plants need not be
considered here because they will be incurred regardless of which new plant Hardgrave
decides to open – that is, they are sunk costs.
The question(s) facing Hardgrave is this: Which of the new locations, in combination
with the existing plants and warehouses, will yield the lowest cost? Note that the (total)
unit cost of shipping from each plant to each warehouse includes both the shipping costs
and the corresponding production costs. In addition, the solution needs to consider the
monthly fixed costs of operating the new facility. Formulate a mixed integer linear
programming (MILP) that Hardgrave will use to solve this problem. Define your decision
variables carefully; write the objective function and all relevant constraints.
Table 1: Hardgrave Machine’s Demand and Supply Data
1
WAREHOUSE
MONTHLY
DEMAND
(UNITS)
Detroit
Houston
New York
Los Angeles
Total
10,000
12,000
15,000
9,000
46,000
PRODUCTION
PLANT
MONTHLY
SUPPLY
COST TO
PRODUCE
ONE UNIT
15,000
6,000
14,000
$48
$50
$52
Cincinnati
Kansas City
Pittsburgh
Total
35,000
Note: Supply needed from new plant = 46,000 - 35,000 = 11,000 units per month.
ESTIMATED PRODUCTION COST PER UNIT AT PROPOSED PLANTS
Seattle
$53
Birmingham
$49
Table 2: Hardgrave Machine’s Shipping Costs
FROM
DETROIT
TO
HOUSTON
NEW YORK
Cincinnati
Kansas City
Pittsburgh
Seattle
Birmingham
$25
$35
$36
$60
$35
$55
$30
$45
$38
$30
$40
$50
$26
$65
$41
LOS
ANGELES
$60
$40
$66
$27
$50
Question #2. Airline Scheduling
Alpha Airline wishes to schedule no more than one flight out of a given airport to each of
the following cities: C, D, L, and N. The available departure slots are 8 A.M., 10 A.M.,
and 12 NOON. Alpha leases the airplanes at the cost of $5000 before and including 10
A.M. and $3000 after 10 A.M., and is able to lease at most two per departure slot. Also, if
a flight leaves for location N in a time slot, there must be a flight leaving for location L in
the same time slot. The expected profit (in $1000) contribution before rental costs per
flight is shown in the table below. Formulate a model for a profit-maximizing (after
deducting rental cost) schedule. Define your decision variables carefully; write the
objective function and all relevant constraints.
Time Slot
8 A.M.
10 A.M.
12 Noon
C
10
6
6
2
D
L
N
9
14
18
10
11
15
9
10
10
Question #3.
A local consulting firm currently has 5 jobs that have to be completed by its contract
employees. After analyzing the time that it would take each specific employee to
complete each specific job, and, the wages each specific employee is paid per hour, it
determined how much it would cost if specific employees were assigned to specific jobs.
These costs were calculated to be:
Cost ($)
Employee
A
B
C
D
E
F
G
1
1,200
1,175
1,080
1,190
1,250
1,200
1,075
2
1,850
1,920
1,885
1,905
1,900
1,950
1,820
Job
3
1,375
1,540
1,475
1,400
1,390
1,465
1,480
4
3,500
3,375
3,400
3,600
3,520
3,400
3,250
5
1,900
2,050
2,010
2,100
1,975
1,950
1,920
a. If each employee can only be assigned at most one job and each job requires only one
employee, which employees should be assigned to which jobs? For remaining parts,
the changes are separately made based on the base model in part a.
b. Because of some unprofessional conduct of employee C when he was previously
assigned to job 1, he can no longer be assigned to job 1. Now which employees
should be assigned to which jobs? Who benefits and who loses because of this
unprofessional conduct?
c. Because employees B and E belonged to “designated” groups, these two employees
must be assigned jobs. Now, which employees should be assigned to which jobs?
Who benefits and who loses because of this designation?
d. Because the consulting firm has a policy that states that only one relative can be
assigned a job, employees A and C cannot both be assigned jobs. Now which
employees should be assigned to which jobs? Who benefits and who loses because of
this company policy?
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e. Employees F and G took the consulting firm to court as the firm would only allow
them to work one job, therefore, the firm now is forced to allow F and G to work up
to two jobs. Given this new fact, which employees should be assigned to which jobs?
Note: This question is not a standard Assignment problem of the network models. It should
be regarded as an integer programming problem with binary decision variables. Binary
constraints are needed in both model formulation and solver solutions.
Question #4.
The board of directors of General Wheels Co. is considering seven large capital
investments. Each investment can be made only once. These investments differ in the
estimated long-run profit (net present value) that they will generate as well as in the amount
of capital required, as shown by the following table.
Investment
opportunity
1
2
3
4
5
6
7
Estimated profit
($million)
$17
$10
$15
$19
$7
$13
$9
Capital required
($million)
$43
$28
$34
$48
$17
$32
$23
The total amount of capital available for these investments is $100 million. Investment
opportunities 1 and 2 are mutually exclusive (i.e., they cannot be chosen simultaneously),
and so are 3 and 4. Furthermore, 5 can be undertaken only if both 1 and 3 are taken.
Opportunity 7 has to be chosen if both 2 and 4 are selected, and Opportunity 7 cannot be
invested unless at least one of 5 and 6 is invested. The objective is to select the combination
of capital investments that will maximize the total estimated long-run profit (net present
value). Formulate this problem as an integer programming model.
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