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

WAREHOUSE	MONTHLY DEMAND (UNITS)	PRODUCTION PLANT	MONTHLY SUPPLY	COST TO PRODUCE ONE UNIT
Detroit	10,000	Cincinnati	15,000	$48
Houston	12,000	Kansas City	6,000	$50
New York	15,000	Pittsburgh	14,000	$52
Los Angeles	9,000
Total	46,000	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

	TO
FROM	DETROIT	HOUSTON	NEW YORK	LOS ANGELES
Cincinnati	$25	$55	$40	$60
Kansas City	$35	$30	$50	$40
Pittsburgh	$36	$45	$26	$66
Seattle	$60	$38	$65	$27
Birmingham	$35	$30	$41	$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
D	9	10	9
L	14	11	10
N	18	15	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 ($)	Job
Employee	1	2	3	4	5
A	1,200	1,850	1,375	3,500	1,900
B	1,175	1,920	1,540	3,375	2,050
C	1,080	1,885	1,475	3,400	2,010
D	1,190	1,905	1,400	3,600	2,100
E	1,250	1,900	1,390	3,520	1,975
F	1,200	1,950	1,465	3,400	1,950
G	1,075	1,820	1,480	3,250	1,920

1. 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?Define your decision variables carefully; write the objective function and all relevant constraints.

For the remaining parts, the changes are separately made based on the base model in part a.For each part, only write the changed or newly added constraints, no need to re-do the whole formulation.

1. 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?
2. 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?
3. 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?
4. 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	Estimated profit ($million)	Capital required ($million)
1	$17	$43
2	$10	$28
3	$15	$34
4	$19	$48
5	$7	$17
6	$13	$32
7	$9	$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.