The Dorton University president has asked the OM department to assign eight biology professors (A, B, C, D, E, F, G, and H) to eight offices (numbered 1 to 8 in the diagram) in the new biology building.

The Dorton University president has asked the OM department to assign eight biology professors (A, B, C, D, E, F, G, and H) to eight offices (numbered 1 to 8 in the diagram) in the new biology building.

The following distances and two-way flows are given:

	DISTANCES BETWEEN OFFICES (FEET)
	1	2	3	4	5	6	7	8
1	—	10	20	30	15	18	25	34
2		—	10	20	18	15	18	25
3			—	10	25	18	15	18
4				—	34	25	18	15
5					—	10	20	30
6						—	10	20
7							—	10
8								—

	TWO-WAY FLOWS (UNITS PER PERIOD)
	A	B	C	D	E	F	G	H
A	—	4	0	0	6	0	0	0
B		—	0	0	0	5	0	5
C			—	0	0	0	0	6
D				—	1	0	0	0
E					—	3	0	0
F						—	2	0
G							—	1
H								—

a.	If there are no restrictions (constraints) on the assignment of professors to offices, how many alternative assignments are there to evaluate?
	40,320

The biology department has sent the following information and requests to the OSCM department:

Offices 1, 4, 5, and 8 are the only offices with windows.

A must be assigned Office 1.

D and E, the biology department co-chairpeople, must have windows.

H must be directly across the courtyard from D.

A, G, and H must be in the same wing.

F must not be next to D or G or directly across from G.

b.	How many possible solutions (assignments of faculty to offices) satisfy all of the stated constraints?

Number to possible solutions

Assume the solution is as follows:

A-1	B-2	G-3	H-4
COURTYARD
E-5	F-6	C-7	D-8

c.	What is the total flow-distance cost measure of this solution?

Total flow-distance cost

Explanation:

a.

For any problem with an equal number of assignees and slots (call this “n”), the number of possible assignments is n! = n × n − 1 × n − 2 × n − 3 × … × 1.Therefore, in this problem, the number of possible assignments is 8! = 40,320.

b.

There are two solutions that satisfy the constraints:

A-1	C-2	G-3	H-4
COURTYARD
E-5	F-6	B-7	D-8

A-1	B-2	G-3	H-4
COURTYARD
E-5	F-6	C-7	D-8

c.

Generally speaking, for all 28 pairs of offices, multiply the distance between the offices times the two-way flows between the faculty that occupy those offices. Then sum up the results to get the total flow-distance measure. Here you only need to worry about those office pairs where there is travel between the assigned faculty members (non-zero flow), significantly reducing your computational work.