There are two vacancies on each of three committees C1 C2 C3

There are two vacancies on each of three committees C1, C2, C3. Adams and Smith are to be assigned two vacancies each, Brown and Jones to one vacancy each. In order to decide how The assignments are to be made, each person is asked to list The committees in order of their preference with 1 for most desirable, 2 for The next most desirable, and 3 for The least desirable. Their Set up The problem of assigning The person to The committees as a assignment/matching problem. You do not need to solve it. Suppose that Adams says The committees 1 and 2 are equally attractive and assigns them both a number 1. How would you change The problem formulation. Continue with (b), suppose that Brown says that committees 2 and 3 are equally attractive and assigns them both number 1. Set up The corresponding problem.

Solution

There are two vacancies on each of three committees C1, C2 and C3 which means in all there are six positions to be filled up. Let me define these vacancies as V1, V2 on C1 ; V3, V4 on C2 ;. and V5, V6 on C3

Adams and Smith are to be assigned two vacancies each and Brown and Jones to one vacancy each which means there are six candidates to fill up the above mentioned vacancies. Let me represent these candidates as A1, A2 (Adams for two); S1, S2 (Smith for two) B1 for Brown and J1 for Jones

For the above mentioned vacancies and candidates the matrix of preferences is defined as follows with the objective of Minimization of the total preferences as most preferred given value as 1 and least preference as 3

In the case That Adams says the committees 1 and 2 are equally attractive and assigns them a number 1, then the first two rows are changed as

Similarly, in case Brown says that committees 2 and 3 are equally attractive and assigns them both number 1, then the fifth row will be replaced by

Person\\ Vacancy	V1	V2	V3	V4	V5	V6
A1 Adams	1	1	2	2	3	3
A2 Adams	1	1	2	2	3	3
S1 Smith	1	1	3	3	2	2
S2 Smith	1	1	3	3	2	2
B1 Brown	3	3	2	2	1	1
J1 Jones	2	2	1	1	3	3