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NTU Management Review Vol. 33 No. 3 Dec. 2023

Table 2 The Case Violating Preference Constraint
(j, t) p 1,j,t x 1,j,t residual capacity w jt x in (12) x in (13)
1,j,t
1,j,t
(1, 1) 0.1 0 1 1 0 0
(1, 2) 0.2 0.7 0.3 1 0 1
(2, 1) 0.3 0 0.3 1 1 1
(2, 2) 0.4 0.3 1 0 1 1

The above solution (x 1,1,2 =x 2,2,2 >0) cannot be a valid equilibrium outcome is due to the
fact that there exists an available facility-session pair that is more preferred than one that
is chosen by some customers (i.e., p 1,2,1 >p 1,1,2 while (2, 1) still has residual capacity). For

this example, the only valid equilibrium that satisfies all preference constraints is listed
in Table 3. In this case, the customers choose to go to facility 1 in session 2 and facility
2 in both sessions with proportion 0.4, 0.3, and 0.3, respectively. Similarly, according to

constraint (11), the value of z is 0.2, and no constraint is violated.
1
In short, our formulation guarantees that if there is still any available facility that a
customer prefers more, the customer will go to the more preferred one instead of others.
Therefore, each customer will act to maximize her/his preference.

Table 3 The Case Satisfying Preference Constraint
(j, t) p 1,j,t x 1,j,t residual capacity w jt x in (12) x in (13)
1,j,t
1,j,t
(1, 1) 0.1 0 1 1 0 0
(1, 2) 0.2 0.4 0.6 1 1 1
(2, 1) 0.3 0.3 0 0 1 1
(2, 2) 0.4 0.3 0 0 1 1

3.4 A Note on the Activity Sessions
While the sets of customers I and facilities J are pretty much given, the set of activity
sessions T is artificially determined by the decision maker. One may wonder how a
practitioner may determine the time unit and number of activity sessions T in a time unit

when applying this model. We briefly discuss this issue in this section to provide a guide
for practitioners.
One basic rule is that a time unit should be chosen so that a customer rarely wants
to visit a facility more than once in a day. For example, if customers are residents and

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