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is 1 implies that either facility is out of capacity in activity session in
̅

���
equilibrium or that facility-session pair is the most preferred out of all available ones.
With such modifications, we propose a new model that integrates both customer
and facility capacity as
max � � � �
� �
s.t. (2) – (4),

(9)
� � � � �
�� ��
� �
� � � �� � � � ��� , � ��� ≥0 ∀ , , (8)
� � � � � ≤ �� ∀ , , (10)
� ���
�� ��
� � � � NTU Management Review Vol. 33 No. 3 Dec. 2023
≥ ∀ , , , (11)
�
��� ��
̅ � �1 � �≥ ∀ , , , (12)
̅
��� ��� ��� �
̅ ��� ≥ ∀ , , , (13)
���
��� ≥ 0 ∀ , , , (14)
�0, 1� ∀ , , (15)
��
̅ ��� �0, 1� ∀ , , , (16)
�� �0, 1� ∀ , (17)
where is a positive and sufficiently large constant.
With such modifications, we propose a new model that integrates both customer and
facility capacity as where M is a positive and sufficiently large constant.
The objective function (8) of the model is to maximize the total number of served
The objective function (8) of the model is to maximize the total number of served
customers. Note that the customers served by facility 0 are not counted since facility 0
customers. Note that the customers served by facility 0 are not counted since facility
represents customers staying at home. Constraints (2) – (4) are same as the uncapacitated
0 represents customers staying at home. Constraints (2) – (4) are same as the
ones. Constraint (9) ensures that the customers can only go to the location where a facility
is built. Furthermore, the number of customers from all locations cannot exceed the
uncapacitated ones. Constraint (9) ensures that the customers can only go to the location
capacity of the facility. Constraint (10) sets the value of w to show whether facility j is
jt
where a facility is built. Furthermore, the number of customers from all locations cannot
still available in activity session t. If w =0, constraints (9) and (10) together ensure that
jt
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facility j is fully occupied in activity session t. On the contrary, if w =1, constraint (10)
jt
is relaxed, and facility j still has residual capacity in the activity session t. In constraint
(11), z is set to be the preference of the most preferred available facility for customer i.
i
Constraint (12) sets the value of x according to z . For the facility-session pair that is most
ijt
i
preferred by customer i, we have p =z according to constraint (11), and thus x can be
i
ijt
ijt
1. For all the remaining pairs, x is forced to be 0 by constraint (12). Constraint (13) then
ijt
ensures that the only available facility-session pair that a customer may visit is her/his top-
priority choice. The remaining constraints are nonnegativity and binary constraints.
It is worth mentioning that, as the objective function is to maximize the number of
served customers, it is acceptable for a customer to visit any facility in any activity session
as long as she/he obtains a positive utility (so that visiting the facility is better than staying

at home). In other words, with the current objective function, a decision maker facing this

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