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Optimal Allocation of Capacitated Facilities Considering Time-Dependent User Preference for User Number
Maximization

problem does not need to precisely estimate all the preference levels p . Instead of this,
ijt
the decision maker only needs to estimate whether a given preference level p is positive
ijt
or not. In this study, we still choose to formulate the problem with p as a real number to
ijt
retain the generality of our model.

We may show that the problem is NP-hard by reducing the problem studied in Kang
et al. (2023) to our problem. This is trivial by observing that his problem is a special case
of ours with |T|=1, i.e., there is only one activity session.

3.3 An Illustrative Example
The following example shows how these constraints express the preference rela-
tionship. In this example, there are three built facilities and two activity sessions which
means six facility-session pairs. Suppose that customers all live in the same location, say
location 1, the preferences of customers and residual capacity of facility-session pairs are

listed in Table 1. We assume that the total demand of the customers at location 1 is 1.

Table 1 The Preference and Residual Capacity of the Example
(j, t) p 1,j,t residual capacity
(1, 1) 0.1 1
(1, 2) 0.2 1
(2, 1) 0.3 0.3
(2, 2) 0.4 0.3
(3, 1) -0.5 0.5
(3, 2) -0.6 0.5

In this case, no one will go to facility 3 in any activity session since the customers
hold negative preference over pairs (3, 1) and (3, 2), which is lower than the zero utility

of staying at home. Therefore, facility 3 does not need to be taken into consideration. If
the customers choose facility-session pairs (1, 2) and (2, 2) with proportion 0.7 and 0.3,
the status is presented in Table 2. Since only pair (2, 2) is fully occupied (i.e. w =0),
2,2
according to constraint (11), the preference of the most preferred available pair among

available ones is p 1,2,1 (i.e. z =0.3). Constraint (12) restricts x 1,1,2 to be 0 since it is not
1
the most preferred one. However, this creates a contradiction with the result given by
constraint (13), where x 1,1,2 =1. The above discussion is summarized in Table 2.

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