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

dependent preferences contributes to the literature.
To model time dependency, it is assumed that one day is split into multiple non-
overlapping activity sessions (e.g., morning, afternoon, evening, and midnight). There
are groups of customers with different population sizes locating at various places. A

decision maker plans to choose the locations and scale levels to build facilities from
a set of candidate locations. The scale level determines the number of customers that
may be served at this facility in an activity session. Customers in the same group tend

to have identical preferences over each facility in each activity session. Still, they may
hold different preferences over different facilities or different activity sessions. That is,
among all built facilities that are still having rooms in some activity sessions, a customer
will choose her/his most preferred facility-session pair to visit that facility in that activity
session. Nonetheless, we also assume that a customer always has the option of staying at

home without visiting any facility, which gives her/him a null utility. That is, if one finds
that no facility-session pair may give her/him a positive utility, she/he will choose to stay
at home.

The decision maker acts to maximize the number of served customers subject to
a budget constraint. More specifically, the decision maker builds several facilities with
a constraint that the total construction cost cannot exceed the budget limit. Each of the
customers then self-selects among built facilities subject to the capacity constraint or
chooses to stay at home without visiting any facility. The objective function of the decision

maker is to maximize the number of customers who visit a facility.
To solve the aforementioned decision maker’s problem of building capacitated
service facilities, we develop two different solution approaches. The first one is through

formulating a mixed integer program so that when one seeks for an exact solution, she/he
may obtain it by solving the program using an existing exact algorithm (such as branch and
bound). Kang, Kung, Chiang, and Yu (2023) proposes a bi-level model that incorporates
preference and capacity based on Hanjoul and Petters’ (1987) formulation. Nevertheless,
Kang et al. (2023)’s model has many constraints and high time complexity also causing

the problem becomes harder to be solved in acceptable time. Camacho-Vallejo, Casas-
Ramírez, and Miranda-Gonzalez (2014) propose another formulation to turn a bi-level
model into a single-level one; however, the capacity issue is omitted in their formulation.

Combining these two works, we formulate a single-level mixed integer program for our

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