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

1. Introduction

The facility location problems have been widely discussed for decades. In a typical
facility location problem, a decision maker decides where to build facilities among

some given locations. Typical objectives of the problem include profit maximization,
cost minimization, etc. for the decision maker. Some facility location problems are
uncapacitated, i.e., facilities are not subject to capacity constraints. On the contrary, there
may be capacitated facility location problem, in which facilities have limited amount

of capacity. Applications of facility location problems lie in the fields of supply chain
management (Pirkul and Jayaraman, 1998; Shavarani, Mosallaeipour, Golabi, and I˙zbirak,
2019), logistics (Lu and Bostel, 2007; Hosseini, Dehghanian, and Salari, 2019), operations
management (Harkness and ReVelle, 2003; Filippi, Guastaroba, and Speranza, 2021), and

healthcare (de Vries, van de Klundert, and Wagelmans, 2020).
When it comes to service facilities, several extensions for the facility location
problem have been studied in recent years. One extension is to consider customers, which
is critical when one builds service facilities like retail stores, parks, hospitals, public

bike stations, etc. Naturally, these service facilities are heterogeneous to customers,
i.e., customers would have different preferences toward them. In this case, whether one
customer would visit one specific facility or not cannot be determined directly by the
decision maker. Technically speaking, in a traditional facility location problem, a customer

can be “assigned” to a facility (e.g., a retail store can be assigned to a warehouse for
replenishment), but in a service facility location problem, a customer cannot be “assigned”
to a facility (e.g., a retail chain owner cannot require a citizen to visit any specific
retail store) and can only be “attracted” to a facility. Each of the customers will choose

facilities to visit based on her/his preference. Applications of facility location problem
with customer can be in many industries. For example, Shih, Chang, and Peng (2002)
verify the relationship among customers’ pick-up demands, chain store locations, and
store performances with empirical studies. Hiassat (2017) presents a model for deciding

facility locations and allocating healthcare resources according to different customer types.
Furthermore, the relative problems are also studied for emergency response facilities (Li,
Zhao, Zhu, and Wyatt, 2011; Abdullah, Adawiyah, and Kamal, 2018), retail stores (Hanjoul
and Petters, 1987), power stations (Abdel-Basset, Gamal, Chakrabortty, and Ryan, 2021),

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