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

Casas-Ramírez, Camacho-Vallejo, and Martínez-Salazar (2018) use a cross entropy
method to solve the upper-level problem and a greedy randomized adaptive procedure to
solve the lower-level one. Drezner, Drezner, and Zerom (2018), though do not directly
model customers, assume that the facilities’ attractiveness may be randomly distributed.

Calvete, Galé, Iranzo, Camacho-Vallejo, and Casas-Ramírez (2020) add the cardinality
constraint into a facility location problem with preference by limiting the maximum
number of customer points that can be assigned to each facility point. Notably, these

works either ignore the capacity issue or only impose a weaker version of the capacity
constraint (e.g., the cardinality constraint). We contribute to the literature by incorporating
the capacity and preference issues in a single model.
Our goal is to build several finite-capacity facilities under a budget constraint and
to maximize the total number of customers with time-dependent preferences. To the

best of our knowledge, Kang et al. (2023) is so far the only work that explicitly includes
both customer and facility capacity in a single model (though excludes customers’ time
preferences). He proposes a greedy algorithm for solving that NP-hard problem. In each

iteration, the benefit evaluation problem is transformed into a maximum flow problem, and
the location with the highest benefit-to-cost ratio is selected. In our study, we extend the
formulation and revise the algorithm to incorporate the time factor.

3. Problem Description and Formulation

In this section, we provide the statement and formulation of our capacitated facility
location problem with time-dependent user preference.

3.1 Uncapacitated Facilities with Customer Preference
We consider a decision maker deciding where to build facilities along with the scale
levels but without capacity constraints. Let J={1,2,3,...,|J|} denote the set of locations where
a facility may be built, and K={1,2,3,...,|K|} represent the set of scale levels that for each

facility decision maker may choose from. The parameter f represents the fixed cost of
jk
building the facility at location j with scale level k. Without loss of generality, we assume

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