Article
Globally Optimal Facility Locations for Continuous-Space
Facility Location Problems
Xuehong Gao
1
, Chanseok Park
2,
* , Xiaopeng Chen
2
, En Xie
3
, Guozhong Huang
1
and Dingli Zhang
1,
*
Citation: Gao, X.; Park, C.; Chen, X.;
Xie, E.; Huang, G.; Zhang, D. Globally
Optimal Facility Locations for
Continuous-Space Facility Location
Problems. Appl. Sci. 2021, 11, 7321.
https://doi.org/10.3390/
app11167321
Academic Editor: Giancarlo Mauri
Received: 17 June 2021
Accepted: 1 August 2021
Published: 9 August 2021
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Attribution (CC BY) license (https://
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4.0/).
1
Research Institute of Macro-Safety Science, School of Civil & Resources Engineering, University of Science
and Technology Beijing, Beijing 100083, China; gaoxh2020@ustb.edu.cn (X.G.);
hjxhuanggz@ustb.edu.cn (G.H.)
2
Department of Industrial Engineering, Pusan National University, Busan 609-735, Korea; xpchen@pusan.ac.kr
3
School of Economics and Management, Nanjing University of Science and Technology,
Nanjing 210094, China; jinyu2016@njust.edu.cn
* Correspondence: cp@pusan.ac.kr (C.P.); zhangdingli@xs.ustb.edu.cn (D.Z.)
Abstract:
The continuous-space single- and multi-facility location problem has attracted much
attention in previous studies. This study focuses on determining the globally optimal facility locations
for two- and higher-dimensional continuous-space facility location problems when the Manhattan
distance is considered. Before we propose the exact method, we start with the continuous-space
single-facility location problem and obtain the global minimizer for the problem using a statistical
approach. Then, an exact method is developed to determine the globally optimal solution for
the two- and higher-dimensional continuous-space facility location problem, which is different from
the previous clustering algorithms. Based on the newly investigated properties of the minimizer, we
extend it to multi-facility problems and transfer the continuous-space facility location problem to
the discrete-space location problem. To illustrate the effectiveness and efficiency of the proposed
method, several instances from a benchmark are provided to compare the performances of different
methods, which illustrates the superiority of the proposed exact method in the decision-making of
the continuous-space facility location problems.
Keywords: facility location problem; mathematical programming; global optimization
1. Introduction
Over the last five decades, the facility location problem, also known as location analy-
sis, has attracted much attention in mathematical science [
1
]. A large number of researchers
have investigated both the formulations and the algorithms for diverse applications in
the private and public sectors [
2
–
4
]. Concerning the private sector, organizations have
to continuously search for new ways to reduce costs, improve customer satisfaction, and
increase profitability due to global competition. Logistics operations, especially facility
locations, such as industrial plants, bicycle-sharing stations, banks, distribution centers,
warehouses, and fourth-generation/fifth-generation (4G/5G) base stations, have tradi-
tionally been an opportune field for cost-saving. Similarly, for the public sector, choosing
appropriate locations for facilities, such as hospitals, ambulance stations, post stations,
transport terminals, medical service centers, and relief centers, enables them to improve
the service level and efficiency. Generally, a better option needs to be done to look for
compromises on behalf of different goals.
As the travel cost and time can be analyzed by discrete or continuous aspects in
the space, the facility location problem is commonly divided into two types, namely
(i) the discrete-space facility location problem and (ii) the continuous-space facility location
problem. Practically, for the discrete-space type, the locations of potential facilities can
just be located among the given specific points, whereas in the continuous-space facility
Appl. Sci. 2021, 11, 7321. https://doi.org/10.3390/app11167321 https://www.mdpi.com/journal/applsci
