# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 1 Introduction  
Over the last five decades, the facility location problem, also known as location analysis, 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, 3, 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 traditionally 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 facilitylocation problem, the facilities are allowed to be located anywhere in the planning area [5]. As a consequence, different resolution strategies need to be adopted for different problems.  
Since there is an infinite number of possible locations for the facilities in the continuous-space facility location problem, it is impossible to determine the globally optimal facility locations. Besides, most of the previous studies focused on two-dimensional multi-facility location problems [6; 7]. In practice, there are many three-dimensional facility location problems [8; 9], such as Wi-Fi stations or server rooms in a manion building. However, these three-dimensional multi-facility location problems have received insufficient attention in previous studies. Accordingly, there is a need for a new strategy to determine the optimal facility locations for the high-dimensional continuous-space multi-facility location problem.  
To fill the above research gap, this study focuses on determining the globally optimal facility locations for the continuous-space facility location problem. First of all, we start with the continuous-space single-facility location problem and obtain the global minimizer for the problem using a statistical approach. Secondly, although some studies [10; 11; 12] also obtained the minimizer for the problem, the properties of the minimizer were not fully investigated. Thirdly, based on the newly investigated properties of the minimizer, we proposed an exact method to determine the globally optimal facility locations for the continuous-space facility location problem. Finally, this study provides several illustrative instances from a benchmark to compare the performances of different methods, which indicates the superiority of the proposed exact method in the decision-making for the continuous-space multi-facility location problem.  
The rest of this paper is organized as follows. Section 2 reviews the previous methods to solve the continuous-space multi-facility location problem, where the research gaps are identified. Section 3 provides the methodology to determine the globally optimal facility locations for the continuous-space multi-facility location problem. The illustrative instances from a benchmark are presented to compare the performances of different methods in Section 4. At last, Section 5 concludes this study with contributions and future directions.