# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 2 Literature Review  
The facility location problem is an important issue in supporting physical distribution as it contributes significantly to the travel time or cost in logistics systems. Since the facility location problem can be divided into two types, namely discrete-space and continuous-space facility location problems, this study focuses on the continuous-space facility location problem. For the continuous-space facility location problem, anywhere in the plane can be considered as the facility location [5]. Thus, there is an infinite number of potential locations for the facilities, thus making it quite difficult to solve [13]. [PERSON] and [PERSON] [14] have tried to locate a single industry or firm to minimize the transportation cost in the plane, known as the Weber problem, which has received much attention in the literature. Moreover, many studies have been developed for locating the facilities in the continuous space to minimize or maximize objective criteria from different perspectives [15; 16; 17; 18; 19; 20]. However, due to an infinite number of possible locations for the facilities, it is a big challenge computationally, unless some appropriate approaches or algorithms can be developed. To solve the above problem, many approaches have been developed previously [21; 22], e.g., by [PERSON] [23]. The earliest approach is an iterative procedure that was offered by [PERSON] [24], which was followed by some other variants [25; 26; 27]. The geometry-based approach is also used for solving continuous-space facility location problems [28]. Since the 1990s, the Voronoi diagram heuristic approach has been applied for solving the \(p\)-center problem in a continuous space [29; 30]. Rather than searching for the optimal facility locations from an infinite number of possible locations, making a list of some good places that are considered as potential facility candidates helps solve the problem. Instead of large-scale computation to obtain the optimal solution, it is necessary to identify a finite set that contains the optimal solution, which is also known as the finite dominating set. As a consequence, search effort can be reduced dramatically so that large-scale computation is avoided. In this research area, the nodes of the network are identified as the finite dominating set for the \(p\)-median problem [31; 32]. However, when the finite dominating set is very large, it is still computationally burdensome to find the optimal solution. To overcome the above difficulty, more recently, the facility location problem is defined as a special clustering problem, where the sets of customers served by the same facility are considered as clusters [33]. The clustering-based algorithms, such as k-means clustering and fuzzy C-means (FCM) clustering algorithms, have been widely applied in previous studies. Although the clustering-based algorithms cannot guarantee the execution time and solution quality, they have proven useful in practice. [PERSON] [34] applied the FCM clustering algorithm to minimize the mean squared distance from each data point to its nearest center. [PERSON] [35; 36] developed different versions of hybrid fuzzy clustering algorithms to group customer demands. [PERSON] and [PERSON] [37] presented a fuzzy clustering-based hybrid method for a multi-facility location problem where the capacity of each facility was unlimited. [PERSON], [PERSON] [38] considered the Euclidean distance in searching for potential locations of temporary emergency medical centers using a clustering-based algorithm. [PERSON] and [PERSON] [39] applied the k-means clustering algorithm so solve a multi-facility location-allocation problem. [PERSON], [PERSON] [40] also developed a clustering-based genetic algorithm for a multi-facility location problem, where the Manhattan distance was considered. After clustering the customers or clients into several groups, the facility locations need to be identified in each of the groups.