# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 2 Literature Review
In this sense, the center-of-gravity (CG) method is one of the most widely used approaches in previous studies [41; 42; 43], which is also known as the weighted mean. In the previous studies using the CG methods, the goals are usually to minimize an objective function involving squared Euclidean distance, Euclidean distance, or Manhattan distance from the facilities to the demand points. For instance, [PERSON] [44] used the CG method so that the average squared Euclidean distance can be minimized from the facility to the demand points. [PERSON] and [PERSON] [37] and [PERSON], [PERSON] [45] also applied the CG method to obtain the optimal facility location with the consideration of Euclidean distance. [PERSON] [46] also applied the CG method to minimize the total weighted Manhattan distance and squared Euclidean distance. However, the CG method can only be applied to minimize the total weighted squared Euclidean distance rather than the total weighted Manhattan distance. Accordingly, the CG method is invalid anymore to minimize the total weighted Manhattan distance. In addition to the above discussions, we also provide a comprehensive review in Table 1, where relevant studies about this topic are summarized. The differences with previous studies are summarized in three aspects. Particularly, many efforts have been dedicated to various continuous-space multi-facility problems considering Euclidean distance. In this study, the Manhattan distance is considered and the optimal facility locations are determined. Besides, most of the previous studies applied a variety of clustering algorithms to obtain the facility locations, which can be considered as a generalized multi-Weber problem and referred to as an uncapacitated multi-facility location-allocation problem as stated by [PERSON] [47]. Moreover, the problem can be interpreted as an enumeration of the Voronoi partitions of the customer set, which has been proven to be an NP-hard problem [48; 49] and cannot guarantee the global optimum of the solutions. As a consequence, the importance of obtaining the globally optimal solution remains an urgent problem, posing the question: How do we find the set of facility locations in a continuous plane that is the global optimum so that the objective function can be minimized? However, determining the global optimum of the facility locations in a continuous plane is quite difficult because of the infinite number of potential locations and continuous distance. Furthermore, different weights at demand points make the multi-facility determination challenging. Neither strategic approaches nor quantitative models to overcome the pitful clustering algorithms in handling the continuous-space facility location problems have been investigated before. Therefore, we focus on determining the global optimum of the solutions for the continuous-space multi-facility location problem, which is different from the previous studies. Therefore, we propose an exact method to transfer the continuous-space multi-facility location problem to the discrete-space multi-facility location problem when the Manhattan distance is considered, which has never been studied to the best of our knowledge.