## 2 Background
### Datasets  
The primary inputs for the FLP problem are the set of demand centers and candidate facilities. The data sets used in this study include the Swain and London datasets (referred to as the benchmark in the FLP literature), which are depicted in Figures 1 and 2, respectively. The coordinates of the demand centers are displayed in each figure, with the size of each point proportional to the demand value. Each point also signifies the potential location to open a facility. The weighted Euclidean distance was used to calculate the total distance between a demand center and an open facility. The Swain dataset, comprising 55 nodes with location coordinates and user population data, has been a valuable resource for researchers in the transportation network optimization domain. This dataset, which provides an approximation of air traveler distribution based on origin and destination data in the Washington, DC area during the 1960s [14], has been utilized in recent studies to carry out simulations and solve the FLP. The Swain dataset has been a popular choice among researchers due to its ability to support a search tree guided process, making it well-suited for benchmarking and evaluating algorithmic solutions to the FLP. For instance, [PERSON] et al. [PERSON] et al. (2015) recently used the Swain dataset in proposing several approaches to address the p-median problem, comparing their results to the Swain dataset. The continued use of this dataset in recent research highlights its enduring value and utility in the scientific community. The use of Swain and London datasets is still relevant in ongoing research due to several reasons. First, these datasets serve as benchmark problems in optimization and machine learning, allowing researchers to evaluate and compare the performance of different algorithms and models. The Swain dataset in particular has been used to study several optimization problems, such as facility location, clustering, and graph partitioning. Second, the Swain dataset is a well-known benchmark problem in the field of community detection, which is the task of identifying densely connected subgraphs in a network. This problem is of great importance in various applications, such as social network analysis,  
Figure 1: Swain’s data set is one of the most used sets in the literature and benchmarks. Figure 2: London data set is one of the most used sets in the literature to represent problems with large instances of facilities. epidemiology, and transportation planning. Due to its large size and heterogeneous structure, the Swain dataset provides a realistic testbed for community detection algorithms. Third, the Swain dataset, as well as the London dataset, is often used in transportation planning research to model the flow of people and goods between different locations. Despite being based on data from the 1960s, these datasets still provide valuable insights into the fundamental principles of transportation systems and can be used to develop and test new models and algorithms. The Swain (Figure 1) and London (Figure 2) datasets remain relevant in ongoing research due to their importance as benchmark problems in optimization, community detection, and transportation planning. Their use allows researchers to develop and evaluate new algorithms and models and to gain insights into the underlying principles of complex systems. The Swain and London datasets were utilized in our study due to their ability to provide simulations of populations with varying levels of demand and population density. These datasets contain location coordinates denoted by 'x' and 'y' values, which can be substituted with actual geospatial data such as building, hospital, or park locations for analysis. Additionally, the datasets include a 'weight' field that represents population density, with higher weights indicating a greater population concentration in a given demand center. We utilized Euclidean distance calculations to simulate travel costs for individuals or groups seeking to access facilities located within demand centers.