## 4 A Case Study  
We present a case study to show how the CDP described by [PERSON] et al. (2018) can be combined with vehicular networks in real-world scenarios to efficiently allocate facilities in a city. This study aims to maximize the minimum distances between any pair of open facilities where each facility has a known capacity and the total capacity of open facilities must exceed a user-defined threshold. The CDP can be defined more formally on a complete, weighted, and undirected graph \(G(V,E)\), where \(V\) is a set of facilities and \(E\) is the set of edges connecting these facilities. Each edge \((i,j)\in E\) has a distance \(d_{i,j}>0\) that satisfies the triangle inequality, considering \(i,j\in V\), with \(i\
eq j\). All distances are symmetric, i.e., \(d_{ij}=d_{ji}\). Each facility \(i\in V\) has a predetermined, known capacity \(c_{i}>0\). As a threshold, collected service capacity \(b\) is needed. The CDP's goal is to find a subset \(O\subset V\) of facilities with a collected capacity greater than \(b\) and maximized the shortest possible distance between any two facilities \(i,j\in S\). The threshold is a minimum collected capacity \(b\) that represents a portion \(m\) of the total facility capacities \(b=m\cdot\sum_{i\in V}c_{i}\). The data used to test are obtained from Open Data BCN, specifically, bicing station in the city of Barcelona ([[https://opendata-ajuntament.barcelona.cat/data/en/dataset/bicing/resource/f59e276c-1a1e-4](https://opendata-ajuntament.barcelona.cat/data/en/dataset/bicing/resource/f59e276c-1a1e-4) fa5-8c89-8a56e56b34]([https://opendata-ajuntament.barcelona.cat/data/en/dataset/bicing/resource/f59e276c-1a1e-4](https://opendata-ajuntament.barcelona.cat/data/en/dataset/bicing/resource/f59e276c-1a1e-4) fa5-8c89-8a56e56b34), accessed on 26 December 2022) is used to model CDP. The dataset contains 496 unique locations of bicycle stations and 45 of them, particularly for the electric bicycle charging station. The distribution of electric bicycle charging stations throughout the Barcelona metropolitan area determines the facility's locations, and the capacity of each facility is the capacity of each electric bicycle charging station. Figure 7 depicts the distribution of potential facilities concerning red flags, and each facility has a known capacity. According to the traffic data ([[https://opendata-ajuntament.barcelona.cat/data/en/dataset/trams](https://opendata-ajuntament.barcelona.cat/data/en/dataset/trams)]([https://opendata-ajuntament.barcelona.cat/data/en/dataset/trams](https://opendata-ajuntament.barcelona.cat/data/en/dataset/trams)), accessed on 26 December 2022), the traffic situation is classified into various states, such as no data, very-fluid, fluid, dense, very-dense, congestion, and cut-off. Here we consider very-fluid, fluid, dense, and very-dense situations and apply different sizes of threshold \(b\) to evaluate the performance of [PERSON] et al. (2018) constructive heuristic model in a realistic scenario where the solution is constructed by adding promising elements one by one until the required capacity was reached and compare it to a random scenario where the solution is randomly selected from a list of elements until all the required capacity is covered. In this section, we demonstrate the outcome of a real numerical experiment using the constructive heuristic methodology [PERSON] et al. (2018), which we then compared to a random scenario. The experiment was conducted on a standard computer. A single instance with 45 unique facility locations and a different capacity threshold was run 30 times with  
Figure 7: Potential Facilities Location. Furthermore, in order to evaluate the effectiveness of the constructive heuristics in real scenarios, Table 2 compares the performance of the random scenario and the constructive heuristic scenario with various thresholds based on the city's traffic situation. The instance was run 30 times with different random seeds and the computational time of both algorithms was less than one second.