# A Hybrid IVFF-AHP and Deep Reinforcement Learning Framework for an ATM Location and Routing Problem
## 4 A Real Case Study
### Determining the District with Clustering  
A new decision-making methodology is proposed, including clustering, Fermatean, fuzzy logic, and MCDM approaches, to select the most suitable ATM location in a risky district. First, the clustering algorithm chooses the districts with the highest risk among the 23 districts of Istanbul. For this selection, 12 input variables (Population, Socioeconomic status (SES) score, Number of vehicles, Average speed, Traffic density, Number of traffic lights, Number of traffic cameras, Urban function index, Education level, Annual income, Intentional murder rate, and Armed incident rate) are considered [12; 13; 14; 21]. ATMs in densely populated areas provide access to more users, increase transaction volume, and increase revenue. Additionally, traffic density may make people who want to use the ATM less vulnerable to theft or attack. Crime rates in the settlement can be an important indicator of ATM security. In areas with high crime rates, ATMs may be at greater risk, and the risk of theft or assault may increase. Missing security measures can indicate that the ATM is at risk. The social and economic situation of the region is important for the security of the ATM. Factors such as low income levels, poverty, and unemployment may cause an increase in ATM theft and other crimes. If ATM theft, assault, or other security incidents have occurred in the settlement before, it may indicate that this location is risky. ATMs in densely urbanized areas generally offer more security measures and higher quality of service. The numerical values are compiled from publicly available sources. Possible cluster values are calculated with the Elbow method, as seen in Figure 2. The most convenient values for k are found as 3, 4, and 5. For clustering, K-means and SOM algorithms with different cluster values are used. As seen in Table 3, the Silhouette validity index of both algorithms is calculated, and the cluster belonging to the algorithm with the optimal Silhouette index is selected. The Silhouette index is one of the internal clustering validation criteria in the literature [22]. This index assesses the clustering findings by averaging the cluster validity of a single sample and potentially ignoring the shared traits of all items. The Silhouette indexes obtain the optimal cluster number 4 in the K-means algorithm in Figure 3.  
Therefore, the classification given in Table 4 has been accepted. All districts in the clusters are also shown on the district map of the European side of Istanbul in Figure 4. \begin{table}
\begin{tabular}{l c c} \hline \hline Cluster 1 & Buyukcekmeec—Catalca—Eyupsultan—Fatih—Kagthane--Kucükcekmeec—Sultangazi & Avclar—Bahceliveler—Basaksehir—Beylikdüzu—Esenler—Gaziosmanpasa & Gaziosmanpasa \\ Cluster 3 & Bagcular—Esenyurt & Arnavutköy—Bakräköy—Bayrampasa—Besiktas—Beyoglu—Güngoren—Saryer—Silivri—Sisli–Zeytinburnu & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The optimal clusters of districts. Figure 3: The cluster representation for k = 4 (Cluster1: Purple, Cluster2: Blue, Cluster3: Green, Cluster4: Yellow, and the red dots represent the centers of the clusters). Figure 2: The Elbow method for the possible number of clusters. After clustering, which clusters to be selected are decided by looking at the average values of risk factors (Table 5). The experts decide that for a district to be considered particularly high-risk, it should have the lowest SES score, a high Intentional murder rate, and a high rate of Armed incidents. It is emphasized that other risk factors would have the same effect on selecting the district with high risk. Therefore, the focus is on these three risk factors. Figure 4: District map of the European side of Istanbul (Cluster1: Purple, Cluster2: Blue, Cluster3: Green, Cluster4: Yellow).