## 3 Prototype Description
### Data Handling
This process can be summarized by:  
\[s(i)=\frac{b(i)-a(i)}{\max a(i),\ b(i)}, \tag{2}\]  
which results in a number between \(-1\) and \(1\), where \(-1\) is a bad categorization of the object \(i\) (not matching its current cluster) and \(1\) is an optimal categorization. To obtain the quality of the clustering in general, the average of \(s(i)\) is obtained for all objects \(i\) in the dataset. The variance rate criterion (VRC) considers the number of observations/data \(n\) and the number of clusters \(k\). When this index is used, an attempt is made to maximize the result as the value of \(k\) is changed. The VRC index is given by:  
\[VRC=\ \frac{BOSS}{k-1}/\frac{WOSS}{n-k}. \tag{3}\]  
where BOSS is the between-group sum of squares that portrays the variance between clusters taking into account the distance from their centroids to the global centroid, and WGSS is the within-group sum of squares that portrays the variance within clusters taking into account the distances from the points in a cluster to its centroid [51]. Figure 1: Representation of customers. The goal of the index is to define a cluster separation measure \(R(S_{i},S_{j},M_{ij})\) that allows for the computation of the average similarity of each cluster with its most similar (neighboring) cluster, the lowest possible value would be the optimal result [52]. With \(S_{i}\) being the dispersion measure of cluster \(i\), \(S_{j}\) being the dispersion measure of cluster \(j\), and \(M_{ij}\) being the distance between clusters \(i\) and \(j\), according to:  
\[R_{ij}\equiv\ \frac{S_{i}+S_{j}}{M_{ij}}\ \ \ R\equiv\ \frac{1}{N}\sum_{i=1}^{N}R_{i}. \tag{4}\]  
\(R_{ij}\) is obtained for all clusters, that is, the ratio of inter- and intra-cluster distances between cluster \(i\) and \(j\). After that, \(R_{i}\) (the highest value of \(R_{ij}\)) is obtained by identifying for each cluster, the neighboring cluster to which it is most similar. The index itself is calculated \(\bar{R}\), this being the total sum of the similarities of \(N\) clusters with their closest neighbors. Eight segmentation solutions were generated with the K-means algorithm, starting from \(k=3\) to \(k=10\). After that, the best results among the \(k\) solutions according to each index were obtained. According to Figure 2, the silhouette index suggested four clusters, while Calinski-Harabasz suggested eight and Davies-Bouldin three. It should be noted that in the interpretation of the silhouette and Calinski-Harabasz index, the highest value is chosen, while in the Davies-Bouldin index, the lowest value is selected. The suggestions for the number of clusters from the indices showed high variability, causing great uncertainty in detecting the number of clusters. This result is common among datasets that do not have naturally occurring clusters. The consumer data typically does not contain natural segments, making it difficult to obtain the optimal number of clusters from internal validation indices [40]. Various features in the data distribution can affect the internal validation indices. The silhouette and Davies-Bouldin indices suffer from close clusters, and Calinski-Harabasz performs poorly on unequal-size distributions [53]. All these cited characteristics are present when viewing the distribution of the data in Figure 3, which in addition to showing different sizes in the possible clusters demonstrates a clustering of data on a specific side of the distribution and a low-density in areas of a high monetary attribute. Figure 2: Internal validation indices (red dots are the optimal number of clusters according to each method).