# Cluster-Based Approaches toward Developing a Customer
## 3 Materials and Methods
### Proposal and Evaluation  
The loyalty program is designed to identify the segment of customers that are most valuable to the company. As previously defined and explained, the most valuable customers are those with the highest gross revenue. Therefore, the program aims to benefit the most valuable customers with rewards and privileges to be determined by the company's managers.  
Determining the number of clusters is crucial for managing these groups, and increasing this number makes the task more challenging [18]. According to the same authors, the number of customers is smaller in business markets compared to consumer markets, and their behaviors vary less than those of consumer market customers. Therefore, to find the most valuable customer groups that can be better managed by the different departments involved in the program, the authors of this study proposed creating three types of segments of the most valuable customer groups.  
The proposed clusters were based on the results of the k-means clustering technique with different numbers of K. According to [PERSON] [7], validation can be based on external criteria (evaluating the results in relation to a pre-specified structure) or internal criteria (evaluating the results in relation to information related to the data alone). The current study used both criteria, using (a) the silhouette coefficient as an internal criterion for evaluating the optimal number of K, and using (b) a questionnaire previously applied to respondents with the aim of finding which was the most valuable customer cluster and which was best managed by the departments involved as an external criterion. The latter was applied after the results were found using the silhouette coefficient metric and will be analyzed and discussed in the subsequent section.  
There are different metrics for the internal evaluation of clusters. [PERSON] [7] used three different metrics to evaluate the results of the clustering techniques applied in his study: the Davies-Bouldin Index, the silhouette coefficient, and the Dunn Index. [PERSON] et al. [4], on the other hand, used [PERSON]'s Bayesian Criterion (BIC) as a criterion for evaluating the clusters found. As previously mentioned, the present study used the silhouette coefficient metric for evaluating the clusters.  
The silhouette coefficient ranges between \(-1\) (indicative of poor clustering) and \(+1\) (indicative of good clustering), with the goal being maximizing the values. The following equation shows how the silhouette coefficient is measured:  
\[S(i)=\frac{b(i)\ -\ a(i)}{max\{a(i),\ b(i)\}} \tag{1}\]  
where a(i) is the average distance of point i from all other points in its cluster, and b(i) is the smallest average distance of i from all points in any other cluster. To clarify, b(i) is found by measuring the average distance of i from each point in cluster A, finding the average distance of i from each point in cluster B, and taking the resulting smallest value.  
Furthermore, this paper makes a comparison of the results obtained for the RFM variables as part of the process of evaluating the results found in the implementation of k-means clustering. The methodology and application were carried out for the dataset created from the variables indicated as main variables, listed in Table 7, by the interviewees. This comparison will be presented in the Section 4 and discussed later.