Intuitionistic Fuzzy Clustering Indices Based on Generalized Fuzzy Orthopartitions
摘要
This article presents two novel indices for capturing how much clusters represented by intuitionistic fuzzy sets are mutually disjoint and cover the initial universe. They are derived from the axioms defining the so-called generalized fuzzy orthopartitions, which extend classical partitions by incorporating both fuzziness and uncertainty. Given a family O of intuitionistic fuzzy sets, the proposed indices, denoted as \(M_d(O)\) and \(M_c(O)\) , respectively quantify the extent of cluster overlap and the degree to which the clusters fail to cover the entire universe. Their mathematical properties are formally analyzed, and it is demonstrated that they take values in the unit interval [0,1], where lower values indicate a clustering structure that more closely aligns with the ideal properties of a generalized fuzzy orthopartition. A clustering is deemed optimal when both indices reach zero, ensuring that the classification is both mutually exclusive and fully exhaustive. To better understand the theoretical framework, an illustrative example is provided, showcasing how different intuitionistic fuzzy clustering configurations can be assessed using these indices. In this case, intuitionistic fuzzy sets represent segments of customers who are classified by the cost of the products they usually buy.