Complex Fermatean fuzzy sets (CFFSs) integrate the ideas of complex fuzzy sets and Fermatean fuzzy sets, where the membership, non-membership, and hesitancy degrees are all complex numbers, allowing the express uncertain information more flexibly and comprehensively. However, how to reasonably measure the discrepancies between CFFSs in decision-making remains an open task. This paper presents a series of new distance measures of CFFSs and their weighted versions based on Hamming, Euclidean, Hausdorff, and Hellinger distances. On this basis, we explore some outstanding properties that the proposed measures satisfy (i.e., boundedness, nondegeneracy, symmetry, and triangular inequality) and demonstrate their effectiveness through several examples. Furthermore, we design a decision-making algorithm as well as a clustering algorithm based on the proposed measures and verify the performance of the proposed measures through several applications.

Currently, many studies have developed distance or divergence measures between intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IvFSs). As a generalization of IFSs, picture fuzzy sets (PFSs) provide a more nuanced representation of uncertain and ambiguous information. Interval-valued picture fuzzy sets (IvPFSs) combine the concepts of IvIFSs and PFSs, providing a highly effective means of representing and processing uncertain, ambiguous and incomplete information. How to better measure the differences between PFSs and IvPFSs is still an open issue. This paper proposes some novel α-divergence measures for PFSs and IvPFSs, respectively. We demonstrate the basic properties of the proposed divergence measures, including non-negativity, non-degeneracy and symmetry. Besides, we analyze some special cases of the proposed divergence measures that degenerate into or are related to several well-known divergences. Then, we construct some numerical examples to demonstrate the effectiveness of the proposed measures concerning existing measures. Finally, the proposed α-divergence measures are applied to pattern recognition, multi-attribute decision-making (MADM) and clustering, demonstrating that these measures possess a high confidence level and can produce trustworthy results, especially in comparable situations.