A Novel Similarity-Based Multi-Attribute Group Decision-Making Method in a Probabilistic Hesitant Fuzzy Environment

Probabilistic hesitant fuzzy sets (PHFSs), a significant expansion of hesitant fuzzy sets (HFSs), were suggested and intensively explored in order to address the problem of missing preference information. Based on previous research on PHFS, we discovered that there are still unresolved issues in the probabilistic hesitant fuzzy environment, such as (i) the similarity between probabilistic hesitant fuzzy elements (PHFEs) has not been studied, (ii) in the study of entropy, the uncertainty resulting from the inner hesitancy of decision makers (DMs) has been neglected; in addition, DMs may obtain different decision results by using different entropy formulas; (iii) the relationship between the similarity and entropy of PHFE has not been researched, and (iv) there is no multi-attribute group decision-making (MAGDM) method that uses similarity in a probabilistic hesitant fuzzy environment. In order to address the aforementioned issues, in this study, we attempt to incorporate similarity into a probabilistic hesitant fuzzy environment and offer a novel similarity-based MAGDM method. First, we define the similarity in probabilistic hesitant fuzzy environments and present some similarity formulas. Furthermore, considering the limitations of entropy presented by other researchers, we redefine the entropy of PHFEs and discuss the relationship between similarity and entropy in probabilistic hesitant fuzzy settings for the first time. Based on the similarity measure and entropy, we offer a new method for MAGDM with unknown attribute weights, which can be effectively applied to the assessment of small and medium-sized enterprises' (SMEs) credit risk. Finally, we demonstrated the effectiveness and robustness of the proposed decision-making process.