Li and Chen (Cognit Comput. 2018; 10:496–505) proposed the concept of D-intuitionistic hesitant fuzzy set as well a method for comparing two sets. have by introducing degree belief decision maker regarding opinion an expert in existing definition intuitionistic set. In future, other researchers may use Chen’s their research work. However, after deep study, it is observed that fails to different...

To investigate implicative filters of hoops furthermore, we apply n-fold theory to (?, ?)-hesitant fuzzy and introduce the notion (?,?)-hesitant filters, obtain some conditions for a filter be filter. We also study preimage image filters.

A sophisticated three-way decision model utilizing a multi-granularity probabilistic hesitant fuzzy rough set is introduced to mitigate the issue of information loss arising from sets when characterizing information. Initially, properties are examined, subsequently proposing distance measure and function tailored these sets. Following this, multi-attribute group decision-making incorporating es...