Generalized extended Bonferroni means for isomorphic membership grades

The generalized extended Bonferroni mean (GEBM) is a powerful tool for modeling the complex process of aggregating information, whether it is homogeneously or heterogeneously connected, within a composite aggregation structure. It maintains several favorable characteristics and effectively captures the diverse and interconnected nature of expert opinions or criteria, which is commonly observed in various decision-making contexts. This research expands upon the existing GEBM framework by applying it to the specific domains of q-rung orthopair fuzzy sets (q-ROFSs) and extended q-rung orthopair fuzzy sets (Eq-ROFSs). Furthermore, it examines the transformation processes among different variants of GEBMs. To facilitate the development of generalized aggregation functions, the de Morgan triplets for q-ROFSs and Eq-ROFSs are established. By introducing an isomorphism, the transformation relationship between the aggregation functions for q-ROFSs and Eq-ROFSs is analyzed. Based on this foundation, the Bonferroni mean de Morgan triplet-based GEBMs for q-ROFSs and Eq-ROFSs are proposed, and the keeping-order relations for these proposed GEBMs are discussed. Finally, several special cases of the GEBMs for q-ROFSs and Eq-ROFSs are obtained, and several relevant theorems are verified.