Constructing the geometric Bonferroni mean from the generalized Bonferroni mean with several extensions to linguistic 2-tuples for decision-making

The geometric Bonferroni mean (GeoBM), which was designed as a primitive attempt to extend the Bonferroni mean (BM), has enriched the family of BM-type aggregation functions in a variety of decision-making contexts to model homogeneous relations among certain attributes. Although it is difficult to dispute the versatility of GeoBM-based information fusion techniques, a key disadvantage of existing GeoBM developments is that they provide no specification regarding how the GeoBM should be designed, constructed, and applied, which has led to the existence of disparate GeoBM definitions and non-unified and unsound generalizations. This has resulted in some confusion within the GeoBM field and has thus presented significant barriers to researchers hoping to apply the latest developments in their specific areas of interest. In this study, we propose a novel GeoBM construction based on the generalized BM (GenBM) by selecting suitable components of the GenBM to provide a decomposable descriptive formulation that is easy to understand and interpret. A modified GeoBM (MGeoBM) is also proposed to enhance the capacity for modeling real-life situations. Furthermore, we demonstrate that the concept of generating the GeoBM from the GenBM can be readily adopted in a generalized extended BM (EBM) framework to construct an extended GeoBM (EGeoBM) that can model heterogeneous relations among certain attributes. The proposed calibrated EBM is an accurate update to the standard EBM, as can be seen on noting that independent attributes can be modeled by the arithmetic mean rather than by the power mean. Several natural extensions of the EGeoBM, e.g., the extended MGeoBM (EMGeoBM) and its weighted form, together with their degenerate forms under various assumptions, are discussed and finally applied to a linguistic 2-tuple environment. Comparisons among several existing linguistic 2-tuple aggregation functions demonstrate that the proposed EMGeoBM performs more robustly than the EBM, providing relatively stable alternative rankings.