Algebraic representations of the weighted mean

We introduce a general framework for the algebraic representation of the weighted mean in the case in which priorities and values refer to a common aggregation domain, for instance in hierarchical multicriteria decision models of the AHP type. The general framework proposed is based on the semifield structure of open interval domains and provides a natural algebraic description of weighted mean aggregation, including the essential mechanism of normalization which transforms priorities into weights. Such description necessarily involves two operations, addition (abelian semigroup) and multiplication (abelian group), which generalize the role of addition and multiplication in P = (0, ∞). In this sense, the semifield framework extends recent work by Cavallo, D'Apuzzo, and Squillante on the multiplicative group structure on the basis of the representation of pairwise comparison matrices and their associated priority vectors. We consider open interval domains S ⊆ R whose semifield structures are generated by bijections ϕ : S ⊆ R → P. Continuous (thus strictly monotonic) bijections play a central role. In such case, continuous strict triangular conorms/norms and uninorms emerge naturally in the representation of the semifield structure and, in their weighted version, they also provide the representation of the weighted mean, in both the arithmetic and geometric forms.