About a consistency index for pairwise comparison matrices over a divisible alo-group

Pairwise comparison matrices (PCMs) over an Abelian linearly ordered (alo)-group {article}{amssymb}{empty}{document}G=(G, , ≤){document} have been introduced to generalize multiplicative, additive and fuzzy ones and remove some consistency drawbacks. Under the assumption of divisibility of {article}{amssymb}{empty}{document}G{document}, for each PCM A=(aij), a ⊙-mean vector {article}{amssymb}{empty}{document}wₘ(A){document} can be associated with A and a consistency measure {article}{amssymb}{empty}{document}IG(A){document}, expressed in terms of ⊙-mean of {article}{amssymb}{empty}{document}G{document}-distances, can be provided. In this paper, we focus on the consistency index {article}{amssymb}{empty}{document}IG(A){document}. By using the notion of rational power and the related properties, we establish a link between {article}{amssymb}{empty}{document}wₘ(A){document} and {article}{amssymb}{empty}{document}IG(A){document}. The relevance of this link is twofold because it gives more validity to {article}{amssymb}{empty}{document}IG(A){document} and more meaning to {article}{amssymb}{empty}{document}wₘ(A){document}; in fact, it ensures that if {article}{amssymb}{empty}{document}IG(A){document} is close to the identity element then, from a side A is close to be a consistent PCM and from the other side {article}{amssymb}{empty}{document}wₘ(A){document} is close to be a consistent vector; thus, it can be chosen as a priority vector for the alternatives.