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

Pairwise comparisonmatrices (PCMs) over an Abelian linearly ordered (alo)- group G = (G, circle dot,=) have been introduced to generalize multiplicative, additive and fuzzy ones and remove some consistency drawbacks. Under the assumption of divisibility of G, for each PCM A = (aij), a circle dot- mean vector (w) under bar (m) (A) can be associated with A and a consistency measure I(G)(A), expressed in terms of circle dot-mean of G-distances, can be provided. In this paper, we focus on the consistency index I(G)(A). By using the notion of rational power and the related properties, we establish a link between (w) under bar (m) (A) and I(G)(A). The relevance of this link is twofold because it gives more validity to I(G)(A) and more meaning to (w) under bar (m) (A); in fact, it ensures that if I(G)(A) is close to the identity element then, from a side A is close to be a consistent PCM and from the other side (w) under bar (m) (A) is close to be a consistent vector; thus, it can be chosen as a priority vector for the alternatives. (C) 2011 Wiley Periodicals, Inc.