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

Pairwise comparison matrices (PCMs) over an Abelian linearly ordered (alo)‐group \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal{G}=(G, \odot, \leq)$\end{document} have been introduced to generalize multiplicative, additive and fuzzy ones and remove some consistency drawbacks. Under the assumption of divisibility of \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal{G}$\end{document}, for each PCM A=(aij), a ⊙‐mean vector \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\underline{w}_{m}(A)$\end{document} can be associated with A and a consistency measure \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I_{\mathcal{G}}(A)$\end{document}, expressed in terms of ⊙‐mean of \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal{G}$\end{document}‐distances, can be provided. In this paper, we focus on the consistency index \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I_{\mathcal{G}}(A)$\end{document}. By using the notion of rational power and the related properties, we establish a link between \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\underline{w}_{m}(A)$\end{document} and \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I_{\mathcal{G}}(A)$\end{document}. The relevance of this link is twofold because it gives more validity to \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I_{\mathcal{G}}(A)$\end{document} and more meaning to \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\underline{w}_{m}(A)$\end{document}; in fact, it ensures that if \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I_{\mathcal{G}}(A)$\end{document} is close to the identity element then, from a side A is close to be a consistent PCM and from the other side \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\underline{w}_{m}(A)$\end{document} is close to be a consistent vector; thus, it can be chosen as a priority vector for the alternatives. © 2011 Wiley Periodicals, Inc.