Reciprocal transitive matrices over abelian linearly ordered groups: Characterizations and application to multi-criteria decision problems

A further discussion of “A Semiring-based study of judgment matrices: properties and models” [Information Sciences 181 (2011) 2166–2176]

“…The paper shows that abelian linearly ordered group represents an algebraic structure more convenient than semiring for dealing with PCMs; in [3][4][5][6][7][8] several results about PCMs over alo-group are already obtained, but further investigation is going on.…”

“…In Ref. [11], the author proposes two special semirings to discuss PCMs; the approach is summarized in Section 2.1; in Section 2.2, abelian linearly ordered groups are shown for dealing with PCMs (see [3][4][5][6][7][8] for further details). In Section 3 a comparison between the two approaches is performed in order to choose the more convenient one.…”

Important Facts and Observations about Pairwise Comparisons (the special issue edition)

“…In [20], a unied framework for both multiplicative and additive reciprocity and consistency was started by a general notion of a reciprocal PC matrix over an abelian linearly ordered group. This approach has been continued in [21] and [45] recently. In [45], among other results, notions of reciprocal and consistent PC matrices over a group were introduced.…”

“…However, the xed scale makes AHP a subset of PC; PC is more general as it does not assume a particular scale, and allows for non-numerical rankings. For instance, the non-numerical rankings of [33] are relations, the scales in [45] are arbitrary groups, and abelian linearly ordered groups (alo-groups) are employed in [20,21,50].…”

Group decision making based on multiplicative consistent reciprocal preference relations

“…In order to unify different approaches on consistency of preference relations with different forms, Cavallo and D'Apuzzo [5] introduced pairwise comparison matrix over an Abelian linearly ordered group. In addition, Cavallo and D'Apuzzo [6] analyzed a condition of transitivity for a reciprocal matrix over an Abelian linearly ordered group and proposed corresponding tools for checking the transitivity and reliability of a weighting vector. Xia and Chen [49] proposed a consistency index is defined by constructing the nearest consistent pairwise comparison matrix on Abelian linearly ordered group from an inconsistent one, together with two consistency improving methods.…”