A general unified framework for pairwise comparison matrices in multicriterial methods

Abstract

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

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

“…We also consider multiplicative consistency for M and additive consistency for ln(M ). 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.…”

“…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].…”

Incomplete preference matrix with elements from an Alo-group and its application to ranking of alternatives

“…|K| denotes the cardinality of K. Notice, thatconsistency index of the matrix A e (K) = {a e ij } K is defined by (6) as I (A e (K)). Minimization in (P2) is carried out with respect to the identity element e.…”

“…[24] and is easy to calculate it in the additive, multiplicative and fuzzy cases. This setting is based on the works of [6], [7], and [24].…”

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

“…12 the authors introduce PCMs, whose entries belong to an Abelian linearly ordered group (alo-group) (G, , ≤). In this way, the reciprocity and consistency conditions, and the notion of consistent vector, are expressed in terms of ; thus, equalities (2)- (8) and (10) are found again.…”