A general unified framework for pairwise comparison matrices in multicriterial methods

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.