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. (C) 2009 Wiley Periodicals, Inc.