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

We consider reciprocal matrices over an abelian linearly ordered group; in this way we provide a general framework including multiplicative, additive and fuzzy matrices. In a multi-criteria decision making context, a pairwise comparison matrix A=(aij) is a reciprocal matrix that represents a useful tool for determining a weighting vector w for a set X of decision elements; but, when A is inconsistent, the weighting vector, usually proposed in literature, may provide a ranking on X that does not agree with the expressed preference intensities aij, thus, it is unreliable. We analyze a condition of transitivity for a reciprocal matrix A=(aij) over an abelian linearly ordered group, that, whenever A is a pairwise comparison matrix, allows us to state a qualitative dominance ranking on X and obtain ordinal evaluation vectors; in this way, we get a first tool for checking the reliability of a weighting vector. We also provide tools to check the transitivity.