Consistency and consensus improving methods for pairwise comparison matrices based on Abelian linearly ordered group

The aim of a valued pairwise comparison matrix is to derive the priority structure over a set of criteria (or alternatives) in decision making. The consistency and consensus of a pairwise comparison matrix should be measured and improved to avoid a misleading priority structure. The basic entries of a pairwise comparison matrix can be described in different forms; hence, different consistency and consensus methods should be developed for different types of matrices. To provide a general framework, the pairwise comparison matrix based on Abelian linearly ordered group is first introduced. A consistency index is defined by constructing the nearest consistent pairwise comparison matrix from an inconsistent one, and two consistency improving methods are introduced. A group pairwise comparison matrix is derived, a consensus index of individual pairwise comparison matrices is defined and two consensus improving methods are developed by introducing a general aggregation operator based on Abelian linearly ordered group. The proposed consistency and consensus methods are convergent and can provide a general framework for existing methods.