On the optimal consistent approximation to pairwise comparison matrices Original Research Article

Consistency retrieval from a biased relative preference table is an imperative task in decision theory. This paper considers the least squares approximation of a pairwise comparison matrix by consistent matrices. It is observed that the highly nonlinear manifold of consistent matrices can be changed into a linear subspace by the componentwise logarithmic transformation. A first order optimality condition therefore can be described in terms of coordinates in the linear subspace. This approach facilitates the otherwise much more complicated optimality condition if working with the variables in the original manifold. Fast nonlinear equation solvers can be employed to solve the problem efficiently.