The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices

Abstract

Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197-208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.

A characterization of the Logarithmic Least Squares Method

“…Third, the Logarithmic Least Squares Methods has been extended to the incomplete case when certain elements of the pairwise comparison matrix are unknown (Bozóki et al, 2010). Axiomatization on this more general domain seems to be promising and within reach, as revealed by Bozóki and Tsyganok (2017), although LLSM sometimes behaves strangely on this general domain (Csató and Rónyai, 2016).…”

Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom

“…Finally, an extension to the incomplete case, when some elements of the pairwise comparison matrix may be missing, deserves a thorough investigation. Row geometric mean method has been defined on this more general domain by Bozóki et al (2010) on the basis of optimization problem (1), without affecting at least one desirable property of the procedure (Bozóki and Tsyganok, 2017).…”

Inconsistency indices for incomplete pairwise comparisons matrices

“…It is easy to observe that when the PC matrix is consistent then (5) and (6) are also equalities. Hence, the consistent PC matrix takes the form:…”

“…They both use the Koczkodaj triad index K ijk (14). The indices are designed as the average of all possible K ijk given as follows 5 :…”

“…The geometric consistency index (12) is directly based on observation (5) i.e. w(a i )/w(a k ) ≈ c ik , where w is the priority vector obtained from GMM [15,1].…”