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Constructive Consistent Approximations in Pairwise Comparisons

Smarzewski Ryszard, Kozera Ryszard

Advances in Science and Technology. Research Journal

2022

Vol. 16, no. 4

243--255

In this paper we investigate groups which admit the existence of weighted consistent approximations for pairwise comparisons matrices. These approximations are defined by extending the classical matrix projection for R_{+} to abstract weighted projections on the non-linear sets of transitive group-valued matrices. It is of interest that all of them are represented by general explicit formulae dependent on an abstract logarithmic function. This general approach is applied to the groups Z^{∗}_{p} and F^{∗}_{2m} which are of fundamental importance in in cryptography. Finally, we use our unified mathematical model of pairwise comparisons for continuous one-parameter unitary groups, which play a fundamental role in physics.

On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices

Koczkodaj Waldemar W., Smarzewski Ryszard, Szybowski Jacek

Fundamenta Informaticae

2020

Vol. 172, nr 4

379--397

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.