arXiv subjects

J. Szybowski

Publications and source records attributed to J. Szybowski.

6 recordsLinked to original sources

arXiv2020

On the use of group theory to generalize elements of pairwise comparisons matrix: a cautionary note

This paper examines the constricted use of group theory in the studies of pairwise comparisons. The presented approach is based on the application of the famous Levi Theorems of 1942 and 1943 for orderable groups. The theoretical foundation for multiplicative (ratio) pairwise comparisons has been provided. Counterexamples have been provided to support the theory. In our opinion, the scientific community must be made aware of the limitations of using the group theory in pairwise comparisons. Groups, which are not torsion free, cannot be used for ratios by Levi's theorems.

math.HO↗

arXiv2020

On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices

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.

cs.OH↗

arXiv2017

Inconsistency indicator maps on groups for pairwise comparisons

This study presents an abelian group approach to analyzing inconsistency in pairwise comparisons. However, it wrongly assumes that an inconsistency indicator can take values in any abelian linearly ordered group. The followup publication (On normalization of inconsistency indicators in pairwise comparisons, a collaboration which includes two of three authors of this publication) shows that any inconsistency indicator should be normalized for a number of practical and theoretical reasons. The publication below fails to demonstrate even one example of a non trivial group (other than a group consisting of real numbers. However, some obtained results (under erroneous assumptions) are still valid.

cs.DM↗

arXiv2017

On normalization of inconsistency indicators in pairwise comparisons

In this study, we provide mathematical and practice-driven justification for using $[0,1]$ normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described.

cs.DM↗

arXiv2015

The key properties of inconsistency indicators for a triad in pairwise comparison matrices

Processing information, acquired by subjective assessments, involves inconsistency analysis in most (if not all) applications of which some are of considerable importance at a national level (see, Koczkodaj/Kulakowski/Ligenza, Scientometrics, 99(3): 911-926, 2014)A triad inconsistency axiomatization in pairwise comparisons was informally proposed in Koczkodaj/Szwarc, FUNDAMENTA INFORMATICAE, 132(4): 485-500, 2014. This study, rectifies it by the use of the distance and theoretical proofs. Three key properties of the indicator are presented in this study and illustrated by several examples.

cs.DM↗

arXiv2013

Pairwise Comparisons Simplified

This study examines the notion of generators of a pairwise comparisons matrix. Such approach decreases the number of pairwise comparisons from $n\cdot (n-1)$ to $n-1$. An algorithm of reconstructing of the PC matrix from its set of generators is presented.

cs.DM↗