Papers by Jacek Szybowski
arXiv: History and Overview, 2019
Compelling evidence against a heedless group theory generalization of pairwise comparisons elemen... more Compelling evidence against a heedless group theory generalization of pairwise comparisons elements is provided by means of counter-examples and mathematical reasoning. The lack of acceptable semantics for selected groups (with negative and complex numbers) and implications are analyzed. This study also provides examples and mathematical reasoning indicating why inconsistency indicators for pairwise comparisons require normalization. Methodological inconsistencies in using group theory for pairwise comparisons should be made public to the scientific community for further discussion and research.
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This study presents an abelian group approach to analyzing inconsistency in pairwise comparisons.... more This study presents an abelian group approach to analyzing inconsistency in pairwise comparisons. A notion of an inconsistency indicator map on a group, taking values in an abelian linearly ordered group, is introduced. For it, metrics and generalized metrics are utilized. Every inconsistency indicator map generates an inconsistency indicator of a pairwise comparisons matrix.
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International Journal of Approximate Reasoning, 2020
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Fundamenta Informaticae, 2020
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International Journal of Approximate Reasoning, 2018
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International Journal of Approximate Reasoning, 2018
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International Journal of Approximate Reasoning, 2017
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International Journal of Applied Mathematics and Computer Science, 2016
This study provides a proof that the limit of a distance-based inconsistency reduction process is... more This study provides a proof that the limit of a distance-based inconsistency reduction process is a matrix induced by the vector of geometric means of rows when a distance-based inconsistent pairwise comparisons matrix is transformed into a consistent PC matrix by stepwise inconsistency reduction in triads. The distance-based inconsistency indicator was defined by Koczkodaj (1993) for pairwise comparisons. Its convergence was analyzed in 1996 (regretfully, with an incomplete proof) and finally completed in 2010. However, there was no interpretation provided for the limit of convergence despite its considerable importance. This study also demonstrates that the vector of geometric means and the right principal eigenvector are linearly independent for the pairwise comparisons matrix size greater than three, although both vectors are identical (when normalized) for a consistent PC matrix of any size.
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Fundamenta Informaticae, 2016
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International Journal of Approximate Reasoning, 2016
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Processing information, acquired by subjective assessments, involves inconsistency analysis in mo... more 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.
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DESCRIPTION This study investigates a powerful model, targeted to subjective assessments, based o... more DESCRIPTION This study investigates a powerful model, targeted to subjective assessments, based on pairwise comparisons. It provides a proof that a distance-based inconsistency reduction transforms an inconsistent pairwise comparisons (PC) matrix into a consistent PC matrix which is generated by the geometric means of rows of a given inconsistent PC matrix. The distance-based inconsistency indicator was de�ned in 1993 for pairwise comparisons. Its convergence was analyzed in 1996 (regretfully, with an incomplete proof; �nally completed in 2010). However, there was no clear interpretation of the convergence limit which is of considerable importance for applications and this study does so. This study �nally ends the ongoing (since 1984) dispute on the approximation method for the inconsistent pairwise comparisons. The convergence limit is the vector of geometric means. It is not the principal right eigenvector of a given PC matrix.
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This study, based on contemporary mathematical foundations, presents an abelian group approach to... more This study, based on contemporary mathematical foundations, presents an abelian group approach to analyzing inconsistency in pairwise comparisons. A general and precise notion of an inconsistency indicator map on a group, taking values in an abelian linearly ordered group, is introduced. For it, metrics and generalized metrics are investigated. Every inconsistency indicator map generates an inconsistency index of a pairwise comparisons matrix. The inconsistency analysis in pairwise comparisons has broad applications in multi-criteria decision making.
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Applied Mathematics and Computation, 2015
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Fundamenta Informaticae, 2015
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Fundamenta Informaticae, 2015
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Fundamenta Informaticae, 2015
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Applied Mathematics and Computation, 2015
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Papers by Jacek Szybowski