Papers by Jacek Szybowski
arXiv (Cornell University), Apr 6, 2023
arXiv (Cornell University), Mar 27, 2023
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.
This study presents empirical evidence of the fast convergence of the distance-based inconsistenc... more This study presents empirical evidence of the fast convergence of the distance-based inconsistency for pairwise comparisons. It is a follow up of the theoretical proof of the inconsistency convergence. The convergence has been proven by the functional analysis method. As most mathematical proofs, computing the number of iterations has not even been considered. Our empirical research shows that the algorithm converges a fast rate. Keywords pairwise comparisons • distance-based inconsistency • convergence • knowledge management 1 Introduction Strict sciences have put more emphasis on processing quantitative (or objective) data than qualitative (subjective) data, which we use more frequently in daily life. However, empirical software engineering often deals with subjective attributes
ArXiv, 2015
This study proposes axioms for inconsistency indicators in pairwise comparisons. The new observat... more This study proposes axioms for inconsistency indicators in pairwise comparisons. The new observation (by Szybowski), that "no $PC$ submatrix may have a worse inconsistency indicator than the given $PC$ matrix" is an essential simplification of the axiomatization. The goal of formulating axioms for all future definitions of new inconsistency indicators is difficult and as illusive as the inconsistency concept itself. This study improves the axiomatization proposed by Koczkodaj and Szwarc in 2014. As a side product, the new axiom allows to prevent approximation error aberrations of an arbitrarily large value in the input data.
This study investigates a powerful model, targeted to subjective assessments, based on pairwise c... more 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 dened 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
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.
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.
We present an algebraic construction of the coboundary of a given cycle as a simpler alternative ... more We present an algebraic construction of the coboundary of a given cycle as a simpler alternative to the geometric one introduced in [M. Allili, T. Kaczynski, Geometric construction of a coboundary of a cycle, Discrete Comput. Geom. 25 (2001), 125–140, T. Kaczynski, Recursive coboundary formula for cycles in acyclic chain complexes, Topol. Methods Nonlinear Anal. 18 (2001), 351–371].
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, [9]) A triad inconsistency axiomatization in pairwise comparisons was informally proposed in [10]. 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.
Organization of a tender is not easy. Preparation of the relevant specification, taking into acco... more Organization of a tender is not easy. Preparation of the relevant specification, taking into account the non-price criteria, implementation of the objective and fair assessment procedure, and last but not least, selecting a satisfactory offer are in practice a considerable challenge. In meeting this challenge appropriate multi-criteria assessment models can help. Models that can cope with different kinds of tangible and intangible criteria. The paper presents the hierarchical bid assessment (HBA) model of making decision in a tender procedure based on the pairwise comparisons method. It combines structural elements known from AHP with the Heuristic Rating Estimation approach. Two different schemes of rating tangible and intangible attributes are proposed. The notion of the success of the customer is defined and the practical method for its use is proposed. Theoretical considerations are illustrated in the relevant example.
ArXiv, 2015
This study, based on contemporary mathematical foun- dations, presents an abelian group approach ... more This study, based on contemporary mathematical foun- dations, presents an abelian group approach to analyzing inconsis- tency 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 general- ized metrics are investigated. Every inconsistency indicator map generates an inconsistency index of a pairwise comparisons matrix. The inconsistency analysis in pairwise comparisons has broad ap- plications in multi-criteria decision making.
Proceedings of the International Symposium on the Analytic Hierarchy Process: the 15th ISAHP conference. Publication date: July 2018, Jul 15, 2018
This article analyzes the problem of reliability of data collected in the pairwise comparison mat... more This article analyzes the problem of reliability of data collected in the pairwise comparison matrices used in the Analytic Hierarchy Process for prioritization of alternatives. The hierarchy obtained from the inappropriate input data may be faulty. Basing on the consistency index defined by Saaty in 1977 we introduce a method which allows to search for the least reliable entries of a pairwise comparison matrix and replace them with their estimations.
International Journal of Approximate Reasoning, 2017
In this study, we provide mathematical and practice-driven justification for using [0, 1] normali... more 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, is presented. A new type of paradox of infinity is described.
We construct the Conley index over a phase space for flows. Our definition is an alternative for ... more We construct the Conley index over a phase space for flows. Our definition is an alternative for the Conley index over a base defined in [M. Mrozek, J.F. Reineck and R. Srzednicki, The Conley index over a base , Trans. Amer. Math. Soc. 352 (2000), 4171-4194]. We also compare it to other Conley-type indices and prove its continuation property.
International Journal of Approximate Reasoning
Procedia Computer Science
Fundamenta Informaticae
ABSTRACT This study presents theoretical proof and empirical evidence of the re-duction algorithm... more ABSTRACT This study presents theoretical proof and empirical evidence of the re-duction algorithm convergence for the distance-based inconsistency in pairwise comparisons. Our empirical research shows that the convergence very quick. It usually takes less than 10 reductions to bring the inconsistency of the pairwise comparisons matrix below the assumed threshold of 1/3 (sufficient for most appli-cations). We believe that this is the first Monte Carlo study demonstrating such results for the convergence speed of inconsistency reduction in pairwise comparisons. Keywords pairwise comparisons · distance-based inconsistency · convergence · knowledge management (to be published in 2015 so Vol and pages are not known yet)
Mathematical Social Sciences
Fundamenta Informaticae
In this study, the orthogonalization process for different inner products is applied to pairwise ... more 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.
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Papers by Jacek Szybowski