arXiv subjects

W. W. Koczkodaj

Publications and source records attributed to W. W. Koczkodaj.

17 recordsLinked to original sources

arXiv2023

Axiomatization of Inconsistency Indicators for Pairwise Comparisons

This study proposes revised axioms for defining inconsistency indicators in pairwise comparisons. It is based on the new findings that "PC submatrix cannot have a worse inconsistency indicator than the PC matrix containing it" and that there must be a PC submatrix with the same inconsistency as the given PC matrix. This study also provides better reasoning for the need of normalization. It is a revision of axiomatization by Koczkodaj and Szwarc, 2014 which proposed axioms expressed informally with some deficiencies addressed in this study.

cs.OH↗

arXiv2022

Combating harmful Internet use with peer assessment and differential evolution

Harmful Internet use (HIU) is a term coined for the unintended use of the Internet. In this study, we propose a more accurate HIU measuring method based on the peer assessment and differential evolution approach. The sample data comprises a juvenile population in Poland; 267 subjects assessed 1,513 peers. In addition to classic statistical analysis, differential evolution has been employed. Results indicate that there may be a substantially higher rate of HIU than other studies have indicated. More accurate measurement of the adolescent population influx affected by HIU is needed for healthcare and welfare system planning. Presented in Prague, Czech Republic, 20-22 July 2022.

cs.CY↗

arXiv2021

Pairwise Comparisons Matrix Decomposition into Approximation and Orthogonal Component Using Lie Theory

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons. Keywords: approximate reasoning, subjectivity, inconsistency, consistency-driven, pairwise comparison, matrix Lie group, Lie algebra, approximation, orthogonality, decomposition.

math.GM↗

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↗

arXiv2017

How to reduce the number of rating scale items without predictability loss?

Rating scales are used to elicit data about qualitative entities (e.g., research collaboration). This study presents an innovative method for reducing the number of rating scale items without the predictability loss. The "area under the receiver operator curve method" (AUC ROC) is used. The presented method has reduced the number of rating scale items (variables) to 28.57\% (from 21 to 6) making over 70\% of collected data unnecessary. Results have been verified by two methods of analysis: Graded Response Model (GRM) and Confirmatory Factor Analysis (CFA). GRM revealed that the new method differentiates observations of high and middle scores. CFA proved that the reliability of the rating scale has not deteriorated by the scale item reduction. Both statistical analysis evidenced usefulness of the AUC ROC reduction method.

stat.ME↗

arXiv2015

Pairwise Comparisons Rating Scale Paradox

This study demonstrates that incorrect data are entered into a pairwise comparisons matrix for processing into weights for the data collected by a rating scale. Unprocessed rating scale data lead to a paradox. A solution to it, based on normalization, is proposed. This is an essential correction for virtually all pairwise comparisons methods using rating scales. The illustration of the relative error currently, taking place, is discussed.

cs.OH↗

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↗

arXiv2015

A different perspective on a scale for pairwise comparisons

One of the major challenges for collective intelligence is inconsistency, which is unavoidable whenever subjective assessments are involved. Pairwise comparisons allow one to represent such subjective assessments and to process them by analyzing, quantifying and identifying the inconsistencies. We propose using smaller scales for pairwise comparisons and provide mathematical and practical justifications for this change. Our postulate's aim is to initiate a paradigm shift in the search for a better scale construction for pairwise comparisons. Beyond pairwise comparisons, the results presented may be relevant to other methods using subjective scales. Keywords: pairwise comparisons, collective intelligence, scale, subjective assessment, inaccuracy, inconsistency.

cs.AI↗

arXiv2015

Managing Null Entries in Pairwise Comparisons

This paper shows how to manage null entries in pairwise comparisons matrices. Although assessments can be imprecise, since subjective criteria are involved, the classical pairwise comparisons theory expects all of them to be available. In practice, some experts may not be able (or available) to provide all assessments. Therefore managing null entries is a necessary extension of the pairwise comparisons method. It is shown that certain null entries can be recovered on the basis of the transitivity property which each pairwise comparisons matrix is expected to satisfy.

cs.OH↗

arXiv2015

An LP-based inconsistency monitoring of pairwise comparison matrices

A distance-based inconsistency indicator, defined by the third author for the consistency-driven pairwise comparisons method, is extended to the incomplete case. The corresponding optimization problem is transformed into an equivalent linear programming problem. The results can be applied in the process of filling in the matrix as the decision maker gets automatic feedback. As soon as a serious error occurs among the matrix elements, even due to a misprint, a significant increase in the inconsistency index is reported. The high inconsistency may be alarmed not only at the end of the process of filling in the matrix but also during the completion process. Numerical examples are also provided.

cs.OH↗

arXiv2015

An Algorithm for the Optimal Consistent Approximation to a Pairwise Comparisons Matrix by Orthogonal Projections

The algorithm for finding the optimal consistent approximation of an inconsistent pairwise comparisons matrix is based on a logarithmic transformation of a pairwise comparisons matrix into a vector space with the Euclidean metric. Orthogonal basis is introduced in the vector space. The orthogonal projection of the transformed matrix onto the space formed by the images of consistent matrices is the required consistent approximation.

cs.OH↗

arXiv2015

A Monte Carlo Study of Pairwise Comparisons

Consistent approximations obtained by geometric means ($GM$) and the principal eigenvector ($EV$), turned out to be close enough for 1,000,000 not-so-inconsistent pairwise comparisons matrices. In this respect both methods are accurate enough for most practical applications. As the enclosed Table 1 demonstrates, the biggest difference between average deviations of $GM$ and $EV$ solutions is 0.00019 for the Euclidean metric and 0.00355 for the Tchebychev metric. For practical applications, this precision is far better than expected. After all we are talking, in most cases, about relative subjective comparisons and one tenth of a percent is usually below our threshold of perception.

cs.OH↗

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↗

arXiv2013

On Axiomatization of Inconsistency Indicators for Pairwise Comparisons

We examine the notion of inconsistency in pairwise comparisons and propose an axiomatization which is independent of any method of approximation or the inconsistency indicator definition (e.g., Analytic Hierarchy Process, AHP). It has been proven that the eigenvalue-based inconsistency (proposed as a part of AHP) is incorrect.

cs.DM↗