Jacek Szybowski

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 ? ( n - 1 ) to n - 1 . An algorithm of reconstructing of the PC matrix from its set of generators is presented.

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.

On normalization of inconsistency indicators in pairwise comparisons

Abstract 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.

Fast Convergence of Distance-based Inconsistency in Pairwise Comparisons

This study presents theoretical proof and empirical evidence of the reduction 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 applications. We believe that this is the first Monte Carlo study demonstrating such results for the convergence speed of inconsistency reduction in pairwise comparisons.

The New Triad based Inconsistency Indices for Pairwise Comparisons

Abstract Pairwise comparisons are widely recognized method supporting decision making process based on the subjective judgments. The key to this method is the notion of inconsistency that has a significant impact on the reliability of results. Inconsistency is expressed by means of inconsistency indices. Depending on their construction, such indices may pay attention to different aspects of the set of pairwise comparisons. The family of indices proposed in this article tries to combine the advantages coming from different indices, thereby increases the expressiveness of the family elements. The newly introduced notion of equivalence can help in comparing the indices and identifying their common properties.

Tender with Success – The Pairwise Comparisons Approach☆

Abstract 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.

The Cycle Inconsistency Index in Pairwise Comparisons Matrices

The method of pairwise comparisons is widely applied in the decision making process. The inconsistency of data may significantly affect the final result. Since the notion of consistency is based on triads or cycles, there is a great need for defining the measure of a triad or cycle inconsistency.In the paper a set of properties of a good cycle inconsistency index is proposed. Two construction methods of a cycle-based inconsistency index for a pairwise comparisons matrix are introduced. All is supported by the examples.

The Limit of Inconsistency Reduction in Pairwise Comparisons

Abstract 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.

The improvement of data in pairwise comparison matrices

Abstract This article analyzes the question of reliability of entries collected in the pairwise comparison matrices used for prioritization of alternatives. The inappropriate input data may result in an invalid output hierarchy. Basing on the Geometric Mean Method introduced by Crawford and Williams and the geometric consistency index defined by Aguaron and Moreno-Jimenez we introduce an algorithm which allows to search for the least reliable pairwise comparison matrix elements and improve them by means of their estimations.

Inconsistency of special cases of pairwise comparisons matrices

This study presents special cases of inconsistent pairwise comparisons PC matrices and analysis of their eigenvalue-based inconsistency index using mathematical methods. All studied special cases of PC matrices are Toeplitz matrices with only three different entries