<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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 limit of inconsistency reduction in pairwise comparisons

Department of Mathematics and Computer Science, Laurentian University, 935 Ramsey Lake Road, Sudbury, ON P3E 2C6,

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków,

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"The limit of inconsistency reduction in pairwise comparisons"}

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...

The limit of inconsistency reduction in pairwise comparisons

Published Online: Sep 29, 2016

Page range: 721 - 729

Received: Dec 23, 2015

Accepted: May 15, 2016

DOI: https://doi.org/10.1515/amcs-2016-0050

Keywords
pairwise comparison, inconsistency reduction, convergence limit, decision making

© 2016 Waldemar W. Koczkodaj et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The limit of inconsistency reduction in pairwise comparisons

Published Online: Sep 29, 2016

Page range: 721 - 729

Received: Dec 23, 2015

Accepted: May 15, 2016

DOI: https://doi.org/10.1515/amcs-2016-0050

Keywordspairwise comparison, inconsistency reduction, convergence limit, decision making

© 2016 Waldemar W. Koczkodaj et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
pairwise comparison, inconsistency reduction, convergence limit, decision making