The limit of inconsistency reduction in pairwise comparisons

Waldemar W. Koczkodaj, and Jacek Szybowski. "The limit of inconsistency reduction in pairwise comparisons." International Journal of Applied Mathematics and Computer Science 26.3 (2016): 721-729. <http://eudml.org/doc/286731>.

@article{WaldemarW2016,
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.},
author = {Waldemar W. Koczkodaj, Jacek Szybowski},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {pairwise comparison; inconsistency reduction; convergence limit; decision making; pairwise comparisons; inconsistency; inconsistency indicator map; generalized metric; group theory; linearly ordered group},
language = {eng},
number = {3},
pages = {721-729},
title = {The limit of inconsistency reduction in pairwise comparisons},
url = {http://eudml.org/doc/286731},
volume = {26},
year = {2016},
}

TY - JOUR
AU - Waldemar W. Koczkodaj
AU - Jacek Szybowski
TI - The limit of inconsistency reduction in pairwise comparisons
JO - International Journal of Applied Mathematics and Computer Science
PY - 2016
VL - 26
IS - 3
SP - 721
EP - 729
AB - 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.
LA - eng
KW - pairwise comparison; inconsistency reduction; convergence limit; decision making; pairwise comparisons; inconsistency; inconsistency indicator map; generalized metric; group theory; linearly ordered group
UR - http://eudml.org/doc/286731
ER -