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.
@article{bwmeta1.element.bwnjournal-article-amcv26i3p721bwm,
author = {Waldemar W. Koczkodaj and Jacek Szybowski},
title = {The limit of inconsistency reduction in pairwise comparisons},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {26},
year = {2016},
pages = {721-729},
zbl = {06524454},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv26i3p721bwm}
}
Waldemar W. Koczkodaj; Jacek Szybowski. The limit of inconsistency reduction in pairwise comparisons. International Journal of Applied Mathematics and Computer Science, Tome 26 (2016) pp. 721-729. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv26i3p721bwm/
[000] Aczel, J. (1948). On means values, Bulletin of the American Mathematical Society 18(4): 443-454, DOI: 10.2478/v10006-008-0039-2.
[001] Aczel, J. and Saaty, T. (1983). Procedures for synthesizing ratio judgements, Journal of Mathematical Psychology 27(1): 93-102. | Zbl 0522.92028
[002] Arrow, K. (1950). A difficulty in the concept of social welfare, Journal of Political Economy 58(4): 328-346.
[003] Bauschke, H. and Borwein, J. (1996). Projection algorithms for solving convex feasibility problems, SIAM Review 38(3): 367-426. | Zbl 0865.47039
[004] Dong, Y., Xu, Y., Li, H. and Dai, M. (2008). A comparative study of the numerical scales and the prioritization methods in AHP, European Journal of Operational Research 186(1): 229-242. | Zbl 1138.91373
[005] Faliszewski, P., Hemaspaandra, E. and Hemaspaandra, L. (2010). Using complexity to protect elections, Communications of the ACM 53(11): 74-82.
[006] Holsztynski, W. and Koczkodaj, W. (1996). Convergence of inconsistency algorithms for the pairwise comparisons, Information Processing Letters 59(4): 197-202. | Zbl 0875.68472
[007] Jensen, R. (1984). An alternative scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 28(3): 317-332.
[008] Kendall, M. and Smith, B. (1940). On the method of paired comparisons, Biometrika 31: 324-345. | Zbl 0023.14803
[009] Koczkodaj, W. (1993). A new definition of consistency of pairwise comparisons, Mathematical and Computer Modelling 18(7): 79-84. | Zbl 0804.92029
[010] Koczkodaj, W., Kosiek, M., Szybowski, J. and Xu, D. (2015). Fast convergence of distance-based inconsistency in pairwise comparisons, Fundamenta Informaticae 137(3): 355-367. | Zbl 1335.68262
[011] Koczkodaj, W. and Szarek, S. (2010). On distance-based inconsistency reduction algorithms for pairwise comparisons, Logic Journal of the IGPL 18(6): 859-869. | Zbl 1201.68114
[012] Koczkodaj, W. and Szybowski, J. (2015). Pairwise comparisons simplified, Applied Mathematics and Computation 253: 387-394. | Zbl 1338.15075
[013] Llull, R. (1299). Ars Electionis (On the Method of Elections), Manuscript.