Recent Advances on Inconsistency Indices for Pairwise Comparisons : A Commentary

• Autorzy: Brunelli M.

Identyfikatory

DOI

10.3233/FI-2016-1338

• Języki publikacji: EN

Abstrakty

EN

This paper recalls the definition of consistency for pairwise comparison matrices and briefly presents the concept of inconsistency index in connection to other aspects of the theory of pairwise comparisons. By commenting on a recent contribution by Koczkodaj and Szwarc, it will be shown that the discussion on inconsistency indices is far from being over, and the ground is still fertile for debates.

Słowa kluczowe

EN

pairwise comparisons; valued preference relations; consistency; inconsistency index; analytic hierarchy process (AHP)

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2016
• Tom: Vol. 144, nr 3/4
• Strony: 321--332
• Opis fizyczny: Bibliogr. 32 poz., wykr.

Twórcy

autor

Brunelli M.

• Systems Analysis Laboratory, Department of Mathematics and Systems Analysis, Aalto University, Otakaari 1 F, 02150 Espoo, Finland

Bibliografia

• [1] Saaty TL. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 1977;15(3):234–281. doi:10.1016/0022-2496(77)90033-5.
• [2] Brunelli M, Canal L, Fedrizzi M. Inconsistency indices for pairwise comparison matrices: a numerical study. Annals of Operations Research. 2013;211(1):493–509. doi:10.1007/s10479-013-1329-0.
• [3] Kułakowski K. Notes on order preservation and consistency in AHP. European Journal of Operational Research. 2015;245(1):333–337. doi:10.1016/j.ejor.2015.03.010.
• [4] Brunelli M, Critch A, Fedrizzi M. A note on the proportionality between some consistency indices in the AHP. Applied Mathematics and Computation. 2013;219(14):7901–7906. doi:10.1016/j.amc.2013.01.036.
• [5] Kułakowski K, Szybowski J. The new triad based inconsistency indices for pairwise comparisons. Procedia Computer Science. 2014;35:1132–1137. Available from: http://creativecommons.org/licenses/by-nc-nd/3.0/. doi:10.1016/j.procs.2014.08.205.
• [6] Barzilai J. Consistency measures for pairwise comparison matrices. Journal of Multi-Criteria Decision Analysis. 1998;7(3):123–132. Available from: http://hdl.handle.net/10222/43854. doi:10.1002/(SICI)1099-1360(199805)7:3¡123::AID-MCDA181¿3.0.CO;2-8.
• [7] Gass SI, Rapcsák T. Singular value decomposition in AHP. European Journal of Operational Research. 2004;154(3):573–584. DBLP, http://dblp.uni-trier.de. Available from: http://dx.doi.org/10.1016/S0377-2217(02)00755-5.
• [8] Koczkodaj WW, Szwarc R. On axiomatization of inconsistency indicators in pairwise comparisons. Fundamenta Informaticae. 2014;132(4):485–500. doi:10.3233/FI-2014-1055.
• [9] Aguarón J, Moreno-Jiménez JM. The geometric consistency index: Approximated thresholds. European Journal of Operational Research. 2003;147(1):137–145. doi:10.1016/S0377-2217(02)00255-2.
• [10] Crawford G, Williams C. A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology. 1985;29(4):387–405. doi:10.1016/0022-2496(85)90002-1.
• [11] Bozóki S, Fülöp J, Rónyai L. On optimal completion of incomplete pairwise comparison matrices. Mathematical and Computer Modelling. 2010;52(1):318–333. doi:10.1016/j.mcm.2010.02.047.
• [12] Shiraishi S, Obata T, Daigo M, Nakajima N. Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: computational experiments. In: ISAHP; 1999.
• [13] Koczkodaj WW, Herman MW, Orlowski M. Managing null entries in pairwise comparisons. Knowledge and Information Systems. 1999;1(1):119–125. doi:10.1007/BF03325094.
• [14] Lamata MT, Peláez JI. A method for improving the consistency of judgements. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2002;10(6):677–686. ISSN:1793-6411. doi:10.1142/S0218488502001727.
• [15] Ishizaka A, Labib A. Review of the main developments in the analytic hierarchy process. Expert Systems and Applications. 2011;38(11):14336–14345. doi:10.1016/j.eswa.2011.04.143.
• [16] Ureña R, Chiclana F, Morente-Molinera JA, Herrera-Viedma E. Managing incomplete preference relations in decision making: a review and future trends. Information Sciences. 2015;302:14–32. doi:10.1016/j.ins.2014.12.061.
• [17] Dong Y, Herrera-Viedma E. Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relation. IEEE Transactions on Cybernetics. 2015;45(4):780–792. doi:10.1109/TCYB.2014.2336808.
• [18] Dong Y, Li CC, Herrera F. An optimization-based approach to adjusting unbalanced linguistic preference relations to obtain a required consistency level. Information Sciences. 2015;292:27–38. doi:10.1016/j.ins.2014.08.059.
• [19] De Baets B, De Meyer H, De Schuymer B, Jenei S. Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare. 2006;26(2):217–238. ISSN:1432-217X. doi:10.1007/s00355-006-0093-3.
• [20] Tanino T. Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems. 1984;12(2):117–131.
• [21] Fishburn PC. Preference Relations and their numerical representations. Theoretical Computer Science. 1999; 217(2):359–383. doi:10.1016/S0304-3975(98)00277-1.
• [22] Ji P, Jiang R. Scale transitivity in the AHP. Journal of the Operational Research Society. 2003;54(8):896–905. Available from: http://www.jstor.org/stable/4101660.
• [23] Yuen KKF. Pairwise opposite matrix and its cognitive prioritization operators: comparisons with pairwise reciprocal matrix and analytic prioritization operators. Journal of the Operational Research Society. 2012; 63(3):322–338. Available from: http://ssrn.com/abstract=1997751.
• [24] Cavallo B, D’Apuzzo L. A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems. 2009;24(4):377–398. doi:10.1002/int.20329.
• [25] Chiclana F, Mata F, Martinez L, Herrera-Viedma E, Alonso S. Integration of a consistency control module within a consensus model. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2008;16(supp01):35–53. doi:10.1142/S0218488508005236.
• [26] Dong Y, Zhang G, Hong WC, Xu Y. Consensus models for AHP group decision making under row geometric mean prioritization method. Decision Support Systems. 2010;49(3):281–289. doi:10.1016/j.dss.2010.03.003.
• [27] Brunelli M, Fedrizzi M. Axiomatic properties of inconsistency indices for pairwise comparisons. Journal of the Operational Research Society. 2015;66(1):1–15. ArXiv:1306.6852 [cs.AI]. doi:10.1057/jors.2013.135.
• [28] Kingman J. A convexity property of positive matrices. The Quarterly Journal of Mathematics. 1961;12(1): 283–284. doi:10.1093/qmath/12.1.283.
• [29] Aupetit B, Genest C. On some useful properties of the Perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process. European Journal of Operational Research. 1993;70(2):263–268. doi:10.1016/0377-2217(93)90044-N.
• [30] Duszak Z, Koczkodaj WW. Generalization of a new definition of consistency for pairwise comparisons. Information Processing Letters. 1994;52(5):273–276. doi:10.1016/0020-0190(94)00155-3.
• [31] Koczkodaj WW. A new definition of consistency of pairwise comparisons. Mathematical and Computer Modelling. 1993;18(7):79–84. doi:10.1016/0895-7177(93)90059-8.
• [32] Horn RA, Johnson CR. Matrix Analysis. Cambridge University Press; 1985. ISBN:9780521386326.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.