On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

• Autorzy: Kułakowski K.

Identyfikatory

DOI

10.3233/FI-2015-1240

• Języki publikacji: EN

Abstrakty

EN

The pairwise comparisons method can be used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular implementations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and error of output. An error describes to what extent the obtained results correspond to the single expert’s assessments. On the basis of the inconsistency and the error, two properties of the weight deriving procedure are formulated. These properties are proven for eigenvector method and Koczkodaj’s inconsistency index. Several estimates using Koczkodaj’s inconsistency index for a principal eigenvalue, Saaty’s inconsistency index and the Condition of Order Preservation are also provided.

Słowa kluczowe

EN

paired comparisons; eigenvectors; least squares; decision making; parameter estimation

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2015
• Tom: Vol. 139, nr 4
• Strony: 403--419
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Kułakowski K.

• AGH University of Science and Technology Department of Applied Computer Science Al. Mickiewicza 30, 30-059 Cracow, Poland

Bibliografia

• [1] C. A. Bana e Costa and J. Vansnick. A critical analysis of the eigenvalue method used to derive priorities in AHP. European Journal of Operational Research, 187(3):1422–1428, June 2008.
• [2] S. Bozóki, J. Fülöp, and L. Rónyai. On optimal completion of incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52(1–2):318 – 333, 2010.
• [3] S. Bozóki and T. Rapcsák. On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization, 42(2):157–175, 2008.
• [4] J. M. Colomer. Ramon Llull: from ‘Ars electionis’ to social choice theory. Social Choice and Welfare, 40(2):317–328, October 2011.
• [5] P. Faliszewski, E. Hemaspaandra, L. A. Hemaspaandra, and J. Rothe. Llull and Copeland Voting Computationally Resist Bribery and Constructive Control. J. Artif. Intell. Res. (JAIR), 35:275–341, 2009.
• [6] G. T. Fechner. Elements of psychophysics, volume 1. Holt, Rinehart and Winston, New York, 1966.
• [7] M. Fedrizzi and S. Giove. Incomplete pairwise comparison and consistency optimization. European Journal of Operational Research, 183(1):303–313, 2007.
• [8] J. Fichtner. On deriving priority vectors from matrices of pairwise comparisons. Socio-Economic Planning Sciences, 20(6):341–345, 1986.
• [9] J. Fülöp. A method for approximating pairwise comparison matrices by consistent matrices. Journal of Global Optimization, 42(3):423–442, June 2008.
• [10] J. Fülöp, W. W. Koczkodaj, and S. J. Szarek. On some convexity properties of the least squares method for pairwise comparisons matrices without the reciprocity condition. J. Global Optimization, 54(4):689–706, 2012.
• [11] S. Greco, B. Matarazzo, and R. Słowi´nski. Dominance-based rough set approach on pairwise comparison tables to decision involving multiple decision makers. In JingTao Yao, Sheela Ramanna, Guoyin Wang, and Zbigniew Suraj, editors, Rough Sets and Knowledge Technology, volume 6954 of Lecture Notes in Computer Science, pages 126–135. Springer Berlin Heidelberg, 2011.
• [12] William Ho. Integrated analytic hierarchy process and its applications - A literature review. European Journal of Operational Research, 186(1):18–18, March 2008.
• [13] A. Ishizaka and A. Labib. Analytic hierarchy process and expert choice: Benefits and limitations. OR Insight, 22(4):201–220, 2009.
• [14] A. Ishizaka and A. Labib. Review of the main developments in the analytic hierarchy process. Expert Systems with Applications, 38(11):14336–14345, October 2011.
• [15] R. Janicki and Y. Zhai. On a pairwise comparison-based consistent non-numerical ranking. Logic Journal of the IGPL, 20(4):667–676, 2012.
• [16] W. W. Koczkodaj. A new definition of consistency of pairwise comparisons. Math. Comput. Model., 18(7):79–84, October 1993.
• [17] W. W. Koczkodaj and S. J. Szarek. On distance-based inconsistency reduction algorithms for pairwise comparisons. Logic Journal of the IGPL, 18(6):859–869, October 2010.
• [18] W.W. Koczkodaj and R. Szwarc. On axiomatization of inconsistency indicators for pairwise comparisons. Fundamenta Informaticae, 4(132):485–500, 2014.
• [19] K. Kułakowski. Heuristic Rating Estimation Approach to The Pairwise Comparisons Method. Fundamenta Informaticae, 133:367–386, 2014.
• [20] K. Kułakowski. Notes on Order Preservation and Consistency in AHP. European Journal of Operational Research, Elsevier 245:333–337, 2015.
• [21] K. Kułakowski, K. Grobler-De˛bska, and J.Wa˛s. Heuristic rating estimation: geometric approach. Journal of Global Optimization, 2014.
• [22] M. J. Liberatore and R. L. Nydick. The analytic hierarchy process in medical and health care decision making: A literature review. European Journal of Operational Research, 189(1):14–14, August 2008.
• [23] L. Mikhailov. Deriving priorities from fuzzy pairwise comparison judgements. Fuzzy Sets and Systems, 134(3):365–385, March 2003.
• [24] O. Perron. Zur Theorie der Matrices. Mathematische Annalen, 64(2):248–263, June 1907.
• [25] F. S. Roberts. Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences. Encyclopedia of mathematics and its applications. Cambridge University Press, 1985.
• [26] T. L. Saaty. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3):234 – 281, 1977.
• [27] T. L. Saaty. The analytic hierarchy and analytic network processes for the measurement of intangible criteria and for decision-making. In Multiple Criteria Decision Analysis: State of the Art Surveys, volume 78 of International Series in Operations Research and Management Science, pages 345–405. Springer New York, 2005.
• [28] T. L. Saaty. On the Measurement of Intangibles. A Principal Eigenvector Approach to Relative Measurement Derived from Paired Comparisons. Notices of the American Mathematical Society, 60(02):192, February 2013.
• [29] J. E. Smith and D. Von Winterfeldt. Anniversary article: decision analysis in management science. Management Science, 50(5):561–574, 2004.
• [30] N. Subramanian and R. Ramanathan. A review of applications of Analytic Hierarchy Process in operations management. International Journal of Production Economics, 138(2):215–241, August 2012.
• [31] L. L. Thurstone. A law of comparative judgment, reprint of an original work published in 1927. Psychological Review, 101:266–270, 1994.
• [32] O. S. Vaidya and S. Kumar. Analytic hierarchy process: An overview of applications. European Journal of Operational Research, 169(1):1–29, February 2006.
• [33] K. K. F. Yuen. Fuzzy cognitive network process: Comparison with fuzzy analytic hierarchy process in new product development strategy. Fuzzy Systems, IEEE Transactions on, PP(99):1–1, 2013.