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Fuzzy eigenvector method for deriving normalized fuzzy priorities from fuzzy multiplicative pairwise comparison matrices

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Abstract

The eigenvector method is one of the most used methods for deriving priorities of objects from multiplicative pairwise comparison matrices in Analytic Hierarchy Process (AHP). Fuzzy extension of AHP has been of much attention in order to capture uncertainty stemming from subjectivity of human thinking and from incompleteness of information that are integral to multi-criteria decision-making problems. Various fuzzy extensions of the eigenvector method have been introduced in order to derive fuzzy priorities of objects from fuzzy multiplicative pairwise comparison matrices. These fuzzy extensions are critically reviewed in this paper, and it is showed that (i) they violate multiplicative reciprocity of the related pairwise comparisons, (ii) they are not invariant under permutation of objects, (iii) the fuzzy maximal eigenvectors are not normalized, or (iv) a given normalized fuzzy maximal eigenvector does not consist of normalized maximal eigenvectors obtainable from multiplicative pairwise comparison matrices forming the fuzzy multiplicative pairwise comparison matrices. Afterwards, a new fuzzy extension of the eigenvector method based on the constrained fuzzy arithmetic is introduced and it is shown that it satisfies all four desirable properties.

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References

  • Blaquero, R., Carrizosa, E., & Conde, E. (2006). Inferring efficient weights from pairwise comparison matrices. Mathematical Methods of Operations Research, 64(2), 271.

    Article  MathSciNet  MATH  Google Scholar 

  • Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(2), 387.

    Article  MATH  Google Scholar 

  • Csutora, R., & Buckley, J. J. (2001). Fuzzy hierarchical analysis: The lambda-max method. Fuzzy Sets and Systems, 120(2), 181.

    Article  MathSciNet  MATH  Google Scholar 

  • Dijkstra, T. K. (2013). On the extraction of weights from pairwise comparison matrices. Central European Journal of Operational Research, 21(1), 103.

    Article  MathSciNet  MATH  Google Scholar 

  • Enea, M., & Piazza, T. (2004). Project selection by constrained fuzzy AHP. Fuzzy Optimization and Decision Making, 3(1), 39.

    Article  MATH  Google Scholar 

  • Fedrizzi, M., & Krejčí, J. (2015). A note on the paper ‘Fuzzy analytic hierarchy process: Fallacy of the popular methods’. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23(6), 965.

    Article  MathSciNet  MATH  Google Scholar 

  • Fichtner, J. (1986). On deriving priority vectors from matrices of pairwise comparisons. Socio-Economic Planning Sciences, 20(6), 341.

    Article  Google Scholar 

  • Goyal, R. K., & Kaushal, S. (2017). Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-017-9267-y.

  • Ishizaka, A. (2014). Comparison of fuzzy logic, AHP, FAHP and hybrid fuzzy AHP for new supplier selection and its performance analysis. International Journal of Integrated Supply Management, 9(1–2), 1.

    Article  Google Scholar 

  • Kaufmann, A., & Gupta, M. M. (1991). Introduction to fuzzy arithmetic: Theory and applications. Boston: International Thomson Computer Press.

    MATH  Google Scholar 

  • Klir, G. J., & Pan, Y. (1998). Constrained fuzzy arithmetic: Basic questions and some answers. Soft Computing, 2(2), 100.

    Article  Google Scholar 

  • Krejčí, J. (2017a). Additively reciprocal fuzzy pairwise comparison matrices and multiplicative fuzzy priorities. Soft Computing, 21(12), 3177. https://doi.org/10.1007/s00500-015-2000-2.

    Article  MATH  Google Scholar 

  • Krejčí, J. (2017b). Fuzzy eigenvector method for obtaining normalized fuzzy weights from fuzzy pairwise comparison matrices. Fuzzy Sets and Systems, 315(1), 26.

    Article  MathSciNet  MATH  Google Scholar 

  • Krejčí, J., Pavlačka, O., & Talašová, J. (2017). A fuzzy extension of analytic hierarchy process based on the constrained fuzzy arithmetic. Fuzzy Optimization and Decision Making, 16(1), 89.

    Article  MathSciNet  MATH  Google Scholar 

  • Laarhoven, P. J. M. V., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1–3), 199.

    MathSciNet  MATH  Google Scholar 

  • Liao, H. C., Mi, X. M., Xu, Z. S., Xu, J. P., & Herrera, F. (2018). Intuitionistic fuzzy analytic network process. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2017.2788881.

  • Saaty, T. L. (1980). The Analytic Hierarchy Process. New York: McGraw Hill.

    MATH  Google Scholar 

  • Saaty, T. L. (2006). There is no mathematical validity for using fuzzy number crunching in the analytic hierarchy process. Journal of Systems Science and Systems Engineering, 15(4), 457.

    Article  Google Scholar 

  • Saaty, T. L., & Hu, G. (1998). Ranking by eigenvector versus other methods in the analytic hierarchy process. Applied Mathematics Letters, 11(4), 121.

    Article  MathSciNet  MATH  Google Scholar 

  • Saaty, T. L., & Vargas, L. G. (1984). Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Mathematical Modelling, 5(5), 309.

    Article  MathSciNet  MATH  Google Scholar 

  • Urena, R., Chiclana, F., Morente-Molinera, J. A., & Herrera-Viedma, F. (2015). Managing incomplete preference relations in decision making: A review and future trends. Information Sciences, 302(1), 14.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, Y. M., & Chin, K. S. (2006). An eigenvector method for generating normalized interval and fuzzy weights. Applied Mathematics and Computation, 181(2), 1257.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, Y. M., & Elhag, T. M. S. (2006). On the normalization of interval and fuzzy weights. Fuzzy Sets and Systems, 157(18), 2456.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, Z. S., & Liao, H. C. (2017). Intuitionistic fuzzy analytic hierarchy process. IEEE Transactions on Fuzzy Systems, 22(4), 749.

    Article  Google Scholar 

  • Zhü, K. (2014). Fuzzy analytic hierarchy process: Fallacy of the popular methods. European Journal of Operational Research, 236(1), 209.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Industrial Engineering, University of Trento, Via Sommarive 9, 38122, Trento, Italy

    Jana Siebert

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  1. Jana Siebert
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Correspondence to Jana Siebert.

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Siebert, J. Fuzzy eigenvector method for deriving normalized fuzzy priorities from fuzzy multiplicative pairwise comparison matrices. Fuzzy Optim Decis Making 18, 175–197 (2019). https://doi.org/10.1007/s10700-018-9291-6

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