On Computing the Variance of a Fuzzy Number

  • Juan Carlos Figueroa-GarcíaEmail author
  • Miguel Alberto Melgarejo-Rey
  • José Jairo Soriano-Mendez
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 833)


This paper presents a comparison of three well known methods for computing the variance of a fuzzy number and a proposal based on the Yager index for convex fuzzy sets: the Carlsson-Fullér, Mendel-Wu, and the sample variance of a fuzzy set. Some considerations about the obtained results are provided and some recommendations are given.


Fuzzy numbers \(\alpha \)-cuts Fuzzy variance Yager index 


  1. 1.
    Carlsson, C., Fullér, R.: On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets Syst. 122(1), 315–326 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chaudhuri, B., RosenFeld, A.: A modified hausdorff distance between fuzzy sets. Inf. Sci. 118, 159–171 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chiao, K.P.: A new ranking approach for general interval type-2 fuzzy sets using extended alpha cuts representation. In: International Conference on Intelligent Systems and Knowledge Engineering, vol. 1, pp. 594–597. IEEE (2015)Google Scholar
  4. 4.
    Chiao, K.P.: Ranking type-2 fuzzy sets using parametric graded mean integration representation. In: International Conference on Machine Learning and Cybernetics, vol. 1, pp. 606–611. IEEE (2016)Google Scholar
  5. 5.
    Figueroa-García, J.C., Chalco-Cano, Y., Román-Flores, H.: Yager index and ranking for interval type-2 fuzzy numbers. IEEE Trans. Fuzzy Syst. (1), 1–9 (2017, in Press)Google Scholar
  6. 6.
    Figueroa-García, J.C., Pachon-Neira, D.: On ordering words using the centroid and Yager index of an interval Type-2 fuzzy number. In: Proceedings of the 2015 Workshop on Engineering Applications (WEA), vol. 1, pp. 1–6. IEEE (2015)Google Scholar
  7. 7.
    Figueroa-García, J.C., Pachon-Neira, D.: A comparison between the centroid and the Yager index rank for type reduction of an interval Type-2 fuzzy number. Rev. Ing. Univ. Dist. 2, 225–234 (2016)Google Scholar
  8. 8.
    Figueroa-García, J.C., Soriano-Mendez, J.J., Melgarejo-Rey, M.A.: On the variance of a fuzzy number based on the Yager index. In: Proceedings of IEEE ColCaCi 2018, pp. 1–6. IEEE (2018)Google Scholar
  9. 9.
    Hung, W.L., Yang, M.S.: Similarity measures between type-2 fuzzy sets. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12(6), 827–841 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Upper Saddle River (1992)zbMATHGoogle Scholar
  11. 11.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River (1995)zbMATHGoogle Scholar
  12. 12.
    Mendel, J.M., Wu, D.: Cardinality, fuzziness, variance and skewness of interval type-2 fuzzy sets. In: Proceedings of FOCI 2007, pp. 375–382. IEEE (2007)Google Scholar
  13. 13.
    Salazar-Morales, O., Serrano-Devia, J.H., Soriano-Mendez, J.J.: Centroid of an interval type-2 fuzzy set: continuous vs. discrete. Revista Ingeniería 16(2), 1–9 (2011)Google Scholar
  14. 14.
    Salazar-Morales, O., Serrano-Devia, J.H., Soriano-Mendez, J.J.: A short note on the centroid of an interval type-2 fuzzy set. In: Proceedings of 2012 Workshop on Engineering Applications (WEA), pp. 1–6. IEEE (2012)Google Scholar
  15. 15.
    Wu, D., Mendel, J.M.: A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets. Inf. Sci. 179(1), 1169–1192 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yager, R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24(1), 143–161 (1981)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universidad Distrital Francisco José de CaldasBogotáColombia

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