Fuzzy Weighted Attribute Combinations Based Similarity Measures

  • Giulianella Coletti
  • Davide Petturiti
  • Barbara Vantaggi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10369)


Some similarity measures for fuzzy subsets are introduced: they are based on fuzzy set-theoretic operations and on a weight capacity expressing the degree of contribution of each group of attributes. For such measures, the properties of dominance and T-transitivity are investigated.


Fuzzy subset Capacity Similarity measure T-transitivity 



This work was partially supported by INdAM-GNAMPA through the Project 2015 U2015/000418 and the Project 2016 U2016/000391 and by the Italian Ministry of Education, University and Research, under grant 2010FP79LR_003.


  1. 1.
    Angilella, S., Greco, S., Lamantia, F., Matarazzo, B.: The application of fuzzy integrals in multicriteria decision making. Eur. J. Oper. Res. 158(3), 734–744 (2004)CrossRefMATHGoogle Scholar
  2. 2.
    Baioletti, M., Coletti, G., Petturiti, D.: Weighted attribute combinations based similarity measures. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 299, pp. 211–220. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31718-7_22 CrossRefGoogle Scholar
  3. 3.
    Bouchon-Meunier, B., Coletti, G., Lesot, M.-J., Rifqi, M.: Towards a conscious choice of a fuzzy similarity measure: a qualitative point of view. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS (LNAI), vol. 6178, pp. 1–10. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14049-5_1 CrossRefGoogle Scholar
  4. 4.
    Chateauneuf, A., Jaray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of möbius inversion. Math. Soc. Sci. 17(3), 263–283 (1989)CrossRefMATHGoogle Scholar
  5. 5.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    De Baets, B., Janssens, S., Meyer, H.D.: On the transitivity of a parametric family of cardinality-based similarity measures. Int. J. Approximate Reasoning 50(1), 104–116 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    De Baets, B., Meyer, H.D.: Transitivity-preserving fuzzification schemes for cardinality-based similarity measures. Eur. J. Oper. Res. 160(3), 726–740 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Denneberg, D.: Non-Additive Measure and Integral, Series B: Mathematical and Statistical Methods, vol. 27. Kluwer Academic Publishers, Dordrecht (1994)CrossRefMATHGoogle Scholar
  9. 9.
    Grabisch, M.: The application of fuzzy integrals in multicriteria decision making. Eur. J. Oper. Res. 69(3), 279–298 (1995)MathSciNetMATHGoogle Scholar
  10. 10.
    Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst. 89(3), 445–456 (1996)MathSciNetMATHGoogle Scholar
  11. 11.
    Grabisch, M., Kojadinovic, I., Meyer, P.: A review of methods for capacity identification in choquet integral based multi-attribute utility theory: applications of the kappalab R package. Eur. J. Oper. Res. 186(2), 766–785 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jaccard, P.: Nouvelles recherches sur la distribution florale. Bull. Soc. Vaud. Sci. Nat. 44, 223–270 (1908)Google Scholar
  13. 13.
    Klement, E., Mesiar, R., Pap, E.: Triangualr Norms, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)CrossRefMATHGoogle Scholar
  14. 14.
    Marichal, J.: An axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria. IEEE Trans. Fuzzy Syst. 8(6), 800–807 (2000)CrossRefGoogle Scholar
  15. 15.
    Scozzafava, R., Vantaggi, B.: Fuzzy inclusion and similarity through coherent conditional probability. Fuzzy Sets Syst. 160(3), 292–305 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wilbik, A., Keller, J.M., Alexander, G.: Similarity evaluation of sets of linguistic summaries. Int. J. Intell. Syst. 27, 226–238 (2012)CrossRefGoogle Scholar
  17. 17.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefMATHGoogle Scholar
  18. 18.
    Yu, X., Zhang, Q.: An extension of cooperative fuzzy games. Fuzzy Sets Syst. 161, 1614–1634 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Giulianella Coletti
    • 1
  • Davide Petturiti
    • 2
  • Barbara Vantaggi
    • 3
  1. 1.Dip. Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Dip. EconomiaUniversità di PerugiaPerugiaItaly
  3. 3.Dip. S.B.A.I.Università di Roma “La Sapienza”RomeItaly

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