Aggregation Operators to Evaluate the Relevance of Classes in a Fuzzy Partition

  • Fabián CastiblancoEmail author
  • Camilo Franco
  • Javier Montero
  • J. Tinguaro Rodríguez
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)


In this paper we propose a comparison process that allows evaluating the relevance of a class in a fuzzy partition. This process starts by forming a commutative group structure based on a fuzzy classification system. On this structure, proximity relations are proposed to establish the comparison process. As our proposal is based on a fuzzy classification system, i.e., recursive De Morgan triples, we study the properties that the operators of such triples must satisfy. Therefore, we propose the study of a property that we have called weakly self-dual.


Fuzzy classification systems Relevance Fuzzy partition Weakly Self-dual 



This research has been partially supported by the Government of Spain (grant TIN2015-66471-P), the Government of Madrid (grant S2013/ICCE-2845), Complutense University (UCM Research Group 910149) and Gran Colombia University (grant JCG2018-CEAC-03).


  1. 1.
    Amo, A., Montero, J., Biging, G., Cutello, V.: Fuzzy classification systems. Eur. J. Oper. Res. 156, 495–507 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Del Amo, A., Gómez, D., Montero, J., Biging, G.: Relevance and redundancy in fuzzy classification systems. Mathware Soft Comput. 8, 203–216 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dombi, J.: A general class of fuzzy operators, the Demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)CrossRefGoogle Scholar
  5. 5.
    Bustince, H., Fernandez, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions. Nonlinear Anal. Theory Methods Appl. 72, 1488–1499 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gómez, D., Rodríguez, J.T., Montero, J., Bustince, H., Barrenechea, E.: N-Dimensional overlap functions. Fuzzy Sets Syst. 287, 57–75 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bustince, H., Pagola, M., Mesiar, R., Hüllermeier, E., Herrera, F.: Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans. Fuzzy Syst. 20, 405–415 (2012)CrossRefGoogle Scholar
  8. 8.
    Bedregal, B., Dimuro, G.P., Bustince, H., Barrenechea, E.: New results on overlap and grouping functions. Inf. Sci. (Ny) 249, 148–170 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Castiblanco, F., Montero, J., Rodríguez, J.T., Gómez, D.: Quality assessment of fuzzy classification: an application to solvency analysis. Fuzzy Econ. Rev. 22, 19–31 (2017)CrossRefGoogle Scholar
  10. 10.
    Castiblanco, F., Franco, C., Montero, J., Rodríguez, J.T.: Relevance of classes in a fuzzy partition. A study from a group of aggregation operators. In: Barreto, G.A., Coelho, R. (eds.) Fuzzy Information Processing, pp. 96–107. Springer, Cham (2018)Google Scholar
  11. 11.
    Calvo, T., Kolesarova, A., Komornikova, M., Mesiar, R.: Aggregation operators, properties, classes and construction methods. In: Calvo, T., et al. (eds.) Aggregation Operators. New Trends and Applications, pp. 3–104. Physica-Verlag, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Bustince, H., De Baets, B., Fernandez, J., Mesiar, R., Montero, J.: A generalization of the migrativity property of aggregation functions. Inf. Sci. (Ny) 191, 76–85 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gómez, D., Rojas, K., Montero, J., Rodríguez, J.T., Beliakov, G.: Consistency and stability in aggregation operators: an application to missing data problems. Int. J. Comput. Intell. Syst 7(3), 595–604 (2014)CrossRefGoogle Scholar
  14. 14.
    Bustince, H., Montero, J., Barrenechea, E., Pagola, M.: Semiautoduality in a restricted family of aggregation operators. Fuzzy Sets Syst. 158, 1360–1377 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Janiš, V., Král, P., Renčová, M.: Aggregation operators preserving quasiconvexity. Inf. Sci. (Ny) 228, 37–44 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cutello, V., Montero, J.: Recursive connective rules. Int. J. Intell. Syst. 14, 3–20 (1999)CrossRefGoogle Scholar
  17. 17.
    Castiblanco, F., Gómez, D., Montero, J., Rodríguez, J.T.: Aggregation tools for the evaluation of classifications. In: Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IFSA-SCIS), 2017 Joint 17th World Congress of International, pp. 1–5, Otsu, Japan. IEEE (2017)Google Scholar
  18. 18.
    Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Upper Saddle River (1988)zbMATHGoogle Scholar
  19. 19.
    Silvert, W.: Symmetric summation: a class of operations on fuzzy sets. IEEE Trans. Man Cybern. 9, 657–659 (1979)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support Theory and Decision Library Series D : System Theory, Knowledge Engineering and Problem Solving (1994)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Economic, Administrative and Accounting SciencesGran Colombia UniversityBogotáColombia
  2. 2.Department of Industrial EngineeringAndes UniversityBogotáColombia
  3. 3.Department of StatisticsComplutense UniversityMadridSpain

Personalised recommendations