On the Aggregation of Zadeh’s Z-Numbers Based on Discrete Fuzzy Numbers

  • Sebastia Massanet
  • Juan Vicente Riera
  • Joan Torrens
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 581)

Abstract

The accurate modelling of natural language is one of the main goals in the theory of computing with words. Based on this idea, Zadeh in 2011, introduced the concept of Z-number which has a great potential not only from the theoretical point of view but also for many possible applications such as in economics, decision analysis, risk assessment, etc. Recently, the authors proposed a new vision of Zadeh’s Z-numbers based on discrete fuzzy numbers that simplifies the computations and maintains the flexibility of the original model from the linguistic point of view. Following with this novel interpretation, in this paper, algebraic structures in the set of Zadeh’s Z-numbers are studied. In this framework, we propose a method to construct aggregation functions from couples of discrete aggregation functions. In particular, t-norms and t-conorms are built. Finally, an application to reach a final decision on a decision making problem is given.

Keywords

Zadeh’s Z-numbers Discrete fuzzy numbers Aggregation functions 

References

  1. 1.
    Akif, M., Karaal, F., Mesiar, R.: Medians and nullnorms on bounded lattices. Fuzzy Sets Syst. 289, 74–81 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aliev, R., Alizadeh, A., Huseynov, O.: The arithmetic of discrete Z-numbers. Inf. Sci. 290, 134–155 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.A.: The Arithmetic of Z-Numbers: Theory and Applications. World Scientific Publishing, River Edge (2015)CrossRefMATHGoogle Scholar
  4. 4.
    Aliev, R.A., Mraiziq, D., Huseynov, O.H.: Expected utility based decision making under Z-information and its application. Comput. Intell. Neurosci. Article ID 364512, 11 pages (2015)Google Scholar
  5. 5.
    Birkhoff, G.: Lattices and their applications. Bull. Am. Math. Soc. 44, 793–800 (1940)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Casasnovas, J., Riera, J.V.: Lattice properties of discrete fuzzy numbers under extended min and max. In: Proceedings of IFSA-EUSFLAT, Lisbon, pp. 647–652 (2009)Google Scholar
  7. 7.
    Casasnovas, J., Riera, J.V.: Extension of discrete t-norms and t-conorms to discrete fuzzy numbers. Fuzzy Sets Syst. 167(1), 65–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cornelis, C., Deschrijver, G., Kerre, E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int. J. Approx. Reason. 35, 55–95 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    De Baets, B.: Model implicators and their characterization. In: Steele, N. (ed.) Proceedings of ISFL 95, First ICSC International Symposium on Fuzzy Logic ICSC, pp. A42–A49. Academic Press (1995)Google Scholar
  10. 10.
    De Baets, B.: Coimplicators, the forgotten connectives. Tatra MT. 12, 229–240 (1997)MathSciNetMATHGoogle Scholar
  11. 11.
    De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets Syst. 104(1), 61–75 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Galatos, N., Jipsen, P., Kowalski, T.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2007)MATHGoogle Scholar
  13. 13.
    Grätzer, G.: General Lattice Theory. Academic Press, Cambridge (1978)CrossRefMATHGoogle Scholar
  14. 14.
    Herrera, F., Herrera-Viedma, E., Martínez, L.: A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114, 43–58 (2000)CrossRefMATHGoogle Scholar
  15. 15.
    Herrera, F., Martínez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 8(6), 746–752 (2000)CrossRefGoogle Scholar
  16. 16.
    Herrera-Viedma, E., Riera, J.V., Massanet, S., Torrens, J.: Some remarks on the fuzzy linguistic model based on discrete fuzzy numbers. In: Angelov, P., et al. (eds.) Intelligent Systems 2014. AISC, pp. 319–330. Springer International Publishing, Cham (2015)Google Scholar
  17. 17.
    Huynh, V., Nakamori, Y.: A satisfactory-oriented approach to multiexpert decision-making with linguistic assessments. IEEE Trans. Syst. Man Cybernet. 35, 184–196 (2005)CrossRefGoogle Scholar
  18. 18.
    Jenei, S., De Baets, B.: On the direct decomposability of t-norms on product lattices. Fuzzy Sets Syst. 139, 699–707 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jiang, Y., Fan, Z., Ma, J.: A method for group decision making with multigranularity linguistic assessment information. Inf. Sci. 178, 1098–1109 (2008)CrossRefMATHGoogle Scholar
  20. 20.
    Lizasoain, I., Moreno, C.: OWA operators defined on complete lattices. Fuzzy Sets Syst. 224, 36–52 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mas, M., Monserrat, M., Torrens, J.: Kernel aggregation functions on finite scales. Constructions from their marginals. Fuzzy Sets Syst. 241, 27–40 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Massanet, S., Riera, J.V., Torrens, J.: A new vision of Zadeh’s Z-numbers. In: Carvalho, J., Lesot, M.J., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol. 611. Springer, Cham (2016)Google Scholar
  23. 23.
    Massanet, S., Riera, J.V., Torrens, J., Herrera-Viedma, E.: A new linguistic computational model based on discrete fuzzy numbers for computing with words. Inf. Sci. 258, 277–290 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Morente-Molinera, J., Pérez, I., Ureña, M., Herrera-Viedma, E.: On multi-granular fuzzy linguistic modeling in group decision making problems: a systematic review and future trends. Knowl.-Based Syst. 74, 49–60 (2015)CrossRefMATHGoogle Scholar
  25. 25.
    Ono, H.: Substructural logics and residuated lattices - an introduction. In: Hendricks, V.F., Malinowski, J. (eds.) Trends in Logic: 50 Years of Studia Logica, pp. 193–228. Kluver Academic Publishers, Netherlands (2003)CrossRefGoogle Scholar
  26. 26.
    Pal, S.K., Banerjee, R., Dutta, S., Sarma, S.: An insight into the Z-number approach to CWW. Fundamentae Informaticae 124, 197–229 (2013)MathSciNetGoogle Scholar
  27. 27.
    Patel, P., Khorasani, E.S., Rahimi, S.: Modeling and implementation of Z-numbers. Soft. Comput. 20(4), 1341–1364 (2016)CrossRefGoogle Scholar
  28. 28.
    Riera, J.V., Massanet, S., Herrera-Viedma, E., Torrens, J.: Some interesting properties of the fuzzy linguistic model based on discrete fuzzy numbers to manage hesitant fuzzy linguistic information. Appl. Soft Comput. 36, 383–391 (2015)CrossRefGoogle Scholar
  29. 29.
    Riera, J.V., Torrens, J.: Aggregation of subjective evaluations based on discrete fuzzy numbers. Fuzzy Sets Syst. 191, 21–40 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Riera, J.V., Torrens, J.: Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations. Fuzzy Sets Syst. 241, 76–93 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Riera, J.V., Torrens, J.: Using discrete fuzzy numbers in the aggregation of incomplete qualitative information. Fuzzy Sets Syst. 264, 121–137 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Roselló, L., Sánchez, M., Agell, N., Prats, F., Mazaira, F.: Using consensus, distances between generalized multi-attribute linguistic assessments for group decision-making. Inf. Fusion 32, 65–75 (2011)Google Scholar
  33. 33.
    Voxman, W.: Canonical representations of discrete fuzzy numbers. Fuzzy Sets Syst. 118(3), 457–466 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Yager, R.: On Z-valuations using Zadeh’s Z-numbers. Int. J. Intell. Syst. 27, 259–278 (2012)CrossRefGoogle Scholar
  35. 35.
    Zadeh, L.: Fuzzy logic = computing with words. IEEE Trans. Fuzzy Syst. 4, 103–111 (1996)CrossRefGoogle Scholar
  36. 36.
    Zadeh, L.: A note on Z-numbers. Inf. Sci. 9(1), 43–80 (2011)CrossRefGoogle Scholar
  37. 37.
    Zhang, D.: Triangular norms on partially ordered sets. Fuzzy Sets Syst. 181, 2923–2932 (2005)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Sebastia Massanet
    • 1
  • Juan Vicente Riera
    • 1
  • Joan Torrens
    • 1
  1. 1.SCOPIA Research Group, Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain

Personalised recommendations