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Generalized Discrete Fuzzy Number and Application in Risk Evaluation

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Abstract

In this paper, the definition of generalized discrete fuzzy numbers is introduced, and a representation theorem of such fuzzy numbers is obtained. Based on the representation theorem, it is shown that the usual addition and multiplication which are defined by Zadeh’s extension principle do not preserve the closeness of the operation, and a new addition operation and a new multiplication operation are defined, which not only preserve the closeness of the operation, but also are easy to calculate. Then some weak orders on the generalized discrete fuzzy number space are defined, and their properties are investigated. At last, a practical example is given to show the application of the algorithmic version to rank uncertain or imprecise discrete quantity.

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References

  1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang, S.S.L., Zadeh, L.A.: On fuzzy mappings and control. IEEE Trans. Syst. Man Cybernet 2, 30–34 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dubois, D., Prade, H.: Towards fuzzy differential calculus, part 1: integration of fuzzy mapping. Fuzzy Sets Syst. 8, 1–17 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, S.J., Lee, C.H.: New methods for students’ evaluation using fuzzy sets. Fuzzy Sets Syst. 104(2), 209–218 (1999)

    Article  MathSciNet  Google Scholar 

  6. Wang, G., Shi, P., Messenger, P.: Representation of uncertain multichannel digital signal spaces and study of pattern recognition based on metrics and difference values on fuzzy cell number spaces. J. IEEE Trans. Fuzzy Systemsvol. 17(2), 421–439 (2009)

    Article  Google Scholar 

  7. Wang, G., Shi, P., Wang, B., Zhang, J.: Fuzzy-ellipsoid numbers and representations of uncertain multichannel digital information. J. IEEE Trans. Fuzzy Syst. 22(5), 1113–1126 (2014)

    Article  MathSciNet  Google Scholar 

  8. Tian, Z., Hu, L., Greenhalgh, D.: Perturbation analysis of fuzzy linear systems. Inf. Sci. 180, 4706–4713 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Esi, A., Açıkgöz, M.: Some new classes of sequences of fuzzy numbers. Int. J. Fuzzy Syst. 13(3), 218–224 (2011)

    MathSciNet  Google Scholar 

  10. Murshid, A.M., Loan, S.A., Abbasi, S.A., Alamoud, A.R.M.: A novel VLSI architecture for a fuzzy inference processor using triangular-shaped membership function. Int. J. Fuzzy Syst. 14(3), 345–360 (2012)

    Google Scholar 

  11. Dat, L.Q., Yu, V.F., Chou, S.Y.: An improved ranking method for fuzzy numbers based on the centroid-index. Int. J. Fuzzy Syst. 14(3), 413–419 (2012)

    MathSciNet  Google Scholar 

  12. Chu, T.C., Charnsethikul, P.: Ordering alternatives under fuzzy multiple criteria decision making via a fuzzy number dominance based ranking approach. Int. J. Fuzzy Syst. 15(3), 263–273 (2013)

    MathSciNet  Google Scholar 

  13. Moreno-Garcia, J., Linares, L.J., Rodriguez-Benitez, L., et al.: Fuzzy numbers from raw discrete data using linear regression. Inf. Sci. 233, 1–14 (2013)

    Article  MATH  Google Scholar 

  14. Hong, D.H.: The law of large numbers and renewal process for T-related weighted fuzzy numbers on \(R^p\). Inf. Sci. 228, 45–60 (2013)

    Article  MATH  Google Scholar 

  15. Nan, J.X., Zhang, M.J., Li, D.F.: Intuitionistic fuzzy programming models for matrix games with payoffs of trapezoidal intuitionistic fuzzy numbers. Int. J. Fuzzy Syst. 16(4), 444–456 (2014)

    MathSciNet  Google Scholar 

  16. Arotaritei, D., Ionescu, F.: Fuzzy Voronoi diagram for disjoint fuzzy numbers of dimension two. J. Intell. Fuzzy Syst. 26, 1253–1262 (2014)

    MATH  MathSciNet  Google Scholar 

  17. Coroianu, L., Gagolewski, M., Grzegorzewski, P.: Nearest piecewise linear approximation of fuzzy numbers. Fuzzy Sets Syst. 233, 26–51 (2014)

    Article  MathSciNet  Google Scholar 

  18. Voxman, W.: Canonical representations of discrete fuzzy numbers. Fuzzy Sets Syst. 118, 457–466 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, G., Wu, C., Zhao, C.: Representation and operations of discrete fuzzy numbers. Southeast Asian Bull. Math 28, 1003–1010 (2005)

    MathSciNet  Google Scholar 

  20. Casasnovas, J., Riera, J.V.: On the addition of discrete fuzzy numbers.” WSEAS Trans. Math, pp. 549–554 (2006)

  21. Casasnovas, J., Riera, J.V.: Discrete fuzzy numbers defined on a subset of natural numbers. Theor. Adv. Appl. Fuzzy Logic Soft Comput. Adv. Soft Comput. 42, 573–582 (2007)

    Google Scholar 

  22. Casasnovas, J., Riera, J.V.: Maximum and minimum of discrete fuzzy numbers. Front. Artif. Intell. Appl. 163, 273–280 (2007)

    Google Scholar 

  23. Casasnovas, J., Riera, J.V.: Lattice properties of discrete fuzzy numbers under extended min and max. Proceedings IFSA-EUSFLAT, pp. 647–652 (2009)

  24. Casasnovas, J., Riera, J.V.: Extension of discrete t-norms and tconorms to discrete fuzzy numbers. Fuzzy Sets Syst. 167(1), 65–81 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Riera, J.V., Torrens, J.: Aggregation of subjective evaluations based on discrete fuzzy number. Fuzzy Sets Syst. 191(16), 21–40 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Riera, J.V., Torrens, J.: Residual implications on the set of discrete fuzzy numbers. Inf. Sci. 247, 131–143 (2013)

    Article  MathSciNet  Google Scholar 

  27. Xie, Y., Wang, G., Wang, B.: The operation of 2-dimensional discrete fuzzy numbers. ICIC Express Lett. 7(11), 2921–2925 (2013)

    Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. Massanet, S., Riera, J.V., Torrens, J., Viedma, E.H.: A new linguistic computational model based on discrete fuzzy numbers for computing with words. Inf. Sci. 258, 277–290 (2014)

    Article  Google Scholar 

  30. Wang, H., Chen, S.: Evaluating studentsanswerscripts using fuzzy numbers associated with degrees of confidence. IEEE Trans. Fuzzy Syst. 16(2), 403–415 (2008)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the Nature Science Foundation of China (Nos. 61273077 and 61433001), the Nature Science Foundation of Zhejiang Province, China (No. LY12A01001).

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Correspondence to Guixiang Wang.

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Wang, G., Wang, J. Generalized Discrete Fuzzy Number and Application in Risk Evaluation. Int. J. Fuzzy Syst. 17, 531–543 (2015). https://doi.org/10.1007/s40815-015-0038-z

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  • DOI: https://doi.org/10.1007/s40815-015-0038-z

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