Advertisement

Order Relations and a Monotone Convergence Theorem in the Class of Fuzzy Sets on ℝn

  • Masami Kurano
  • Masami Yasuda
  • Jun-ichi Nakagami
  • Yuji Yoshida
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 73)

Abstract

Concerning with the topics of fuzzy decision processes, a brief survey on ordering of fuzzy numbers on ℝ is presented and an extension to that of fuzzy sets(numbers) on ℝ n are considered. This extension is a pseudo order ≼ K defined by a non-empty closed convex cone K and characterized by the projection into its dual cone K +. Especially a structure of the lattice is presented on the class of pyramid-type fuzzy sets. Moreover, we study the convergence of a sequence of fuzzy sets on ℝ n which is monotone w.r.t. the order ≼ K . Our study is carried out by restricting the class of fuzzy sets into the subclass in which the order ≼ K becomes a partial order so that a monotone convergence theorem is proved. This restricted subclass of fuzzy sets is created and characterized in the concept of a determining class. These results are applied to obtain the limit theorem for a sequence of fuzzy sets defined by the dynamic fuzzy system with a monotone fuzzy relation. Several figures are illustrated to comprehend our results.

Keywords

Partial Order Fuzzy Number Binary Relation Closed Convex Cone Dual Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Congxin, W., Cong, C. (1997) The supremum and infimum of the set of fuzzy numbers and its application. J. Math. Anal. Appl. 210, 499–511CrossRefGoogle Scholar
  2. 2.
    Diamond, P., Kloeden, P. (1994) Metric Spaces of Fuzzy Sets, Theory and Applications. World ScientificGoogle Scholar
  3. 3.
    Dubois, D., Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications. Academic PressGoogle Scholar
  4. 4.
    Furukawa, N. (1993) An order relation and fundamental operations on fuzzy numbers and their applications to fuzzy linear programming Bulletin Infor. Cybernetics 25, 161–179Google Scholar
  5. 5.
    Furukawa, N. (1993) A parametric total order on fuzzy numbers and a fuzzy shortest route problem. Optimization 30, 367–377CrossRefGoogle Scholar
  6. 6.
    Furukawa, N. (1997) Parametric orders on fuzzy numbers and their roles in fuzzy optimization problems. Optimization 40, 171–192CrossRefGoogle Scholar
  7. 7.
    Kaufmann, A., Gupta, M. M. (1988) Fuzzy Mathematical Models in Engineering and Management Science. North-HollandGoogle Scholar
  8. 8.
    Kurano, M., Yasuda, M., Nakagami, J., Yoshida, Y. (1992) A limit theorem in some dynamic fuzzy systems. Fuzzy Sets and Systems 51, 83–88CrossRefGoogle Scholar
  9. 9.
    Kurano, M., Yasuda, M., Nakagami, J., Yoshida, Y. (1996) Markov-type fuzzy decision processes with a discounted reward on a closed interval. European Journal of Operations Research 92, 649–662CrossRefGoogle Scholar
  10. 10.
    Kurano, M., Yasuda, M., Nakagami, J., Yoshida, Y. (1999) The time average reward for some dynamic fuzzy systems. Computers and Mathematics with Applications 37, 77–86CrossRefGoogle Scholar
  11. 11.
    Kurano, M., Yasuda, M., Nakagami, J., Yoshida, Y. (2000) Ordering of fuzzy sets — A brief survey and new results. Journal of Operations Research Society of Japan 43, 138–148CrossRefGoogle Scholar
  12. 12.
    Kuroiwa, D., Tanaka, T., Ha, T. X. D. (1997) On cone convexity of set-valued maps. Nonlinear Analysis, Theory and Applications 30, 1487–1496Google Scholar
  13. 13.
    Li, H. X., Yen, V. C. (1995) Fuzzy Sets and Fuzzy Decision-making. CRC PressGoogle Scholar
  14. 14.
    Nanda, S. (1989) On sequences of fuzzy numbers. Fuzzy Sets and Systems 33, 123–126CrossRefGoogle Scholar
  15. 15.
    Novak, V. (1989) Fuzzy Sets and Their Applications. Adam Hilder, Bristol-BostonGoogle Scholar
  16. 16.
    Ramík, J., Rimanek, J. (1985) Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets and Systems 16, 123–138CrossRefGoogle Scholar
  17. 17.
    Stoer, J., Witzgall, C. (1970) Convexity and Optimization in Finite Dimensions I. Springer-Verlag, Berlin and New YorkCrossRefGoogle Scholar
  18. 18.
    Stowinski, R.:editor (1998) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Kluwer Academic PublishersGoogle Scholar
  19. 19.
    Syau, Y. R. (1997) Sequences in a fuzzy metric space. Computers and Mathematics with Applications 33, 73–76CrossRefGoogle Scholar
  20. 20.
    Yoshida, Y., Yasuda, M., Nakagami, J., Kurano, M. (1993) A potential of fuzzy relations with a linear structure: The contractive case. Fuzzy sets and Systems 60, 283–294CrossRefGoogle Scholar
  21. 21.
    Yoshida, Y., Yasuda, M., Nakagami, J., Kurano, M. (1994) A potential of fuzzy relations with a linear structure: The unbounded case. Fuzzy sets and Systems 66, 83–96CrossRefGoogle Scholar
  22. 22.
    Yoshida, Y. (1998) A time-average fuzzy reward criterion in fuzzy decision processes. Information Science 110, 103–112CrossRefGoogle Scholar
  23. 23.
    Yoshida, Y., Yasuda, M., Nakagami, J., Kurano, M. (1998) A limit theorem in dynamic fuzzy systems with a monotone property. Fuzzy sets and Systems 94, 109–119CrossRefGoogle Scholar
  24. 24.
    Zadeh, L. A. (1965) Fuzzy sets. Information and Contro l8, 338–353CrossRefGoogle Scholar
  25. 25.
    Zhang, K., Hirota, K. (1997) On fuzzy number lattice (R, 4). Fuzzy sets and Systems 92, 113–122CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Masami Kurano
    • 1
  • Masami Yasuda
    • 1
  • Jun-ichi Nakagami
    • 1
  • Yuji Yoshida
    • 2
  1. 1.Chiba UniversityChibaJapan
  2. 2.Kitakyushu UniversityKitakyushuJapan

Personalised recommendations