Dynamical Aspects in Fuzzy Decision Making pp 187-212 | Cite as

# Order Relations and a Monotone Convergence Theorem in the Class of Fuzzy Sets on ℝ^{n}

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- 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.Diamond, P., Kloeden, P. (1994) Metric Spaces of Fuzzy Sets, Theory and Applications. World ScientificGoogle Scholar
- 3.Dubois, D., Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications. Academic PressGoogle Scholar
- 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.Furukawa, N. (1993) A parametric total order on fuzzy numbers and a fuzzy shortest route problem. Optimization
**30**, 367–377CrossRefGoogle Scholar - 6.Furukawa, N. (1997) Parametric orders on fuzzy numbers and their roles in fuzzy optimization problems. Optimization
**40**, 171–192CrossRefGoogle Scholar - 7.Kaufmann, A., Gupta, M. M. (1988) Fuzzy Mathematical Models in Engineering and Management Science. North-HollandGoogle Scholar
- 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.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.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.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.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.Li, H. X., Yen, V. C. (1995) Fuzzy Sets and Fuzzy Decision-making. CRC PressGoogle Scholar
- 14.Nanda, S. (1989) On sequences of fuzzy numbers. Fuzzy Sets and Systems
**33**, 123–126CrossRefGoogle Scholar - 15.Novak, V. (1989) Fuzzy Sets and Their Applications. Adam Hilder, Bristol-BostonGoogle Scholar
- 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.Stoer, J., Witzgall, C. (1970) Convexity and Optimization in Finite Dimensions I. Springer-Verlag, Berlin and New YorkCrossRefGoogle Scholar
- 18.Stowinski, R.:editor (1998) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Kluwer Academic PublishersGoogle Scholar
- 19.Syau, Y. R. (1997) Sequences in a fuzzy metric space. Computers and Mathematics with Applications
**33**, 73–76CrossRefGoogle Scholar - 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.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.Yoshida, Y. (1998) A time-average fuzzy reward criterion in fuzzy decision processes. Information Science
**110**, 103–112CrossRefGoogle Scholar - 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.Zadeh, L. A. (1965) Fuzzy sets. Information and Contro
**l8**, 338–353CrossRefGoogle Scholar - 25.Zhang, K., Hirota, K. (1997) On fuzzy number lattice
*(R*, 4). Fuzzy sets and Systems**92**, 113–122CrossRefGoogle Scholar