DOI QR코드

DOI QR Code

THE AUTOCONTINUITY OF MONOTONE INTERVAL-VALUED SET FUNCTIONS DEFINED BY THE INTERVAL-VALUED CHOQUET INTEGRAL

  • Received : 2007.04.26
  • Accepted : 2008.03.03
  • Published : 2008.03.25

Abstract

In a previous work [18], the authors investigated autocontinuity, converse-autocontinuity, uniformly autocontinuity, uniformly converse-autocontinuity, and fuzzy multiplicativity of monotone set function defined by Choquet integral([3,4,13,14,15]) instead of fuzzy integral([16,17]). We consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [18]. These integrals, which can be regarded as interval-valued aggregation operators, have been used in [10,11,12,19,20]. In this paper, we investigate some characterizations of monotone interval-valued set functions defined by the interval-valued Choquet integral such as autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity.

Keywords

monotone interval-valued set functions;interval-valued functions;fuzzy measures;Choquet integrals

References

  1. J. Aubin, Set-valued analysis, 1990, Birkauser Boston.
  2. R. J. Aumann, Integrals of set-valued Junctions, J. Math. Anal. Appl. 12 (1965), 1-12. https://doi.org/10.1016/0022-247X(65)90049-1
  3. M.J. Bilanos, L.M. de Campos and A. Gonzalez, Convergence properties of the monotone expectation and its application to the extension of fuzzy measures, Fuzzy Sets and Systems 33 (1989), 201-212. https://doi.org/10.1016/0165-0114(89)90241-8
  4. L.M. de Campos and M.J. Bilanos, Characterization and comparison of Sugeno and Choquet integrals, Fuzzy Sets and Systems 52 (1992), 61-67. https://doi.org/10.1016/0165-0114(92)90037-5
  5. J. Fan and W. Xie, Distance measure and induced fuzzy entropy, Fuzzy Sets and Systems 52 (1992), 61-67. https://doi.org/10.1016/0165-0114(92)90037-5
  6. L. C. Jang, B.M. Kil, YK. Kim and J. S. Kwon, Some properties of Choquet integrals of set-valued Junctions, Fuzzy Sets and Systems 91 (1997), 95-98. https://doi.org/10.1016/S0165-0114(96)00124-8
  7. L. C. Jang and J. S. Kwon, On the representation of Choquet integrals of set-valued Junctions and null sets, Fuzzy Sets and Systems 112 (1), 233-239. https://doi.org/10.1016/S0165-0114(98)00184-5
  8. L.C. Jang, T. Kim and J.D. Jeon, On set-valued Choquet intgerals and convergence theorems, Advanced Studies and Contemporary Mathematics 6(1) (2003), 63-76.
  9. L.C. Jang, T. Kim and J.D. Jeon, On set-valued Choquet intgerals and convergence theorems (II), Bull. Korean Math. Soc. 40(1) (2003), 139-147. https://doi.org/10.4134/BKMS.2003.40.1.139
  10. L.C. Jang, Interval-valued Choquet integrals and their applications, J. of Applied Mathematics and computing 16(1-2) (2004), 429-445.
  11. L.C. Jang, The application of interval-valued Choquet integrals in multicriteria decision aid, J. of Applied Mathematics and computing 20(1-2) (2006), 549-556.
  12. L.C. Jang, A note on the monotone interval-valued set Junction defined by interval-valued Choquet integral, Commun. Korean Math. Soc. 22(2) (2007), ***- *** https://doi.org/10.4134/CKMS.2007.22.2.227
  13. T. Murofushi and M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems 29 (1989), 201-227. https://doi.org/10.1016/0165-0114(89)90194-2
  14. T. Murofushi and M. Sugeno, A theory of Fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. and Appl. 159 (1991), 532-549. https://doi.org/10.1016/0022-247X(91)90213-J
  15. T.Murofushi and M. Sugeno, Some quantities represented by Choquet integral, Fuzzy Sets and Systems 56 (1993), 229-235. https://doi.org/10.1016/0165-0114(93)90148-B
  16. Z. Wang, The autocontinuity of set Junction and the fuzzy integral, J. of Math. Anal. Appl. 99 (1984), 195-218. https://doi.org/10.1016/0022-247X(84)90243-9
  17. Z. Wang, On the null-additivity and the autocontinuity of fuzzy measure, Fuzzy Sets and Systems 45 (1992), 223-226. https://doi.org/10.1016/0165-0114(92)90122-K
  18. Z. Wang, G.J. Klir and W. Wang, Monotone set Junctions defined by Choquet integral, Fuzzy measures defined by fuzzy integral and their absolute continuity, Fuzzy Sets and Systems 81 (1996), 241-250. https://doi.org/10.1016/0165-0114(95)00181-6
  19. W. Zeng and H. Li, Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems 157 (2006), 1477-1484. https://doi.org/10.1016/j.fss.2005.11.020
  20. D. Zhang, C.Guo and D. Liu, Set-valued Choquet integrals revisited, Fuzzy Sets and Systems 147 (2004), 475-485. https://doi.org/10.1016/j.fss.2004.04.005