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Fuzzy Measures and Comonotonicity on Multisets

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6820))

Abstract

Fuzzy measures on multisets are studied. We show that a class of multisets can be represented as a subset of positive integers. Comonotonicity for multisets are defined. We show that a fuzzy measure on multisets with some comonotonicity condition can be represented by generalized fuzzy integral.

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References

  1. Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 1329–1379. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  2. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953-1954)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dellacherie, C.: Quelques commentaires sur les prolongements de capacités, Séminaire de Probabilités 1969/1970. Lecture Notes in Mathematics, Strasbourg, vol. 191, pp. 77–81 (1971)

    Google Scholar 

  4. Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92(2), 167–189 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo, C., Zhang, D.: On set-valued fuzzy measures. Information Sciences 160, 13–25 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hickman, J.L.: A note on the concept of multiset. Bulletin of the Australian Mathematical Society 22, 211–217 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  8. Klement, E.P., Mesiar, R., Pap, E.: Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval. Int. J. of Unc., Fuzziness and Knowledge Based Systems 8(6), 701–717 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ling, C.H.: Representation of associative functions. Publ. Math. Debrecen 12, 189–212 (1965)

    MathSciNet  MATH  Google Scholar 

  10. Marichal, J.-L., Roubens, M.: Entropy of discrete fuzzy measures. Int. J. of Unc., Fuzz. and Knowledge Based Systems 8(6), 625–640 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Miyamoto, S.: Generalizations of multisets and rough approximations. Int. J. of Intel. Syst. 19, 639–652 (2004)

    Article  MATH  Google Scholar 

  12. Murofushi, T., Sugeno, M.: An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets and Systems 29, 201–227 (1989)

    Google Scholar 

  13. Narukawa, Y., Torra, V.: Multidimensional generalized fuzzy integral. Fuzzy Sets and Systems 160, 802–815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ralescu, D., Adams, G.: The fuzzy integral. J. Math. Anal. Appl. 75, 562–570 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sugeno, M.: Theory of fuzzy integrals and its applications, Ph. D. Dissertation, Tokyo Institute of Technology (1974)

    Google Scholar 

  16. Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Torra, V., Narukawa, Y.: Modeling decisions: information fusion and aggregation operators. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  18. Torra, V., Stokes, K., Narukawa, Y.: Fuzzy Measures on Multisets (submitted)

    Google Scholar 

  19. Yager, R.R.: On the theory of bags. Int. J. of General Systems 13, 23–37 (1986)

    Article  MathSciNet  Google Scholar 

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Narukawa, Y., Stokes, K., Torra, V. (2011). Fuzzy Measures and Comonotonicity on Multisets. In: Torra, V., Narakawa, Y., Yin, J., Long, J. (eds) Modeling Decision for Artificial Intelligence. MDAI 2011. Lecture Notes in Computer Science(), vol 6820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22589-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-22589-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22588-8

  • Online ISBN: 978-3-642-22589-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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