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Fuzzy Measures and Integrals: Recent Developments

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Fifty Years of Fuzzy Logic and its Applications

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 326))

Abstract

This paper surveys the basic notions and most important results around fuzzy measures and integrals, as proposed independently by Choquet and Sugeno, as well as recent developments. The latter includes bases and transforms on set functions, fuzzy measures on set systems, the notion of horizontal additivity, basic Choquet calculus on the nonnegative real line introduced by Sugeno, the extension of the Choquet integral for nonmeasurable functions, and the notion of universal integral.

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Notes

  1. 1.

    A chain from \(\varnothing \) to \(X\) is a sequence \(\varnothing =A_0,A_1,\ldots , A_q=X\) of sets in \({\mathcal F}\) such that \(A_0\subset A_1\subset \cdots \subset A_q\). Its length is \(q\), and the chain is maximal if no other chain from \(\varnothing \) to \(X\) contains it.

References

  1. Pap, E.: Null-Additive Set Functions. Kluwer Academic (1995)

    Google Scholar 

  2. Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic (1994)

    Google Scholar 

  3. Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer (2009)

    Google Scholar 

  4. Pap, E. (ed.): Handbook of Measure Theory. Elsevier (2002)

    Google Scholar 

  5. Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals. Theory and Applications (edited volume). Studies in Fuzziness. Physica Verlag (2000)

    Google Scholar 

  6. Grabisch, M., Labreuche, Ch.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann. Oper. Res. 175, 247–286 (2010). doi:10.1007/s10479-009-0655-8

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. Choquet, G.: Theory of capacities. Ann. de l’Institut Fourier 5, 131–295 (1953)

    Article  MathSciNet  Google Scholar 

  9. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dubois, D., Prade, H.: Possibility Theory. Plenum Press (1988)

    Google Scholar 

  11. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)

    Google Scholar 

  12. Stanley, R.: Two poset polytopes. Discrete Comput. Geom. 1, 9–23 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Radojevic, D.: The logical representation of the discrete Choquet integral. Belg. J. Oper. Res. Stat. Comput. Sci. 38, 67–89 (1998)

    MATH  MathSciNet  Google Scholar 

  14. Faigle, U., Grabisch, M.: Linear transforms, values and least square approximation for cooperation systems. working paper (2014)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pp. 307–317. Princeton University Press (1953)

    Google Scholar 

  17. Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (III): interaction index. In: 9th Fuzzy System Symposium, pp. 693–696. Sapporo, Japan (1993) (In Japanese)

    Google Scholar 

  18. Grabisch, M.: The application of fuzzy integrals in multicriteria decision making. Eur. J. Oper. Res. 89, 445–456 (1996)

    Article  MATH  Google Scholar 

  19. de Wolf, R.: A brief introduction to Fourier analysis on the Boolean cube. Theory Comput. Libr. Grad. Surv. 1, 1–20 (2008)

    Google Scholar 

  20. O’Donnell, R.: Analysis of Boolean functions, draft 2.0, ch. 1–3. http://www.cs.cmu.edu/odonnell11/boolean-analysis (2007)

  21. Hurst, S., Miller, D., Muzio, J.: Spectral Techniques in Digital Logic. Academic Press, London (1985)

    Google Scholar 

  22. Grabisch, M., Miranda, P.: Exact bounds of the Möbius inverse of monotone set functions. working paper (2013)

    Google Scholar 

  23. Miranda, P., Grabisch, M.: Optimization issues for fuzzy measures. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 7(6), 545–560 (1999)

    Google Scholar 

  24. Miranda, P., Grabisch, M., Gil, P.: \(p\)-symmetric fuzzy measures. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 10(Suppl.), 105–123 (2002)

    Google Scholar 

  25. Miranda, P., Grabisch, M.: \(p\)-symmetric bi-capacities. Kybernetika 40(4), 421–440 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Faigle, U., Kern, W.: The Shapley value for cooperative games under precedence constraints. Int. J. Game Theory 21, 249–266 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Algaba, E., Bilbao, J.M., Borm, P., López, J.J.: The position value for union stable systems. Math. Meth. Oper. Res. 52, 221–236 (2000)

    Article  MATH  Google Scholar 

  28. Faigle, U., Grabisch, M.: A discrete Choquet integral for ordered systems. Fuzzy Sets Syst. 168, 3–17 (2011). doi:10.1016/j.fss.2010.10.003

    Article  MATH  MathSciNet  Google Scholar 

  29. Faigle, U., Grabisch, M., Heyne, M.: Monge extensions of cooperation and communication structures. Eur. J. Oper. Res. 206, 104–110 (2010). doi:10.1016/j.ejor.2010.01.043

    Article  MATH  MathSciNet  Google Scholar 

  30. Honda, A., Grabisch, M.: An axiomatization of entropy of capacities on set systems. Eur. J. Oper. Res. 190, 526–538 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lange, F., Grabisch, M.: Values on regular games under Kirchhoff’s laws. Math. Soc. Sci. 58, 322–340 (2009). doi:10.1016/j.mathsocsci.2009.07.003

    Article  MATH  MathSciNet  Google Scholar 

  32. Barthélemy, J.-P.: Monotone functions on finite lattices: an ordinal approach to capacities, belief and necessity functions. In: Fodor, J., De Baets, B., Perny, P. (eds.) Preferences and Decisions under Incomplete Knowledge, pp. 195–208. Physica Verlag (2000)

    Google Scholar 

  33. Zhou, C.: Belief functions on distributive lattices. Artif. Intell. 201, 1–31 (2013)

    Article  Google Scholar 

  34. Šipoš, J.: Integral with respect to a pre-measure. Math. Slovaca 29, 141–155 (1979)

    MATH  MathSciNet  Google Scholar 

  35. Grabisch, M.: The Möbius function on symmetric ordered structures and its application to capacities on finite sets. Discrete Math. 287(1–3), 17–34 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Grabisch, M.: The symmetric Sugeno integral. Fuzzy Sets Syst. 139, 473–490 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Singer, I.: Extensions of functions of 0–1 variables and applications to combinatorial optimization. Numer. Funct. Anal. Optim. 7(1), 23–62 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  38. Marichal, J.-L.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Trans. Fuzzy Syst. 8(6), 800–807 (2000)

    Article  MathSciNet  Google Scholar 

  39. Marichal, J.-L., Mathonet, P., Tousset, E.: Mesures floues définies sur une échelle ordinale. working paper (1996)

    Google Scholar 

  40. Mesiar, R.: \(k\)-order pan-additive discrete fuzzy measures. In: 7th IFSA World Congress, pp. 488–490. Prague, Czech Republic (1997)

    Google Scholar 

  41. Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 1329–1379. Elsevier Science (02002)

    Google Scholar 

  42. Couceiro, M., Marichal, J.-L.: Axiomatizations of Lovász extensions of pseudo-boolean functions. Fuzzy Sets Syst. 181, 28–38 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  43. Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  44. de Campos, L., Bolaños, M.J.: Characterization and comparison of Sugeno and Choquet integrals. Fuzzy Sets Syst. 52, 61–67 (1992)

    Article  MATH  Google Scholar 

  45. Murofushi, T., Sugeno, M., Machida, M.: Non-monotonic fuzzy measures and the Choquet integral. Fuzzy Sets Syst. 64, 73–86 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  46. Couceiro, M., Marichal, J.-L.: Axiomatizations of quasi-Lovász extensions of pseudo-boolean functions. Aequat. Math. 82, 213–231 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  47. Couceiro, M., Marichal, J.-L.: Discrete integrals based on comonotonic modularity. Axioms 3, 390–403 (2013)

    Article  Google Scholar 

  48. Marichal, J.-L.: On Sugeno integral as an aggregation function. Fuzzy Sets Syst. 114, 347–365 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  49. Sugeno, M.: A note on derivatives of functions with respect to fuzzy measures. Fuzzy Sets Syst. 222, 1–17 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  50. Sugeno, M.: A way to choquet calculus. IEEE Trans. Fuzzy Systems, to appear

    Google Scholar 

  51. Weber, S.: \(\bot \)-decomposable measures and integrals for archimedean t-conorms \(\bot \). J. Math. Anal. Appl. 101, 114–138 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  52. Kruse, R.: Fuzzy integrals and conditional fuzzy measures. Fuzzy Sets Syst. 10, 309–313 (1983)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  54. Murofushi, T., Sugeno, M.: Fuzzy t-conorm integrals with respect to fuzzy measures : generalization of Sugeno integral and Choquet integral. Fuzzy Sets Syst. 42, 57–71 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  56. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy integral I. Kybernetika 41, 397–422 (2005)

    MATH  MathSciNet  Google Scholar 

  57. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy integral II. Kybernetika 41, 469–496 (2005)

    MATH  MathSciNet  Google Scholar 

  58. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy integral III. Kybernetika 41, 497–518 (2005)

    MATH  MathSciNet  Google Scholar 

  59. Shilkret, N.: Maxitive measure and integration. Nederl. Akad. Wetensch. Proc. Ser. A 74, 109–116 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  60. Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for choquet and sugeno integral. IEEE Trans. Fuzzy Syst. 18, 178–187 (2010)

    Article  Google Scholar 

  61. Klement, E.P., Mesiar, R., Spizzichino, F., Stupňanová, A.: Universal integrals based on copulas. Fuzzy Optim. Decis. Making 13, 273–286 (2014)

    Article  Google Scholar 

  62. Azrieli, Y., Lehrer, E.: Extendable cooperative games. J. Public Econ. Theor. 9, 1069–1078 (2007)

    Article  Google Scholar 

  63. Lehrer, E., Teper, R.: The concave integral over large spaces. Fuzzy Sets Syst. 159, 2130–2144 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  64. Lehrer, E.: A new integral for capacities. Econ. Theor. 39, 157–176 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  65. Even, Y., Lehrer, E.: Decomposition-integral: Unifying choquet and the concave integrals. Econ. Theor. 56, 33–58 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Paris School of Economics, University of Paris I, 106-112, Bd. de L’Hôpital, 75013, Paris, France

    Michel Grabisch

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  1. Michel Grabisch
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Correspondence to Michel Grabisch .

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  1. Department of Computer Science, Texas State University, San Marcos, Texas, USA

    Dan E. Tamir

  2. School of Computing and Information Sciences, Florida International University, Miami, Florida, USA

    Naphtali D. Rishe

  3. School of Computing and Information Sciences, The University of South Florida, Tampa, Florida, USA, and Florida International University, Miami, Florida, USA

    Abraham Kandel

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Grabisch, M. (2015). Fuzzy Measures and Integrals: Recent Developments. In: Tamir, D., Rishe, N., Kandel, A. (eds) Fifty Years of Fuzzy Logic and its Applications. Studies in Fuzziness and Soft Computing, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-19683-1_8

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  • DOI: https://doi.org/10.1007/978-3-319-19683-1_8

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