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Aggregation Based on Integrals: Recent Results and Trends

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Aggregation Operators

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 97))

Abstract

The paper presents an overview of recent results related to the use of non additive integrals (Choquet and Sugeno integrals) as aggregation operators. We essentially address two main issues, which are the treatment of negative numbers, and the case of ordinal information to aggregate. We do not explicitely presuppose any particular form of aggregation problem (e.g. multicriteria decision making), and remain at an abstract level, although decision making in general can be considered as the main motivation of the work.

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References

  1. P. Benvenuti and R. Mesiar. Integrals with respect to a general fuzzy measure. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 205–232. Physica Verlag, 2000.

    Google Scholar 

  2. C. Berge. Principles of Combinatorics. Academic Press, 1971.

    Google Scholar 

  3. T. Calvo, R. Mesiar, and J. Martin. Integral based aggregation of real data. In 6th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 58–62, Granada, Spain, July 1996.

    Google Scholar 

  4. K. Cao-Van and B. De Baets. Decomposition of Choquet and Sugeno integrals and the inverse problem. In 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 402409, Madrid, Spain, July 2000.

    Google Scholar 

  5. G. Choquet. Theory of capacities. Annales de l’Institut Fourier, 5: 131–295, 1953.

    Article  MathSciNet  Google Scholar 

  6. L. de Campos and M.J. Bola nos. Characterization and comparison of Sugeno and Choquet integrals. Fuzzy Sets ê4 Systems, 52: 61–67, 1992.

    Article  MATH  Google Scholar 

  7. D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic, 1994.

    Google Scholar 

  8. D. Denneberg and M. Grabisch. Interaction transform of set functions over a finite set. Information Sciences, 121: 149–170, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Denneberg and M. Grabisch. Functions and measures with linear ordinal scales, (a-)symmetric Sugeno integral and ordinal Ky Fan distance. In FUR X, Torino, Italy, May 2001.

    Google Scholar 

  10. D. Dubois and H. Prade. Possibility Theory. Plenum Press, 1988.

    Google Scholar 

  11. D. Dubois and H. Prade. Possibility theory: qualitative and quantitative aspects. In D.M. Gabbay and Ph. Smets, editors, Handbook of Defeasible Reasoning and Uncertainty Management Systems, pages 169–226. Kluwer Academic Publishers, 1998.

    Google Scholar 

  12. D. Dubois, H. Prade, and R. Sabbadin Qualitative decision theory with Sugeno integrals. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals–Theory and Applications, pages 314–332. Physica Verlag, Heidelberg, 2000.

    Google Scholar 

  13. M. Grabisch. Fuzzy integral in multicriteria decision making. Fuzzy Sets ê4 Systems, 69: 279–298, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Grabisch. The application of fuzzy integrals in multicriteria decision making. European J. of Operational Research, 89: 445–456, 1996.

    Article  MATH  Google Scholar 

  15. M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. Int. J. of Uncertainty, Fuzziness, and Knowledge Based Systems, 5: 587–607, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Grabisch. Fuzzy measures and integrals for decision making and pattern recognition. Tatra Mountains Mathematical Publications, 13: 7–34, 1997.

    MathSciNet  MATH  Google Scholar 

  17. M. Grabisch. k-additive and k-decomposable measures. In Proc. of the Linz Seminar, pages 88–94, Linz, Austria, February 1997.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. M. Grabisch. On the representation of k-decomposable measures. In 7th IFSA World Congress, Prague, Czech Republic, June 1997.

    Google Scholar 

  20. M. Grabisch. Fuzzy integral as a flexible and interpretable tool of aggregation. In B. Bouchon-Meunier, editor, Aggregation of evidence under fuzziness, pages 51–72. Physica Verlag, 1998.

    Google Scholar 

  21. M. Grabisch. A graphical interpretation of the choquet integral. IEEE Tr. on Fuzzy Systems, 8: 627–631, 2000.

    Article  Google Scholar 

  22. M. Grabisch. The interaction and Möbius representations of fuzzy measures on finite spaces, k-additive measures: a survey. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals - Theory and Applications, pages 70–93. Physica Verlag, 2000.

    Google Scholar 

  23. M. Grabisch. Symmetric and asymmetric fuzzy integrals: the ordinal case. In 6th Int. Conf. on Soft Computing (Iizuka’2000), Iizuka, Japan, October 2000.

    Google Scholar 

  24. M. Grabisch and Ch. Labreuche. To be symmetric or asymmetric? A dilemna in decision making. In J. Fodor, B. De Baets, and P. Perny, editors, Preferences and Decisions under Incomplete Knowledge, pages 179–194. Physica Verlag, 2000.

    Google Scholar 

  25. M. Grabisch and Ch. Labreuche. The symmetric and asymmetric Choquet integrals on finite spaces for decision making. Statistical Letters,submitted.

    Google Scholar 

  26. M. Grabisch, Ch. Labreuche, and J.C. Vansnick. On the extension of pseudo-Boolean functions for the aggregation of interacting bipolar criteria. Mathematical Social Sciences,submitted.

    Google Scholar 

  27. M. Grabisch, J.L. Marichal, and M. Roubens. Equivalent representations of a set function with applications to game theory and multicriteria decision making. Mathematics of Operations Research, 25 (2): 157–178, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Grabisch, H.T. Nguyen, and E.A. Walker. Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic, 1995.

    Google Scholar 

  29. J.L. Marichal. Aggregation operators for multicriteria decision aid. PhD thesis, University of Liège, 1998.

    Google Scholar 

  30. J.L. Marichal, P. Mathonet, and E. Tousset. Mesures floues définies sur une échelle ordinale. working paper, 1996.

    Google Scholar 

  31. R. Mesiar.k-order pan-additive discrete fuzzy measures. In 7th IFSA World Congress, pages 488–490, Prague, Czech Republic, June 1997.

    Google Scholar 

  32. R. Mesiar. Generalization of k-order additive discrete fuzzy measures. Fuzzy Sets and Systems, 102: 423–428, 1999.

    Article  MathSciNet  Google Scholar 

  33. R. Mesiar. k-order additive fuzzy measures. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 7: 561–568, 1999.

    Article  Google Scholar 

  34. R. Mesiar. Three alternative definitions of k-order additive fuzzy measures. BUSEFAL, 83: 57–62, 2000.

    Google Scholar 

  35. P. Miranda and M. Grabisch. Characterizing k-additive measures. In 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’2000), pages 1063–1070, Madrid, Spain, July 2000.

    Google Scholar 

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

    Google Scholar 

  37. T. Murofushi and M. Sugeno. Fuzzy t-conorm integrals with respect to fuzzy measures: generalization of Sugeno integral and Choquet integral. Fuzzy Sets ê4 Systems, 42: 57–71, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Murofushi and M. Sugeno. Some quantities represented by the Choquet integral. Fuzzy Sets £i Systems, 56: 229–235, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  39. F.S. Roberts. Measurement Theory. Addison-Wesley, 1979.

    Google Scholar 

  40. D. Schmeidler. Integral representation without additivity. Proc. of the Amer. Math. Soc., 97 (2): 255–261, 1986.

    Article  Google Scholar 

  41. G. Shafer. A Mathematical Theory of Evidence. Princeton Univ. Press, 1976.

    Google Scholar 

  42. L.S. Shapley. A value for n-person games. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pages 307–317. Princeton University Press, 1953.

    Google Scholar 

  43. M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.

    Google Scholar 

  44. M. Sugeno and T. Murofushi. Fuzzy measure theory, volume 3 of Course on fuzzy theory. Nikkan Kógyb, 1993. In Japanese.

    Google Scholar 

  45. A. Tversky and D. Kahneman. Advances in prospect theory: cumulative representation of uncertainty. J. of Risk and Uncertainty, 1992.

    Google Scholar 

  46. J. Sipos. Integral with respect to a pre-measure. Math. Slovaca, 29: 141–155, 1979.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. LIP6, Université de Paris VI, 8, rue du Capitaine Scott, 75015, Paris, France

    Michel Grabisch

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  1. Michel Grabisch
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of the Balearic Islands, 07071, Palma de Mallorca, Spain

    Tomasa Calvo  & Gaspar Mayor  & 

  2. Department of Computer Science, University of Alcalá, 28871, Alcalá de Henares, Spain

    Tomasa Calvo

  3. Faculty of Civil Engineering, Slovak Technical University, Radlinského 11, 81368, Bratislava, Slovakia

    Radko Mesiar

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© 2002 Physica-Verlag Heidelberg

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Grabisch, M. (2002). Aggregation Based on Integrals: Recent Results and Trends. In: Calvo, T., Mayor, G., Mesiar, R. (eds) Aggregation Operators. Studies in Fuzziness and Soft Computing, vol 97. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1787-4_2

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  • DOI: https://doi.org/10.1007/978-3-7908-1787-4_2

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-662-00319-0

  • Online ISBN: 978-3-7908-1787-4

  • eBook Packages: Springer Book Archive

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