Caracterizing k-Additive Fuzzy Measures

  • Pedro Miranda
  • Michel Grabisch
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 90)


Recently, Grabisch has proposed the concept of k-additive measures to cope with the complexity problem involved by the use of fuzzy measures [8]. The concept has proven to be useful in multicriteria decision making, since it brings a model which is both flexible and simple to use.


Binary Relation Aggregation Operator Fuzzy Measure Weak Order Ordered Weighted Average 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    T. Calvo and B. De Baets. Aggregation operators defined by k-order additive/maxitive fuzzy measures. Int. J. of uncertainty, Fuzziness and Knowledge-Based Systems, (6): 533–550, 1998.MATHCrossRefGoogle Scholar
  2. [2]
    A. Chateauneuf. Modeling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research, (52): 3–20, 1994.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    A. Chateauneuf and J. Y. Jaffray. Characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences, 1989.Google Scholar
  4. [4]
    G. Choquet. Theory of capacities. Annales de l’Institut Fourier, (5): 131–295, 1953.MathSciNetCrossRefGoogle Scholar
  5. [5]
    T. Gajdos. Measuring inequalities without linearity in envy: Choquet integral for symmetric capacities. (Working paper).Google Scholar
  6. [6]
    M. Grabisch. Pattern classification and feature extraction by fuzzy integral. In 3d European Congr. on Intelligent Techniques and Soft Computing (EUFIT), pages 1465–1469, Aachen (Germany), August 1995.Google Scholar
  7. [7]
    M. Grabisch. The application of fuzzy integrals in multicriteria decision making. European J. of Operational Research, (89): 445–456, 1996.MATHCrossRefGoogle Scholar
  8. [8]
    M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, (92):167–189, 1996.MathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Grabisch. k-additive measures: Recent issues and challenges. In 5th Int. Conf. on Soft Computing and Information/Intelligent Systems, pages 394–397, Izuka (Japan), October 1998.Google Scholar
  10. [10]
    M. Grabisch. On lower and upper approximation of fuzzy measures by k-order additive measures. In 7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), pages 1577–1584, Paris (France), July 1998.Google Scholar
  11. [11]
    G.H. Hardy, J.E. Littlewood, and G. Pólya. Inequalities. Cambridge Univ. Press, Cambridge (UK ), 1952.MATHGoogle Scholar
  12. [12]
    K. Kao-Van and B. De Baets. A decomposition of k-additive Choquet and k-maxitive Sugeno integrals. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems,to appear.Google Scholar
  13. [13]
    T. Murofushi and M. Sugeno. Some quantities represented by the Choquet integral. Fuzzy Sets and Systems, (56): 229–235, 1993.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    E. Ben Porath and I. Gilboa. Linear measures, the Gini index, and the income-equality trade-off. Journal of Economic Theory, (64): 443–467, 1994.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    G. C. Rota. On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift fiir Wahrscheinlichkeitstheorie and Verwandte Gebiete, (2): 340–368, 1964.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    D. Schmeidler. Integral representation without additivity. Proc. of the Amer. Math. Soc., (97(2)): 255–261, 1986.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.Google Scholar
  18. [18]
    J. A. Weymark. Generalized Gini inequality indices. Mathematical Social Sciences, (1): 409–430, 1981.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    R. R. Yager. On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Systems, Man e4 Cybern., (18): 183–190, 1988.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Pedro Miranda
    • 1
  • Michel Grabisch
    • 2
  1. 1.Departamento de EstadísticaI.O. y D.M.OviedoSpain
  2. 2.Université Pierre et Marie Curie- LIP6ParisFrance

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