Skip to main content

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 90))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.

    Article  MATH  Google Scholar 

  2. A. Chateauneuf. Modeling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research, (52): 3–20, 1994.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. T. Gajdos. Measuring inequalities without linearity in envy: Choquet integral for symmetric capacities. (Working paper).

    Google Scholar 

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

    Article  MATH  Google Scholar 

  8. M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, (92):167–189, 1996.

    Article  MathSciNet  Google Scholar 

  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. 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. G.H. Hardy, J.E. Littlewood, and G. Pólya. Inequalities. Cambridge Univ. Press, Cambridge (UK ), 1952.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  18. J. A. Weymark. Generalized Gini inequality indices. Mathematical Social Sciences, (1): 409–430, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Miranda, P., Grabisch, M. (2002). Caracterizing k-Additive Fuzzy Measures. In: Bouchon-Meunier, B., Gutiérrez-Ríos, J., Magdalena, L., Yager, R.R. (eds) Technologies for Constructing Intelligent Systems 2. Studies in Fuzziness and Soft Computing, vol 90. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1796-6_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-7908-1796-6_17

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-2504-6

  • Online ISBN: 978-3-7908-1796-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics