A Generalization of Hukuhara Difference

  • Luciano Stefanini
Part of the Advances in Soft Computing book series (AINSC, volume 48)

Abstract

We propose a generalization of the Hukuhara difference. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is seggested to attempt a generalization of division for real intervals and fuzzy numbers.

Keywords

Hukuhara difference Interval and fuzzy arithmetic Fuzzy numbers Invertible fuzzy operations Interval and fuzzy calculus 

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References

  1. 1.
    Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 151, 581–599 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bouchon-Meunier, B., Kosheleva, O., Kreinovich, V., Nguyen, H.T.: Fuzzy numbers are the only fuzzy sets that keep invertible operations invertible. Fuzzy Sets Syst. 91, 155–163 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Diamond, P., Kloeden, P.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific Publishing Co., Inc, River Edge (1994)MATHGoogle Scholar
  4. 4.
    Hukuhara, M.: Integration des applications measurables dont la valeur est un compact convexe. Fukc Ekvacioj 10, 205–223 (1967)MATHMathSciNetGoogle Scholar
  5. 5.
    Laksmikantham, V., Mohapatra, R.N.: Theory of Fuzzy Differential Equations and Inclusions. Taylor and Francis, New York (2003)Google Scholar
  6. 6.
    Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs (1974)MATHGoogle Scholar
  7. 7.
    Stefanini, L.: A generalization of Hukuhara difference for interval and fuzzy arithmetic. Working Paper EMS Series, University of Urbino (2008), www.repec.org

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Luciano Stefanini
    • 1
  1. 1.Department of Economics and Quantitative Methods (DEMQ) and Faculty of conomicsUniversity of UrbinoUrbinoItaly

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