Re: Order on Fuzzy Numbers

From: Ulrich Bodenhofer (ulrich.bodenhofer@scch.at)
Date: Fri Jan 12 2001 - 18:15:34 MET

  • Next message: Uwe Wagner: "Order on Fuzzy Numbers"

    Hi,

    although your question, IMHO, does not fit to the original thread, here is
    my
    answer: Of course, there exists a rather simple approach that fulfills
    all the three classical axioms of reflexivity, transitivity, and
    antisymmetry -
    just demand the usual interval ordering

       [a,b]<=[c,d] <==> a<=b & c<=d

    for each \alpha-cut of the fuzzy numbers (which corresponds to the extension
    of
    this interval ordering to fuzzy intervals by means of Zadeh's extension
    principle).
    Be aware, that this ordering is not linear and that antisymmetry gets lost
    if you
    do no longer assume convexity of the fuzzy sets under consideration (fuzzy
    numbers are usually defined to be convex anyway). Beside this simple
    approach
    there is a vast number of other approaches - ranging from totally primitive
    ones
    to quite strange exotic ones. Some are based on defuzzification (and
    therefore
    enforcing linearity while having a serious problem with antisymmetry),
    some on fuzzy relations, etc.

    For more information, I would like to recommend the following literature:

    @article{BortolanDegani:85,
      author = {G. Bortolan and R. Degani},
      title = {A Review of Some Methods for Ranking Fuzzy Subsets},
      journal = {Fuzzy Sets and Systems},
      volume = {15},
      pages = {1--19},
      year = {1985}
    }

    @article{KoczyHirota:93,
      author = {L. T. K\'oczy and K. Hirota},
      title = {Ordering, distance and closeness of fuzzy sets},
      journal = {Fuzzy Sets and Systems},
      volume = {59},
      number = {3},
      pages = {281--293},
      year = {1993}
    }

    @incollection{WangKerre:96,
      author = {X. Wang and E. E. Kerre},
      title = {On the Classification and the Dependencies of the
                      Ordering Methods},
      booktitle = {Fuzzy Logic Foundations and Industrial Applications},
      editor = {D. Ruan},
      publisher = {Kluwer Academic Publishers},
      address = {Boston},
      series = {International Series in Intelligent Technologies},
      year = {1996},
      pages = {73--90}
    }

    @inproceedings{Bodenhofer:00b,
      author = {U. Bodenhofer},
      title = {A General Framework for Ordering Fuzzy Alternatives
                      with Respect to Fuzzy Orderings},
      booktitle = {Proc. IPMU 2000},
      address = {Madrid},
      month = {July},
      volume = {II},
      pages = {1071--1077},
      year = {2000}
    }

    Regards,
    Ulrich

    "Uwe Wagner" <wagner@ipd.info.uni-karlsruhe.de> wrote in message
    news:93ke3f$ge0$1@news.rz.uni-karlsruhe.de...
    > Does anybody now, if there exists a partial order of fuzzy numbers? A
    > mathematical Partial Order is a Relation @ that is
    >
    > reflexsiv a@b
    > transitiv a@b and b@c => a@c
    > antisymetric a@b and b@a => a=b
    >
    > Thank you
    >
    > Uwe
    >
    >

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