How to aggregate Fuzzy Intervals with Fuzzy Arithmetic?

atik (atik@ptw.tu-darmstadt.de)
Tue, 24 Aug 1999 00:37:54 +0200 (MET DST)

Hello to all,

I have a question about arithmetic operations with fuzzy sets. Let me note
first, that I am familiar with the application of the extension principle to
LR sets, e.g. triangular or trapezoid. Hovewer, what I'm trying to do is
summing up fuzzy sets, which are achieved as result of an inference process
(max/min or max/prod) and to divide them by the no. of fuzzy sets (the
latter step should not be a problem). Now, depending on the inference
method applicated and the number of rules activated, the resulting fuzzy
sets have at least five charactersitic points (max/min) or seven points
(max/prod) instead of normally four (m,n,a,b) or three (m=n,a,b). Please
note that the five linguistic terms are defined by trapezoid membership
functions. In addition, in the case of max/prod-inference we may also have
the problem of the missing convexity of the resulted fuzzy set.

Motivation:
As a result of the inference process I get 10 results, first, represented as
fuzzy sets. Since these are too many results to draw conclusions by the
decision maker, I have to aggregate them in a further step to an overall
result, also as a fuzzy set.

The idea ist to perform the aggregation without defuzzification of the ten
interim results mentioned (=>hierarchical fuzzy expert system for decision
support), mainly to avoide the loss of information bout the fuzziness in the
results. The defuzzification should be done as the final step (overall
single score). A second reason is that without the fuzzy arithmetic
operation desribed, I do not see any possibility to determine the fuzziness
of the overall result of the rule based calculation, but only of the interim
results, what is, according to the goal of the task, not sufficient.

Does anyone know, how to perform arithmetic operations in the case desribed?
I would be very grateful for any help. Please feel free to give any
comments, hints concerning papers, etc.

Thanks for reading in advance.
Warm regards
Alp

-------------------------------------------------------------
Alp Atik
Darmstadt University of Technology
Institute for Production Engineering and Machine Tools (PTW)
Petersenstr. 30
D-64287 Darmstdt, Germany
--------------------------------------------------------------

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