Re: Fuzzy Arithmetic

Scott Ferson (scott@ramas.com)
Sun, 26 Jul 1998 15:56:45 +0200 (MET DST)

Siler doesn't emphasize that fuzzy arithmetic
generalizes ordinary interval analysis. You can
think of a fuzzy number as a nested stack of
intervals at each of many presumption levels
(alpha) over [0,1]. Fuzzy arithmetic can be
then identified with interval analysis at each
of these alpha values. The extension principle
and this iterated interval analysis give the same
answers so long as your inputs are both fuzzy
numbers or fuzzy intervals, that is, so long as
each operand is a fuzzy set on the reals that
has only one hump and reaches 1 for at least
some values. For proofs, see the book by
Kaufmann and Gupta, "Introduction to Fuzzy
Arithmetic".

For an introductory explanation about why it might
be useful to use fuzzy arithmetic in risk analysis
instead of, say, some Monte Carlo approach, see
http://www.ramas.com/Rcal-faq.htm#fuzzy or
http://www.ramas.com/Rcal-faq.htm#fdiff

For the sake of clarity, we should mention that
the fuzzy number depicted below is what Siler
refers to as "fuzzy 5" and Jeschke's software
represents as [5,5,2,1]. I usually represent this
fuzzy number as [3,5,6] because I think it's a
more natural generalization of the interval
notation that doesn't require mental addition
and subtraction.

1.0 - b
.8 - b b
.6 - b b
.4 - b b
.2 - b b
.0 - b b
--------------------------------------------
3 4 5 6

Scott Ferson <scott@ramas.com>
Applied Biomathematics, 516-751-4350, fax -3435

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