## Ranking Fuzzy Numbers

### Ranking Fuzzy Numbers

What is the best way to rank fuzzy numers with overalping membership
functions to decide the best alternative?
i.e. to find out the min or max of a group of fuzzy numbers with
overalping membership functions

### Ranking Fuzzy Numbers

Quote:>What is the best way to rank fuzzy numers with overalping membership
>functions to decide the best alternative?
>i.e. to find out the min or max of a group of fuzzy numbers with
>overalping membership functions

x < y and x = y on fuzzy numbers are (as expected) fuzzy propositions.
So max / min in crisp sense may simply not exist.

The one thing you can do is to evaluate for each number the
possibility/necessity that this given number is greater or equal than
all others from the set. I.e. to make a sort of classification of the
given numbers as max of the set. To get exactly one number you could
defuzzify this classification, but this would be rather empiric.

Another approach is to consider the set of the fuzzy number as a whole
and to try to find its upper boundary (appropriately defined). The
result would be also a fuzzy number.

---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de

### Ranking Fuzzy Numbers

I read that some methods use the Center of Gravity of the [Fuzzy
Number/Membership Function] Curve to defuzzify the fuzzy number.
I don't understand how a fuzzy number can be difuzzified by the same
value when the membership function is represented by an equilateral
triangle, a rectangle, or a trabizoid that have a common base.

> >What is the best way to rank fuzzy numers with overalping membership
> >functions to decide the best alternative?
> >i.e. to find out the min or max of a group of fuzzy numbers with
> >overalping membership functions

> x < y and x = y on fuzzy numbers are (as expected) fuzzy propositions.
> So max / min in crisp sense may simply not exist.

> The one thing you can do is to evaluate for each number the
> possibility/necessity that this given number is greater or equal than
> all others from the set. I.e. to make a sort of classification of the
> given numbers as max of the set. To get exactly one number you could
> defuzzify this classification, but this would be rather empiric.

> Another approach is to consider the set of the fuzzy number as a whole
> and to try to find its upper boundary (appropriately defined). The
> result would be also a fuzzy number.

> ---
> Regards,
> Dmitry Kazakov
> www.dmitry-kazakov.de

### Ranking Fuzzy Numbers

Quote:>I read that some methods use the Center of Gravity of the [Fuzzy
>Number/Membership Function] Curve to defuzzify the fuzzy number.

The method depends on the meaning of the membership function. If its
values are possibilities, then center of gravity is meaningless.

Quote:>I don't understand how a fuzzy number can be difuzzified by the same
>value when the membership function is represented by an equilateral
>triangle, a rectangle, or a trabizoid that have a common base.

Well, perhaps it just has no sense. In many cases defuzzification is
purely empiric, so it is in vain to ask why it gives this or that
result. Just because! (:-)) Avoid defuzzification if you can.

---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de

### Ranking Fuzzy Numbers

"Dmitry A. Kazakov" wrote

Quote:> Well, perhaps it just has no sense. In many cases defuzzification is
> purely empiric, so it is in vain to ask why it gives this or that
> result. Just because! (:-)) Avoid defuzzification if you can.

I have created and implemented a scheme that allows a fuzzy qualitative
value, represented by a 2-tuple [rank, confidence factor], to be mapped to
a range on a quantitative/ratio scale and a function to consolidate
qualitative values:
http://protodesign-inc.com/files/TPgtc001.pdf

I am not sure if it relates to what you are discussing in this thread,
though.

--
Best Regards,
Greg Chien
e-mail: remove n.o.S.p.a.m.
http://protodesign-inc.com

### Ranking Fuzzy Numbers

> "Dmitry A. Kazakov" wrote
>> Well, perhaps it just has no sense. In many cases defuzzification is
>> purely empiric, so it is in vain to ask why it gives this or that
>> result. Just because! (:-)) Avoid defuzzification if you can.

> I have created and implemented a scheme that allows a fuzzy qualitative
> value, represented by a 2-tuple [rank, confidence factor], to be mapped to
> a range on a quantitative/ratio scale and a function to consolidate
> qualitative values:
> http://protodesign-inc.com/files/TPgtc001.pdf

> I am not sure if it relates to what you are discussing in this thread,
> though.

Yes it does. As far as I undersood the paper, your numbers are ordered
through their ranks, so the original question gets answered.

The rank you introduce can be viewed as a linguistic variable, with the
membership function of the shape you define in a special way. As such it
would be then a fuzzy number. From this point of view, the confidence
factor in a tuple is a classification of the observed thing x (also a fuzzy
number in the most general case). So x is (good, 0.3) can be interpreted as
x is a subset (instance, member whatsoever) of good with the confidence
0.3. Analoguosly:

A. In the possibility theory it would be Pos(good|x) = 0.3.

B. In the probability theory (Bayesian approach): Pr(good|x) = 0.3.

So everything depends on the interpretation of the confidence factors, which
in turn determines the rules one uses to combine them. This is what your
consolidation function does, which is in general neither A nor B, but
something else.

Further in your model fuzzy numbers become comparable, because you strictly
limit the possible shapes of the linguistic variables and use the rules
which retain the shape. Which of course leads to usual questions (for all
*-norms):

1. How intuitive, generic, are the axioms of the confidence factors a
posteriory induced by the consolidation rule? Does it cover all interesting
cases?

2. Same about the chosen subset of the membership functions shapes.

These questions cannot be answered, for the fuzzy community (:-)) is not
ready to answer that. We are trying this and that, but we are definitely
not ready to roll up a final set of axioms and to say that is!

--
Regards,
Dmitry A. Kazakov
www.dmitry-kazakov.de

Dear All,

Currently, I am doing some studies in fuzzy number ranking.
Just wondering anybody have any good articles related to this
approach-maximizing and minimizing sets in fuzzy number ranking...

Cheers,
chin wen cheong

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