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.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

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

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

The method depends on the meaning of the membership function. If itsQuote:>I read that some methods use the Center of Gravity of the [Fuzzy

>Number/Membership Function] Curve to defuzzify the fuzzy number.

values are possibilities, then center of gravity is meaningless.

Well, perhaps it just has no sense. In many cases defuzzification isQuote:>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.

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

"Dmitry A. Kazakov" wrote

I have created and implemented a scheme that allows a fuzzy qualitativeQuote:> 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.

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

> "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.

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

1. fuzzy number ranking-maximizing and minimizing sets approach

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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