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**Comparison of fuzzy numbers based on the probability measure of fuzzy events.**
*(English)*
Zbl 0654.60008

As a ranking method between fuzzy numbers, the authors propose the mean m(A) and the dispersion G(A) of a fuzzy number A relative to a probability measure p, i.e.
\[
m(A)=\int x \mu_ Adp/\int \mu_ Adp,\quad G(A)=[\int (x-m(A))^ 2\mu_ Adp/\int \mu_ Adp]^{-1/2},
\]
where \(\mu_ A\) denotes the membership function of A. Using several p’s, they overcome the difficulty that one fixed ranking method does not discriminate in some cases.

In the reviewer’s opinion, the approach is somewhat artificial. The formal use of probability measures and their means (why not their modes, their medians?) and dispersions (why not absolute deviations, entropy measures?) presents, indeed, a large number of different ranking methods. But in a real situation, a fruitful choice of - maybe several - ranking methods must be made in a more context-dependent way. Also, the connection between random fuzzy numbers and fuzzy numbers of type 2 seems to be somewhat diffuse.

In the reviewer’s opinion, the approach is somewhat artificial. The formal use of probability measures and their means (why not their modes, their medians?) and dispersions (why not absolute deviations, entropy measures?) presents, indeed, a large number of different ranking methods. But in a real situation, a fruitful choice of - maybe several - ranking methods must be made in a more context-dependent way. Also, the connection between random fuzzy numbers and fuzzy numbers of type 2 seems to be somewhat diffuse.

Reviewer: W.Näther

### MSC:

60A99 | Foundations of probability theory |

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

03E72 | Theory of fuzzy sets, etc. |

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\textit{E. S. Lee} and \textit{R. J. Li}, Comput. Math. Appl. 15, No. 10, 887--896 (1988; Zbl 0654.60008)

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