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

Lee, E. S.; Li, R.-J.

Comput. Math. Appl. 15, No. 10, 887-896 (1988).

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.

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.

Keywords:

ranking method between fuzzy numbers; membership function; ranking methods

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