Fuzzy specifications and random fuzzy events considered as basic tools for statistical prediction (Q1116149)

The authors introduce a concept of a random fuzzy event with applications to performance evaluation of a manufacturing process. Let B be a fuzzy set, called a fuzzy specification, on \({\mathbb{R}}\) and x a random variable defined on a probability space (\(\Omega\),\({\mathcal A},p)\). Then \(Y(\omega)=B(x(\omega))\in [0,1]\) is a random fuzzy event provided Y is Borel measurable. Note that this notion is different from the concept of a fuzzy event as introduced by \textit{L. A. Zadeh} [J. Math. Anal. Appl. 23, 421-427 (1968; Zbl 0174.490)]. Furthermore, some properties of random fuzzy events are given which are analogous to those of probability theory. As the authors point out: ``We have for random fuzzy events definitions which are formally identical to the classical definitions in probability but have a different fuzzy interpretation''. Finally, the authors indicate a generalization in case B is an L-fuzzy set on a set E. For applications of fuzzy sets to performance evaluation in the non-random case see also the reviewer in M. G. Singh (ed.), Systems and control encyclopedia. (1989), pp. 4764-4766, and references therein.