In many cases, fuzziness and randomness simultaneously appear in a system. Hybrid variable is a tool to describe this phenomena. Fuzzy random variable and random fuzzy variable are instances of hybrid variable. In order to measure hybrid event, a concept of chance measure is proposed in this paper. Furthermore, several useful properties about this measure are proved such as self-duality, subadditivity and semicontinuity. Some concepts are also presented such as chance distribution, expected value, variance, moments, critical values, entropy, distance and sequence convergences.
In renewal processes, fuzziness and randomness often coexist intrinsically. Based on the random fuzzy theory, a delayed renewal process with random fuzzy interarrival times is proposed in this paper. Relations between the renewal number and interarrival times in such a process are investigated. Useful theorems such as the elementary renewal theorem, the Blackwell renewal theorem and the Smith key renewal theorem in a conventional delayed renewal process are extended to their counterparts for random fuzzy delayed renewal processes.
This paper proposes an axiomatic frameworkfrom which we develop the theory of type-2 (T2) fuzziness,called fuzzy possibility theory. First, we introduce theconcept of a fuzzy possibility measure in a fuzzy possibilityspace (FPS). The fuzzy possibility measure takes onregular fuzzy variable (RFV) values, so it generalizes thescalar possibility measure in the literature. One of theinteresting consequences of the FPS is that it leads to a newdefinition of T2 fuzzy set on the Euclidean space