Uncertain Fuzzy Models and Their Applications: I. Uncertain, Fuzzy, and Uncertain Fuzzy Elements and Sets

摘要

This is the first of a series of papers that consider mathematical methods and tools for constructing mathematical models of complex objects and means for observing, measuring, and recording them, on the one hand, and for representing subjective statements about the reliability of these models and conclusions based on them, on the other hand. Such models, which are said to be uncertain fuzzy (UF), are usually based on complicated, inaccurate, contradictory, and unreliable information. In the UF models under consideration, the inaccuracy and fuzziness of statements inherent in models of complex objects and means for examining them are characterized in terms of values of possibility and (or) necessity measures on an ordinal scale where only the <, >, and = relations have meaningful interpretations [1]. Accordingly, the reliability of statements (which cannot be absolute, because the knowledge of the properties of the objects to be modeled and means for studying them is incomplete in principle) is characterized in terms of the values of plausibility and (or) belief measures, also on an ordinal scale; the values of these measures characterize subjective statements concerning the adequacy of certain aspects of the model under consideration according to their imprecision and uncertainty. This paper considers the notions of fuzzy, uncertain, and uncertain fuzzy elements and sets, UF functions, UF events, etc. Plausibility measures of possibility and integrals are defined. Expressions for the plausibility of the possibility of simplest events are given. The next paper of the series will consider the theory of a plausibility measure of possibility and an integral, and the third, concluding, paper will suggest methods for optimal estimation and decision making.