Fuzzy number theory to obtain conservative results with respect to probability

Abstract

Fuzzy number and possibility theories are used for problems where uncertainties in the definition of input data do not allow for a treatment by means of probabilistic methods. Starting from a scarce/uncertain body of information, fuzzy numbers are used to define possibility distributions as well as upper and lower bounds for a wide class of probability distributions compatible with available data. It is investigated if relations between possibility distributions and probability measures are preserved also when the fuzzy number represents an output variable computed making use of extended fuzzy operations. General real one-to-one and binary operations are considered. Asymptotic expressions (for small/large fractiles) for the membership function of the fuzzy number and for CDFs given by probability theory are obtained. It is shown that fuzzy number theory gives conservative bounds (with respect to probability) for characteristic values corresponding to prescribed occurrence expectations. These results are of special interest for computational applications. In fact, it is easier to define fuzzy variables than random variables when no or few statistical data are available (as in the case of structural design stages). Moreover, extended fuzzy operations are much simpler than analogous operations required in the framework of probability, especially when several variables are involved.