Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster–Shafer theory

There are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or Boolean value. In this paper, we use the Dempster–Shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number. The complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach. These relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the Dempster–Shafer theory. A natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples.