Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition

The main aim of this paper is to characterize the set of real parameters associated to a fuzzy number, represented in a general form which include the most important characteristics, with the following property: for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter. The uniqueness of the nearest trapezoidal fuzzy number with this property is proved, the average Euclidean distance being considered. As an important property, each resulting trapezoidal approximation operator is continuous. The main results are illustrated by examples.