Approximations by Interval, Triangular and Trapezoidal Fuzzy Numbers

Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These researches can be grouped into two classes: the Euclidean distance class and the non-Euclidean distance class. Most approximations in the Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. In this paper, we study interval, triangular, and trapezoidal approximations under a weighted Euclidean distance which generalize all approximations in the Euclidean distance class. First, we embed fuzzy numbers into a Hilbert space, and then introduce these weighted approximations by means of best approximations from closed convex subsets of the Hilbert space. Finally, we apply the reduction principle to simplify calculations of these approximations. Keywords— weighted trapezoidal approximation, triangular fuzzy number, Hilbert space