Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity (Q448953)

The paper shows how to approximate a fuzzy number in such a way that an average Euclidean distance between the original fuzzy number and its approximation is minimized under the condition of preservation of ambiguity. Here the ambiguity of a fuzzy number \(A\) is expressed in the form NEWLINE\[NEWLINE\int^1_0 \alpha(A_U(\alpha)- A_L(\alpha))\,d\alphaNEWLINE\]NEWLINE with \(A_U\) and \(A_L\) computed as \(A_U(\alpha)= \sup\{x\in \mathbb R\mid A(x)\geq \alpha\}\) and \(A_L(\alpha)= \text{inf}\{x\in \mathbb R\mid A(x)\geq \alpha\}\).NEWLINENEWLINE The study provides detailed formulas in case of approximation realized by (a) intervals, (b) triangular fuzzy numbers, and (c) trapezoidal fuzzy numbers.