## Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity.(English)Zbl 1258.03079

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 $\int^1_0 \alpha(A_U(\alpha)- A_L(\alpha))\,d\alpha$ with $$A_U$$ and $$A_L$$ computed as $$A_U(\alpha)= \sup\{x\in \mathbb R\mid A(x)\geq \alpha\}$$ and $$A_L(\alpha)= \text{inf}\{x\in \mathbb R\mid A(x)\geq \alpha\}$$.
The study provides detailed formulas in case of approximation realized by (a) intervals, (b) triangular fuzzy numbers, and (c) trapezoidal fuzzy numbers.

### MSC:

 3e+72 Theory of fuzzy sets, etc. 2.6e+51 Fuzzy real analysis
Full Text:

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