Abstract
The problem of the piecewise linear approximation of fuzzy numbers giving outputs nearest to the inputs with respect to the Euclidean metric is discussed. The results given in Coroianu et al. (Fuzzy Sets Syst 233:26–51, 2013) for the 1-knot fuzzy numbers are generalized for arbitrary n-knot (nge 2) piecewise linear fuzzy numbers. Some results on the existence and properties of the approximation operator are proved. Then, the stability of some fuzzy number characteristics under approximation as the number of knots tends to infinity is considered. Finally, a simulation study concerning the computer implementations of arithmetic operations on fuzzy numbers is provided. Suggested concepts are illustrated by examples and algorithms ready for the practical use. This way, we throw a bridge between theory and applications as the latter ones are so desired in real-world problems.
Highlights
A family of fuzzy numbers constitutes an important subclass of fuzzy sets having countless applications in all cases where imprecise real values are modeled by their fuzzy counterparts
The problem of the piecewise linear approximation of fuzzy numbers by the so-called 1-knot fuzzy numbers was considered in Coroianu et al (2013)
We give there practical approximation algorithms and illustrative examples but we provide a simulation study on the approximation accuracy of the computer calculations on fuzzy numbers and stability of some fuzzy number characteristics
Summary
A family of fuzzy numbers constitutes an important subclass of fuzzy sets having countless applications in all cases where imprecise real values are modeled by their fuzzy counterparts. The problem of the piecewise linear approximation of fuzzy numbers by the so-called 1-knot fuzzy numbers was considered in Coroianu et al (2013) Each such 1-knot fuzzy number is completely characterized by six points on the real line. Instead of six points on the real line and piecewise linear sides each consisting of two segments that characterize 1-knot fuzzy numbers, we consider n-knot fuzzy numbers (where n ≥ 2) which enables to quantify the uncertainty at n intermediate levels between 0 and 1. Such fuzzy numbers were already introduced in paper Báez-Sánchez et al (2012) and were called polygonal fuzzy numbers. Some open problems and directions for the further research are sketched there
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