Percentage Comparison of Fuzzy Numbers Using a Newly Presented Method in the Context of Surrogate Modeling

Abstract

The O-index presented here allows a statement about the percentage deviation between two fuzzy numbers. For this purpose, one fuzzy number is defined s the reference. This fuzzy number is described with the help of its core value and its support. The deviation of the other fuzzy number, which is defined as the comparison number, is then quantified via the area differences of the left and right limits of the two numbers. In addition, the case when decomposed fuzzy numbers are involved, which are given as alpha \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} -cuts is taken into account. Thus, the O-index- alpha \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} can be used to calculate a separate percentage deviation for each alpha \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} -cut and thus generate additional knowledge. The O-index then allows a very detailed description of the deviation between two fuzzy numbers. One application of the O-index is the estimation of the accuracy of a surrogate model in relation to a reference model in the context of uncertainty quantification. This is illustrated by a mechanical example, a bending beam.