Ranking of Multidimensional Uncertain Information Based on Metrics on the Fuzzy Ellipsoid Number Space

Abstract

On the usual high-dimensional fuzzy number space, the usual supremum metric $D$ (derived by the Hausdorff metric between the level sets of usual $n$ -dimension fuzzy numbers) and the “ $p$ ” metric $\rho _p$ are the most common metrics. However, due to the complexity of the level sets of usual $n$ -dimensional fuzzy numbers, the two kinds of metrics not only have a tendency to be rougher, but also are difficult to give concrete expression formulas (this affects their theory and application research). In this paper, some new metrics on fuzzy ellipsoid number space are introduced, which not only can better reveal the difference between two different fuzzy ellipsoid numbers, but also have concrete expression formulas (expressed with the level set functions of fuzzy ellipsoid numbers). And the properties of the new introduced metrics and the relationships between the new metrics and the usual metrics ( $D$ and $\rho _p$ ) are studied, and some results are obtained. Then, we give the concept of supremum (infimum) of bounded subsets of fuzzy ellipsoid number space, and obtain its concrete calculation formula. And then, by using the obtained results, we propose a method to rank multidimensional uncertain information, and give a practical example to show the application and the rationality of the proposed techniques.