Fuzzy Number Metrics Research Based On The Structured Element Theory

For the objective fact that elements with different membership degree should have different contribution to the metric measure between fuzzy numbers, this paper presents two types of fuzzy number metrics weighted by structured element-E-Hausdorff metric and E-Lp metric. Firstly, we define two kinds of metrics weighted by structured element dHE and dpE, on the family (B[-1,1]) of all the same monotone and standard bounded functions on closed interval [-1,1], and discuss the completeness and separability of those two metric spaces; Next, using the fuzzy functional induced by normal fuzzy structured element, we give out a method that the metric of the closed bounded fuzzy number space is induced by the metric on function space B[-1,1]. Furthermore, two types of fuzzy number metrics dNHE and dNpE, weighted by structured element which both are induced by dNE and dpE, are presented,and analyze completeness and separability of the two induced fuzzy number metric spaces; Lastly, the difference and relationship between dNHE and dNpE, and the metrics defined by traditional method are shown. The results show that two types of fuzzy number metrics weighted by structured element considers of the contribution of the degree of membership to the fuzzy number distance, due to the contact between metric of fuzzy numbers and metric of functions, the study of properties of fuzzy number space can be transformed into the study of properties of function space, this opens up a way to the research on fuzzy number.