Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1]

<p>In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space ε1 and the space B[−1, 1], which defined as a set of standard monotonic bounded functions with monotonicity on interval [−1, 1]. Furthermore, a new approach based upon the monotonic bounded functions has been proposed to create fuzzy numbers and represent them by suing fuzzy structured element. In order to make two different metrics based space in B[−1, 1], Hausdorff metric and Lp metric, which both are classical functional metrics, are adopted and their topological properties are discussed. In addition, by the means of introducing fuzzy functional to space B[−1, 1], we present two new fuzzy number’s metrics. Finally, according to the proof of homeomorphism between fuzzy number space ε1 and the space B[−1, 1], it’s argued that not only does it give a new way to study the fuzzy analysis theory, but also makes the study of fuzzy number space easier.</p>

Lp-Type of Weighted Fuzzy Number Metrics Induced by Fuzzy Structured Element

For the objective fact that elements with different membership degrees should have different contribution to the metric measure between fuzzy numbers, this paper presents Lp-type of fuzzy number metrics weighted by structured element. Firstly, we define a metric weighted by structured element 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 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, a fuzzy number metric weighted by structured element which is induced by is presented ,and analyze completeness and separability of the induced fuzzy number metric spaces; Lastly, the difference and relationship between and the metric defined by traditional method are shown.

Solving the Fuzzy BiLevel Linear Programming with Multiple Followers through Structured Element Method

The optimal solution of fuzzy bilevel linear programming with multiple followers (MFFBLP) model is shown to be equivalent to the optimal solution of the bilevel linear programming with multiple followers by using fuzzy structured element theory. The optimal solution to this model is found out by adopting the Kuhn-Tucker approach. Finally, an illustrative numerical example for this model is also provided to demonstrate the feasibility and efficiency of the proposed method.