Moments Of Reduced Fuzzy Variables With Applications

Type-2 fuzzy variable is a measurable map from a fuzzy possibility space to the real number space, and it is an appropriate tool to describe type-2 fuzziness. The possibility that a type-2 fuzzy variable takes its value is a regular fuzzy variable. So, it is easy to determine the possibility distribution of a type-2 fuzzy variable.This thesis first defines three kinds of equivalent values (EVs) for a regular fuzzy variable by L-S integral, and proposes the EV-based reduction method. Since the reduced fuzzy variable is characterized by a parametric possibility distribution, it is more flexible and robust than a fuzzy variable in practical applications. Then this thesis defines three kinds of nth moments for a reduced fuzzy variable via L-S integral to measure the variation of its parametric possibility distribution about ex-pected value, and discusses the convexity of the second moment about the decision vector. After that, this thesis applies the reduction method and the second moment to portfolio selection problems with type-2 fuzzy returns, and develops two classes of generalized fuzzy mean-moment models. When security returns are characterized by type-2 trapezoidal fuzzy variables, the proposed models are equivalent to parametric quadratic convex programming problems, which can be solved by conventional opti-mization methods or general purpose softwares. Finally, two numerical experiments are provided to illustrate the modeling ideas and the effectiveness of the proposed models and solution methods.The main contribution of this thesis contains the following four aspects:(f) three kinds of EVs for a regular fuzzy variable are defined; (2) the EV-based re-duction method is proposed, and the parametric possibility distributions of reduced fuzzy variables are presented; (3) for a reduced fuzzy variable, three kinds of nth moments are defined, and two classes of generalized fuzzy portfolio selection models are established; (4) two numerical experiments are shown to illustrate the modeling ideas and the effectiveness of the proposed models and solution methods.