The Basic Theory And Applications Of N-dimensional Fuzzy Sets

This dissertation proposes the concept of n-dimensional fuzzy sets for the first time, which is the generalization of Zadeh fuzzy sets, intuitionistic (interval-valued) fuzzy sets and interval-valued intuitionistic fuzzy sets. Based on the concept of n-dimensional fuzzy sets, we establish the basic theory and method on n-dimensional fuzzy sets, including the basic theory of n-dimensional fuzzy sets, n-dimensional fuzzy vector subspaces, n-dimensional convex fuzzy sets and theory of n-dimensional fuzzy numbers, convex mappings and programming based on n-dimensional fuzzy sets, and the similarity theory of n-dimensional fuzzy sets and its application in fuzzy risk analysis. The idiographic works are summarized as follows:1. Chapter2, introduces the concept of n-dimensional fuzzy sets and the relations among the Zadeh fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets,3-dimensional fuzzy sets, and n-dimensional fuzzy sets; defines the cut sets on n-dimensional fuzzy sets as n+1-valued fuzzy sets, which is exactly the generalization of cut sets on Zadeh fuzzy sets, and establishes the corresponding decomposition theorems and representation theorems; the theory of vector-valued cut sets on Zadeh fuzzy sets is introduced dually; at the end, the rationality of above conclusions is illuminated from the viewpoint of category theory.2. Chapter3, generalizes the concept of fuzzy vector subspaces and convex fuzzy sets to the case of n-dimension. Based on the theory of cut sets on n-dimensional fuzzy sets and the neighborhood relations between a fuzzy point and a n-dimensional fuzzy set, the definition of (α,β)-n-dimensional fuzzy subspace is given and three kinds of significant (α,β)-n-dimensional fuzzy subspace are obtained by applying the n+1-valued Lukasiewicz implication, according to which, the (s,t)-n dimensional fuzzy subspace is derived and the operations properties of the (s, t)-n dimensional fuzzy subspace are obtained; similarly, the definition of (α,β)-n-dimensional convex fuzzy set is given, the (∈,∈)-n-dimensional convex fuzzy sets and (∈,∈∨q)-n-dimensional convex fuzzy sets, which is the significant ones, are discussed, and some significant results are obtained; based on the research of n-dimensional convex fuzzy sets, the concept of n-dimensional (closed) fuzzy number is given, the corresponding operation properties and the parameter representation theorem are obtained, the ordering and operations of the n-dimensional fuzzy numbers are discussed.3. Chapter4, based on the research of n-dimensional fuzzy numbers and their operations, the basic theory of (generalized) n-dimensional convex fuzzy mappings is established; the properties and operations of n-dimensional convex fuzzy mappings, and some important concepts such as positively homogeneous, infimal convolution, and right scalar multiplication are given; the n-dimensional convex fuzzy programming is researched preliminarily.4. Chapter5, combines the theory of n-dimensional fuzzy sets with the theory of pattern recognition and synthesis analysis, the axiom definition of n-dimensional fuzzy sets similarity is given; takes the3-dimensional fuzzy numbers as an example, an algorithm and the similarity formula are given, furthermore, the above results are applied to the research of fuzzy risk analysis.