n维模糊集的基础理论及其应用
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摘要
本文首次提出了n维模糊集的概念,它是Zadeh模糊集、直觉(区间值)模糊集和区间值直觉模糊集的推广.由此,我们建立了n维模糊集的基本理论和方法,主要内容包括：n维模糊集基础理论、n维模糊向量子空间、凸n维模糊集与n维模糊数理论、基于n维模糊集的凸映射及数学规划、n维模糊集相似度理论及其在模糊风险分析中的应用.具体工作可概括如下：
1.第2章,给出n维模糊集的基本概念及其与模糊集、区间值模糊集、直觉模糊集和三维模糊集之间的关系；将n维模糊集的截集定义为n+1值模糊集,而它恰是Zadeh模糊集截集的推广,由此建立了相应的分解定理和表现定理；在此基础上,对偶地给出了Zadeh模糊集的向量值截集理论；最后从范畴论的角度说明结论的合理性.
2.第3章,将模糊向量子空间与凸模糊集的概念推广到n维情形.根据n维模糊集截集理论和模糊点与n维模糊集的邻属关系,并利用n+1-值Lukasiewicz蕴涵,给出α,β)-n维模糊向量子空间的定义,得到三种有意义的α,β)-n维模糊向量子空间形式,并将其统一推广为(s／]-n维模糊向量子空间,研究了(s,t]-n维模糊向量子空间的运算性质；类似地,给出(a,β)-凸n维模糊集的定义,对(∈,∈)-凸n维模糊集和(∈,∈v力-凸n维模糊集这两种非常有意义的凸n维模糊集进行了讨论,得到一些有意义的结果；在凸n维模糊集研究的基础上,给出n维(闭)模糊数的概念,得到相应的运算性质和参数表示定理,并讨论了n维模糊数的序关系和运算.
3.第4章,以n维模糊数及其运算的研究为基础,建立了(广义)凸n维模糊映射的基本理论；给出了凸n维模糊映射的性质和运算,以及正齐次性、下卷积、右数乘等重要概念；并对凸n维模糊规划进行了初步研究.
4.第5章,将n维模糊集理论与模式识别理论和综合分析理论相结合,给出n维模糊集相似度的公理化定义；以三维梯形模糊数为例,给出三维梯形模糊数的一种运算法则及相似度公式,并将所得的结果应用到模糊风险分析的研究中.
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.
1.第2章,给出n维模糊集的基本概念及其与模糊集、区间值模糊集、直觉模糊集和三维模糊集之间的关系；将n维模糊集的截集定义为n+1值模糊集,而它恰是Zadeh模糊集截集的推广,由此建立了相应的分解定理和表现定理；在此基础上,对偶地给出了Zadeh模糊集的向量值截集理论；最后从范畴论的角度说明结论的合理性.
2.第3章,将模糊向量子空间与凸模糊集的概念推广到n维情形.根据n维模糊集截集理论和模糊点与n维模糊集的邻属关系,并利用n+1-值Lukasiewicz蕴涵,给出α,β)-n维模糊向量子空间的定义,得到三种有意义的α,β)-n维模糊向量子空间形式,并将其统一推广为(s／]-n维模糊向量子空间,研究了(s,t]-n维模糊向量子空间的运算性质；类似地,给出(a,β)-凸n维模糊集的定义,对(∈,∈)-凸n维模糊集和(∈,∈v力-凸n维模糊集这两种非常有意义的凸n维模糊集进行了讨论,得到一些有意义的结果；在凸n维模糊集研究的基础上,给出n维(闭)模糊数的概念,得到相应的运算性质和参数表示定理,并讨论了n维模糊数的序关系和运算.
3.第4章,以n维模糊数及其运算的研究为基础,建立了(广义)凸n维模糊映射的基本理论；给出了凸n维模糊映射的性质和运算,以及正齐次性、下卷积、右数乘等重要概念；并对凸n维模糊规划进行了初步研究.
4.第5章,将n维模糊集理论与模式识别理论和综合分析理论相结合,给出n维模糊集相似度的公理化定义；以三维梯形模糊数为例,给出三维梯形模糊数的一种运算法则及相似度公式,并将所得的结果应用到模糊风险分析的研究中.
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.
引文
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