【作者】 刘自新；

【导师】 冯恩民；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2007， 博士

【摘要】 模糊集理论为人类处理模糊信息提供了一套有效的方法。1983年，保加利亚学者Atanassov将模糊集理论加以推广，提出了直觉模糊集的概念，它从隶属度和非隶属度两方面描述不确定信息，为事物属性的描述提供了更多的选择方式，在处理不确定信息时具有更强的表现能力。直觉模糊集已经在模式识别、神经网络、群决策、模糊优化等领域得到了很好的应用。本文在直觉模糊集的基础上，研究了直觉模糊子空间，凸直觉模糊映射和直觉模糊规划问题，所取得的结果进一步丰富了模糊集理论的内容，也为决策者解决不确定问题提供了更有效的方法。本文取得的主要结果概括如下：1．第3章对直觉模糊子空间理论进行了研究。给出了直觉模糊点，直觉模糊子空间，直觉模糊仿射集的定义并讨论了直觉模糊子空间和直觉模糊仿射集的关系，利用直觉模糊点研究了直觉模糊仿射集与直觉模糊子空间，证明了向量空间E上的直觉模糊集是E上的直觉模糊子空间或直觉模糊仿射集的充要条件。给出了直觉模糊线性变换和直觉模糊仿射变换的定义，证明了线性映射将直觉模糊子空间映射为直觉模糊子空间的定理，讨论了直觉模糊线性变换和直觉模糊仿射变换之间的关系，从而进一步推广了模糊空间和模糊变换的基本理论。2．第4章对凸直觉模糊映射进行了研究。首先在凸模糊集有关理论的基础上，讨论了凸直觉模糊集、严格凸直觉模糊集、半严格凸直觉模糊集、拟凸直觉模糊集、严格拟凸直觉模糊集、半严格拟凸直觉模糊集之间的关系；其次在区间数和模糊数的定义及运算的基础上，给出了直觉模糊数的定义及其四则运算，给出了直觉模糊数的Hamming距离及其极大极小的定义，并由此讨论了直觉模糊数的一种比较方法；最后给出了凸直觉模糊映射、（φ₁，φ₂）-凸直觉模糊映射、Preinvex直觉模糊映射、quasi-preinvex直觉模糊映射的定义并讨论了它们之间的关系。3．第5章讨论了直觉模糊规划问题。首先将直觉模糊集理论应用于实际的不确定问题中，用直觉模糊集的隶属函数和非隶属函数描述不确定问题的目标函数和约束条件，推广了Verdegay和Werners的模糊线性规划模型，并给出了数值算例。其次利用凸直觉模糊映射给出了凸直觉模糊规划模型并讨论了其相应的性质，这些研究进一步丰富了模糊数学规划的内容，为解决带有不确定参数的数学规划问题提供了有力的工具。更多还原

【Abstract】 The theories of fuzzy sets provide a kind of efficient method to deal with fuzzyinformation. In 1983, Atanassov introduced the concept of intuitionistic fuzzy set whichextended the theories of fuzzy sets. It processes the information of membership andnon-membership, provides more choices for the attribute description of an object andexpresses uncertainty better than fuzzy sets. It has been applied in the area of patternrecognition, neural network, group decision-making, fuzzy optimization, etc. This paper aimsat studying Intuitionistic fuzzy subspace, convex intuitionistic fuzzy mappings andintuitionistic fuzzy programming on the basis of intuitionistic fuzzy set, The results obtainedin it enrich the theories of fuzzy sets and provide an efficient method for decision makerswhen they face uncertainty problem. The main results obtained in this dissertation aresummarized as follows:1. In Chapter 3, the theories of intuitionistic fuzzy subspace are studied. The concepts ofintuitionistic fuzzy point, intuitionistic fuzzy subspace and intuitioistic fuzzy affine set areproposed. The relation between intuitionistic fuzzy supspace and intuitionistic fuzzy affine setis also discussed. The paper using intuitionistic fuzzy point investigates the intuitionisticfuzzy affine sets and intuitionistic fuzzy subspaces, and proves a necessary and sufficientcondition for an intuitionistic fuzzy set is an intuitionistic fuzzy subspace or an intuitionisticfuzzy affine set of vector space E. The concepts of intuitionistic fuzzy linear transformationand intuitionistic fuzzy affine transformation are proposed and their relation is discussed too.These results further extend the theories of fuzzy sets.2. In Chapter 4, the convex intuitionistic fuzzy mappings are studied. Based on thetheories of fuzzy sets, the relations of convex intuitionistic fuzzy set, strictly convexintuitionistic fuzzy set, semi-strictly convex intuitionistic fuzzy set, quasi-convexintuitionistic fuzzy set, strictly quasi-convex intuitionistic fuzzy set and semi-strictlyquasi-convex intuitionistic fuzzy set are studied. Then the concept of intuitionistic fuzzynumber and its operations are presented based on the concept and operations of intervalnumbers and fuzzy numbers. The maximal and minimal of intuitionistic fuzzy numbers andhamming distance are proposed, a method of comparing intuitionistic fuzzy numbers is giventoo. At last, the concepts and relations of convex intuitionistic fuzzy mapping, （φ₁,φ₂）-convex intuitionistic fuzzy mapping, preinvex intuitionistic fuzzy mapping,quasi-preinvex intuitionistic fuzzy mapping are given.3. In Chapter 5, the models of intuitionistic fuzzy programming are discussed. Firstly,Applying the theory of intuitionistic fuzzy set in uncertainty problems, the paper describes theobject and constraints by membership function and non-membership function of intuitionisticfuzzy sets, extends the model of fuzzy linear programming introduced by Verdegay andWerners and gives a numerical example. Secondly, it presents the model of convexintuitionistic fuzzy programming by using convex intuitionistic fuzzy mapping and discussesits properties. These researches enrich the contents of fuzzy mathematical programming andprovide a good method for mathematical programming with uncertainty parameters.更多还原