【作者】 邵迎超；

【导师】 秦克云；

【作者基本信息】 西南交通大学 ， 交通信息工程及控制， 2013， 博士

【摘要】 美国控制论专家Zadeh于1965年提出的模糊集理论、波兰数学家Pawlak于1982年提出的粗糙集理论和俄国数学家Molodtsov于1999年提出的软集理论在处理不确定性问题时都显示出各自的优势与特色。这三种理论相辅相成,具有紧密的联系。目前,软集理论研究主要集中在三个方面：一是对其基本理论的进一步完善；二是将软集理论与其它研究领域融合；三是软集理论的应用研究。模糊集理论、粗糙集理论和软集理论作为主要的处理不确定性问题的三种数学工具,其融合研究是目前的研究热点之一众所周知,城市交通拥堵是各国发展中普遍存在的问题。随着交通需求量的增加,城市道路拥堵现象越来越严重,所造成的损失是难以估量的。如何准确地预测出即将形成的交通拥堵状态,并采取及时有效的措施,是近年来城市交通拥堵问题研究的热点之一。本文在相关研究的基础上从软模糊粗糙集理论的角度探讨预测城市交通拥堵的问题。本文旨在系统研究软集理论与模糊集理论以及格论的融合问题,探讨相关模型的性质,以及特定运算之下构成的代数结构,并在此基础上研究软集理论在城市交通拥堵预测中的应用。本文的主要研究成果如下：(1)提出了软格(soft lattice)的概念,证明了软格对于软集运算∩ε,n,U和八具有封闭性；基于∈-软集(∈-soft sets)和q-软集(q-soft sets)的概念,提出了理想软格(idealistic soft lattices)和滤子软格(filteristic soft lattices)的概念并讨论了它们的基本性质；基于Maji等提出的模糊软集(fuzzy soft sets)的概念,提出了软模糊格(soft fuzzy lattice)和软模糊理想(soft fuzzy ideal)的概念,并探讨它们构成的格结构；提出了生成模糊子格(generated fuzzy sublattice)的概念,给出了求一个格的任意子集的生成模糊子格的方法,并由此诱导出模糊子格之间的运算∪和U,证明了一个格的所有模糊子格在运算U和∩下构成一个格,一个格上的所有软模糊子格在运算U和∩下构成一个格。(2)将软集理论应用于RSL-代数,提出了软RSL-代数(soft RSL-algebra)的概念,讨论了软RSL-代数的基本性质,定义了软RSL-子代数之间的运算U,Uε,∩和nε,证明了软RSL-代数在同态映射下具有保序性。(3)基于Aktas和Cagman提出的软群概念和Acar提出的软环(soft ring)概念,研究了软群和软环所构成的代数结构,基于交换群的子群之间的并运算∪,诱导出交换群上软群之间的运算∪,证明了交换群的子群和交换群上的软群分别在运算∪和∪下的封闭性；证明了交换群上的软群在运算∪和∩下构成一个分配格。(4)将软集理论应用于粗糙集理论,建立了一种软粗糙集的模型并讨论了它的代数结构,证明了特定论域上的所有软粗糙集可以构成一个格。提出一个基于软集的软模糊粗糙集模型,给出了软模糊粗糙集之间的运算∪,∪ε,∩和∩ε；证明了特定论域上的所有软模糊粗糙集关于运算∪,∩ε,以及∪ε,∩下分别构成分配格。(5)基于软模糊粗糙集理论,研究了一种预测城市交通拥堵的方法。更多还原

【Abstract】 The fuzzy set theory introduced by American Cybernetics expert Zadeh,the rough set theory introduced by Polish mathematician Pawlak in1982and the soft set theory introduced by Russian mathematician Molodtsov have shown their own advantages and characteristic in dealing with uncertainty. The three kinds of theories complement each other and they are closely related.Now the research on the soft set theory focuses on three major aspects:the first is perfecting the soft theory, the second is merging it with the other studied fields; the third is on its application. As three kinds of major mathematical tool dealing with uncertainty, the study merging the fuzzy set theory, the rough set theory and the soft set theory is one of the hot spots at present.It is well known that the urban traffic congestion is a general problem in the development of every country in the world. With the increase of traffic demand, urban traffic congestion becomes more serious. The direct and indirect economic loss caused by traffic congestion is amazing. How to predict accurately the forthcoming congestion and to take measures timely is one of the hot spots in the study on urban traffic congestion.The intent of this paper is studying on the problems merging with the soft set theory,the fuzzy set theory, the rough set theory and the lattice theory and discussing the properties of the related models and the algebraic structures under the related operators. Based on the results, this paper presents a method predicting urban traffic congestion.The main results and innovations in this thesis are summarized as follows:1. The notion of the soft lattice is presented and the properties that the soft latt∩ice is closed under the operators∩ε,∩,∪and∧are derived. By means of∈-soft sets and q-soft sets, some characterizations of idealistic soft lattices and filteristic soft lattices are investigated. Based on the notion of the soft fuzzy set presented by Maji et al., the notions of the soft fuzzy lattice and the soft fuzzy ideal on a lattice are presented and their algebraic structures are discussed. The notion of generated fuzzy sublattices of an arbitrary subset of a lattice is presented and the method getting the generated fuzzy sublattices of an arbitrary subset is presented. The operator U between the fuzzy sublattices is inducted. The results are derived that the set of all fuzzy sublattices of a lattice forms a lattice on the operators∪and∩and that the set of all soft fuzzy sublattices forms a lattice on the operators U and∩.2. Applying the soft set theory to the RSL-algebra,the notion of the soft RSL-algebra is presented and some related properties are derived, and the operators between the soft RSL-subalgebras∪,∩,∪ε and∩ε are defined. The result is derived that the RSL-algebra is isotonic under a homomorphic mapping.3. Based on the notions of the soft groups presented by Aktas and Cagman and the soft rings presented by Acar, their algebraic structures are discussed, respectively. Based on the operator∪between the subgroups of a commutative group, the operator U between the soft groups on a commutative group is inducted and the results are derived that the subgroups of a commutative group and the soft groups on a commutative group are closed under the operators U and U, respectively. The result is derived that the set of all soft groups over a commutative group on the operators∩and∪forms a lattice.4. Applying the soft set theory to the the rough set theory, the model of the soft rough set is constructed and its algebraic structures are discussed. The result is derived that the set of soft rough sets over a given initial universe forms a lattice. A model of soft fuzzy rough set based on a soft set is constructed, and the operators∪,∪ε,∩and∩ε between the soft fuzzy rough sets are presented. The results are derived that the set of soft fuzzy rough sets over a given initial universe under the operators∩and∪ε,∩ε and∪, forms a distributive lattice, respectively.5. Based on the soft fuzzy rough set theory, a method predicting on urban traffic congestion is discussed.更多还原