模糊集合

Firstly, we analyze the deficiencies ofexisting ones, extend domains of fuzzy logical operators from the cap product withtwo sets to them with arbitrary (finite, enumerable, or non-enumerable) number ofsets to form two general mappings so as to conform them to the fuzzy conjunctionand infimum, and to fuzzy disjunction and supremum, respectively, and fuzzify theset inclusion of fuzzy sets. Similar to the classical adjunction, the concept of fuzzyadjunction is introduced.

首先分析了原有"模糊形态学"框架的缺陷，进而将模糊合取和模糊析取算子的定义域从两个集合的笛卡儿乘积推广到任意多个集合的笛卡儿乘积，形成两个一般的映射，使得模糊合取与下确界、模糊析取与上确界保持一致，以此将模糊集的包含关系一般化。

Based upon RMTong's thought of regarding fuzzy model as the set of fuzzy conditional proposition by converting the accurate value into a fuzzy one, the structure of a fuzzy system formed by alloy element adding amount and material characteristic norm is determined, and then the fuzzy model of the system is built.

根据RMTong把模糊模型看作是模糊条件命题的集合这一基本思想，通过精确量的模糊化，确定了合金元素加入量与材料性能指标组成的模糊系统的结构，建立了该系统的模糊模型，为研究合金加入量对材料性能的影响提供了一种普遍意义下的新方法

Chapter 4 establishes the theory of convex fuzzy mappings: The concepts, such as Jensen's inequality, positively homogeneous, infimal convolution, right scalar multiplication and convex hull are introduced. The corresponding theorems are demonstrated by using the parametric representations of fuzzy numbers. In anti-fuzzy number space, the conjugate mapping of convex fuzzy mapping is concerned, and convexities of conjugate set and conjugate mapping of convex fuzzy mapping are proved. The notions of subgradient, subdifferential, differential with respect to convex fuzzy mappings are investigated, which provides the basis of the theory of fuzzy extremum problems.

在第4章中，建立了有关凸模糊映射的理论：建立了关于凸模糊映射的Jensen不等式、模糊正齐次映射、凸模糊映射的下卷积、右数乘和凸包等概念，利用模糊数的参数化表示，给出了相应的定理；在反模糊数空间，对凸模糊映射的共轭也作了探讨，证明了凸模糊映射的共轭集合和共轭映射都是凸的；最后对凸模糊映射的次梯度、次微分和微分等概念进行了研究，为模糊极值理论打下了基础。

In this thesis, we defined duality Fuzzy relation in a pointwise Fuzzy set by a set of duality Fuzzy point. Then we defined Fuzzy congruence and Fuzzy rough set in Fuzzy semigroups.

本文把二维模糊点作成的集合定义为一个点态化模糊集上的模糊二元关系，进一步定义模糊同余关系，并通过它又定义点态化模糊集上模糊粗糙子集，并研究了它们的一些性质。

Based on the method of multi-objective fuzzy optimization and the overhead traveling crane design, a mathematical model of fuzzy optimization for main box beam section is established. A fuzzy goal anthology is constructed. The fuzziness of multi-objective function and constraints is defined. The transformation of the fuzzy constraints set to normal constraints set in real number domain is realized by applying optimal level cut set method. The optimal solution and optimal constraint value of sub-objective function in the feasible field are found using the genetic algorithms.

应用多目标模糊优化方法，以工程设计中桥式起重机箱形主梁设计为实例，建立了桥式起重机箱形主梁多目标模糊优化设计的数学模型，构造了模糊目标集，确定了多目标函数的模糊性和约束的模糊性，并应用最优水平截集法实现了模糊约束到实数论域上的集合转换，从而用遗传算法寻求到子目标函数在可行域空间的最优解和约束最优值。

The notion of random fuzzy variable was first introduced by Baoding Liu in 2002. A fuzzy random variable is a function from a probability space to the set of fuzzy variables.

模糊随机变量定义为从概率空间到模糊变量集合的函数，而随机模糊变量定义为从可信性空间到随机变量集合的函数。

Some statistical properties of product assembly are obtained on the basis of industrial investigation of assembly procedure and analysis of assembly operations of typical products; concept of fuzzy quotient set is introduced to the study of assembly sequence planning and a theoretical frame and algorithm for assembly sequence planning based on product fuzzy quotient set is presented; theory and method of fuzzy automata and fuzzy decision are used in the investigation of sensor based assembly sequence planning and a theoretical frame and algorithm for assembly sequence planning based on information sensing of assembly process are presented; an assembly model for product is put forward; study of product disassemblability is carried on.

论文通过对产品装配工艺的工厂调研及典型产品的装配作业分析得到了有关产品装配的一些统计特征；将模糊商集合的概念引入装配顺序规划的研究中，提出了基于产品模糊商集合的装配顺序规划的理论框架和算法；将模糊自动机及模糊决策的理论和方法应用到基于感知的装配顺序规划的研究中，提出了基于装配过程信息感知的装配顺序规划的理论框架和算法；提出了一种产品装配模型；进行了产品可拆卸性的研究。

Secondly,the experts'power weight was related to two factors:the importanceof the decision makers EID〓 and the relative uniformity or consensus of the evaluation of thedecision makers RAD〓Lastly,based on the study of the fuzzy maximal and minimalaggregation,the evaluation method and criteria on the fuzzy utility were given as a solution.

基于模糊极大集合与模糊极小集合概念，对模糊数表示的方案效用排序问题给出了对模糊效用的评判方法和评判准则

At the early 1980s, is an effective mathematic tool to deal with the uncertain or fuzzy knowledge. Rough rule theory has been widely used in information retrieval, digital library, software engineering and knowledge discovery. Its main idea includes acquisition of rough decision rules by reduction of data under the condition of keeping an invariability of classification capacity. And it basically represents a concept by a pair of approximation sets according to the partition on the given data based on the indiscernibility relation. Rough sets theory is well suited to deal with inconsistency of the rough approximations. So it is a new mathematical tool dealing with vagueness and uncertainty, and has found its applications in many areas such as AI, KDD, pattern recognition and classification and fault diagnosis.

主要完成了以下几个方面的工作：(1)对于高速移动机器人车道线跟踪问题中的规则知识获取，首先阐述了模糊逻辑系统的基本特点、工作原理和基本作法，分析了模糊规则知识获取过于依赖专家经验知识、对于每一个细节几乎都需要人工主观经验的参与、难以处理大数据集样本等方面的不足，进而阐述了粗糙集合理论的基本特点、基于粗糙集合理论的规则获取的关键技术和工作原理，为后续章节中经典粗糙集合模型的扩展及粗糙决策规则集的性能分析做好基础；(2)在经典粗糙集理论及其一般扩展的可变精度粗糙集VPRS的基础上，提出了非均匀可变精度粗糙集模型k-VPRS。

The pattern recognition method mainly counts pattern-recognition method, sentence structure pattern- recognition method, fuzzy set to discern the pattern-recognitions of the method, artificial neural network, etc.

图像识别的方法主要有统计模式识别方法、句法模式识别方法、模糊集合识别方法、人工神经网络的模式识别等。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。