广义犹豫模糊软集的相似度量及其应用

在广义模糊软集和犹豫模糊软集的基础上给出广义犹豫模糊软集的概念,并研究广义犹豫模糊软集的相似度量。首先利用三种犹豫模糊集合的包含度,构造犹豫模糊集间的相似度量公式。然后在犹豫模糊集相似度基础上给出广义犹豫模糊软集相似度量的公理化定义,并构造广义犹豫模糊软集的相似度量公式,这些公式可以计算参数集不同时两个广义犹豫模糊软集间的相似度。最后利用广义犹豫模糊软集相似度量方法构造了一种决策方法,并将这个决策方法应用于环境治理问题中。通过实例验证了所提出方法的可行性和有效性。     

区间二型模糊相似度与包含度

相似度与包含度是模糊集合理论中的两个重要概念,但对于二型模糊集合的研究还较为少见．鉴于此,提出了新的区间二型模糊相似度与包含度．首先选择了二者的公理化定义;然后基于公理化定义提出了新的计算公式,并讨论了二者的相互转换关系;最后通过实例来验证二者的性能,并将区间二型模糊相似度与Yang-Shih聚类方法相结合,用于高斯区间二型模糊集合的聚类分析,得到了合理的层次聚类树．仿真实例表明新测度具有一定的实用价值．     

二型模糊相似度及其应用

二型模糊集可以直接处理高度不确定性,并且具有很强的实际应用背景。基于二型模糊相似度的公理化定义,给出了新的二型模糊相似度计算公式。进一步,将二型模糊相似度与Yang-Shih方法相结合,用于二型模糊数据的聚类分析,聚类结果与Yang-Lin的结果进行了比较,实例表明新的相似度更合理。此外,基于二型模糊相似度,讨论了二型模糊信息系统的属性约简问题,给出了相应约简的分辨函数法,并通过实例说明了该方法的具体计算步骤。     

基于包含度的区间二型模糊粗糙集

基于区间二型模糊包含度的公理化定义,给出了新的区间二型模糊包含度计算公式.进一步,通过包含度定义了区间二型模糊粗糙集,并讨论了它的一些基本性质.最后,利用区间二型模糊粗糙集研究了连续域决策信息系统的属性约简,给出了新的约简方法.实例说明了该约简方法的具体计算步骤,并且通过实验验证了该算法的有效性和可行性.     

On generalized similarity measures for Pythagorean fuzzy sets and their applications to multiple attribute decision-making

In this paper, we develop a new and flexible method for Pythagorean fuzzy decision-making using some trigonometric similarity measures. We first introduce two new generalized similarity measures between Pythagorean fuzzy sets based on cosine and cotangent functions and prove their validity. These similarity measures include some well-known Pythagorean fuzzy similarity measures as their particular and limiting cases. The measures are demonstrated to satisfy some very elegant properties which prepare the ground for applications in different areas. Further, the work defines a generalized hybrid trigonometric Pythagorean fuzzy similarity measure and discuss its properties with particular cases. Then, based on the generalized hybrid trigonometric Pythagorean fuzzy similarity measure, a method for dealing with multiple attribute decision-making problems under Pythagorean fuzzy environment is developed. Finally, a numerical example is given to demonstrate the flexibility and effectiveness of the developed approach in solving real-life problems.     

Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions

In this paper, we deal with the problem of classification of interval type-2 fuzzy sets through evaluating their distinguishability. To this end, we exploit a general matching algorithm to compute their similarity measure. The algorithm is based on the aggregation of two core similarity measures applied independently on the upper and lower membership functions of the given pair of interval type-2 fuzzy sets that are to be compared. Based on the proposed matching procedure, we develop an experimental methodology for evaluating the distinguishability of collections of interval type-2 fuzzy sets. Experimental results on evaluating the proposed methodology are carried out in the context of classification by considering interval type-2 fuzzy sets as patterns of suitable classification problem instances. We show that considering only the upper and lower membership functions of interval type-2 fuzzy sets is sufficient to (i) accurately discriminate between them and (ii) judge and quantify their distinguishability.     

Uncertainty measures for interval type-2 fuzzy sets

Fuzziness (entropy) is a commonly used measure of uncertainty for type-1 fuzzy sets. For interval type-2 fuzzy sets (IT2 FSs), centroid, cardinality, fuzziness, variance and skewness are all measures of uncertainties. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions use a Representation Theorem for IT2 FSs. Formulas for computing the cardinality, fuzziness, variance and skewness of an IT2 FS are derived. These definitions should be useful in IT2 fuzzy logic systems design using the principles of uncertainty, and in measuring the similarity between two IT2 FSs.     

Some results on information measures for complex intuitionistic fuzzy sets

Complex intuitionistic fuzzy sets (CIFSs), modeled by complex-valued membership and nonmembership functions with codomain the unit disc in a complex plane, handle two-dimensional information in a single set. Under this environment, the primary objective of the present study is to introduce some novel formulae of information measures (similarity measures, distance measures, entropies, and inclusion measures) and discuss the transformation relationships among them. To demonstrate the efficiency of the proposed similarity measures, we apply it to pattern recognition problem and a detailed comparative analysis is conducted with some of the existing measures. Further, algorithms based on proposed measures are developed for handing multicriteria decision-making problems and their working is illustrated with the help of an example. Besides this, the practicality of the proposed similarity measure is demonstrated by developing a clustering algorithm under CIFS environment.     

Ⅱ-型模糊集截集的定义及特征研究

经典模糊集的截集概念是经典模糊集合与经典集合联系的桥梁,对于II-型模糊集,该文在分析II-型模糊集、区间值II-型模糊集、I-型模糊集以及经典集合之间关系的基础上,定义了II-型模糊集的截集概念,分析了II-型模糊集截集的特征,仿真证明了II-型模糊集截集的有效性,为基于II-型模糊集的决策、聚类等实际应用提供了新的方法。     

Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications

In this paper we propose an entropy measure for interval-valued intuitionistic fuzzy sets, which generalizes three entropy measures defined independently by Szmidt, Wang and Huang, for intuitionistic fuzzy sets. We also give an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets. In particular, the proposed entropy measure for interval-valued intuitionistic fuzzy sets can yield a similarity measure. Several illustrative examples are given to demonstrate the practicality and effectiveness of the proposed formulas. We apply the similarity measure to solve problems on pattern recognitions, multi-criteria fuzzy decision making and medical diagnosis.     

Ⅱ型模糊控制综述

Ⅱ型模糊集合是传统Ⅰ型模糊集合的扩展,其特征是隶属度值本身为模糊集合.基于Ⅱ型模糊集合的Ⅱ型模糊控制器可以同时有效地处理语言和数据不确定性,在高小确定场合具有明显超过相应Ⅰ型控制器的性能表现.本文首先对Ⅱ型模糊集合及系统理论进行了概述,然后对Ⅱ型非自适应模糊控制器Ⅱ型自适应模糊控制器和Ⅱ型自组织模糊控制器的研究进展分别...     

The generalized Dice similarity measures for Pythagorean fuzzy multiple attribute group decision making

The Pythagorean fuzzy set (PFS) is characterized by two functions expressing the degree of membership and the degree of nonmembership, which square sum of them is equal or less than 1. It was proposed as a generalization of a fuzzy set to deal with indeterminate and inconsistent information. In this study, we shall present some novel Dice similarity measures of PFSs and the generalized Dice similarity measures of PFSs and indicates that the Dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized Dice similarity measures in some parameter values. Then, we propose the generalized Dice similarity measures-based multiple attribute group decision-making models with Pythagorean fuzzy information. Then, we apply the generalized Dice similarity measures between PFSs to multiple attribute group decision making. Finally, an illustrative example is given to demonstrate the efficiency of the similarity measures for selecting the desirable ERP system.     

II型模糊集合与系统研究进展

Ⅱ型模糊集合是传统Ⅰ型模糊集合的扩展,其本质是模糊集合中隶属度值的再次模糊化表示.Ⅱ型模糊集合可以直接处理模糊规则的不确定性,是解决现实环境高不确定性问题的有效手段.本文首先简要给出了Ⅱ型模糊集合与系统的基本概念,然后分别回顾了广义和区间Ⅱ型模糊理论的发展历史.接着分别讨论了广义和区间Ⅱ型模糊系统的计算复杂性问题研究进展,并进一步介绍了基于区间Ⅱ型模糊集合的词计算理论发展状况.最后给出了本文的结论和进一步研究问题的展望.     

New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule

The centroid of an interval type-2 fuzzy set (IT2 FS) provides a measure of the uncertainty of such a FS. Its calculation is very widely used in interval type-2 fuzzy logic systems. In this paper, we present properties about the centroid of an IT2 FS. We also illustrate many of the general results for a T2 fuzzy granule (FG) in order to develop some understanding about the uncertainty of the FG in terms of its vertical and horizontal dimensions. At present, the T2 FG is the only IT2 FS for which it is possible to obtain closed-form formulas for the centroid, and those formulas are in this paper.     

Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems

In this paper, we propose some distance measures between type-2 fuzzy sets, and also a new family of utmost distance measures are presented. Several properties of different proposed distance measures have been introduced. Also, we have introduced a new ranking method for the ordering of type-2 fuzzy sets based on the proposed distance measure. The proposed ranking method satisfies the reasonable properties for the ordering of fuzzy quantities. Some properties such as robustness, order relation have been presented. Limitations of existing ranking methods have been studied. Further for practical use, a new method for selecting the best alternative, for group decision making problems is proposed. This method is illustrated with a numerical example.     

犹豫模糊集的α-截集及其应用

经典截集是联系模糊集和清晰集的桥梁。犹豫模糊集作为经典模糊集的拓展,它的相关理论研究还不够深入,特别是它与经典Ⅰ型模糊集以及其他模糊集之间的关系还缺少讨论。通过分析犹豫模糊集与Ⅰ型模糊集、区间Ⅱ型模糊集之间的关系,引入了犹豫模糊集的α-截集的概念并讨论其性质,根据该截集推导出犹豫模糊集的分解（表示）定理和更普适的扩展原则。通过分析相关性质及仿真实例,说明了犹豫模糊集的截集概念的合理性,为犹豫模糊多属性决策和聚类分析等问题提供了新的方法。这些结果也极大丰富了犹豫模糊集的相关基础理论。     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义,使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合,并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型：离散、半连通、连通及复合型,并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support,CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合,并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后,探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.     

一种应用软粗糙模糊集的多属性决策方法

将软粗糙模糊集应用于多属性决策问题,用软粗糙模糊集分析模糊知识表达系统,定义了软模糊决策系统、决策分类模糊软集依赖度、条件双射软集对决策分类模糊软集的重要性、软模糊决策系统的约简、软模糊决策系统的决策规则等概念,借助这些概念给出了一种基于软粗糙模糊集的多属性决策算法,通过实例分析说明了该算法的可行性。