基于信息测度的勾股模糊决策模型及其应用

相对于直觉模糊集，勾股模糊集能够更为全面和有效地表达描述复杂问题中的不确定和非一致信息，使其受到了广泛研究。对于属性评价值为勾股模糊数并且属性指标权重信息数据完全未知的多属性决策问题，以提出的勾股模糊信息测度为基础，设计了新的多属性决策模型。该模型运用对数函数设计了一种新的勾股模糊数信息熵计算方法；引入了勾股模糊相似度概念，并结合对数行数提出勾股模糊数相似度的衡量方法，随后挖掘出勾股模糊数的信息熵和相似度之间的内在联系；运用提出的勾股模糊熵和相似度计算方法，构建新的多属性决策模型，并进行应用研究。实验结果表明，提出的模型合理有效，同时拓展了模型的使用范围。     

基于勾股模糊语言幂加权平均算子的多属性群体决策方法

针对多属性群决策问题,采用能够方便专家参考语言集信息进行评价并且取值灵活的勾股模糊语言集进行了处理。首先,基于语言集和勾股模糊集的距离测度给出了勾股模糊语言数距离测度的定义与相关性质;然后,以勾股模糊语言数的距离测度作为幂均（PA）算子的距离度量,提出了勾股模糊语言幂加权平均（PFLPWA）算子用以对群决策过程中不同专家评价矩阵进行融合,并同时在融合过程中考虑专家评价的差异性;最后,基于PFLPWA算子构建了勾股模糊语言环境下的群体决策新方法,并通过案例分析检验了PFLPWA算子应用于群决策中的有效性和适用性。     

基于勾股模糊集评价的三支决策方法

针对勾股模糊三支决策概率阈值难以确定的问题,文中提出基于优化表示的勾股模糊三支决策概率阈值确定方法.首先从优化的视角研究一对对偶模型,利用KKT条件证明该对偶模型与决策粗糙集模型的等价性.然后,在确定勾股模糊集评价的三支决策概率阈值时引入对偶模型,基于勾股模糊数非线性排序法建立一对非线性规划模型,证明模型最优解的存在性与唯一性.最后,采用优化技术搜索模型最优解,并提出基于勾股模糊集评价的三支决策方法.算例及对比分析表明文中方法能有效克服现有方法难以确定勾股模糊三支决策概率阈值的不足.     

基于模糊数模糊集的模式识别

讨论了模糊数模糊集的模式识别问题.首先,引入了模糊数的两种排序方法及模糊数模糊集的概念,并以此给出了模糊数fuzzy集的隶属原则;其次,给出了模糊数模糊集的模式识别的实例;最后,为了度量所得结论的可靠程度,引进了可信度的概念,并提出了可信度的计算公式.     

基于同构Frank t-模与s-模的勾股模糊 Frank集结算子及其应用

运用单位区间上的自同构构造一种适用于勾股模糊环境下的同构Frank t-模与其对偶s-模,进而定义勾股模糊集的广义运算法则,并探究新法则的相关性质.应用新的运算法则提出勾股模糊Frank加权平均(PFFWA)算子与勾股模糊Frank加权几何(PFFWG)算子,证明算子的相关性质.利用PFFWA与PFFWG算子提出一种解决勾股模糊多属性决策问题的新方法.通过解决航空公司服务质量评估问题,对比分析新方法与现存的决策方法,进而表明新方法的可行性和灵活性, 并验证了新方法具有反馈决策者态度特征的能力.     

直觉模糊集（数）间相似度量新方法

直觉模糊集概念产生于模糊集概念,自Atanassov提出这个概念以来,已得到了众多研究者的关注并被应用到不同的领域。作为直觉模糊理论的一个重要研究内容,研究者已在不同文献中提出多种不同的直觉模糊集相似度量方法,但这些方法在一些特殊情况下并不总是有效。指出了影响直觉模糊集（数）相似度量的因素,分析了现有方法存在不足的原因,提出了一种综合考虑隶属度、非隶属度、犹豫度、核及其相互影响后的新的相似度量方法,指出并证明了该方法所具有的新的性质。数字实例表明该方法可以克服现存几种方法的缺陷,结果符合人们直觉,具有更强的区分数据能力。     

勾股模糊偏好关系及其在群体决策中的应用

以区间模糊偏好关系(IVFPR)和直觉模糊偏好关系(IFPR)的理论框架为依据,将勾股模糊数(PFN)引入偏好关系中,定义勾股模糊偏好关系(PFPR)和加性一致性PFPR.然后,提出标准化勾股模糊权重向量(PFWV)的概念,并给出构造加性一致性PFPR的转换公式.为获取任意给定的PFPR的权重向量,建立以给定的PFPR与构造的加性一致性PFPR偏差最小为目标的优化模型.针对多个勾股模糊偏好关系的集结,利用能够有效处理极端值并满足关于序关系单调的勾股模糊加权二次(PFWQ)算子作为集结工具.进一步,联合PFWQ算子和目标优化模型提出一种群体决策方法.最后,通过案例分析表明所提出方法的实用性和可行性.     

勾股模糊H平均算法应用于多媒体图像系统选择

提出了一种勾股模糊H平均算法应用多媒体图像系统选择问题。首先,定义了基于t-模和t-余模的勾股模糊数运算;讨论了勾股模糊Heronian平均算法的三个特征性质和经常使用的特例;然后构建了新的勾股模糊决策模型,该模型在构建过程中能够挖掘输入数据之间关联性,还提高了决策的使用范围;最后,将构建的模型应用于多媒体图像系统选择案例来验证有效性。     

勾股模糊H平均决策算法及应用

研究了勾股模糊数信息环境下属性值间存在内在关联性的多属性决策问题。首先定义了基于t-模和t-余模的勾股模糊数运算；将Heronian平均融入到聚合算子的构建过程中；讨论了勾股模糊Heronian平均算法的3个特征性质和经常使用的特例。然后构建了改进的勾股模糊决策模型，该模型在考虑输入属性值之间关联性的同时，提高了决策的使用范围。最后通过多属性决策实例验证了改进的决策模型合理有效。     

一种新的直觉模糊集距离及其在决策中的应用

首先针对直觉模糊集距离中是否包含直觉模糊集通过隶属度、非隶属度以及犹豫度这三种信息,以及直觉模糊集距离是否满足相应距离度量的条件对其进行了详细分析,发现现有方法都是直接将犹豫度直接引入到直觉模糊集距离中,从而会产生不一致性。鉴于此,定义了一种新的直觉模糊集距离度量方法,其不仅考虑隶属度和非隶属度信息,同时还考虑犹豫度对隶属度和非隶属度的分配,从而间接地将犹豫度也引入到直觉模糊集距离中。其次,证明了所提距离度量满足距离度量条件,并结合实例将其与现有距离度量方法进行比较分析,说明了新方法的合理性。最后,将所提出方法应用于多准则模糊决策中,进一步说明了新方法的有效性和可行性。     

Pythagorean Fuzzy Clustering Analysis: A Hierarchical Clustering Algorithm with the Ratio Index‐Based Ranking Methods

The Pythagorean fuzzy set introduced by R. R. Yager in 2014 is a useful tool to model imprecise and ambiguous information appearing in decision and clustering problems. In this study, we present a general type of distance measure for Pythagorean fuzzy numbers (PFNs) and propose a novel ratio index‐based ranking method of PFNs. The novel ranking method of PFNs has more powerful ability to discriminate the magnitude of PFNs than the existing ranking methods for PFNs, which is further extended to compare the magnitude of interval‐valued Pythagorean fuzzy numbers (IVPFNs). The IVPFN is a new extension of PFN, which is parallel to interval‐valued intuitionistic fuzzy number. We introduce a general type of distance measure for IVPFNs. Afterwards, we study a kind of clustering problems in Pythagorean fuzzy environments in which the evaluation values are expressed by PFNs and/or IVPFNs and develop a novel Pythagorean fuzzy agglomerative hierarchical clustering approach. In the proposed clustering method, we define the concept of the dissimilarity degree between two clusters for each criterion and introduce the clustering procedure in the criteria level. To take all the criteria into account, we also introduce the overall clustering procedure, which is based on the overall dissimilarity degrees for a fixed aggregation operator such as the commonly used weighted arithmetic average operator or the ordered weighted averaging operator. In the overall clustering process, (1) we present a deviation degree‐based method to derive the weights of criteria and further obtain the overall clustering results if the weights of criteria are completely unknown; (2) we employ the ratio index‐based ranking method of IVPFNs to obtain the overall clustering results if the weights of criteria are given in advance and are expressed by IVPFNs. The salient feature of the proposed clustering method is that it not only can address the clustering problems in which the weights of criteria are not given precisely in advance but also can manage simultaneously the PFNs and IVPFNs data.     

A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making

The Pythagorean fuzzy set, as a new extension of intuitionistic fuzzy set, has recently been developed to manage the complex uncertainty in practical group decision problems. The purpose of this article is to develop a new decision method based on similarity measure to address multiple criteria group decision making problems within Pythagorean fuzzy environment based on Pythagorean fuzzy numbers (PFNs). The contribution of this article is fivefold: (1) An accuracy function of PFNs is defined and a new ranking method for PFNs is proposed; (2) new Pythagorean fuzzy aggregating operators are developed; (3) a novel similarity measure for PFNs is presented, and some desirable properties are discussed; (4) a simple and effective Pythagorean fuzzy group decision method is introduced; and (5) The proposed method is applied to address the selection problem of photovoltaic cells.     

Pythagorean Fuzzy Information Measures and Their Applications

Pythagorean fuzzy set (PFS), originally proposed by Yager, is more capable than intuitionistic fuzzy set (IFS) to handle vagueness in the real world. The main purpose of this paper is to investigate the relationship between the distance measure, the similarity measure, the entropy, and the inclusion measure for PFSs. The primary goal of the study is to suggest the systematic transformation of information measures (distance measure, similarity measure, entropy, inclusion measure) for PFSs. For achieving this goal, some new formulae for information measures of PFSs are introduced. To show the efficiency of the proposed similarity measure, we apply it to pattern recognition, clustering analysis, and medical diagnosis. Some illustrative examples are given to support the findings and also demonstrate their practicality and effectiveness of similarity measure between PFSs.     

Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision-making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision-making with PFSs.     

Pythagorean fuzzy TOPSIS for multicriteria group decision-making with unknown weight information through entropy measure

In this study, a new technique for order preference by similarity to ideal solution (TOPSIS)-based methodology is proposed to solve multicriteria group decision-making problems within Pythagorean fuzzy environment, where the information about weights of both the decision makers (DMs) and criteria are completely unknown. Initially, generalized distance measure for Pythagorean fuzzy sets (PFSs) is defined and used to initiate a new Pythagorean fuzzy entropy measure for computing weights of the criteria. In the decision-making process, at first, weights of DMs are computed using TOPSIS through the geometric distance model. Then, weights of the criteria are determined using the entropy weight model through the newly defined entropy measure for PFSs. Based on the evaluated criteria weights, TOPSIS is further applied to obtain the score value of alternatives corresponding to each decision matrix. Finally, the score values of the alternatives are aggregated with the calculated DMs’ weights to obtain the final ranking of the alternatives to avoid the loss of information, unlike other existing methods. Several numerical examples are considered, solved, and compared with the existing methods.     

Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making

The main feature of Pythagorean fuzzy sets is that it is characterized by five parameters, namely membership degree, nonmembership degree, hesitancy degree, strength of commitment about membership, and direction of commitment. In this paper, we first investigate four existing comparison methods for ranking Pythagorean fuzzy sets and point out by examples that the method proposed by Yager, which considers the influence fully of the five parameters, is more efficient than the other ones. Later, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the five parameters of Pythagorean fuzzy sets. Based on the proposed distance measures, we present some similarity measures of Pythagorean fuzzy sets. Furthermore, a multiple criteria Pythagorean fuzzy group decision‐making approach is proposed. Finally, a numerical example is provided to illustrate the validity and applicability of the presented group decision‐making method.     

Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

The Pythagorean fuzzy set (PFS) which is an extension of intuitionistic fuzzy set, is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in a variety of fields. Whereas, how to measure PFSs’ distance appropriately is still an open issue. It is well known that the square root of Jensen–Shannon divergence is a true metric in the probability distribution space which is a useful measure of distance. On account of this point, a novel divergence measure between PFSs is proposed by taking advantage of the Jensen–Shannon divergence in this paper, called as PFSJS distance. This is the first work to consider the divergence of PFSs for measuring the discrepancy of data from the perspective of the relative entropy. The new PFSJS distance measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of PFSs. Then, numerical examples demonstrate that the PFSJS distance can avoid generating counter-intuitive results which is more feasible, reasonable and superior than existing distance measures. Additionally, a new algorithm based on the PFSJS distance measure is designed to solve the problems of medical diagnosis. By comparing the different methods in the medical diagnosis application, it is found that the new algorithm is as efficient as the other methods. These results prove that the proposed method is practical in dealing with the medical diagnosis problems.     

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.     

The Properties of Continuous Pythagorean Fuzzy Information

In practical decision‐making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support (membership) degree and the against (nonmembership) degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets (PFS), as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers (PFNs), which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail.