Entropy and Cross‐entropy for Generalized Hesitant Fuzzy Information and Their Use in Multiple Attribute Decision Making

In this paper, we present the entropy, cross‐entropy, and similarity measure for generalized hesitant fuzzy information and discuss their desirable properties. Some measure formulas are developed, and the relationships among them are investigated. We show that the similarity measure and entropy for generalized hesitant fuzzy information can be transformed by each other based on their axiomatic definitions. Then we develop two approaches for solving multiple attribute decision making, in which the attribute values are given in the form of generalized hesitant fuzzy elements (GHFEs). In the first approach, the attribute weight vector is determined by the generalized hesitant fuzzy entropies, and the optimal alternative is obtained by comparing the generalized hesitant fuzzy cross‐entropies between alternatives and positive‐ideal or negative‐ideal solutions; in the second approach, the attribute weight vector is derived from the maximizing deviation method and optimal alternative is obtained by using the technique for order preference by similarly to ideal solution (TOPSIS) method. Finally, an example is provided to illustrate the practicality and effectiveness of the developed approaches.     

Novel hesitant fuzzy linguistic entropy and cross-entropy measures in multiple criteria decision making

In this contribution, we mainly investigate how new entropy and cross entropy measures of hesitant fuzzy linguistic term sets (HFLTSs) can be designed by using the counterparts proposed for linguistic term sets (LTSs). In this circumstance, we intend to point out some drawbacks of the existing entropies, and then extend the theory of entropy and cross entropy measures of HFLTSs by constructing a number of new entropies. Furthermore, we compare the results of the approach being proposed based on the new entropy and cross entropy measures with that of the weight-determining method and the hesitant fuzzy linguistic alternative queuing method (HFL-AQM).     

Uncertain linguistic hesitant fuzzy sets and their application in multi‐attribute decision making

To denote the quantitative and qualitative fuzzy information simultaneously, this paper introduces a new type of fuzzy sets called uncertain linguistic hesitant fuzzy sets, which are denoted by an uncertain linguistic variable with several possible interval membership degrees. Considering the application of this type of fuzzy sets, several basic operational laws are defined, and several properties are studied. Meanwhile, an ordered relationship is introduced. Then, two types of uncertain linguistic hesitant fuzzy aggregation operators are defined. One uses additive measures, and the other is based on λ‐fuzzy measures. Then, a similarity measure is presented, by which models for the optimal weight vector are constructed. After that, an approach to uncertain linguistic hesitant fuzzy multi‐attribute decision making is developed. Finally, an illustrative example for evaluating corporate environmental performance is offered to show the concrete practicality of the procedure.     

基于符号距离和交叉熵的概率犹豫模糊多属性决策方法

针对概率犹豫模糊环境下属性权重完全未知的多属性决策问题,提出基于符号距离和交叉熵的多属性决策方法.首先,定义用于测量决策者犹豫程度的3种概率犹豫模糊元的犹豫度:数值犹豫度,信息不完全度和总犹豫度,基于3种犹豫度提出概率犹豫模糊符号距离;然后,为了避免人为添加元素,定义调和概率犹豫模糊元,并结合信息不完全度提出概率犹豫模糊元的交叉熵;最后,根据概率犹豫模糊元的符号距离和交叉熵构建多属性决策模型,并通过算例验证了该模型的有效性和合理性.     

参数化犹豫模糊熵及其应用

犹豫模糊熵是刻画犹豫模糊集不确定程度的重要工具。针对现有犹豫模糊熵的一些不足,首先基于犹豫模糊集提出犹豫模糊熵的公理化定义,并构造出参数化犹豫模糊熵;其次,通过一些具体数值算例,将新提出的参数化犹豫模糊熵与现有犹豫模糊熵进行对比分析,结果显示所研究的熵能够更加灵活有效地描述信息的未知程度;然后,探究了参数化犹豫模糊熵在多属性决策问题中的应用,使用该熵来确定属性的权重,并借助逼近于理想解排序法（TOPSIS）以及分数函数,提出了一种解决最优方案选取问题的方法;最后,通过具体实例,验证了参数化犹豫模糊熵与所给决策方法具有一定的实用性和可行性。     

区间犹豫模糊熵和区间犹豫模糊相似度

基于犹豫模糊熵的概念,提出了区间犹豫模糊熵和相似度的概念,同时研究了它们之间的相互关系。给出了区间犹豫模糊熵的公理化定义,在此基础上构造了两种形式的熵测度公式,并且证明了它们满足区间犹豫模糊熵的四条公理化准则;依据区间犹豫模糊熵引入了区间犹豫模糊加权熵的概念;提出了区间犹豫模糊相似度的概念,并且研究了区间犹豫模糊环境下的熵和相似度之间的关系。     

指数型犹豫模糊熵在多属性决策中的应用

提出一种指数型犹豫模糊熵,并基于熵权法给出犹豫模糊多属性决策模型.首先,给出犹豫模糊元熵的公理化定义,构造犹豫模糊元的指数型犹豫模糊熵测度公式,并证明指数型犹豫模糊熵测度公式满足犹豫模糊元熵的公理化定义基本准则.在此基础上,引入犹豫模糊集的熵定义和熵测度公式,并证明犹豫模糊集的指数型犹豫模糊熵测度公式同样满足犹豫模糊集...     

基于对称交互熵的犹豫模糊信息相似度及聚类应用

研究面向犹豫模糊信息的聚类方法。首先,定义犹豫模糊相对熵、对称交互熵,并基于信息论的角度提出一个新的犹豫模糊相似度公式;然后,利用相似度公式构造相似系数矩阵,基于编网聚类方法对犹豫模糊集进行聚类;最后,通过算例验证了所提出方法的有效性。     

犹豫模糊信息相似性测度与群一致性测度及群决策应用

针对目前关于犹豫模糊运算与测度的研究中存在的不足,首先给出犹豫模糊熵函数的定义,并将其作为犹豫模糊信息不确定性测度,进而提出犹豫模糊信息特征向量概念,以信息特征向量为出发点对犹豫模糊距离测度和相似性测度展开研究;为优化群决策过程,提出基于完全优先关系的群一致性测度概念并研究其性质;最后,提出基于相似性测度和群一致性测度的群决策方法并结合算例验证所提出方法的有效性.     

Pythagorean犹豫模糊交叉熵的多属性决策方法

针对决策信息为Pythagorean犹豫模糊数的多属性群决策问题,提出一种基于Pythagorean犹豫模糊交叉熵的多属性群决策方法。引入Pythagorean犹豫模糊交叉熵的概念。以Pythagorean犹豫模糊交叉熵作为决策信息差异程度的度量,提出专家权重和属性权重的确定模型。提出一种基于Pythagorean犹豫模糊熵的TOPSIS方法,并通过光伏电站选址案例说明了该方法的可行性和有效性。     

基于概率犹豫模糊熵的多属性决策方法

针对概率犹豫模糊元的多个隶属度和其概率各不相同的特点,提出基于概率犹豫模糊熵的多属性决策方法.首先,定义3种新的概率犹豫模糊熵:模糊熵、犹豫熵和总熵,以分别测量概率犹豫模糊元的模糊性、犹豫性和整体不确定性;然后给出3种熵测度的公理化定义和表达式;最后,根据概率犹豫模糊元的3种熵,构建能够解决属性权重完全未知的多属性决策模型,并通过案例和对比分析验证所提模型的有效性和合理性.     

Inclusion measure for typical hesitant fuzzy sets,the relative similarity measure and fuzzy entropy

Typical hesitant fuzzy sets (THFSs), possessing a finite-set-valued fuzzy membership degrees called typical hesitant fuzzy elements (THFEs), is a special kind of hesitant fuzzy sets. Fuzzy inclusion relationship, as the order structure in fuzzy mathematics, plays an elementary role in the theoretical research and practical applications of fuzzy sets. In this paper, a new partial order for THFEs is defined via the disjunctive semantic meaning of a set, based on which fuzzy inclusion relationship is defined for THFSs. Furthermore, inclusion measures are defined to present the quantitative ranking of every two THFEs and THFSs and different inclusion measures are constructed. The related similarity measure, distance and fuzzy entropy of THFSs are presented and their relationship with inclusion measures are investigated. Finally, an example is given to show that the inclusion measure can be applied effectively in hesitant fuzzy multi-attribute decision making.     

基于指数熵加权的降维犹豫模糊兰氏距离测度及应用

首先提出几种基于兰氏距离的犹豫模糊集距离测度.然后针对两个犹豫模糊数中的隶属度个数不相等问题,提出新的犹豫模糊数降维方案.该方案不需要反复添加最大最小隶属度数值到犹豫模糊数中,不仅很好地保留了数据的原始信息,而且减少了计算距离时的计算量.针对属性权重信息完全未知的情况,采用实际数据信息构造犹豫模糊指数熵,并利用信息熵最小化原则计算得到属性权重.最后利用指数熵加权的降维犹豫模糊兰氏距离测度,结合实际的医疗诊断数据进行实例分析.结果表明,所提出的基于指数熵加权的降维犹豫模糊兰氏距离测度不仅在$\lambda$取不同值时诊断结果一致,而且减少了计算量,提高了诊断效率,对实时、有效的医疗诊断具有一定的应用价值.     

Interval‐Valued Intuitionistic Fuzzy Multiattribute Group Decision Making Based on Cross Entropy Measure and Choquet Integral

In this paper, a new operator called the arithmetic interval‐valued intuitionistic fuzzy Choquet aggregation (AIVIFCA) operator is defined. Since interactions between elements might exist in all their combinations, the generalized Shapley AIVIFCA (GSAIVIFCA) operator is introduced. Further, to simplify the complexity of solving a fuzzy measure, the 2‐additive generalized Shapley AIVIFCA (2AGSAIVIFCA) operator is presented. Moreover, a decision procedure to interval‐valued intuitionistic fuzzy multiattribute group decision making is developed. When the weight vectors on attribute set and expert set are not exactly known, the models for obtaining the optimal fuzzy measures are established by using the defined cross entropy measure and the Shapley function. Finally, a numerical example is provided to illustrate the developed procedure.     

Correlation Coefficients of Interval‐Valued Hesitant Fuzzy Sets and Their Application Based on the Shapley Function

Interval‐valued hesitant fuzzy sets permit the membership degree of an element to have several possible interval values in [0, 1] rather than real numbers, which can well deal with inherent hesitancy and uncertainty in the human decision‐making process. In this paper, we first point out the issue of the existing correlation coefficients of interval‐valued hesitant fuzzy sets and then define several new ones, which do not have to consider the lengths of interval‐valued hesitant fuzzy elements and the arrangement of their possible interval values. Since the assumption that the elements in a set are independent is usually violated, we further define several Shapley weighted correlation coefficients, which consider their interactions. To deal with the situations where the elements are correlative and the weight formation is incompletely known, models for the optimal fuzzy measures on a feature set and on an attribute set are established, respectively. Finally, a procedure to pattern recognition and multiattribute decision making with incomplete weight information and interactive conditions is developed. Meanwhile, the corresponding examples are provided to show the practicality and feasibility of the proposed procedures.     

On Typical Hesitant Fuzzy Prioritized “or” Operator in Multi‐Attribute Decision Making

Since hesitant fuzzy set was proposed, multi‐attribute decision making (MADM) with hesitant fuzzy information, which is also called hesitant fuzzy MADM, has been a hot research topic in decision theory. This paper investigates a special kind of hesitant fuzzy MADM problems in which the decision data are expressed by several possible values, and the evaluative attributes are in different priority levels. Firstly, we introduce the definitions of hesitant fuzzy t‐norm and t‐conorm by extending the notions of t‐norm and t‐conorm to the hesitant fuzzy environment and explore their constructions by means of t‐norms and t‐conorms. Then motivated by the prioritized “or” operator (R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 2008;48:263–274), we develop the typical hesitant fuzzy prioritized “or” operator based on the developed hesitant fuzzy t‐norms and t‐conorms. In this operator, the degree of satisfaction of each alternative in each priority level is derived from a hesitant fuzzy t‐conorm to preserve trade‐offs among the attributes in the same priority level, and the priority weights of attributes are induced by a hesitant fuzzy t‐norm to model the prioritization relationship among attributes. Furthermore, we apply the developed typical hesitant fuzzy prioritized “or” operator to solving the MADM problems in which the decision data are expressed by several possible values and the attributes are in different priority levels. In addition, two numerical examples are given to, respectively, illustrate the applicability and superiority of the developed aggregation operator by comparative analyses with previous research.     

对偶犹豫模糊集的相关系数及其应用

对偶犹豫模糊集因其可以给决策者提供更多的决策信息成为模糊决策的热点研究问题,相关性指标可以用来度量两个模糊信息之间的相关关系,熵可以用来度量模糊信息的不确定程度。提出了一种基于对偶犹豫模糊集相关系数和熵的模糊多属性群决策方法。定义了对偶犹豫模糊集相关系数的概念,讨论了其基本性质;提出了两种对偶犹豫模糊集的熵,在此基础上,给出了模糊多属性群决策的权重确定方法;基于对偶犹豫模糊集相关系数和熵,提出了一种属性权重完全未知条件下的模糊多属性群决策方法;通过案例分析说明了该方法的有效性和可行性。     

区间值对偶犹豫模糊集的相关系数及其应用

区间值对偶犹豫模糊集因其可能隶属度与可能非隶属度均采用区间的形式而更具有一般性,因而得到广泛的应用。相关系数可以用来度量两个模糊信息之间的相关关系。基于区间值对偶犹豫模糊集相关系数提出了一种新的多属性群决策方法。在对偶犹豫模糊集的基础上给出了区间值对偶犹豫模糊集的定义及其基本运算;给出了区间值对偶犹豫模糊集的相关系数的定义及相应的计算公式;构造了确定权重的优化模型;基于区间值对偶犹豫模糊集的相关系数和确定权重的优化模型,提出一种属性权重部分未知的模糊多属性群决策方法,并通过实例说明该方法的有效性和可行性。     

Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making

The uncertainty measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) is important for information discrimination under intuitionistic fuzzy environment. Although many entropy measures and knowledge measures haven been proposed to depict uncertainty of AIFSs, how to measure the uncertainty of AIFSs is still an open topic. The relation between uncertainty and other measures like entropy measures, fuzziness and intuitionism is not clear. This paper introduces uncertainty measures by using new defined divergence-based cross entropy measure of AIFSs. Axiomatic properties of the developed uncertainty measure are analysis, together with the monotony property of uncertainty degree with respect to fuzziness and intuitionism. To adjust the contribution of fuzzy entropy and intuitionistic entropy on the total uncertainty, the proposed cross entropy and uncertainty measures are parameterized. Numerical examples indicate the effectiveness and agility of the biparametric uncertainty measure in quantifying uncertainty degree. Then we apply the cross entropy and uncertainty measures into an optimal model to determine attribute weights in multi-attribute group decision making (MAGDM) problems. A new method for intuitionistic fuzzy MAGDM problems is proposed to show the efficiency of proposed measures in applications. It is demonstrated by application examples that the proposed measures can get reasonable results coinciding with other existing methods.     

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

Zhu et al. (2012) proposed dual hesitant fuzzy set as an extension of hesitant fuzzy sets which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand these sets are more flexible, and provides much more information about the situation. In this paper, the axiom definition of a similarity measure between dual hesitant fuzzy sets is introduced. A new similarity measure considering membership and nonmembership degrees of dual hesitant fuzzy sets has been presented and also it is shown that the corresponding distance measures can be obtained from the proposed similarity measures. To check the effectiveness, the proposed similarity measure is applied in a bidirectional approximate reasoning systems. Mathematical formulation of dual hesitant fuzzy assignment problem with restrictions is presented. Two algorithms based on the proposed similarity measure, are developed to finds the optimal solution of dual hesitant fuzzy assignment problem with restrictions. Finally, the proposed method is illustrated by numerical examples.