广义正交模糊Maclaurin对称平均算子及其应用*

研究广义正交模糊决策环境下的集结算子及其决策应用。针对在信息集成时,需要考虑多个输入变量之间的相关关系以及专家的评价值为广义正交模糊信息的多属性决策问题,提出一种解决广义正交模糊多属性决策问题的方法。考虑到Maclaurin对称平均算子能够反映多个输入变量之间的相关关系,利用该算子集结广义正交模糊信息,提出了广义正交模糊Maclaurin对称平均算子、广义正交模糊加权Maclaurin对称平均算子,并研究了这些算子的性质和特殊情形。提出了基于广义正交模糊集结算子的多属性决策方法,并通过实例验证了其可行性和优势。     

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。     

加型Pythagorean模糊偏好关系的多属性决策方法

考虑Pythagorean模糊偏好关系的多属性决策问题,提出了加性Pythagorean模糊偏好关系的多属性决策方法.基于加性一致性Pythagorean模糊偏好关系提出一种新的Pythagorean模糊权重确定模型.给出了可接受加性一致性Pythagorean模糊偏好关系的定义,并针对不满足可接受加性一致性的Pyth...     

Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making

In this paper, we define the Choquet integral operator for Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet integral average (PFCIA) operator and Pythagorean fuzzy Choquet integral geometric (PFCIG) operator. The operators not only consider the importance of the elements or their ordered positions but also can reflect the correlations among the elements or their ordered positions. It is worth pointing out that most of the existing Pythagorean fuzzy aggregation operators are special cases of our operators. Meanwhile, some basic properties are discussed in detail. Later, we propose two approaches to multiple attribute group decision making with attributes involving dependent and independent by the PFCIA operator and multi‐attributive border approximation area comparison (MABAC) in Pythagorean fuzzy environment. Finally, two illustrative examples have also been taken in the present study to verify the developed approaches and to demonstrate their practicality and effectiveness.     

毕达哥拉斯模糊优先集成算子及其决策应用

作为直觉模糊集的推广形式,毕达哥拉斯模糊数能更好地刻画现实中的不确定性,此外在某些问题上,方案的属性之间往往具有优先关系,针对此类信息的集成问题,将毕达哥拉斯模糊数与优先集成算子相结合,提出了毕达哥拉斯模糊优先集成算子,包括毕达哥拉斯模糊优先加权平均算子和毕达哥拉斯模糊优先加权几何算子,并讨论了这些算子的性质。在此基础上,提出了毕达哥拉斯模糊优先集成算子的多属性决策方法,最后将其应用于国内四家航空公司服务质量评价中,说明了该算子的有效性和可行性。     

Pythagorean模糊幂Bonferroni集成算子及其决策应用

结合幂平均与Bonferroni平均集成算子的优点,定义了毕达哥拉斯模糊幂Bonferroni平均和毕达哥拉斯模糊加权幂Bonferroni平均集成算子,其不仅考虑了数据信息之间的整体均衡性,还考虑了属性之间可能存在的相互关联关系。研究了这些集成算子的优良性质和特殊情形,并在此基础上提出了一种属性间存在相关性的毕达哥拉斯模糊多属性决策方法。将其应用于国内航空公司的服务质量评价中,并与现有方法进行分析比较,验证了所提方法的有效性和可行性。     

Hesitant Fuzzy Linguistic Maclaurin Symmetric Mean Operators and their Applications to Multi‐Criteria Decision‐Making Problem

Due to the limitation of knowledge and the vagueness of human being thinking, decision makers prefer to use hesitant fuzzy linguistic sets (HFLSs) to estimate alternatives. Some methods of HFLSs have been researched based on the more familiar means such as the arithmetic mean and the geometric mean; however, Maclaurin symmetric mean (MSM) that can be used to reflect the interrelationships among input arguments have not been applied to solve hesitant fuzzy linguistic multi‐criteria decision‐making problems. In this paper, two hesitant fuzzy linguistic harmonic averaging operators are proposed: the hesitant fuzzy linguistic MSM (HFLMSM) operator and the hesitant fuzzy linguistic weighted MSM (HFLWMSM) operator. Furthermore, an approach based on the HFLWMSM operator is proposed. Finally, to verify the validity and feasibility of the proposed approach, an illustrative example and corresponding comparison analysis are presented in the end.     

Some Hesitant Fuzzy Einstein Aggregation Operators and Their Application to Multiple Attribute Group Decision Making

Hesitant fuzzy sets, as a new generalized type of fuzzy set, has attracted scholars’ attention due to their powerfulness in expressing uncertainty and vagueness. In this paper, motivated by the idea of Einstein operation, we develop a family of hesitant fuzzy Einstein aggregation operators, such as the hesitant fuzzy Einstein Choquet ordered averaging operator, hesitant fuzzy Einstein Choquet ordered geometric operator, hesitant fuzzy Einstein prioritized weighted average operator, hesitant fuzzy Einstein prioritized weighted geometric operator, hesitant fuzzy Einstein power weighted average operator, and hesitant fuzzy Einstein power weighted geometric operator. And we also study some desirable properties and generalized forms of these operators. Then, we apply these operators to deal with multiple attribute group decision making under hesitant fuzzy environments. Finally, a numerical example is provided to illustrate the practicality and validity of the proposed method.     

Pythagorean模糊对称交叉熵及加权投影的多属性决策

在Pythagorean模糊多属性决策问题中,以欧式距离等距离测度为基础计算各备选方案与正、负理想解的距离,可能产生与正理想解距离更近的待选方案却与负理想解的距离也更近,导致所得方案排序结果并不能真实反映各备选方案的优劣程度.为有效克服决策结果的逆序问题,提出满足对称性、有界性的Pythagorean模糊对称交叉熵,进...     

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。     

Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

Recently, a new model based on Pythagorean fuzzy set (PFS) has been presented to manage the uncertainty in real‐world decision‐making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision‐making problems with PFSs, we propose an extended technique for order preference by similarity to ideal solution method to deal effectively with them. In this approach, we first propose a score function based comparison method to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. Then, we define a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. Afterward, a revised closeness is introduced to identify the optimal alternative. At length, a practical example is given to illustrate the developed method and to make a comparative analysis.     

FH-OWA算子及其在模糊多属性决策中的应用

现实中,绝大多数的决策是模糊决策,而决策结果很大程度上取决于聚合算子的选取。为了使信息聚合更加科学合理,研究了H-OWA算子（Heronian ordered weighted averaging operator）。鉴于H-OWA算子的优点和局限性,提出了基于三角模糊数的FH-OWA算子（fuzzy Heronian ordered weighted averaging operator）,并研究了其幂等性、单调性、有界性及交替性。最后,将FH-OWA算子应用于模糊多属性决策中,并与原文献进行了比较和分析,结果表明FH-OWA算子在信息聚合时侧重所有决策者意见的“一致性”,而不是个别专家的权威性。     

Symmetric Pythagorean Fuzzy Weighted Geometric/Averaging Operators and Their Application in Multicriteria Decision‐Making Problems

Pythagorean fuzzy sets (PFSs), originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.     

泛在学习中基于模糊多属性决策的学习控制模型

作为一种新型的学习范式,泛在学习具有去计算机化的特性。在这种新型的学习环境下,课堂的组织具有分布式松散的特点,学习者不必受制于地理位置空间和时间的限制,从而拥有更好的学习自主性选择权以及更佳的学习体验,但这也对学习者的学习控制提出了更高的要求。文中提出了一种基于模糊多属性决策的学习控制模型,根据备选知识点的掌握程度、重要程度以及与当前知识点的依赖程度给出备选知识点的排序以供学习者选择,引导学习者完成对知识的掌握。     

Some q‐Rung Orthopai Fuzzy Bonferroni Mean Operators and Their Application to Multi‐Attribute Group Decision Making

In the real multi‐attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of q‐rung orthopair fuzzy sets (q‐ROFs) is that the sum of the qth power of the membership degree and the qth power of the degrees of non‐membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with q‐rung orthopair fuzzy numbers (q‐ROFNs) to propose the q‐rung orthopair fuzzy BM (q‐ROFBM) operator, the q‐rung orthopair fuzzy weighted BM (q‐ROFWBM) operator, the q‐rung orthopair fuzzy geometric BM (q‐ROFGBM) operator, and the q‐rung orthopair fuzzy weighted geometric BM (q‐ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on q‐ROFWBM and q‐ROFWGBM operators are very useful to deal with MAGDM problems.     

A Note on Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

In this note, we point out an error to the proof of Theorem 3.4 in Zhang and Xu (Int J Intell Syst 2014;29(12):1061–1078) by a counterexample. We find that the inequality (i.e., ) with respect to the degrees of indeterminacy of any three Pythagorean fuzzy numbers in the proof of Theorem 3.4 in Zhang and Xu's paper is not valid. A new proof is provided in this note.     

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.     

基于矢量投影法的武器系统模糊多属性决策评估

武器系统效能评估是属于多属性决策问题,一般对武器系统的描述及评判准则通常以语词描述,因此语意不明及模糊不清。为了克服这些问题,文中基于模糊多属性决策法,由武器系统性能规格数据及专家的意见来建立武器系统评判准则和模糊函数,通过将加权后的方案矢量投影到理想解上,再将负理想解投影到方案矢量上,进而在两个投影的基础上构建方案与理想解的相对贴近度,用以确定多属性决策方案的优劣次序。最后以迫击炮系统评估实例来说明评估的过程。     

Projection Model for Fusing the Information of Pythagorean Fuzzy Multicriteria Group Decision Making Based on Geometric Bonferroni Mean

As a new generalization of fuzzy sets, Pythagorean fuzzy sets (PFSs) can availably handle uncertain information more flexibly in the process of decision making. Through synthesizing the Bonferroni mean and the geometric mean, the geometric Bonferroni mean (GBM) captures the interrelationship of the input arguments. Considering the interrelationship among the input arguments, we introduce GBM into Pythagorean fuzzy situations and expand its applied fields. Under the Pythagorean fuzzy environment, we develop the Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni mean (WPFGBM) operators describing the interrelationship between arguments and some special properties of them are also investigated. Then, we employ the WPFGBM operator to fuse the information in the Pythagorean fuzzy multicriteria group decision making (PFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the projection model, we present its extension and further design a new method for the application of PFMCGDM. Finally, an example is given to elaborate on the performance of our proposed method.     

Fuzzy Generalized Prioritized Weighted Average Operator and its Application to Multiple Attribute Decision Making

The prioritized weighted average (PWA) operator was originally introduced by Yager. The prominent characteristic of the PWA operator is that it takes into account prioritization among attributes and decision makers. By combining the idea of generalized mean and PWA operator, we propose a new prioritized aggregation operator called fuzzy generalized prioritized weighted average (FGPWA) operator for aggregating triangular fuzzy numbers. The properties of the new aggregation operator are studied out and their special cases are examined. Furthermore, based on the FGPWA operator, an approach to deal with multiple attribute group decision making problems under triangular fuzzy environments is developed. Finally, a practical example is provided to illustrate the multiple attribute group decision making process.