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.     

Group Decision Making Under Interval‐Valued Multiplicative Intuitionistic Fuzzy Environment Based on Archimedean t‐Conorm and t‐Norm

The main focus of this paper is to investigate group decision‐making (GDM) method under interval‐valued multiplicative intuitionistic fuzzy environment based on Archimedean t‐conorm and t‐norm. First of all, some operations laws are proposed for interval‐valued multiplicative intuitionistic fuzzy elements, which is an extension of multiplicative intuitionistic fuzzy operations developed earlier by other scholars. The effectiveness of these proposed operations is illustrated with some numerical examples. Then, a series of aggregation operators are proposed and the desirable properties are also studied. This paper reveals that some existing multiplicative intuitionistic fuzzy and interval‐valued multiplicative intuitionistic fuzzy aggregation operators are the special cases of the operators proposed in this paper. Finally, a GDM method based on proposed operators under interval‐valued multiplicative intuitionistic fuzzy environment is proposed, and a real case about annual evaluation for personnel of Zhejiang University of Finance and Economics is presented to illustrate the effectiveness of the proposed method.     

A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making

The objective of this article is to extend and present an idea related to weighted aggregated operators from fuzzy to Pythagorean fuzzy sets (PFSs). The main feature of the PFS is to relax the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted averaging (PFEWA), Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA), generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA), and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA), are proposed in this article. Some desirable properties corresponding to it have also been investigated. Furthermore, these operators are applied to decision‐making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality, and effectiveness of the new approach. Finally, a systematic comparison between the existing work and the proposed work has been given.     

Fuzzy Multicriteria Decision‐Making Methods: A Comparative Analysis

Given a multicriteria decision‐making problem, an obvious question emerges: Which method should be used to solve it? Although some efforts had been made, the question remains open. The aim of this contribution is to compare a set of multicriteria decision‐making methods sharing three features: same fuzzy information as input data, the need of a data normalization procedure, and quite similar information processing. We analyze the rankings produced by fuzzy MULTIMOORA, fuzzy TOPSIS (with two normalizations), fuzzy VIKOR, and fuzzy WASPAS with different parameterizations, over 1200 randomly generated decision problems. The results clearly show their similarities and differences, the impact of the parameters settings, and how the methods can be clustered, thus providing some guidelines for their selection and usage.     

Fundamental Properties of Interval‐Valued Pythagorean Fuzzy Aggregation Operators

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems with interval‐valued Pythagorean fuzzy sets (IVPFSs). First, the concept, operational laws, score function, and accuracy function of IVPFSs are defined. Then, based on the operational laws, two interval‐valued Pythagorean fuzzy aggregation operators are developed for aggregating the interval‐valued Pythagorean fuzzy information, such as interval‐valued Pythagorean fuzzy weighted average (IVPFWA) operator and interval‐valued Pythagorean fuzzy weighted geometric (IVPFWG) operator. A series of inequalities of aggregation operators are studied. Later, we develop some interval‐valued Pythagorean fuzzy point operators. Moreover, combining the interval‐valued Pythagorean fuzzy point operators with IVPFWA operator, we present some interval‐valued Pythagorean fuzzy point weighted averaging (IVPFPWA) operators, which can adjust the degree of the aggregated arguments with some parameters. Then, we propose an interval‐valued Pythagorean fuzzy ELECTRE method to solve uncertainty MAGDM problem. Finally, an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its 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。     

A Novel Improved Accuracy Function for Interval Valued Pythagorean Fuzzy Sets and Its Applications in the Decision‐Making Process

The objective of this work is to present an improved accuracy function for the ranking order of interval‐valued Pythagorean fuzzy sets (IVPFSs). Shortcomings of the existing score and accuracy functions in interval‐valued Pythagorean environment have been overcome by the proposed accuracy function. In the proposed function, degree of hesitation between the element of IVPFS has been taken into account during the analysis. Based on it, multicriteria decision‐making method has been proposed for finding the desirable alternative(s). Finally, an illustrative example for solving the decision‐making problem has been presented to demonstrate application of the proposed approach.     

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.     

A Novel Correlation Coefficients between Pythagorean Fuzzy Sets and Its Applications to Decision‐Making Processes

Pythagorean fuzzy set (PFS) is one of the most successful in terms of representing comprehensively uncertain and vague information. Considering that the correlation coefficient plays an important role in statistics and engineering sciences, in this paper, after pointing out the weakness of the existing correlation coefficients between intuitionistic fuzzy sets (IFSs), we propose a novel correlation coefficient and weighted correlation coefficient formulation to measure the relationship between two PFSs. Pairs of membership, nonmembership, and hesitation degree as a vector representation with the two elements have been considered during formulation. Numerical examples of pattern recognition and medical diagnosis have been taken to demonstrate the efficiency of the proposed approach. Results computed by the proposed approach are compared with the existing indices.     

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.     

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.     

A Method for Decision Making Based on Generalized Aggregation Operators

A new method for decision making based on generalized aggregation operators is presented. We use a concept that it is known in the literature as the index of maximum and minimum level (IMAM). This index uses distance measures and other techniques that are very useful for decision making. In this paper, it is suggested a generalization by using generalized and quasi‐arithmetic means. As a result, it is obtained the generalized and quasi‐arithmetic weighted IMAM (GWIMAM and quasi‐WIMAM) and the generalized ordered weighted averaging IMAM (GOWAIMAM) and the quasi‐OWAIMAM operator. The main advantage is that it provides a parameterized family of aggregation operators that includes a wide range of special cases such as the generalized IMAM and the OWAIMAM. Thus, the decision maker may take decisions according to his degree of optimism and considering ideals in the decision process. We also develop an application of the new approach in a decision‐making problem regarding product selection.     

Some q‐Rung Orthopair Fuzzy Aggregation Operators and their Applications to Multiple‐Attribute Decision Making

The q‐rung orthopair fuzzy sets (q‐ROFs) are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic 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. Under these environments, we propose the q‐rung orthopair fuzzy weighted averaging operator and the q‐rung orthopair fuzzy weighted geometric operator to deal with the decision information, and their some properties are well proved. Further, based on these operators, we presented two new methods to deal with the multi‐attribute decision making problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.     

Intuitionistic Fuzzy Geometric Bonferroni Means and Their Application in Multicriteria Decision Making

As a useful aggregation technique, the Bonferroni mean (BM) can capture the interrelationship between input arguments and has been a hot research topic recently. Based on the classic BM, many BM operators have been proposed and developed, such as the weighted BM, the generalized BM, the intuitionistic fuzzy BM, and so on. However, these BM operators are all based on the averaging mean, which is one of the basic aggregation approaches and focuses on the group opinion and another basic one is the geometric mean, which gives more importance to the individual opinions. To combine with the geometric mean and the BM, in this paper, we propose the geometric BM, the weighted geometric BM, and the generalized weighted geometric BM. These new geometric BMs can reflect the geometric interrelationship between the individual criterion and other criteria and keep the main advantage of BM. Furthermore, we investigate the geometric BMs under the intuitionistic fuzzy environment, which is more common phenomenon in modern life and develop three intuitionistic fuzzy geometric Bonferroni mean operators, i.e., the intuitionistic fuzzy geometric Bonferroni mean (IFGBM), the intuitionistic fuzzy weighted geometric Bonferroni mean (IFWGBM), and the intuitionistic fuzzy generalized weighted geometric Bonferroni mean (IFGWGBM) and study their desirable properties, such as idempotency, commutativity, monotonicity, and boundedness. Finally, on the basis of the IFWGBM and IFGWGBM operators, we propose an approach to multicriteria decision making under the intuitionistic fuzzy environment, and a practical example is provided to illustrate our results.     

Support-intuitionistic模糊集集成算子及其决策应用

介绍Support-intuitionistic模糊集(SIFS)的概念,定义SIFS运算法则和客观权重,在此基础上提出SIFS加权算术平均算子和加权几何平均算子并证明相关定理。建立基于SIFS模糊信息的决策方法及定义SIFS得分函数和精确函数,利用SIFS支持度确定目标属性的权重使得属性权重成为客观值,加强集成结果的客观性。通过实例验证了该决策方法适用于处理基于SIFS模糊信息环境的决策问题。     

Multicriteria Decision‐Making Method Based on Similarity Measures under Single‐Valued Neutrosophic Refined and Interval Neutrosophic Refined Environments

In this paper, we propose three similarity measure methods for single‐valued neutrosophic refined (SVNR) sets and interval neutrosophic refined (INR) sets based on Jaccard, Dice, and Cosine similarity measures of SVN‐sets and interval neutrosophic sets. Furthermore, we suggest two multicriteria decision‐making (MCDM) methods under SVNR environment and INR environment, and give applications of proposed MCDM methods. Finally, we suggest a consistency analysis method for proposed similarity measures between INR‐sets and give an application to demonstrate process of the method.     

A Novel Method for Multiattribute Decision Making with Interval‐Valued Pythagorean Fuzzy Linguistic Information

Pythagorean fuzzy set (PFS), proposed by Yager (2013), is a generalization of the notion of Atanassov's intuitionistic fuzzy set, which has received more and more attention. In this paper, first, we define the weighted Minkowski distance with interval‐valued PFSs. Second, inspired by the idea of the Pythagorean fuzzy linguistic variables, we define a new fuzzy variable called interval‐valued Pythagorean fuzzy linguistic variable set (IVPFLVS), and the operational laws, score function, accuracy function, comparison rules, and distance measures of the IVPFLVS are defined. Third, some aggregation operators are presented for aggregating the interval‐valued Pythagorean fuzzy linguistic information such as the interval‐valued Pythagorean fuzzy linguistic weighted averaging (IVPFLWA), interval‐valued Pythagorean fuzzy linguistic ordered weighted averaging (IVPFLOWA) , interval‐valued Pythagorean fuzzy linguistic hybrid averaging, and generalized interval‐valued Pythagorean fuzzy linguistic ordered weighted average operators. Fourth, some desirable properties of the IVPFLWA and IVPFLOWA operators, such as monotonicity, commutativity, and idempotency, are discussed. Finally, based on the IVPFLWA or interval‐valued Pythagorean fuzzy linguistic geometric weighted operator, a practical example is provided to illustrate the application of the proposed approach and demonstrate its practicality and effectiveness.     

A Novel Method of Ranking Hesitant Fuzzy Values for Multiple Attribute Decision‐Making Problems

Despite of several generalizations of fuzzy set theory, the notion of hesitant fuzzy set (HFS), which permits the membership having a set of possible values, is interesting and very useful in modeling real‐life problems with anonymity. In this article, we introduce a new score function for ranking hesitant fuzzy elements (HFEs), which are the fundamental units of HFSs. Comparison with the existing score function shows that the proposed method meets all the well‐known properties of a ranking measure and has no counterintuitive examples. On the basis of the relationships between the aggregation operators for HFEs, we derive a series of interesting properties of the new score function. Finally, we apply the proposed score function to solve the hesitant fuzzy multiattribute decision‐making problems.     

Knowledge Engineering for Rough Sets Based Decision‐Making Models

In this paper, a review of decision‐making models based on the rough set theory is presented. The use of these techniques allows for the presence of uncertainty in computer models that are developed for decision making, and to formulate the decision‐making models using the experiences of previous decisions made. Since the formulation of these models differs from the classical approach of decision‐making models, in this paper, the models are analyzed and a method is proposed for its implementation.