On some correlation coefficients in Pythagorean fuzzy environment with applications

A Pythagorean fuzzy set (PFS) is an extension of an intuitionistic FS that can be extended by relaxing the restriction on the grades of satisfaction and dissatisfaction. PFS is a powerful tool for dealing with uncertainty and vagueness. Correlation analysis of PFSs is a hot research topic in Pythagorean fuzzy (PF) theory and has practical applications in many areas, such as decision-making, pattern recognition, medical diagnosis, engineering, and so forth. In this communication, we introduce some novel correlation coefficients in the PF-environment satisfying the condition that the correlation coefficient of two PFSs is one if and only if the two sets are equal. We discuss the properties and applications of the proposed measures in pattern recognition, medical diagnosis, multicriteria decision-making, and clustering analysis. Furthermore, the superiority of our proposed correlation coefficients over some existing ones is also established. We also extend the correlation coefficients to interval-valued PFSs.     

Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications

In this paper, we presented 10 similarity measures between Pythagorean fuzzy sets (PFSs) based on the cosine function by considering the degree of membership, degree of nonmembership and degree of hesitation in PFSs. Then, we applied these similarity measures and weighted similarity measures between PFSs to pattern recognition and medical diagnosis. Finally, two illustrative examples are given to demonstrate the efficiency of the similarity measures for pattern recognition and medical diagnosis.     

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.     

The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets

Pythagorean fuzzy sets (PFSs) as a new generalization of fuzzy sets (FSs) can handle uncertain information more flexibly in the process of decision making. In our real life, we also may encounter a hesitant fuzzy environment. In view of the effective tool of hesitant fuzzy sets (HFSs) for expressing the hesitant situation, we introduce HFSs into PFSs and extend the existing research work of PFSs. Concretely speaking, this paper considers that the membership degree and the non-membership degree of PFSs are expressed as hesitant fuzzy elements. First, we propose a new concept of hesitant Pythagorean fuzzy sets (HPFSs) by combining PFSs with HFSs. It provides a new semantic interpretation for our evaluation. Meanwhile, the properties and the operators of HPFSs are studied in detail. For the sake of application, we focus on investigating the normalization method and the distance measures of HPFSs in advance. Then, we explore the application of HPFSs to multi-criteria decision making (MCDM) by employing the technique for order preference by similarity to ideal solution (TOPSIS) method. A new extension of TOPSIS method is further designed in the context of MCDM with HPFSs. Finally, an example of the energy project selection is presented to elaborate on the performance of our approach.     

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.     

A new trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted averaging operator and its application for enterprise location

Recently some new models based on Pythagorean fuzzy sets (PFSs) have been proposed to deal with the uncertainty in multiple attribute group decision making (MAGDM) problems. In this paper, considering linguistic variables and entropic, we propose a new trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted averaging operator to solve MAGDM problems. Next, we study some main properties by utilizing some operational laws of the trapezoidal Pythagorean fuzzy linguistic variables. Finally, a numerical example concerning the enterprise location is given to illustrate the practicality and effectiveness of the proposed operator.     

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.     

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.     

Multiattribute group decision-making based on Pythagorean fuzzy Einstein prioritized aggregation operators

Pythagorean fuzzy set (PFS) is a powerful tool to deal with the imprecision and vagueness. Many aggregation operators have been proposed by many researchers based on PFSs. But the existing methods are under the hypothesis that the decision-makers (DMs) and the attributes are at the same priority level. However, in real group decision-making problems, the attribute and DMs may have different priority level. Therefore, in this paper, we introduce multiattribute group decision-making (MAGDM) based on PFSs where there exists a prioritization relationship over the attributes and DMs. First we develop Pythagorean fuzzy Einstein prioritized weighted average operator and Pythagorean fuzzy Einstein prioritized weighted geometric operator. We study some of its desirable properties such as idempotency, boundary, and monotonicity in detail. Moreover we propose a MAGDM approach based on the developed operators under Pythagorean fuzzy environment. Finally, an illustrative example is provided to illustrate the practicality of the proposed approach.     

Novel neutrality operation–based Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis

Pythagorean fuzzy sets (PFSs) accommodate more uncertainties than L_x the intuitionistic fuzzy sets and hence its applications are more extensive. Under the PFS, the objective of this paper is to develop some new operational laws and their corresponding weighted geometric aggregation operators. For it, we define some new neutral multiplication and power operational laws by including the feature of the probability sum and the interaction coefficient into the analysis to get a neutral or a fair treatment to the membership and nonmembership functions of PFSs. Associated with these operational laws, we define some novel Pythagorean fuzzy weighted, ordered weighted, and hybrid neutral geometric operators for Pythagorean fuzzy information, which can neutrally treat the membership and nonmembership degrees. The desirable relations and the characteristics of the proposed operators are studied in details. Furthermore, a multiple attribute group decision-making approach based on the proposed operators under the Pythagorean fuzzy environment is developed. Finally, an illustrative example is provided to show the practicality and the feasibility of the developed approach.     

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.     

Pythagorean fuzzy Bonferroni means based on T-norm and its dual T-conorm

For multiple-attribute decision making problems in Pythagorean fuzzy environment, few existing aggregation operators consider interrelationships among the attributes. To deal with this issue, this article extends the Bonferroni means to Pythagorean fuzzy sets (PFSs) to provide Pythagorean Fuzzy Bonferroni means. We first extend t-norm and its dual t-conorm to propose the generalized operational laws for PFSs, which can be considered as the extensions of the known ones. Based on these new laws, Pythagorean fuzzy weighted Bonferroni mean operator and Pythagorean fuzzy weighted geometric Bonferroni mean operator are developed, both of them can capture the correlations among Pythagorean fuzzy input arguments and their desired properties and special cases are also investigated in detail. At last, a novel approach is proposed based on the developed operators with its effectiveness being proved by an investment selection problem.     

Multicriteria group decision-making method using the distances-based similarity measures between intuitionistic trapezoidal fuzzy numbers

The Hamming and Euclidean distances between intuitionistic trapezoidal fuzzy numbers and the distances-based similarity measures are proposed in this study, then an intuitionistic trapezoidal fuzzy multicriteria group decision-making method is established using the similarity measures and expected weight values, in which linguistic values of intuitionistic trapezoidal fuzzy numbers for linguistic terms are used to assess alternatives with respect to qualitative criteria and criteria weights. We establish simple and exact formulae to solve the multicriteria group decision-making problem based on the similarity measures between the ideal alternative and each alternative, the ranking order of all the alternatives and the best one can be determined by the proposed similarity measures. Finally, an illustrative example demonstrates the implementation process of the technique.     

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.     

Pythagorean Dombi fuzzy aggregation operators with application in multicriteria decision-making

A Pythagorean fuzzy set, an extension of intuitionistic fuzzy sets, is very helpful in representing vague information that occurs in real world scenarios. The Dombi operators with operational parameters, have excellent flexibility. Due to the flexible nature of these Dombi operational parameters, this research paper introduces some new aggregation operators under Pythagorean fuzzy environment, including Pythagorean Dombi fuzzy weighted arithmetic averaging (PDFWAA) operator, Pythagorean Dombi fuzzy weighted geometric averaging (PDFWGA) operator, Pythagorean Dombi fuzzy ordered weighted arithmetic averaging operator and Pythagorean Dombi fuzzy ordered weighted geometric averaging operator. Further, this paper presents several advantageous characteristics, including idempotency, monotonicity, boundedness, reducibility and commutativity of preceding operators. By utilizing PDFWAA and PDFWGA operators, this article describes a multicriteria decision-making (MCDM) technique for solving MCDM problems. Finally, a numerical example related to selection of a leading textile industry is presented to illustrate the applicability of our proposed technique.     

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.     

Pythagorean fuzzy preference ranking organization method of enrichment evaluations

Recently, a new extension of fuzzy sets, Pythagorean fuzzy sets (PFS), has attracted a lot of attention from scholars in various fields of research. Due to PFS’s powerfulness in modeling the imprecision of human perception in multicriteria decision-making (MCDM) problems, this paper aims to extend the classical preference ranking organization method of enrichment evaluations (PROMETHEE) into the Pythagorean fuzzy environment. The proposed method takes not only the weights related to different criteria but also the preference relations as Pythagorean fuzzy numbers, therefore providing a broader range of choices for the decision-maker to express their preferences. Five properties are put forward to regulate the designing of both intuitionistic and Pythagorean fuzzy PROMETHEE (PF-PROMETHEE) preference functions. Furthermore two illustrative examples are given to demonstrate the detailed procedure of PF-PROMETHEE, and comparisons are made to distinguish the differences among our proposed method, the classical PROMETHEE and intuitionistic PROMETHEE. The results show that PF-PROMETHEE is effective, comprehensive, and applicable to a wide range of MCDM problems.     

Vague集的多目标模糊决策方法

在进行了阐述Vague集的相关概念和运算后,对多目标模糊决策进行了定义,给出多目标模糊决策的基本思路,并对已有的基于Vague集的多目标模糊决策方法进行了分析,得出现有Vague集的多目标模糊决策方法存在着缺陷,从而提出了一种新的多目标模糊决策的Vague集方法,该方法利用一个新的评分函数对方案进行排序,选出最优方案.最后给出其相关性质,证明其解决了现有方法的缺陷,并通过实例阐明本文方法的有效性和优越性.     

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。     

Pythagorean fuzzy multicriteria group decision making through similarity measure based on point operators

In this paper, a series of similarity measures based on point operators for Pythagorean fuzzy sets are proposed. Using the proposed similarity measures, two new aggregation operators, viz., Pythagorean fuzzy‐dependent averaging operator and Pythagorean fuzzy‐dependent geometric operator, are developed. The advantage of using these operators is that the influence of unfair arguments of aggregated results could be eliminated, since the associated weights are taken from the aggregated Pythagorean fuzzy arguments. Also, the proposed operators have the capability to adjust the degree of aggregated arguments with the controlling parameters. To establish the application potentiality of those operators, a methodology for solving multicriteria group decision‐making problems having Pythagorean fuzzy arguments is developed. A numerical example is provided to demonstrate the proficiency of the proposed method. The achieved results are compared with the results of other existing technique.