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.     

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 Dombi aggregation operators and its applications in multiple attribute decision-making

The operations of -norm and -conorm, developed by Dombi, were generally known as Dombi operations, which may have a better expression of application if they are presented in a new form of flexibility within the general parameter. In this paper, we use Dombi operations to create a few Pythagorean fuzzy Dombi aggregation operators: Pythagorean fuzzy Dombi weighted average operator, Pythagorean fuzzy Dombi order weighted average operator, Pythagorean fuzzy Dombi hybrid weighted average operator, Pythagorean fuzzy Dombi weighted geometric operator, Pythagorean fuzzy Dombi order weighted geometric operator, and Pythagorean fuzzy Dombi hybrid weighted geometric operator. The distinguished feature of these proposed operators is examined. At that point, we have used these operators to build up a model to remedy the multiple attribute decision-making issues under Pythagorean fuzzy environment. Ultimately, a realistic instance is stated to substantiate the created model and to exhibit its applicability and viability.     

Multicriteria decision-making using Archimedean aggregation operators in Pythagorean hesitant fuzzy environment

In multicriteria decision-making (MCDM), the existing aggregation operators are mostly based on algebraic t-conorm and t-norm. But, Archimedean t-conorms and t-norms are the generalized forms of t-conorms and t-norms which include algebraic, Einstein, Hamacher, Frank, and other types of t-conorms and t-norms. From that view point, in this paper the concepts of Archimedean t-conorm and t-norm are introduced to aggregate Pythagorean hesitant fuzzy information. Some new operational laws for Pythagorean hesitant fuzzy numbers based on Archimedean t-conorm and t-norm have been proposed. Using those operational laws, Archimedean t-conorm and t-norm-based Pythagorean hesitant fuzzy weighted averaging operator and weighted geometric operator are developed. Some of their desirable properties have also been investigated. Afterwards, these operators are applied to solve MCDM problems in Pythagorean hesitant fuzzy environment. The developed Archimedean aggregation operators are also applicable in Pythagorean fuzzy contexts also. To demonstrate the validity, practicality, and effectiveness of the proposed method, a practical problem is considered, solved, and compared with other existing method.     

基于毕达哥拉斯模糊幂加权平均算子的多属性群决策方法

研究了毕达哥拉斯模糊环境下的多属性群决策问题。首先,将毕达哥拉斯模糊信息引入幂平均加权算子,提出毕达哥拉斯模糊幂加权平均（PFPWA） 算子,并研究所提算子的基本性质。然后,在毕达哥拉斯模糊数（PFN） 为信息输入的框架内,提出基于毕达哥拉斯模糊幂加权平均算子的群决策方法。所提出的方法使用毕达哥斯拉信息使得决策者的信息表达更加灵活,并且在信息集结过程中采用幂加权平均算子能够同时考虑专家权威与评估信息的可信度。最后,通过案例分析验证了所提方法的可行性和有效性。     

Some power Maclaurin symmetric mean aggregation operators based on Pythagorean fuzzy linguistic numbers and their application to group decision making

The power average (PA) operator and Maclaurin symmetric mean (MSM) operator are two important tools to handle the multiple attribute group decision‐making (MAGDM) problems, and the combination of two operators can eliminate the influence of unreasonable information from biased decision makers (DMs) and can capture the interrelationship among any number of arguments. The Pythagorean fuzzy linguistic set (PFLS) is parallel to the intuitionistic linguistic set (ILS), which is more powerful to convey the uncertainty and ambiguity of the DMs than ILS. In this paper, we propose some power MSM aggregation operators for Pythagorean fuzzy linguistic information, such as Pythagorean fuzzy linguistic power MSM operator and Pythagorean fuzzy linguistic power weighted MSM (PFLPWMSM) operator. At the same time, we further discuss the properties and special cases of these operators. Then, we propose a new method to solve the MAGDM problems with Pythagorean fuzzy linguistic information based on the PFLPWMSM operator. Finally, some illustrative examples are utilized to show the effectiveness of the proposed method.     

Pythagorean fuzzy power aggregation operators in multiple attribute decision making

In this paper, we utilize power aggregation operators to develop some Pythagorean fuzzy power aggregation operators: Pythagorean fuzzy power average operator, Pythagorean fuzzy power geometric operator, Pythagorean fuzzy power weighted average operator, Pythagorean fuzzy power weighted geometric operator, Pythagorean fuzzy power ordered weighted average operator, Pythagorean fuzzy power ordered weighted geometric operator, Pythagorean fuzzy power hybrid average operator, and Pythagorean fuzzy power hybrid geometric operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean fuzzy multiple attribute decision‐making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

A novel multiple-attribute group decision-making method based on q-rung orthopair fuzzy generalized power weighted aggregation operators

In this paper, a novel approach is developed to deal with multiple-attribute group decision-making (MAGDM) problem under q-rung orthopair fuzzy environment. Firstly, some operators have been proposed to aggregate q-rung orthopair fuzzy information, such as the q-rung orthopair fuzzy generalized power averaging (q-ROFGPA) operator, the q-rung orthopair fuzzy generalized power weighted averaging (q-ROFGPWA) operator, the q-rung orthopair fuzzy generalized power geometric (q-ROFGPG) operator, and the q-rung orthopair fuzzy generalized power weighted geometric (q-ROFGPWG) operator. In addition, some desirable properties and special cases of these operators are discussed. Second, a novel approach is developed to solve MAGDM problem under the q-rung orthopair fuzzy environment based on the proposed q-ROFGPWA and q-ROFGPWG operators. Finally, a practical example is given to illustrate the application of the proposed method, and further the sensitivity analysis and comparative analysis are carried out.     

A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making

As an extension of fuzzy set, a Pythagorean fuzzy set has recently been developed to model imprecise and ambiguous information in practical group decision‐making problems. The aim of this paper is to introduce a novel aggregation method for the Pythagorean fuzzy set and analyze possibilities for its application in solving multiple attribute decision‐making problems. More specifically, a new Pythagorean fuzzy aggregation operator called the Pythagorean fuzzy induced ordered weighted averaging‐weighted average (PFIOWAWA) operator is developed. This operator inherits main characteristics of both ordered weighted average operator and induced ordered weighted average to aggregate the Pythagorean fuzzy information. Some of main properties and particular cases of the PFIOWAWA operator are studied. A method based on the proposed operator for multiple attribute group decision making is developed. Finally, we present a numerical example of selection of research and development projects to illustrate applicability of the new approach in a multiple attribute group decision‐making problem.     

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.     

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.     

基于优先集成算子的直觉乘法偏好关系共识模型*

针对专家之间具有优先关系时直觉乘法偏好关系下群体共识决策问题,提出一种基于优先集成算子的直觉乘法偏好关系共识方法。为了有效集结专家偏好信息,提出直觉乘法优先加权平均(IMPWA)算子和直觉乘法优先加权几何(IMPWG)算子,并研究其相关性质;定义直觉乘法偏好关系的共识度和接近度概念,据此完成非共识偏好信息的识别和修正,构建一种迭代共识算法。案例表明该方法的可行性和有效性。     

Cubic fuzzy Einstein aggregation operators and its application to decision-making

In this paper, we define some Einstein operations on cubic fuzzy set (CFS) and develop three arithmetic averaging operators, which are cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating cubic fuzzy data. The CFEHWA operator generalises both the CFEWA and CFEOWA operators. Furthermore, we develop various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply CFEHWA operator to multiple attribute decision-making with cubic fuzzy data. Finally, a numerical example is constructed to demonstrate the established approach.     

犹豫模糊Einstein几何算子及应用

研究了属性权重信息已知条件下的犹豫模糊信息集结算子及其在多属性群决策问题中的应用。基于Einstein运算定义了犹豫模糊Einstein和、犹豫模糊Einstein积以及犹豫模糊Einstein幂运算,并且研究了犹豫模糊Einstein运算法则间的关系。提出了四种犹豫模糊信息集结算子,即犹豫模糊Einstein加权几何（HFEWG）算子、犹豫模糊Einstein有序加权几何（HFEOWG）算子、犹豫模糊Einstein混合几何（HFEHG）算子和犹豫模糊Einstein诱导有序加权几何（HFEIOWG）算子,并分析了这些算子的性质。给出了基于HFEIOWG算子的犹豫模糊多属性决策方法,并结合投资公司对金融产品的选择来验证提出的决策方法是可行有效的。     

Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making

The power Bonferroni mean (PBM) operator can relieve the influence of unreasonable aggregation values and also capture the interrelationship among the input arguments, which is an important generalization of power average operator and Bonferroni mean operator, and Pythagorean fuzzy set is an effective mathematical method to handle imprecise and uncertain information. In this paper, we extend PBM operator to integrate Pythagorean fuzzy numbers (PFNs) based on the interaction operational laws of PFNs, and propose Pythagorean fuzzy interaction PBM operator and weighted Pythagorean fuzzy interaction PBM operator. These new Pythagorean fuzzy interaction PBM operators can capture the interactions between the membership and nonmembership function of PFNs and retain the main merits of the PBM operator. Then, we analyze some desirable properties and particular cases of the presented operators. Further, a new multiple attribute decision making method based on the proposed method has been presented. Finally, a numerical example concerning the evaluation of online payment service providers is provided to illustrate the validity and merits of the new method by comparing it with the existing methods.     

改进加权支持度的Pythagorean模糊交叉幂均群决策方法

针对毕达哥拉斯环境下的多属性群决策问题,首先,将毕达哥拉斯模糊数和幂均算子相结合,创造性地拓展了一种新的改进加权支持度;然后,基于此提出了改进加权支持度的毕达哥拉斯模糊交叉幂均算子,并讨论了该算子的性质,进而建立一种毕达哥拉斯模糊背景下能够反映决策属性间相互作用的决策方法;最后,将其应用于智慧城市的评价中。实例分析表明,该方法可以解决实际的多属性群决策问题,并可以进一步应用到智慧物流、模式识别、人工智能等领域。     

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.     

Prioritized aggregation operators based on the priority degrees in multicriteria decision-making

We consider the multicriteria decision-making (MCDM) problems where there exists a prioritization relationship over the criteria. We introduce the concept of the priority degree. Then we give three kinds of prioritized aggregation operators based on the priority degrees: the prioritized averaging operator with the priority degrees, the prioritized scoring operator with the priority degrees, and the prioritized ordered weighted averaging operator with the priority degrees. Some desired properties of these prioritized aggregation operators are also investigated. The priority degree plays an important role in the prioritized MCDM problems. We also investigate how to select a proper priority degree according to the giving decision information. By using an illustrative example, we show that the prioritized aggregation operators based on the priority degrees provide the decision-makers more choices and they are more flexible in the process of decision-making.     

毕达哥拉斯犹豫模糊集成算子及其决策应用

针对毕达哥拉斯犹豫模糊多属性决策中,集成算子的重要作用以及集成算子不完善的情况,较为系统地研究了毕达哥拉斯犹豫模糊集成算子。为此,在毕达哥拉斯模糊数的运算和运算法则基础上,定义了毕达哥拉斯犹豫模糊有序加权平均算子（PHFOWA）、广义有序加权平均算子（GPHFOWA）和混合平均算子（PHFHA）,以及毕达哥拉斯犹豫模糊有序加权几何平均算子（PHFOWG）、广义有序加权几何平均算子（GPHFOWG）和混合几何平均算子（PHFHG）,并结合数学归纳法,分别给出了它们的计算公式,讨论了它们的有界性、单调性和置换不变性等性质。建立了基于毕达哥拉斯犹豫模糊集成算子的多属性决策方法,并应用算例和相关方法比较说明了决策方法的可行性与有效性。     

Some q-rung orthopair fuzzy Hamy mean operators in multiple attribute decision-making and their application to enterprise resource planning systems selection

In this paper, the Hamy mean (HM) operator, weighted HM (WHM), dual HM (DHM) operator, and dual WHM (WDHM) operator under the q-rung orthopair fuzzy sets (q-ROFSs) is studied to propose the q-rung orthopair fuzzy HM (q-ROFHM) operator, q-rung orthopair fuzzy WHM (q-ROFWHM) operator, q-rung orthopair fuzzy DHM (q-ROFDHM) operator, and q-rung orthopair fuzzy weighted DHM (q-ROFWDHM) operator and some of their desirable properties are investigated in detail. Then, we apply these operators to multiple attribute decision-making problems. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.