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。     

广义犹豫模糊信息集成及其多属性群决策

在犹豫模糊环境下,主要研究了基于阿基米德范数的广义信息集成算法,并提出了一种新的多属性群决策方法。基于阿基米德T-范数和S-范数,定义了广义犹豫模糊运算法则;运用新定义的广义犹豫模糊运算法则,提出了广义犹豫模糊有序加权平均(G-HFOWA)算子,研究了其优良性质;探讨了在某些特殊情况下,广义犹豫模糊有序加权平均算子将转化为一些常见的犹豫模糊信息集成算子,包括犹豫模糊有序加权平均算子、犹豫模糊Einstein有序加权平均算子、犹豫模糊Hamacher有序加权平均算子以及犹豫模糊Frank有序加权平均算子;基于广义信息集成算子,构建了一种新的犹豫模糊多属性群决策方法,并将其应用于区域经济协调发展研究过程中,以验证提出的决策方法是可行的与有效的。     

Pythagorean hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making

Hamacher product is a t‐norm and Hamacher sum is a t‐conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Hamacher operations to develop some Pythagorean hesitant fuzzy aggregation operators: Pythagorean hesitant fuzzy Hamacher weighted average (PHFHWA) operator, Pythagorean hesitant fuzzy Hamacher weighted geometric (PHFHWG) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average (PHFHOWA) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric (PHFHOWG) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted average (PHFHIOWA) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted geometric (PHFHIOWG) operator, Pythagorean hesitant fuzzy Hamacher induced correlated aggregation operators, Pythagorean hesitant fuzzy Hamacher prioritized aggregation operators, and Pythagorean hesitant fuzzy Hamacher power aggregation operators. The special cases of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean hesitant fuzzy multiple attribute decision making problems. Finally, a practical example for green supplier selections in green supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

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.     

Hesitant interval-valued Pythagorean fuzzy VIKOR method

In this article, we investigate multiple attribute decision-making problems with hesitant interval-valued Pythagorean fuzzy information. First, the concepts of hesitant interval-valued Pythagorean fuzzy set are defined, and the operation laws, the score function, and accuracy function have been developed. Then several distance measures for hesitant interval-valued Pythagorean fuzzy values have been presented including the Hamming distance, Euclidean distance, and generalized distance, and so on. Based on the operational laws, a series of aggregation operators have been developed including the hesitant interval-valued Pythagorean fuzzy weighted averaging (HIVPFWA) operator, the hesitant interval-valued Pythagorean fuzzy geometric weighted averaging (HIVPFGWA) operator, the hesitant interval-valued Pythagorean fuzzy ordered weighed averaging (HIVPFOWA) operator, and hesitant interval-valued Pythagorean fuzzy ordered weighed geometric averaging (HIVPFOWGA) operator. By using the generalized mean operator, we also develop the generalized hesitant interval-valued Pythagorean fuzzy weighed averaging (GHIVPFWA) operator, the generalized hesitant interval-valued Pythagorean fuzzy weighed geometric averaging (GHIVPFWGA) operator, the generalized hesitant interval-valued Pythagorean fuzzy ordered weighted averaging (GHIVPFOWA) operator, and generalized hesitant interval-valued Pythagorean fuzzy ordered weighted geometric averaging (GHIVPFOWGA) operator operator. We further develop several hybrid aggregation operators including the hesitant interval-valued Pythagorean fuzzy hybrid averaging (HIVPFHA) operator and the generalized hesitant interval-valued Pythagorean fuzzy hybrid averaging (GHIVPFHA) operator. Based on the distance measures and the aggregation operators, we propose a hesitant interval-valued Pythagorean fuzzy VIKOR method to solve multiple attribute decision problems with multiple periods. Finally, an illustrative example for evaluating the metro project risk is given to demonstrate the feasibility and effectiveness of the proposed method.     

New exponential operational laws and their aggregation operators for interval‐valued Pythagorean fuzzy multicriteria decision‐making

In this article, we define two new exponential operational laws about the interval‐valued Pythagorean fuzzy set (IVPFS) and their corresponding aggregation operators. However, the exponential parameters (weights) of all the existing operational laws of IVPFSs are crisp values in IVPFS decision‐making problems. As a supplement, this paper first introduces new exponential operational laws of IVPFS, where bases are crisp values or interval numbers and exponents are interval‐valued Pythagorean fuzzy numbers. The prominent characteristic of these proposed operations is studied. Based on these laws, we develop some new weighted aggregation operators, namely the interval‐valued Pythagorean fuzzy weighted exponential averaging operator and the dual interval‐valued Pythagorean fuzzy weighted exponential averaging. Finally, a decision‐making approach is presented based on these operators and illustrated with some numerical examples to validate the developed approach.     

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.     

改进的区间犹豫算子应用于物流企业选择决策

针对现有区间犹豫模糊Hamacher算子存在的缺陷，构建了一种基于改进的区间犹豫模糊Hamacher加权算子的群决策方法。在分析现有区间犹豫模糊Hamacher算子不能满足幂等性的基础上，定义新的区间犹豫模糊Hamacher四则运算；提出两种改进的区间犹豫模糊Hamacher加权算子，包括改进的区间犹豫模糊Hamacher有序加权平均（I-IVHFHOWA）算子和改进的区间犹豫模糊Hamacher有序加权几何（I-IVHFHOWG）算子，并详细探究它们的常用算子形式以及算子之间的内在联系；建立基于I-IVHFHOWA算子和I-IVHFHOWG算子的物流企业选择决策模型，并通过实例说明模型的有效性。     

Some q-rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the $q\mathrm{t}\mathrm{h}$ power of the membership and the $q\mathrm{t}\mathrm{h}$ power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.     

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.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

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.     

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.     

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.     

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.     

Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

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.     

Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision‐making process

In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision‐making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.     

基于三角犹豫Einstein算法的多属性决策模型

针对属性信息为三角犹豫模糊信息的多属性决策问题,结合Einstein运算,构建了一种基于三角犹豫模糊Einstein集成算法的多属性决策方法。首先,考虑到决策信息为三角犹豫模糊数且属性间存在一定的内在联系,基于三角犹豫模糊数的运算法则,提出了三角犹豫模糊Einstein加权平均（THFEWA）算子和三角犹豫模糊Einstein加权几何（THFEWG）算子;其次,针对三角犹豫模糊元的有序位置存在具有不同权重的情况,构建了三角犹豫模糊Einstein有序加权平均（THFEOWA）算子和三角犹豫模糊Einstein有序加权几何（THFEOWG）算子,并讨论了它们相应的基本性质;最后建立了基于THFEOWA算子和THFEOWG算子的多属性决策模型,并通过实例说明提出的决策模型是合理和有效的。