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

In this paper, we investigate multiple attribute decision making (MADM) problems based on Frank triangular norms, in which the attribute values assume the form of hesitant fuzzy information. Firstly, some basic concepts of hesitant fuzzy set (HFS) and the Frank triangle norms are introduced. We develop some hesitant fuzzy aggregation operators based on Frank operations, such as hesitant fuzzy Frank weighted average (HFFWA) operator, hesitant fuzzy Frank ordered weighted averaging (HFFOWA) operator, hesitant fuzzy Frank hybrid averaging (HFFHA) operator, hesitant fuzzy Frank weighted geometric (HFFWG) operator, hesitant fuzzy Frank ordered weighted geometric (HFFOWG) operator, and hesitant fuzzy Frank hybrid geometric (HFFHG) operator. Some essential properties together with their special cases are discussed in detail. Next, a procedure of multiple attribute decision making based on the HFFHWA (or HFFHWG) operator is presented under hesitant fuzzy environment. Finally, a practical example that concerns the human resource selection is provided to illustrate the decision steps of the proposed method. The result demonstrates the practicality and effectiveness of the new method. A comparative analysis is also presented.     

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。     

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.     

犹豫梯形模糊算子在多属性决策中的应用

结合犹豫模糊集和梯形模糊集,提出犹豫梯形模糊集的概念。首先,给出犹豫梯形模糊数的运算法则,探讨犹豫梯形模糊加权平均（HTrFWA）算子和犹豫梯形模糊加权几何（HTrFWG）算子。考虑到犹豫梯形模糊数的有序位置存在具有不同权重的情况,定义了犹豫梯形模糊有序加权平均（HTrFOWA）算子和犹豫梯形模糊有序加权几何（HTrFOWG）算子,并讨论了其相应的运算定理。其次,构建犹豫梯形模糊数的得分函数,并给出犹豫梯形模糊数的排序方法。最后,提出了基于HTrFWA算子和HTrFWG算子的犹豫梯形模糊多属性决策方法,并通过实例进行验证。     

Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making

In this paper, we develop a series of induced generalized aggregation operators for hesitant fuzzy or interval-valued hesitant fuzzy information, including induced generalized hesitant fuzzy ordered weighted averaging (IGHFOWA) operators, induced generalized hesitant fuzzy ordered weighted geometric (IGHFOWG) operators, induced generalized interval-valued hesitant fuzzy ordered weighted averaging (IGIVHFOWA) operators, and induced generalized interval-valued hesitant fuzzy ordered weighted geometric (IGIVHFOWG) operators. Next, we investigate their various properties and some of their special cases. Furthermore, some approaches based on the proposed operators are developed to solve multiple attribute group decision making (MAGDM) problems with hesitant fuzzy or interval-valued hesitant fuzzy information. Finally, some numerical examples are provided to illustrate the developed approaches.     

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.     

毕达哥拉斯模糊Heronian算子的多属性决策方法

针对毕达哥拉斯模糊环境下的多属性决策问题中的信息集成问题,其中方案的属性间相互关联相互影响,提出了一种毕达哥拉斯模糊Heronian算子的多属性决策方法。首先将毕达哥拉斯模糊数与Heronian算子相结合,提出了毕达哥拉斯模糊Heronian算子和毕达哥拉斯模糊加权Heronian算子,并讨论了这些算子的性质,给出了相应的证明。在此基础上,提出了基于毕达哥拉斯模糊Heronian算子的多属性决策方法;最后将其应用到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。     

Some 2-tuple linguistic Pythagorean Heronian mean operators and their application to multiple attribute decision-making

In this paper, we expand the generalised Heronian mean (GHM) operator, generalised weighted Heronian mean (GWHM), geometric Heronian mean (GHM) operator, and weighted geometric Heronian mean (WGHM) operator with 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) to propose generalised 2-tuple linguistic Pythagorean fuzzy Heronian mean (G2TLPFHM) operator, generalised 2-tuple linguistic Pythagorean fuzzy weighted Heronian mean (G2TLPFWHM) operator, 2-tuple linguistic Pythagorean fuzzy geometric Heronian mean (2TLPFGHM) operator, 2-tuple linguistic Pythagorean fuzzy weighted geometric Heronian mean (2TLPFWGHM) operator. Then, the MADM methods are proposed with these operators. In the end, we utilise an applicable example for green supplier selection to prove the proposed 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.     

Some new Pythagorean fuzzy Choquet–Frank aggregation operators for multi‐attribute decision making

Pythagorean fuzzy set (PFS) whose main feature is that the square sum of the membership degree and the non‐membership degree is equal to or less than one, is a powerful tool to express fuzziness and uncertainty. The aim of this paper is to investigate aggregation operators of Pythagorean fuzzy numbers (PFNs) based on Frank t‐conorm and t‐norm. We first extend the Frank t‐conorm and t‐norm to Pythagorean fuzzy environments and develop several new operational laws of PFNs, based on which we propose two new Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet–Frank averaging operator (PFCFA) and Pythagorean fuzzy Choquet–Frank geometric operator (PFCFG). Moreover, some desirable properties and special cases of the operators are also investigated and discussed. Then, a novel approach to multi‐attribute decision making (MADM) in Pythagorean fuzzy context is proposed based on these operators. Finally, a practical example is provided to illustrate the validity of the proposed method. The result shows effectiveness and flexible of the new method. A comparative analysis is also presented.     

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.     

Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making

As a fuzzy set extension, the hesitant set is effectively used to model situations where it is allowable to determine several possible membership degrees of an element to a set due to the ambiguity between different values. We first introduce some new operational rules of hesitant fuzzy sets based on the Hamacher t-norm and t-conorm, in which a family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information. Some basic properties of these proposed operators are given, and the relationships between them are shown in detail. We further discuss the interrelations between the proposed aggregation operators and the existing hesitant fuzzy aggregation operators. Applying the proposed hesitant fuzzy operators, we develop a new technique for hesitant fuzzy multicriteria decision making problems. Finally, the effectiveness of the proposed technique is illustrated by mean of a practical example.     

New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making

The q-rung orthopair fuzzy sets are superior to intuitionistic fuzzy sets or Pythagorean fuzzy sets in expressing fuzzy and uncertain information. In this paper, some partitioned Bonferroni means (BMs) for q-rung orthopair fuzzy values have been developed. First, the q-rung orthopair fuzzy partitioned BM (q-ROFPBM) operator and the q-rung orthopair fuzzy partitioned geometric BM (q-ROFPGBM) operator are developed. Some desirable properties and some special cases of the new aggregation operators have been studied. The q-rung orthopair fuzzy weighted partitioned BM (q-ROFWPBM) operator and the q-rung orthopair fuzzy partitioned geometric weighted BM (q-ROFPGWBM) operator are also developed. Then, a new multiple-attribute decision-making method based on the q-ROFWPBM (q-ROFPGWBM) operator is proposed. Finally, a numerical example of investment company selection problem is given to illustrate feasibility and practical advantages of the new method.     

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。     

Uncertain linguistic hesitant fuzzy sets and their application in multi‐attribute decision making

To denote the quantitative and qualitative fuzzy information simultaneously, this paper introduces a new type of fuzzy sets called uncertain linguistic hesitant fuzzy sets, which are denoted by an uncertain linguistic variable with several possible interval membership degrees. Considering the application of this type of fuzzy sets, several basic operational laws are defined, and several properties are studied. Meanwhile, an ordered relationship is introduced. Then, two types of uncertain linguistic hesitant fuzzy aggregation operators are defined. One uses additive measures, and the other is based on λ‐fuzzy measures. Then, a similarity measure is presented, by which models for the optimal weight vector are constructed. After that, an approach to uncertain linguistic hesitant fuzzy multi‐attribute decision making is developed. Finally, an illustrative example for evaluating corporate environmental performance is offered to show the concrete practicality of the procedure.     

Hesitant Pythagorean fuzzy Maclaurin symmetric mean operators and its applications to multiattribute decision-making process

Hesitant Pythagorean fuzzy (HPF) sets can easily express the uncertain information while Maclaurin symmetric mean (MSM) operator, can capture the interrelationship among the multiattributes, and are suitable for aggregating the information into a single number. By taking the advantages of both, in this paper, we extend the traditional MSM to HPF environment. For this, we develop the HPFMSM operator for aggregating the HPF information. The desirable characteristics, such as idempotency, monotonicity, and boundedness, are studied. Then, we discussed some special cases with respect to the parameter value of the HPFMSM operators and showed that it generalizes the various existing operators. Furthermore, we studied the weighted HPFMSM operator to aggregate the HPF information with different preferences to the input arguments. On the basis of these operators, we solved the multiattribute decision-making problems with HPF information. The practicality and effectiveness of the developed approach are demonstrated through a numerical example.     

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。     

广义犹豫正态模糊信息集成及其多属性群决策

定义了犹豫正态模糊元及其运算法则、得分函数、Euclidean距离等概念;提出了广义犹豫正态模糊有序加权平均算子,并研究其性质,该算子不仅尽可能多地保留决策者的偏好信息,还可依据决策者的主观意愿选择不同的参数和属性权重,使得决策结果达到决策者的期望值;紧接着对属性权重和算子参数赋予不同的数值,获取广义犹豫正态模糊有序加权平均算子的若干种特殊算子,并探讨两个常用算子的大小关系;针对属性权重完全未知的多属性群决策问题,构建一种基于广义犹豫正态模糊有序加权平均算子的群决策方法。该方法利用同一属性下所有方案属性值间的距离求得最优权重,然后将同一方案下各属性值集结成为综合属性值,进而得到方案优劣排序。通过实例分析说明该方法的可行性和有效性。     

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.     

Some induced intuitionistic fuzzy aggregation operators applied to multi-attribute group decision making

In this paper, we define various induced intuitionistic fuzzy aggregation operators, including induced intuitionistic fuzzy ordered weighted averaging (OWA) operator, induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, induced interval-valued intuitionistic fuzzy OWA operator, and induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator. We also establish various properties of these operators. And then, an approach based on I-IFHA operator and intuitionistic fuzzy weighted averaging (WA) operator is developed to solve multi-attribute group decision-making (MAGDM) problems. In such problems, attribute weights and the decision makers' (DMs') weights are real numbers and attribute values provided by the DMs are intuitionistic fuzzy numbers (IFNs), and an approach based on I-IIFHA operator and interval-valued intuitionistic fuzzy WA operator is developed to solve MAGDM problems where the attribute values provided by the DMs are interval-valued IFNs. Furthermore, induced intuitionistic fuzzy hybrid geometric operator and induced interval-valued intuitionistic fuzzy hybrid geometric operator are proposed. Finally, a numerical example is presented to illustrate the developed approaches.