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.     

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.     

Algorithms for complex interval‐valued q‐rung orthopair fuzzy sets in decision making based on aggregation operators,AHP, and TOPSIS

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.     

Multiple attribute decision‐making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under q‐rung orthopair fuzzy environment

A Q‐rung orthopair fuzzy set (q‐ROFS) originally proposed by Yager (2017) is a new generalization of orthopair fuzzy sets, which has a larger representation space of acceptable membership grades and gives decision makers more flexibility to express their real preferences. In this paper, for multiple attribute decision‐making problems with q‐rung orthopair fuzzy information, we propose a new method for dealing with heterogeneous relationship among attributes and unknown attribute weight information. First, we present two novel q‐rung orthopair fuzzy extended Bonferroni mean (q‐ROFEBM) operator and its weighted form (q‐ROFEWEBM). A comparative example is provided to illustrate the advantages of the new operators, that is, they can effectively model the heterogeneous relationship among attributes. We prove that some existing known intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are special cases of the proposed q‐ROFEBM and q‐ROFEWEBM operators. Meanwhile, several desirable properties are also investigated. Then, a new knowledge‐based entropy measure for q‐ROFSs is also proposed to obtain the attribute weights. Based on the proposed q‐ROFWEBM and the new entropy measure, a new method is developed to solve multiple attribute decision making problems with q‐ROFSs. Finally, an illustrative example is given to demonstrate the application process of the proposed method, and a comparison analysis with other existing representative methods is also conducted to show its validity and superiority.     

Some q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making

Multiple-attribute group decision making (MAGDM) under linguistic environment is an important part of modern decision sciences, and information aggregation operator plays an import role in solving MAGDM problems. In this paper, an approach for solving MAGDM problem with q-rung orthopair fuzzy 2-tuple linguistic information is developed. First, the q-rung orthopair fuzzy 2-tuple linguistic weighted averaging (q-ROFTLWA) operator and the q-rung orthopair fuzzy 2-tuple linguistic weighted geometric (q-ROFTLWG) operator are presented. Furthermore, the q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean (q-ROFTLMM) operator and the q-rung orthopair fuzzy 2-tuple linguistic dual Muirhead mean (q-ROFTLDMM) operator are proposed on the basis of Muirhead mean (MM) operator and dual Muirhead mean (DMM) operator. Then, an approach is developed to deal with MAGDM problem under q-rung orthopair fuzzy 2-tuple linguistic environment based on the proposed operators. Finally, a numerical example for selecting desirable emergency alternative(s) in the process of designing emergency preplan is given to illustrate the application of the developed method and demonstrate its effectiveness.     

Some q-rung interval-valued orthopair fuzzy Maclaurin symmetric mean operators and their applications to multiple attribute group decision making

In this paper, according to the Maclaurin symmetric mean (MSM) operator, the dual MSM (DMSM) operator and the q-rung interval-valued orthopair fuzzy set (q-RIVOFS), we develop some novel MSM operators under the q-rung interval-valued orthopair fuzzy environment, such as, the q-rung interval-valued orthopair fuzzy MSM operator, the q-rung interval-valued orthopair fuzzy weighted MSM (q-RIVOFWMSM) operator, the q-rung interval-valued orthopair fuzzy DMSM operator, and the q-rung interval-valued orthopair fuzzy weighted DMSM operator. In addition, some precious properties and numerical examples of these new operators are given in detail. These new operators have the advantages of considering the interrelationship of arguments and can deal with multiple attribute group decision-making problems with q-rung interval-valued orthopair fuzzy information. Finally, a reality example for green suppliers selection in green supply chain management is provided to demonstrate the proposed approach and to verify its rationality and scientific.     

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.     

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.     

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.     

Minkowski‐type distance measures for generalized orthopair fuzzy sets

The generalized orthopair fuzzy set inherits the virtues of intuitionistic fuzzy set and Pythagorean fuzzy set in relaxing the restriction on the support for and support against. The very lax requirement provides decision makers great freedom in expressing their beliefs about membership grades, which makes generalized orthopair fuzzy sets having a wide scope of application in practice. In this paper, we present the Minkowski‐type distance measures, including Hamming, Euclidean, and Chebyshev distances, for q‐rung orthopair fuzzy sets. First, we introduce the Minkowski‐type distances of q‐rung orthopair membership grades, based on which we can rank orthopairs. Second, we propose several distances over q‐rung orthopair fuzzy sets on a finite discrete universe and subsequently discuss their applications to multiattribute decision‐making problems. Then we extend these results to a continuous universe, both bounded and unbounded cases are considered. Some illustrative examples are employed to substantiate the conceptual arguments.     

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.     

Some interval-valued q-rung orthopair weighted averaging operators and their applications to multiple-attribute decision making

Q-rung orthopair fuzzy sets (q-ROFSs), initially proposed by Yager, are a new way to reflect uncertain information. The existing intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets are special cases of the q-ROFSs. However, due to insufficiency in available information, it is difficult for decision makers to exactly express the membership and nonmembership degrees by crisp numbers, and interval membership degree and interval nonmembership degree are good choices. In this paper, we propose the concept of interval-valued q-rung orthopair fuzzy set (IVq-ROFS) based on the ideas of q-ROFSs and some operational laws of q-rung orthopair fuzzy numbers (q-ROFNs). Then, some interval-valued q-rung orthopair weighted averaging operators are presented based on the given operational laws of q-ROFNs. Further, based on these operators, we develop a novel approach to solve multiple-attribute decision making (MADM) problems under interval-valued q-rung orthopair fuzzy environment. Finally, a numerical example is provided to illustrate the application of the proposed method, and the sensitivity analysis is further carried out for the parameters.     

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.     

Multi-attribute group decision-making methods based on q-rung orthopair fuzzy linguistic sets

With the continuous development of the economy and society, decision-making problems and decision-making scenarios have become more complex. The q-rung orthopair fuzzy set is getting more and more attention from researchers, which is more general and flexible than Pythagorean fuzzy set and intuitionistic fuzzy set under complex vague environment. In this study, the concept of q-rung orthopair fuzzy linguistic set (q-ROFLS) is proposed and a new q-rung orthopair fuzzy linguistic method is developed to handle MAGDM problem. Firstly, the conception, operation laws, comparison methods, and distance measure methods of the q-ROFLS are proposed. Secondly, the q-ROFL weighted average operator, q-ROFL ordered weighted average operator, q-ROFL hybrid weighted average operator, q-ROFL weighted geometric operator, q-ROFL ordered weighted geometric operator, and q-ROFL hybrid weighted geometric operator are proposed, and some interesting properties, special cases of these operators are investigated. Furthermore, a new method to cope with MAGDM problem based on q-ROFL weighted average operator (q-ROFL weighted geometric operator) is developed. Finally, a practical example for suppliers selection is provided to verify the practicality of the presented method, and the effectiveness and flexibility of the presented method are illustrated by sensitive analysis and comparative analysis.     

q-Rung orthopair uncertain linguistic partitioned Bonferroni mean operators and its application to multiple attribute decision-making method

A q-rung orthopair uncertain linguistic set can be served as an extension of an uncertain linguistic set (ULS) and a q-rung orthopair fuzzy set, which can also be treated as a generalized form of the existing intuitionistic ULS and Pythagorean ULS. The new linguistic set uses the uncertain linguistic variable to express the qualitative evaluation information and allows decision makers to provide their true views freely in a larger membership grade space. In this paper, we investigate the Bonferroni mean under the q-rung orthopair uncertain linguistic environment, then we propose the q-rung orthopair uncertain linguistic Bonferroni mean and its weighted form. Furthermore, considering the specific partition pattern among the attributes, the q-rung orthopair uncertain linguistic partitioned Bonferroni mean and its weighted form are developed. Meanwhile, we discuss several representative cases and attractive properties of our proposed operators in depth. Subsequently, a novel multi-attribute decision-making method is developed based on the above-mentioned aggregation operators. In the end, a comprehensible case is performed to analyze the superiority of the developed method by comparing with other typical studies.     

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.     

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。     

勾股模糊H平均算法应用于多媒体图像系统选择

提出了一种勾股模糊H平均算法应用多媒体图像系统选择问题。首先,定义了基于t-模和t-余模的勾股模糊数运算;讨论了勾股模糊Heronian平均算法的三个特征性质和经常使用的特例;然后构建了新的勾股模糊决策模型,该模型在构建过程中能够挖掘输入数据之间关联性,还提高了决策的使用范围;最后,将构建的模型应用于多媒体图像系统选择案例来验证有效性。