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The purpose of this study is to offer requisite models for unravelling the complex issues that arise in a supplier selection‐order allocation problem by soothing the risk and disturbances. To do this, a new uncertain environment, interval type‐2 Pythagorean fuzzy set (IT2PFS) is introduced to assist the experts for assuring secure and reliable outcomes in hesitant situations. The operational laws on IT2PFS under Dombi t‐norm and t‐conorm are defined, which are further utilized to develop geometric Bonferroni mean and Bonferroni mean operators based on Choquet integral (CI) under IT2PFS. The aggregation operators can ennoble the pliability of the information blending process through the adjustment of several parameters and can make good interactions among the criteria with their weights. Thereafter, to determine the weights of the criteria, an empirical method, namely the decision making trial and evaluation laboratory (DEMATEL) is amplified by conjoining CI and IT2PFS into it. A multicriteria decision making method, namely, the multiattribute border approximation area comparison (MABAC) and subsequently the CI based grey relational analysis method are then employed to classify the suppliers and derive their corresponding weights with reference to the sustainable criteria. Over and above, a new multiobjective optimization model is exhibited to uphold the purchasing managers for assembling suppliers keeping in mind the sustainability aspects and overall risk of suppliers by following Markowitz portfolio theory. A real‐life supply chain management problem is demonstrated to elucidate the aptness and efficiency of the proposed work. Furthermore, the effectiveness and importance of the study are validated via the comparative analysis with existing approaches. The main contribution of the study is in handling the difficulty and confusion that arise during information gathering, information aggregation, suppliers evaluation and order allocation phases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of fuzzy set, called Pythagorean linguistic sets (PLSs), to address the preferred and nonpreferred degrees of linguistic variables. Moreover, it allows decision makers to offer effectively handle uncertain information more flexible than intuitionistic linguistic sets (ILSs) when one compares two alternatives in the process of decision making. Some of the fundamental operational laws, score, accuracy, and aggregation operators are defined, and their properties are investigated. Preference relation (PR) is a useful and efficient tool for decision making that only requires the decision makers to compare two alternatives at one time. Taking the advantages of PLSs and PRs, this paper also introduces Pythagorean linguistic preference relations (PLPRs) and studies their application. We propose an approach for group decision making using group recommendations based on consistency matrices and feedback mechanism. First, the proposed method constructs the collective consistency matrix, the weight collective PRs, and the group collective PRs. Then, it constructs a consensus relation for each expert and determines the group consensus degree (GCD) for all experts. If the GCD is smaller than a predefined threshold value, then a feedback mechanism is activated to update the PLPRs. Finally, after the GCD is greater than or equal to the predefined threshold value, we calculate the arithmetic mathematical average values of the updated group collective PR to select the most appropriate alternative. [ABSTRACT FROM AUTHOR]

The reliable evaluation of financial performance of universities plays an important role in sustainable development of universities, and it could be regarded as a multicriteria group decision making problem. Based on this, in this paper, with respect to the attribute evaluation values are expressed by interval 2‐tuple Pythagorean fuzzy linguistic variables, we develop an extended MULTIMOORA method. First, we propose the interval 2‐tuple Pythagorean fuzzy linguistic set based on Pythagorean fuzzy set and interval‐valued 2‐tuple linguistic variables, and further introduce the score and accuracy functions and Hamming distance of it. Then, considering that there are some deficiencies in the traditional MULTIMOORA method, such as exist the case of circular reasoning, and so forth. We improve it by proposing a comprehensive integration approach, which can not only consider both the utility values and sort results, but also reflect the preference attitude of decision makings simultaneously. Based on these research findings, an interval 2‐tuple Pythagorean fuzzy linguistic MULTIMOORA method is presented and the calculation process of this method is described in detail. Finally, we give the numerical example concerning the evaluation of financial management performance in universities to illustrate practicability and reliability of the proposed approach and compare the proposed method with different methods to perform its flexibility. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets, as an extension of intuitionistic fuzzy sets to deal with uncertainty, have attracted much attention since their introduction, in both theory and application aspects. In this paper, we investigate multiple attribute decision‐making (MADM) problems with Pythagorean linguistic information based on some new aggregation operators. To begin with, we present some new Pythagorean fuzzy linguistic Muirhead mean (PFLMM) operators to deal with MADM problems with Pythagorean fuzzy linguistic information, including the PFLMM operator, the Pythagorean fuzzy linguistic‐weighted Muirhead mean operator, the Pythagorean fuzzy linguistic dual Muirhead mean operator and the Pythagorean fuzzy linguistic dual‐weighted Muirhead mean operator. The main advantages of these aggregation operators are that they can capture the interrelationships of multiple attributes among any number of attributes by a parameter vector P and make the information aggregation process more flexible by the parameter vector P. In addition, some of the properties of these new aggregation operators are proved and some special cases are discussed where the parameter vector takes some different values. Moreover, we present two new methods to solve MADM problems with Pythagorean fuzzy linguistic information. Finally, an illustrative example is provided to show the feasibility and validity of the new methods, to investigate the influences of parameter vector P on decision‐making results, and also to analyze the advantages of the proposed methods by comparing them with the other existing methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Multicriteria decision making (MCDM) is to select the optimal candidate which has the best quality from a finite set of alternatives with multiple criteria. One important component of MCDM is to express the evaluation information, and the other one is to aggregate the evaluation results associated with different criteria. For the former, Pythagorean fuzzy set (PFS) is employed to represent uncertain information in this paper, and for the latter, the soft likelihood function developed by Yager is used. To address MCDM issues from a new perspective, the likelihood function of PFS is first proposed in this study and, to improve some of its limitations, the ordered weighted averaging (OWA)‐based soft likelihood function is defined, which introduces the attitudinal characteristic to identify decision makers' subjective preferences. In addition, the defined soft likelihood function of PFS is extended by weighted OWA operator considering the importance weight of the argument. Several illustrative cases are provided based on the presented (weighted) OWA‐based soft likelihood functions in Pythagorean fuzzy environment for MCDM problem. [ABSTRACT FROM AUTHOR]

As an extension of Pythagorean fuzzy sets, the q‐rung orthopair fuzzy sets (q‐ROFSs) can easily solve uncertain information in a broader perspective. Considering the fine property of q‐ROFSs, we introduce q‐ROFSs into decision‐theoretic rough sets (DTRSs) and use it to portray the loss function. According to the Bayesian decision procedure, we further construct a basic model of q‐rung orthopair fuzzy decision‐theoretic rough sets (q‐ROFDTRSs) under the q‐rung orthopair fuzzy environment. At the same time, we design the corresponding method for the deduction of three‐way decisions by utilizing projection‐based distance measures and TOPSIS. Then, we extend q‐ROFDTRSs to adapt the group decision‐making (GDM) scenario. To fuse different experts' evaluation results, we propose some new aggregation operators of q‐ROFSs by utilizing power average (PA) and power geometric (PG) operators, that is, q‐rung orthopair fuzzy power average, q‐rung orthopair fuzzy power weighted average (q‐ROFPWA), q‐rung orthopair fuzzy power geometric, and q‐rung orthopair fuzzy power weighted geometric (q‐ROFPWG). In addition, with the aid of q‐ROFPWA and q‐ROFPWG, we investigate three‐way decisions with q‐ROFDTRSs under the GDM situation. Finally, we give the example of a rural e‐commence GDM problem to illustrate the application of our proposed method and verify our results by conducting two comparative experiments. [ABSTRACT FROM AUTHOR]

Multiple attribute linguistic group decision making (MALGDM) is one of the important research topics in decision‐making theory. The material supplier grading involves multiple attributes and forms a typical MALGDM problem. In this paper, based on interval‐valued Pythagorean uncertain linguistic Euclidean distance (IVPUL‐Euclidean distance) and IVPUL aggregation operators, we develop a new outranking sorting approach, which can cope with MALGDM problems in IVPUL environment. The proposed IVPUL‐Euclidean distance defines a novel outranking relation, and it introduces three parameters that can adjust outranking sorting result according to various decision‐making situations. Meanwhile, the IVPUL aggregation operators are introduced to aggregate the IVPUL information. Eventually, we give a numerical example concerning the comprehensive assessment of material supplier to illustrate validity and applicability of the proposed approach and compare the proposed method with different generalized aggregation operators to perform its flexibility. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets (PFSs) accommodate more uncertainties than Lx 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. [ABSTRACT FROM AUTHOR]

Preference relations (PRs) as useful and efficient information description tools have been widely applied in practical multiple‐attribute group decision‐making (MAGDM) problems. Due to fuzziness and complexity of problems, the limitations of the experts' professional knowledge and experience, the provided PRs are usually inconsistent, will lead to unreasonable conclusion. Consequently, we define the concept of trapezoidal Pythagorean fuzzy linguistic PRs (TrPFLPRs), a method for confirm the consistent of PRs and repair inconsistent PFs in this paper. Meanwhile, we propose trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted Minkowski distance operator to aggregate the TrPFLPRs information. Finally, a numerical example is given to illustrate the practicality and effectiveness of the proposed operator. [ABSTRACT FROM AUTHOR]

The operations of t‐norm and t‐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. [ABSTRACT FROM AUTHOR]

Linguistic q‐rung orthopair fuzzy numbers (Lq‐ROFNs) are a qualitative form of q‐rung orthopair fuzzy numbers (q‐ROFNs) where the membership and nonmembership degrees are represented by linguistic variables. The Lq‐ROFNs can describe a broader range of linguistic assessment information flexibly by adjusting the parameter q based on different situations, so they are more superior to the linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers in real application. Based on the Lq‐ROFNs, we introduce the entropy measure which can be used to determine the indefiniteness of the assessment information. Then, based on the linguistic entropy measure, we further propose a method to obtain the attribute weights when the weight information is incomplete known. For aggregating the assessment information, the power average (PA) operator can reduce the influence of extreme data caused by the biased decision‐makers by considering the support degree of different evaluation individuals, and the Muirhead mean (MM) operator can take the interrelationship of different numbers of attributes into account by adjusting the parameter vector based on the real situations. In this paper, based on these two operators, we firstly propose the linguistic q‐rung orthopair fuzzy PA operator and linguistic q‐rung orthopair fuzzy weighted PA operator. Further, for combing the advantages of the MM operator and PA operator, we propose the linguistic q‐rung orthopair fuzzy power MM (PMM) operator and linguistic q‐rung orthopair fuzzy weighted PMM operator, and then investigate some properties of them. Finally, a new multiple‐attribute group decision‐making (MAGDM) method is proposed to process the Lq‐ROFNs, and some practical examples are given to illustrate the effectiveness and superiority of this new method in comparison with other existing MAGDM methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of fuzzy preference structure, called Pythagorean fuzzy preference relations (PFPRs), to describe uncertain evaluation information in group decision‐making process. Moreover, it allows decision makers to offer effectively handle uncertain information more flexibly than intuitionistic fuzzy preference relations when one compares two alternatives in the process of decision making. Using PFPRs, we propose an approach for group decision making based on group recommendations and consistency matrices. First, the proposed approach constructs the collective consistency matrix, the weight collective preference relations (PRs), and the group collective PR. Then, it construct a consensus relation for each expert and determinate the group consensus degree for all experts. If the group consensus degree is smaller than a predefined threshold value, then it identify the consensus values in each consensus relation which are smaller than the group consensus degree and updates the Pythagorean fuzzy preference values corresponding to the identified consensus values. The above process is continued, until the group consensus degree is larger than or equal to the predefined threshold value. Finally, based on the group collective PR, we calculate the row arithmetic mathematical average values and with respect to that values the various methods are applied for ranking the preference order of the alternatives. Numerical example are provided to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

This paper investigates a novel induced ordered weighted averaging (IOWA) distance operator and its application in Pythagorean fuzzy (PF) multiattribute group decision making (MAGDM). First, a new induced aggregated distance operator named the weighted IOWA distance (WIOWAD) operator is developed, which differs from the existing methods in that it considers the dual roles of the order‐inducing variables at the same time. In other words, in addition to inducing the order of the arguments, the order‐inducing variables of the WIOWAD operator also plays an important role in moderating the associated weight vector. Some useful properties and different families of the WIOWAD are also discussed. Then, an extension of the WIOWAD within the PF situation is presented, thus obtaining the PFWIOWAD operator. Furthermore, a MAGDM method based on the PFWIOWAD is introduced. Finally, the practicality and effectiveness of proposed approach are illustrated in a research and development project selection problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The hesitant Pythagorean fuzzy set is frequently considered as a solution for decision making under uncertainty. Whereas the representation of uncertain information might be not sufficient in the hesitant Pythagorean fuzzy environment, thus the concept of interval‐valued hesitant Pythagorean fuzzy sets (IVHPFSs) is proposed. Specifically, we first propose the concept of IVHPFSs and then study the operational rules and distance measures of IVHPFSs in detail. To ease the possible application, we explore two decision‐making processes in the setting of IVHPFSs by drawing support from the technique for order preference by similarity to ideal solution and Choquet integral‐based method. Finally, the selecting processes of project private partner are also presented to demonstrate the decision‐making processes based on IVHPFSs and compared with some similar techniques. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a new conception of linguistic q‐rung orthopair fuzzy number (Lq‐ROFN) is proposed where the membership and nonmembership of the q‐rung orthopair fuzzy numbers (q‐ROFNs) are represented as linguistic variables. Compared with linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers, the Lq‐ROFNs can more fully describe the linguistic assessment information by considering the parameter q to adjust the range of fuzzy information. To deal with the multiple‐attribute group decision‐making (MAGDM) problems with Lq‐ROFNs, we proposed the linguistic score and accuracy functions of the Lq‐ROFNs. Further, we introduce and prove the operational rules and the related properties characters of Lq‐ROFNs. For aggregating the Lq‐ROFN assessment information, some aggregation operators are developed, involving the linguistic q‐rung orthopair fuzzy power Bonferroni mean (BM) operator, linguistic q‐rung orthopair fuzzy weighted power BM operator, linguistic q‐rung orthopair fuzzy power geometric BM (GBM) operator, and linguistic q‐rung orthopair fuzzy weighted power GBM operator, and then presents their rational properties and particular cases, which cannot only reduce the influences of some unreasonable data caused by the biased decision‐makers, but also can take the interrelationship between any two different attributes into account. Finally, we propose a method to handle the MAGDM under the environment of Lq‐ROFNs by using the new proposed operators. Further, several examples are given to show the validity and superiority of the proposed method by comparing with other existing MAGDM methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets are powerful techniques for modeling vagueness in practice. The aim of this article is to investigate an effective means to aggregate uncertain information and then employ it into settling multiple criteria decision making (MCDM) problems within the Pythagorean fuzzy circumstances. To capture the nature of the reality, some special cases should be comprehensively considered. First, though correlation commonly exist among criteria, a deep insight should also be provided into some realistic situations, in which not all the criteria are interrelated to others. Besides, it is more reasonable to take the importance of the input arguments into consideration. Effected by aforementioned point, this article explores a Pythagorean fuzzy partitioned normalized weighted Bonferroni mean (PFPNWBM) operator with the combination of partitioned Bonferroni mean (BM) and normalized weighted BM operators considering Shapley fuzzy measure. Subsequently, in the context of partially known weight information, models are established to identify the optimal Shapley fuzzy measure. Moreover, integrated the PFPNWBM operator with the optimal Shapley fuzzy measure identification model, a Pythagorean fuzzy MCDM approach is designed. Finally, an illustrative example and detailed analyses are performed to demonstrate its feasibility and reliability. [ABSTRACT FROM AUTHOR]

Robots have received considerable attention in many manufacturing companies due to their great capabilities and characteristics. Selecting an appropriate robot for a specific application can be regarded as a challenging multicriteria decision‐making problem. Furthermore, decision makers are inclined to represent their opinions by using linguistic terms owing to their ambiguous thinking. In this regard, we put forward a novel robot selection model by integrating quality function development (QFD) theory and qualitative flexible multiple criteria method (QUALIFLEX) under interval‐valued Pythagorean uncertain linguistic context. For the developed model, the evaluations given by decision makers are presented as interval‐valued Pythagorean uncertain linguistic sets for dealing with the uncertainty and vagueness of decision makers' information. An extended QFD method is used for determining criteria weights from the perspective of customers. A modified QUALIFLEX technique based on closeness degree is utilized to generate the ranking order of alternative robots and determine the most suitable one. Finally, an empirical example of an auto manufacturing company is applied to clarify the effectiveness and accuracy of the proposed robot selection approach. [ABSTRACT FROM AUTHOR]

The aim of this article is to develop a novel multiple criteria decision analysis (MCDA) method using a Pearson‐like correlation‐based Pythagorean fuzzy (PF) compromise approach under complex uncertainty based on PF sets and interval‐valued Pythagorean fuzzy (IVPF) sets. Because of the complexity and ambiguity involved in real‐life decision‐making situations, this article utilizes the theory of Pythagorean fuzziness, which is characterized by flexible degrees of membership, nonmembership, and indeterminacy to describe uncertain information more comprehensively. PF and IVPF sets possess exceptional abilities to accurately reflect the uncertainty, fuzziness, and vagueness inherent in the decision information. However, manipulating PF and IVPF information is a complicated and difficult task for most decision makers. In this regard, this article extends the well‐known and widely used concept of correlation coefficients to develop simple and effective compromise models for solving MCDA problems in PF and IVPF contexts. This article conducts an extended analysis of Pearson‐like correlation coefficients for PF and IVPF sets separately and introduces new concepts of PF and IVPF correlation coefficients to furnish a solid basis for the proposed methodology. Furthermore, this article develops useful concepts of PF and IVPF correlation‐based closeness coefficients to simultaneously measure the relative closeness to the positive‐ideal PF/IVPF solutions and the relative remoteness from the negative‐ideal PF/IVPF solutions. On the basis of the developed concepts, this article proposes a novel Pearson‐like correlation‐based PF/IVPF compromise approach to address uncertain MCDA problems involving PF/IVPF information and determine the ultimate priority orders among competing alternatives. Finally, this article provides an illustrative application about a financing decision of working capital management to verify the developed approach and demonstrate its feasibility and practicality. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop some new logarithm operational laws (LOL) with real number base λ for the Pythagorean fuzzy sets. Some properties of LOL have been studied and based on these, various weighted averaging and geometric operators have been developed. Then, we utilized it to solve the decision‐making problems. The validity of the proposed method is demonstrated with a numerical example and compared the results with the several existing approaches result. Finally, the influences of logarithmic operation and the selection of the logarithmic base λ in practice are discussed. [ABSTRACT FROM AUTHOR]

The q‐rung orthopair set (q‐ROFSs) can serve as a generalization of the existing orthopair fuzzy sets, including intuitionistic fuzzy sets and Pythagorean fuzzy sets. The most desirable characteristic of q‐ROFSs is that they support a greater space of allowable membership grades and provide decision makers more freedom in describing their true opinions. As a classical aggregation operator, Heronian mean (HM) can model the interrelationship between attributes. In this paper, we extend the traditional HM to aggregate q‐rung orthopair fuzzy information and propose the q‐rung orthopair fuzzy HM and its weighted form. Further, to overcome the shortcomings of the traditional HM, considering the possible partition structure in the actual decision situations, we propose the q‐rung orthopair fuzzy partitioned Heronian mean operator and the q‐rung orthopair fuzzy weighted partitioned Heronian mean operator. Then, some special cases and some desirable properties are investigated and discussed. A new multiple attribute group decision‐making(MAGDM) technique is developed based on the proposed q‐rung orthopair fuzzy operators. Finally, a representative example is provided to verify the effectiveness and superiority of the proposed method by comparing with other several existing representative MAGDM methods. [ABSTRACT FROM AUTHOR]

Abstract: q‐Rung orthopair fuzzy set (q‐ROFS) is a powerful tool that attracts the attention of many scholars in dealing with uncertainty and vagueness. The aim of paper is to present a new score function of q‐rung orthopair fuzzy number (q‐ROFN) for solving the failure problems when comparing two q‐ROFNs. Then a new exponential operational law about q‐ROFNs is defined, in which the bases are positive real numbers and the exponents are q‐ROFNs. Meanwhile, some properties of the operational law are investigated. Later, we apply them to derive the q‐rung orthopair fuzzy weighted exponential aggregation operator. Additionally, an approach for multicriteria decision‐making problems under the q‐rung orthopair fuzzy data is explored by applying proposed aggregation operator. Finally, an example is investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q‐rung orthopair fuzzy decision‐making methods, are (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: The need of suitable divergence measures arise as they play an important role in discrimination of two probability distributions. The present communication is devoted to the introduction of one such divergence measure using Jensen inequality and Shannon entropy and its validation. Also, a new dissimilarity measure based on the proposed divergence measure is introduced. Besides establishing validation, some of its major properties are also studied. Further, a new multiple attribute decision making method based on a proposed dissimilarity measure is introduced and is thoroughly explained with the help of an illustrated example. The paper is summed up with an application of the proposed dissimilarity measure in pattern recognition. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the idea of orthopair membership grades and the related idea of general orthopair fuzzy sets. It is noted that these generalize the intuitionistic and Pythagorean fuzzy sets by allowing the support for and against membership to be almost anywhere in [0, 1] × [0, 1], giving systems modelers great freedom in capturing human knowledge. The aggregation of generalized orthopair fuzzy sets is considered with particular concern for the OWA and Choquet aggregation. The concepts of possibility and certainty as well as plausibility and belief are investigated in this general orthopair environment. We study arithmetic operations on general orthopair fuzzy sets. We show how to obtain associated interval valued fuzzy sets from general orthopair fuzzy sets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, two new approaches have been presented to view q‐rung orthopair fuzzy sets. In the first approach, these can viewed as L‐fuzzy sets, whereas the second approach is based on the notion of orbits. Uncertainty index is the quantity H A ( x ) = 1 − ( A + ( x ) ) q − ( A − ( x ) ) q, which remains constant for all points in an orbit. Certain operators can be defined in q‐ROF sets, which affect H A ( x ) when applied to some q‐ROF sets. Operators I δ, M δ , ν, and K δ , ν have been defined. It is studied that how these operators affect H A ( x ) when applied to some q‐ROF set A. [ABSTRACT FROM AUTHOR]

Abstract: The interval‐valued Pythagorean fuzzy sets can easily handle uncertain information more flexibly in the process of decision making. Considering the interrelationship among the input arguments, we extend the Bonferroni mean and the geometric Bonferroni mean to the interval‐valued Pythagorean fuzzy environment and solve its practical application problems. First, we develop the interval‐valued Pythagorean fuzzy Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy Bonferroni mean (WIVPFBM) operators. The properties of these aggregation operators are investigated. Then, we also develop the interval‐valued Pythagorean fuzzy geometric Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy geometric Bonferroni mean (WIVPFGBM) operators and analyze their properties. Third, we utilize the WIVPFBM and WIVPFGBM operators to fuse the information in the interval‐valued Pythagorean fuzzy multicriteria group decision making (IVPFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the linear assignment method, we present its extension and further design a new algorithm for the application of IVPFMCGDM. Finally, an example is given to elaborate our proposed algorithm and validate its excellent performance. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: The Pythagorean fuzzy set introduced by R. R. Yager in 2014 is a useful tool to model imprecise and ambiguous information appearing in decision and clustering problems. In this study, we present a general type of distance measure for Pythagorean fuzzy numbers (PFNs) and propose a novel ratio index‐based ranking method of PFNs. The novel ranking method of PFNs has more powerful ability to discriminate the magnitude of PFNs than the existing ranking methods for PFNs, which is further extended to compare the magnitude of interval‐valued Pythagorean fuzzy numbers (IVPFNs). The IVPFN is a new extension of PFN, which is parallel to interval‐valued intuitionistic fuzzy number. We introduce a general type of distance measure for IVPFNs. Afterwards, we study a kind of clustering problems in Pythagorean fuzzy environments in which the evaluation values are expressed by PFNs and/or IVPFNs and develop a novel Pythagorean fuzzy agglomerative hierarchical clustering approach. In the proposed clustering method, we define the concept of the dissimilarity degree between two clusters for each criterion and introduce the clustering procedure in the criteria level. To take all the criteria into account, we also introduce the overall clustering procedure, which is based on the overall dissimilarity degrees for a fixed aggregation operator such as the commonly used weighted arithmetic average operator or the ordered weighted averaging operator. In the overall clustering process, (1) we present a deviation degree‐based method to derive the weights of criteria and further obtain the overall clustering results if the weights of criteria are completely unknown; (2) we employ the ratio index‐based ranking method of IVPFNs to obtain the overall clustering results if the weights of criteria are given in advance and are expressed by IVPFNs. The salient feature of the proposed clustering method is that it not only can address the clustering problems in which the weights of criteria are not given precisely in advance but also can manage simultaneously the PFNs and IVPFNs data. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

The generalized Heronian mean and geometric Heronian mean operators provide two aggregation operators that consider the interdependent phenomena among the aggregated arguments. In this paper, the generalized Heronian mean operator and geometric Heronian mean operator under the q-rung orthopair fuzzy sets is studied. First, the q-rung orthopair fuzzy generalized Heronian mean (q-ROFGHM) operator, q-rung orthopair fuzzy geometric Heronian mean (q-ROFGHM) operator, q-rung orthopair fuzzy generalized weighted Heronian mean (q-ROFGWHM) operator, and q-rung orthopair fuzzy weighted geometric Heronian mean (q-ROFWGHM) operator are proposed, and some of their desirable properties are investigated in detail. Furthermore, we extend these operators to q-rung orthopair 2-tuple linguistic sets (q-RO2TLSs). Then, an approach to multiple attribute decision making based on q-ROFGWHM (q-ROFWGHM) operator is proposed. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]

Owing to the information insufficiency, it might be difficult for decision makers to precisely evaluate their assessments in real decision-making. As a new extension of the Pythagorean fuzzy sets, the interval-valued Pythagorean fuzzy sets (IVPFSs) can availably provide enough input space for decision makers to evaluate their assessments with interval numbers. By extending the Bonferroni mean to model the heterogeneous interrelationship among attributes, the extended Bonferroni mean (EBM) was examined. Considering the partition structure of relationship among the attributes, we introduce the EBM into the interval-valued Pythagorean fuzzy environment and develop two new aggregation operators, namely, intervalvalued Pythagorean fuzzy extended Bonferroni mean and weighted interval-valued Pythagorean fuzzy extended Bonferroni mean (WIVPFEBM) operators. Meanwhile, some of their special cases and properties are also deeply discussed. Subsequently, by employing theWIVPFEBM operator, we propose an approach for multiple attribute decision making with IVPFSs. Finally, a practical illustration of the E-commerce project selection problem is investigated by our proposed method, which successfully demonstrates the applicability of our results. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: Analyzing demand in environments with incomplete information is a challenging task. This paper proposes a novel agent‐based Pythagorean fuzzy approach for analyzing this kind of demand. First, a Bayesian game is described with a large number of finite players, and this is followed by a Pythagorean fuzzy‐based decision mechanism. Unlike the classical methods in the literature, the proposed method in this paper neither assumes nor forecasts the demand in a system. Instead, it tries to analyze the demand when there is limited availability of input data, or processing data are computationally expensive. The study ends with an application of the proposed system to an electricity grid. Electricity prices used as an incentive to construct an agent‐based system that efficiently reduces the peak amounts in a smart grid by analyzing the demand. Test results provide evidence that the proposed approach is promising to design demand response systems. [ABSTRACT FROM AUTHOR]

Abstract: Decision‐theoretic rough sets (DTRSs), which provide a classical model of three‐way decisions (3WDs), play an important role in risk decision‐making problems. The risk is associated with the loss function of DTRSs, which is evaluated by the decision makers. As a new extension of fuzzy sets, Pythagorean fuzzy sets can handle uncertain information more flexibly than intuitionistic fuzzy sets in the process of decision making and it gives a new measure for the determination of loss functions of DTRSs. More specifically, we take into account the loss functions of DTRSs with Pythagorean fuzzy numbers and propose a Pythagorean fuzzy decision‐theoretic rough set (PFDTRS) model. Some properties of the expected losses are carefully investigated. Then we further design three approaches for deriving 3WDs with the PFDTRS model. The group decision making (GDM) based on the PFDTRS model is also discussed. It provides a novel interpretation for the determination of loss functions. With the aid of the Pythagorean fuzz weighted averaging operator, we aggregate the loss functions, as suggested by the all experts, which support a coherent way of designing information granules in the presence of numerics. An algorithm for 3WDs in GDM based on the PFDTRS model is designed. Then, an example is presented to elaborate on 3WDs with the PFDTRS model. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-known Bonferroni mean and develop its generalized aggregation operators in the Pythagorean fuzzy environment. More specifically, by considering the interrelationship between arguments with Pythagorean fuzzy information, we develop the Pythagorean fuzzy Bonferroni mean (PFBM) and some special properties and cases of them are also discussed. Furthermore, taking the multicriteria decision making environment into consideration, we extend the results of PFBM and develop the weighted Pythagorean fuzzy Bonferroni mean (WPFBM). Meanwhile, we also propose an approach for the application of WPFBM. However, during the application of the WPFBM operator, the calculation is very complex and time consuming. Hence, we introduce the multithreading into the application of the WPFBM operator and develop an accelerative calculating algorithm for it. To validate the performance of the accelerative calculating algorithm, we further design the corresponding experimental analysis. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]