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The purpose of this article is to conduct a main path analysis of 627 articles on the theme of Pythagorean fuzzy sets (PFSs) in the Web of Science (WoS) from 2013 to 2020, to provide a conclusive and comprehensive analysis for researchers in this field, and to provide a study on preliminary understanding of PFSs.

The research topic of Pythagorean fuzzy fields, through keyword extraction and describing the changes in characteristic themes over the past eight years, are firstly examined. Main path analysis, including local and global main paths and key route paths, is then used to reveal the most influential relationships between papers and to explore the trajectory and structure of knowledge transmission.

The application of Pythagorean fuzzy theory to the field of decision-making has been popular, and combinations of the traditional Pythagorean fuzzy decision-making method with other fuzzy sets have attracted widespread attention in recent years. In addition, over the past eight years, research interest has shifted to different types of PFSs, such as interval-valued PFSs.

This paper implicates to investigate the growth in certain trends in the literature and to explore the main paths of knowledge dissemination in the domain of PFSs in recent years.

This paper aims to identify the topics in which researchers are currently interested, to help scholars to keep abreast of the latest research on PFSs.

The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.

In real-world situations, Pythagorean fuzzy numbers are exceptionally useful for representing ambiguous data. The authors look at multi-criteria decision-making issues in which the parameters have a prioritization relationship. The idea of a priority degree is introduced. The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.

The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.

The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined.

The purpose of this paper is to develop the Pythagorean fuzzy extension of CODAS method.

Supplier selection is a critical issue for manufacturing companies since it is a multidimensional problem including several conflicting criteria. A suitable multi criteria decision making (MCDM) method that could consider vagueness and impreciseness in the assessments should be used for this kind of problems. Pythagorean fuzzy sets (PFSs) are characterized by a membership degree and a non-membership degree satisfying the condition that their square sum is equal to or less than 1. PFSs extend the concept of intuitionistic fuzzy sets (IFSs). COmbinative Distance-based Assessment (CODAS) method is relatively a new MCDM technique introduced by Keshavarz Ghorabaee et al. (2016).

Pythagorean fuzzy CODAS gives better results than ordinary fuzzy CODAS since it considers the hesitancy of decision makers and presents a larger space for membership and non-membership definition.

The value of this paper is the proposal of a new method to use for the solutions of MCDM problems under vagueness and impreciseness. To show validity and effectiveness of the proposed method, an application to the supplier selection problem is given.

This is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (IHFN) is relaxed by the membership functions of Pythagorean probabilistic hesitant fuzzy number (PyPHFN), so the range of domain value of PyPHFN is greatly expanded. The paper aims to develop a novel decision-making technique based on aggregation operators under PyPHFNs. For this, the authors propose Algebraic operational laws using algebraic norm for PyPHFNs. Furthermore, a list of aggregation operators, namely Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy weighted geometric (PyPHFWG) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted average (PyPHFOWA) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted geometric (PyPHFOWG) operator, Pythagorean probabilistic hesitant fuzzy hybrid weighted average (PyPHFHWA) operator and Pythagorean probabilistic hesitant fuzzy hybrid weighted geometric (PyPHFHWG) operator, are proposed based on the defined algebraic operational laws. Also, interesting properties of these aggregation operators are discussed in detail.

PyPHFN is not only a generalization of the traditional IHFN, but also a more effective tool to deal with uncertain multi-attribute decision-making problems.

In addition, the authors design the algorithm to handle the uncertainty in emergency decision-making issues. At last, a numerical case study of coronavirus disease 2019 (COVID-19) as an emergency decision-making is introduced to show the implementation and validity of the established technique. Besides, the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique.

Paper is original and not submitted elsewhere.

The purpose of this paper is to develop a methodology based on TODIM (an acronym in Portuguese for interactive and multicriteria decision-making) approach for the selection of the best alternative in the context of multi criteria group decision-making (MCGDM) problems under possibilistic uncertainty in interval-valued Pythagorean fuzzy (IVPF) environment.

In this paper, IVPF-TODIM method is proposed. Some new point operator-based similarity measures (POSMs) for IVPF sets (IVPFSs) are introduced which have the capability to reduce the degree of uncertainty of the elements in the universe of discourse corresponding to IVPFS. Then the newly defined POSMs are used to compute the measure of relative dominance of each alternative over other alternatives in the IVPF-TODIM context. Finally, generalized mean aggregation operator is used to find the best alternative.

As the TODIM method is used to solve the MCGDM problems under uncertainty, POSMs are developed by using three parameters which can control the effect of decision-makers’ psychological perception under risk.

The decision values are used in IVPF numbers (IVPFNs) format.

The proposed method is capable to solve real-life MCGDM problems with not only IVPFNs format but also with interval-valued intuitionistic fuzzy numbers.

As per authors’ concern, no approach using TODIM with IVPFNs is found in literature to solve MCGDM problems under uncertainty. The final judgment values of alternatives using the extended TODIM methodology are highly corroborate in compare to the results of existing methods, which proves its great potentiality in solving MCGDM problems under risk.

Project selection is a critical decision for any organization seeking to commission a large-scale construction project. Project selection is a complex multi-criteria decision-making problem with significant uncertainty and high risks. Fuzzy set theory has been used to address various aspects of project uncertainty, but with key practical limitations. This study aims to develop and apply a novel Pythagorean fuzzy sets (PFSs) approach that overcomes these key limitations.

The study is particular to complex project selection in the context of increasing interest in resilience as a key project selection criterion. Project resilience is proposed and considered in the specific situation of a large-scale construction project selection case study. The case study develops and applies a PFS approach to manage project uncertainty. The case study is presented to demonstrate how PFS is applied to a practical problem of realistic complexity. Working through the case study highlights some of the key benefits of the PFS approach for practicing project managers and decision-makers in general.

The PFSs approach proposed in this study is shown to be scalable, efficient, generalizable and practical. The results confirm that the inclusion of last aggregation and last defuzzification avoids the potentially critical information loss and relative lack of transparency. Most especially, the developed PFS is able to accommodate and manage domain expert expressions of uncertainty that are realistic and practical.

The main novelty of this study is to address project resilience in the form of multi-criteria evaluation and decision-making under PFS uncertainty. The approach is defined mathematically and presented as a six-step approach to decision-making. The PFS approach is given to allow multiple domain experts to focus more clearly on accurate expressions of their agreement and disagreement. PFS is shown to be an important new direction in practical multi-criteria decision-making methods for the project management practitioner.

The application of the traditional failure mode and effects analysis (FMEA) technique has been widely questioned in evaluation information, risk factor weights and robustness of results. This paper develops a novel FMEA framework with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment to solve these problems.

This paper introduces innovatively interval-value Pythagorean fuzzy weighted averaging (IVPFWA) operator, Tchebycheff metric distance and interval-value Pythagorean fuzzy weighted geometric (IVPFWG) operator into the MULTIMOORA submethods to obtain the risk ranking order for emergencies. Finally, an illustrative case is provided to demonstrate the practicality and feasibility of the novel fuzzy FMEA framework.

The feasibility and validity of the proposed method are verified by comparing with the existing methods. The calculation results indicate that the proposed method is more consistent with the actual situation of project and has more reference value.

The research results can provide supporting information for risk management decisions and offer decision-making basis for formulation of the follow-up emergency control and disposal scheme, which has certain guiding significance for the practical popularization and application of risk management strategies in the infrastructure projects.

A novel approach using FMEA with extended MULTIMOORA method is developed under IVPF environment, which considers weights of risk factors and experts. The method proposed has significantly improved the integrity of information in expert evaluation and the robustness of results.

The purpose of this paper is to create a numerical technique to tackle the challenge of selecting software reliability growth models (SRGMs).

A real-time case study with five SRGMs tested against a set of four selection indexes were utilised to show the functionality of TOPSIS approach. As a result of the current research, rating of the different SRGMs is generated based on their comparative closeness.

An innovative approach has been developed to generate the current SRGMs selection under TOPSIS environment by blending the entropy technique and the distance-based approach.

In any multi-criteria decision-making process, ambiguity is a crucial issue. To deal with the uncertain environment of decision-making, various devices and methodologies have been explained. Pythagorean fuzzy sets (PFSs) are perhaps the most contemporary device for dealing with ambiguity. This article addresses novel tangent distance-entropy measures under PFSs. Additionally, numerical illustration is utilized to ascertain the strength and authenticity of the suggested measures.

This study aims to address the inherent uncertainties within closed-loop supply chain (CLSC) networks through the application of a multi-objective approach, specifically focusing on the optimization of integrated production and transportation processes. The primary purpose is to enhance decision-making in supply chain management by formulating a robust multi-objective model.

In dealing with uncertainty, this study uses Pythagorean fuzzy numbers (PFNs) to effectively represent and quantify uncertainties associated with various parameters within the CLSC network. The proposed model is solved using Pythagorean hesitant fuzzy programming, presenting a comprehensive and innovative methodology designed explicitly for handling uncertainties inherent in CLSC contexts.

The research findings highlight the effectiveness and reliability of the proposed framework for addressing uncertainties within CLSC networks. Through a comparative analysis with other established approaches, the model demonstrates its robustness, showcasing its potential to make informed and resilient decisions in supply chain management.

This study successfully addressed uncertainty in CLSC networks, providing logistics managers with a robust decision-making framework. Emphasizing the importance of PFNs and Pythagorean hesitant fuzzy programming, the research offered practical insights for optimizing transportation routes and resource allocation. Future research could explore dynamic factors in CLSCs, integrate real-time data and leverage emerging technologies for more agile and sustainable supply chain management.

This research contributes significantly to the field by introducing a novel and comprehensive methodology for managing uncertainty in CLSC networks. The adoption of PFNs and Pythagorean hesitant fuzzy programming offers an original and valuable approach to addressing uncertainties, providing practitioners and decision-makers with insights to make informed and resilient decisions in supply chain management.

The purpose of this study is to build a consensus model of social network group decision-making (SNGDM) based on improved PageRank algorithm. By objectively and fairly measuring the evaluation ability of participants in the decision-making process, the authors can improve the fairness and authenticity of the weight solution of decision-makers (DM) in the decision-making process. This ensures the reliability of the final group consensus results.

This study mainly includes six parts: preference expression, calculation of DM's weight, preference aggregation, consensus measurement, opinion adjustment and alternative selection. First, Pythagorean fuzzy expression is introduced to express the preference of DMs, which expands the scope of preference expression of DMs. Second, based on the social network structure among DMs, the process of “mutual judgment” among DMs is increased to measure the evaluation ability of DMs. On this basis, the PageRank algorithm is improved to calculate the weight of DMs. This makes the process of reaching consensus more objective and fair. Third, in order to minimize the evaluation difference between groups and individuals, a preference aggregation model based on plant growth simulation algorithm (PGSA) is proposed to aggregate group preferences. Fourth, the consensus index of DMs is calculated from three levels to judge whether the consensus degree reaches the preset value. Fifth, considering the interaction of DMs in the social network, the evaluation value to achieve the required consensus degree is adjusted according to the DeGroot model to obtain the overall consensus. Finally, taking the group preference as the reference, the ranking of alternatives is determined by using the Pythagorean fuzzy score function.

This paper proposes a consensus model of SNGDM based on improved PageRank algorithm to aggregate expert preference information. A numerical case of product evaluation is introduced, and the feasibility and effectiveness of the model are explained through sensitivity analysis and comparative analysis. The results show that this method can solve the problem of reaching consensus in SNGDM.

Different DMs may have different judgment criteria for the same decision-making problem, and the angle and depth of considering the problem will also be different. By increasing the process of mutual evaluation of DMs, the evaluation ability of each DM is judged only from the decision-making problem itself. In this way, the evaluation opinions recognized by most DMs will form the mainstream of opinions, and the influence of corresponding DMs will increase. Therefore, in order to improve the fairness and reliability of the consensus process, this study measures the real evaluation ability of DMs by increasing the “mutual judgment” process. On this basis, the defect of equal treatment of PageRank algorithm in calculating the weight of DMs is improved. This ensures the authenticity and objectivity of the weight of DMs. That is to improve the effectiveness of the whole evaluation mechanism. This method considers both the influence of DMs in the social network and their own evaluation level. The weight of DMs is calculated from two aspects: sociality and professionalism. It provides a new method and perspective for the calculation of DM’s weight in SNGDM.