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Prioritization is usually required in problems involving multi-attribute group decision-making (MAGDM). Several strategies and procedures have been introduced in fuzzy systems to apply prioritization. This study examines the MAGDM problem in a Pythagorean fuzzy (PF) setting with varying amounts of demand for specialists and attributes. We regard the novel Aczel Alsina aggregation operators (AOs) as the most addition to fuzzy mathematics that can deal with uncertainties significantly. We suggest a few PF AOs based on Aczel Alsina t-norm and t-conorm, including the PF-prioritized Aczel Alsina averaging (PFPAAA) and PF-prioritized Aczel Alsina geometric (PFPAAG) operators. It is proven that these AOs fulfil the aggregation criteria by investigating the properties of monotonicity, boundedness, and idempotency. The weights for prioritization are derived from the knowledge of experts, and the proposed operators can capture the phenomenon of prioritization among the aggregated arguments. The proposed AOs are then applied to assess fire extinguishers using a MAGDM technique. The importance of PFPAAA and PFPAAG operators is verified by comparing the proposed AOs with other well-known AOs. [ABSTRACT FROM AUTHOR]

As an important way to cultivate talents, school-enterprise cooperation is highly praised by the educational and economic circles of all countries. Vocational education spans enterprises and schools, and is an education of cross-border innovation. Vocational schools should make the people they cultivate meet the needs of the market and enterprises, and carry out in-depth school-enterprise cooperation is one of the effective ways. The school-enterprise cooperation between vocational colleges and enterprises makes vocational education more targeted and practical, and plays an important role in the training of skilled talents. It is not only an inevitable requirement for enterprise development and participation in competition, but also an important direction of China's vocational education reform, and is the fundamental outlet for the development of China's vocational education. The performance evaluation of school-enterprise cooperation in vocational colleges is classical multiple-attribute group decision-making (MAGDM) issues. Recently, the TODIM and VIKOR method has been used to solve MAGDM issues. The 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) are used as a tool for characterizing uncertain information during the performance evaluation of school-enterprise cooperation in vocational colleges. In this manuscript, we design the 2-tuple linguistic Pythagorean fuzzy TODIM-VIKOR(2TLPF-TODIM-VIKOR) method to solve the MAGDM under 2TLPFSs. In the end, a numerical case study for performance evaluation of school-enterprise cooperation in vocational colleges is given to validate the proposed method. [ABSTRACT FROM AUTHOR]

The striking theory of ELECTRE III approach, being a marvelous strategy to deal with pseudo criterion, prevails over the traditional variants of ELECTRE method and other decision-making approaches for veracious decision-making. The noticeable efficiency and broader space of complex Pythagorean fuzzy model make it more significant and dominant for modeling two dimensional imprecise knowledge. The remarkable contribution of this study is to present a high aptitude variant of ELECTRE method by taking the advantage of the flexible structure of complex Pythagorean fuzzy sets closely following the outranking principles of ELECTRE III method. The proposed complex Pythagorean fuzzy ELECTRE III method is accredited to employ the theory of ELECTRE III technique to excellently deal with pseudo criterion as well as the two dimensional imprecise data for authentic decision-making. The proposed methodology uses three different threshold values, including preference, indifference and veto threshold values, to check the preference relation between alternatives. The presented strategy is applied to a case study for material selection to get the befitting decision. The comparative study with Pythagorean fuzzy ELECTRE III method is also included in this article to verify its decision-making aptitude. [ABSTRACT FROM AUTHOR]

Bonferroni means (BM) operator is the extended form of the arithmetic mean operator, used for simplifying non-dominant and non-feasible problems diagnosed in genuine life scenarios. A lot of aggregation operators are the specific parts of the BM operators under the consideration of different values of parameters which are the main parts of the BM operators. In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called complex Pythagorean fuzzy (CPF) setting, the objective of this scenario is to diagnose the CPF power BM (CPFPBM) operator and utilize their beneficial results with important properties. Moreover, a multi-attribute decision-making (MADM) technique is evaluated in the presence of invented operators for CPF settings. In the last of this study, we diagnosed the superiority and efficiency of the invented works with the help of sensitive analysis and graphical illustrations to enhance the gap of the research works. [ABSTRACT FROM AUTHOR]

This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers. [ABSTRACT FROM AUTHOR]

A sustainable supply chain (SSC) is vital for company's sustainability success, so it is imperative to identify and prioritize SSC's design requirements (DRs) for better SSC planning. For customer-centric markets, the customer requirements (CRs) need to be integrated into SSC's DRs. This paper thus proposes a customer-centric approach based on Analytic Network Process (ANP), Quality Function Deployment (QFD), Grey Relational Analysis (GRA), and Pythagorean Fuzzy Set (PFS) to rank SSC's DRs, considering CRs and information ambiguity. The PFS is combined with ANP, QFD, and GRA to better handle uncertainty in the SSC. The Pythagorean fuzzy ANP is applied to analyze the correlations among the sustainable CRs and determine the corresponding weights. The sustainable CRs are transformed into the DRs using the Pythagorean fuzzy QFD. The relationships among the resulting DRs are analyzed through Pythagorean fuzzy GRA to prioritize DRs. The approach is validated through a case study. The results obtained in this paper shows that the proposed method is efficient to prioritize DRs of SSC with the consideration of sustainable CRs under uncertain environment. The novelties of proposed method are that it not only offers a customer-oriented SSC planning method through the integration of ANP, QFD and GRA, but also can reflect the uncertain information with a broader membership representation space via PFSs. Based on the proposed method, the decision-maker can conduct comprehensive analysis to prioritize DRs and design appropriate SSC to fulfill CRs under uncertain environment. [ABSTRACT FROM AUTHOR]

Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy, parameterized Pythagorean fuzzy entropy, Pythagorean fuzzy cross-entropy, and weighted Pythagorean fuzzy cross-entropy measures are introduced consecutively based on the novel score factors, indeterminacy factors and J-divergence. Then some comparative experiments prove the rationality and effectiveness of the novel entropy measures and cross-entropy measures. Additionally, the Pythagorean fuzzy entropy and cross-entropy measures are designed to solve pattern recognition and multiple criteria decision making (MCDM) problems. The numerical examples, by comparing with the existing ones, demonstrate the applicability and efficiency of the newly proposed models. [ABSTRACT FROM AUTHOR]

Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy, parameterized Pythagorean fuzzy entropy, Pythagorean fuzzy cross-entropy, and weighted Pythagorean fuzzy cross-entropy measures are introduced consecutively based on the novel score factors, indeterminacy factors and J-divergence. Then some comparative experiments prove the rationality and effectiveness of the novel entropy measures and cross-entropy measures. Additionally, the Pythagorean fuzzy entropy and cross-entropy measures are designed to solve pattern recognition and multiple criteria decision making (MCDM) problems. The numerical examples, by comparing with the existing ones, demonstrate the applicability and efficiency of the newly proposed models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Hamacher aggregation operators and Choquet integral with dual Pythagorean hesitant fuzzy information. Then, motivated by the ideal of Hamacher operation and Choquet integral, we have developed some Hamacher correlated operators for aggregating dual hesitant Pythagorean fuzzy information. The prominent characteristic of these proposed operators is studied. Then, we have utilized these two operators to develop some approaches to solve the dual hesitant Pythagorean fuzzy MADM problems. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to evolve a novel area-based Pythagorean fuzzy decision model via an approach-oriented measure and an avoidance-oriented measure in support of multiple criteria decision analysis involving intricate uncertainty of Pythagorean fuzziness. Pythagorean membership grades embedded in a Pythagorean fuzzy set is featured by tensible functions of membership, non-membership, indeterminacy, strength, and direction, which delivers flexibility and adaptability in manipulating higher-order uncertainties. However, a well-defined ordered structure is never popular in real-life issues, seldom seen in Pythagorean fuzzy circumstances. Consider that point operators can make a systematic allocation of the indeterminacy composition contained in Pythagorean fuzzy information. This paper exploits the codomains of the point operations (i.e., the quantities that express the extents of point operators) to launch new measurements of approach orientation and avoidance orientation for performance ratings. This paper employs such measurements to develop an area-based performance index and an area-based comprehensive index for conducting a decision analysis. The applications and comparative analyses of the advanced area-based approach to some decision-making problems concerning sustainable recycling partner selection, company investment decisions, stock investment decisions, and working capital financing decisions give support to methodological advantages and practical effectiveness. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy set (PyFS) is an extension of various fuzzy concepts, such as fuzzy set (FS), intuitionistic FS, and it is enhanced mathematical gizmo to pact with uncertain and vague information. In this article, some drawbacks in the Dombi operational rules for Pythagorean fuzzy numbers (PyFNs) are examined and some improved Dombi operational laws for PyFNs are developed. We also find out that the value aggregated using the existing Dombi aggregation operators (DAOs) is not a PyFN. Furthermore, we developed two new aggregations, improved existing aggregation operators (AOs) for aggregating Pythagorean fuzzy information (PyFI) and are applied to multiple-attribute decision making (MADM). To acquire full advantage of power average (PA) operators proposed by Yager, the Pythagorean fuzzy Dombi power average (PyFDPA) operator, the Pythagorean fuzzy Dombi weighted power average (PyFDWPA) operator, Pythagorean fuzzy Dombi power geometric (PyFDPG) operator, Pythagorean fuzzy Dombi weighted geometric (PyFDPWG) operator, improved the existing AOs and their desirable properties are discussed. The foremost qualities of these developed Dombi power aggregation operators is that they purge the cause of discomfited data and are more supple due to general parameter. Additionally, based on these Dombi power AOs, a novel MADM approach is instituted. Finally, a numerical example is given to show the realism and efficacy of the proposed approach and judgment with the existing approaches is also specified. [ABSTRACT FROM AUTHOR]

The objective of the paper is to present an extension of the ordered weighted average (OWA) operator, probability OWA (POWA) operator, distance measures and Pythagorean membership grades. These new operators are called the Pythagorean membership grade OWA distance (PMGOD) operator and the probabilistic Pythagorean membership grade OWA distance (PPMGOD) operator. This extension includes in one formulation the ability to measure the ideal vs. real situation of the distance operators, the weighting vector and reordering step of the OWA operator and the way in which it includes uncertainty based on the satisfaction of the criteria of the Pythagorean membership degree. These propositions are applied to new business ventures in the city of Colombia, where the decision maker can express his/her concerns about the uncertainty and probabilities of the current business environment and, thus, generate a better decision-making process. [ABSTRACT FROM AUTHOR]

In this paper, a multistage decision-making problem concerning uncertainty and ambiguity is discussed using Pythagorean fuzzy sets. Complement Pythagorean fuzzy membership grades and their properties are also considered. Using the definition of an alpha-level set, we introduce the multistage decision-making problems, where the possibility theory and satisfaction grades are declared with the help of Pythagorean membership grades. Pythagorean multistage decision-making is an uncertain theory, where decision-maker has only one opportunity to choose the scenario under the combination of Pythagorean possibility and satisfaction grades at each stage. According to the selection of criteria, a series of decision points are concluded. The payoff collaborates with these decision points at each stage. The multistage decision-making using Pythagorean fuzzy sets is the scenario-based theory in place of other theories like lottery-based theory etc. The results have been calculated using multistage Pythagorean fuzzy sets in which the decision-maker has only one chance to select the optimal solution. The TOPSIS technique has been applied and the comparison between these two techniques is highlighted. [ABSTRACT FROM AUTHOR]

Complex Pythagorean fuzzy (CPF), a worthwhile generalization of Pythagorean fuzzy set, is a powerful tool to deal with two-dimensional or periodic information. In this paper, we develop two prioritized aggregation operators (AOs) under CPF environment, namely, complex Pythagorean fuzzy prioritized weighted averaging (CPFPWA) operator and complex Pythagorean fuzzy prioritized weighted geometric (CPFPWG) operator. We consider the prioritization relationship among criteria and decision makers (DMs) to make our result more accurate as in real decision making (DM) problems, the criteria and DMs have different priority level. Further, we discuss remarkable properties of our proposed AOs. Moreover, we promote the evolution of MCDM problem by investigating an algorithm in CPF environment with its flow chart. Finally, to check the superiority and validity of proposed operators, we compare the computed results with the different existing techniques. [ABSTRACT FROM AUTHOR]

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model. [ABSTRACT FROM AUTHOR]

A Pythagorean fuzzy soft set, an extension of intuitionistic fuzzy soft set, plays an essential role to handle the vagueness in many real-life problems. We apply this concept to graph theory, and present certain new notions including, perfectly regular Pythagorean fuzzy soft graphs (PFSGs), perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum (O ˆ NDS) and closed neighborhood degree sum (C ˆ NDS) of PFSGs. Finally, we discuss some decision-making problems of PFSGs. [ABSTRACT FROM AUTHOR]

This manuscript provides an advanced mathematical model for the censuses process to reduce the drawbacks of the existing methods. From the living ideas of m-polar fuzzy set (MPFS) and Pythagorean fuzzy set (PFS), we establish a novel concept of Pythagorean m-polar fuzzy set (PMPFS). We introduce some fundamental operations on Pythagorean m-polar fuzzy sets and explain these concepts with the help of illustrations. With this novel perspective, we build up modified forms of Dombi's aggregation operators named as Pythagorean m-polar fuzzy Dombi weighted arithmetic average (PMPFDWAA) operator and Pythagorean m-polar fuzzy Dombi weighted geometric average (PMPFDWGA) operator. We discuss certain properties of the proposed operators based on Pythagorean m-polar fuzzy numbers (PMPFNs). Mathematical modeling on real world problems often implicate multi-factors, multi-attributes and multi-polar information. We discuss a case study for the censuses process to elaborate the proposed algorithm for multi-criteria decision-making (MCDM). We also discuss how the drawbacks of existing methods can be handled by applying this novel perspective. Lastly, we present a comparative analysis, validity of proposed algorithm, influence of operational parameter, convergence and sensitivity analysis to indicate the flexibility and advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In the present communication, a new (R, S)-norm discriminant measure of Pythagorean fuzzy sets has been proposed along with its various properties. Monotonic behavior with respect to the parameters R & S and their proof of validity have also been studied. Methodologies and necessary steps of the algorithms for various decision-making problems viz. pattern recognition problem, medical diagnosis problem and multi criteria decision-making problem have been outlined on the basis of the proposed information measure. For the sake of illustration, numerical example for each case has been provided. A comparative analysis for the considered applications has also been studied in contrast with the popular existing methodologies stating important observations and advantages. [ABSTRACT FROM AUTHOR]

We study, in this paper, some notions related to Pythagorean fuzzy soft sets (PFSSs) along with their algebraic structures. We present operations on PFSSs and their peculiar characteristics and elaborate them with real life examples and tabular representation to develop the affluence of linguistic variables based on Pythagorean fuzzy soft (PFS) information. We present an application of PFSSs to the multi-criteria group decision-making (MCGDM) related to site selection, accompanied by Algorithm and flow chart. We develop PFS TOPSIS method and PFS VIKOR method as extensions of the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) respectively. Finally, we rendered an application in stock exchange investment problem and tackled it by both PFS TOPSIS and PFS VIKOR methods. [ABSTRACT FROM AUTHOR]

To describe the values of criteria and to generate a sort of alternatives are two important issues in multi-criteria decision-making (MCDM). A superior tool for the former issue is Pythagorean hesitant fuzzy number (PHFN) and an effective tool for the latter issue is aggregation operator. So far, a number of aggregation operators of PHFNs have been presented within the academia. Each aggregation operator has its own characteristics and can work well for its specific purpose. But there is not yet an aggregation operator of PHFNs that can provide satisfying generality and flexibility in aggregating the values of criteria and capturing the interactions of criteria. The Archimedean t-conorm and t-norm (ATT) are well-known for having the capability to generate versatile and flexible operational rules for fuzzy numbers, while the Muirhead mean (MM) operator is an all-in-one aggregation operator for capturing the interrelationships of the aggregated arguments. To this end, the MM operator and the ATT for PHFNs are combined to present a Pythagorean hesitant fuzzy Archimedean MM (PHFAMM) operator and a weighted PHFAMM operator and a new MCDM method based on the presented operators is proposed in this paper. Firstly, the generalised expressions of the presented operators are provided. The properties of the operators are explored and proved and their specific expressions based on Algebraic, Einstein, Hamacher, and Frank ATTs are then constructed. Based on these specific expressions, a new method for solving the MCDM problems based on PHFNs is developed. Finally, the developed MCDM method is demonstrated via an example, a set of experiments and qualitative and quantitative comparisons. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Bonferroni mean (BM) operator, generalized Bonferroni mean (GBM), and dual generalized Bonferroni mean (DGBM) operator with interval-valued 2-tuple linguistic Pythagorean fuzzy numbers (IV2TLPFNs) to propose interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (IV2TLPFWBM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (IV2TLPFWGBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (GIV2TLPFWBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (GIV2TLPFWGBM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (DGIV2TLPFWBM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (DGIV2TLPFWGBM) operator. Then the MADM methods are proposed with these operators. In the end, we utilize an applicable example for green supplier selection in green supply chain management to prove the proposed methods. [ABSTRACT FROM AUTHOR]

To describe the values of criteria and to generate a sort of alternatives are two important issues in multi-criteria decision-making (MCDM). A superior tool for the former issue is Pythagorean hesitant fuzzy number (PHFN) and an effective tool for the latter issue is aggregation operator. So far, a number of aggregation operators of PHFNs have been presented within the academia. Each aggregation operator has its own characteristics and can work well for its specific purpose. But there is not yet an aggregation operator of PHFNs that can provide satisfying generality and flexibility in aggregating the values of criteria and capturing the interactions of criteria. The Archimedean t-conorm and t-norm (ATT) are well-known for having the capability to generate versatile and flexible operational rules for fuzzy numbers, while the Muirhead mean (MM) operator is an all-in-one aggregation operator for capturing the interrelationships of the aggregated arguments. To this end, the MM operator and the ATT for PHFNs are combined to present a Pythagorean hesitant fuzzy Archimedean MM (PHFAMM) operator and a weighted PHFAMM operator and a new MCDM method based on the presented operators is proposed in this paper. Firstly, the generalised expressions of the presented operators are provided. The properties of the operators are explored and proved and their specific expressions based on Algebraic, Einstein, Hamacher, and Frank ATTs are then constructed. Based on these specific expressions, a new method for solving the MCDM problems based on PHFNs is developed. Finally, the developed MCDM method is demonstrated via an example, a set of experiments and qualitative and quantitative comparisons. [ABSTRACT FROM AUTHOR]

In this article, we expand the original TODIM(Portuguese acronym for Interactive Multi-Criteria Decision Making) method to the 2-tuple linguistic Pythagorean fuzzy environment to propose the 2TLPFs TODIM method. In the extended method, we use 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) to present the criteria values in multiple attribute decision making (MADM) problems. Firstly, we briefly introduce the definition, operational laws, some aggregation operators and the distance calculating method of 2TLPFNs. Then, the calculation steps of the original TODIM model are presented in simplified form. Thereafter, we extend the original TODIM model to the 2TLPFNs environment to build the 2TLPFNs TODIM model, our proposed method, which is more reasonable and scientific in considering the subjectivity of DM's behaviors and the dominance of each alternative over others. Finally, a numerical example for the safety assessment of a construction project is proposed to illustrate the new method, and some comparisons are also conducted to further illustrate the advantages of the new method. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy Hamacher aggregation operator is one of the aggregation operators in a Pythagorean fuzzy decision making environment. In most cases, it is assumed that all elements in Pythagorean fuzzy set (PFSs) are independent. However, in real life of decision problems, most of the criteria are interrelated. Population size in a solid waste management decision, for example, is related to human health since the two criteria are mutually dependent. This paper aims to introduce an aggregation operator that purposely dealt with the interactions between elements of PFSs. The Choquet integral operator is used to propose an innovation to the Pythagorean fuzzy Hamacher operator. Pythagorean fuzzy Hamacher Choquet integral average operators (PFHCIA) and Pythagorean fuzzy Hamacher Choquet integral geometric operators (PFHCIG) are proposed. The beauty of the proposed operators is it consider the interaction of the criteria in the decision making process. The proposed aggregation operators are the extension to the Pythagorean fuzzy Hamacher aggregation operators where Choquet integral is used to handle interactions between criteria. As the case study is considered, a water resource management problem is presented to portraythe application the proposed approach. Comparable results are also presented to check its feasibility and effectiveness. It is shown that the ranking using the proposed operators are inconsistent with the existing method. The importance of an aggregation function that can capture the problem of interdependence between the criteria is the main contribution of this paper. [ABSTRACT FROM AUTHOR]

In this manuscript we develop a comprehensive model to tackle decision making problems where strong points of view are in the favor and against some projects, entities or plans. Therefore a new approach is adopted to hybrid q-rung orthopair fuzzy sets with notions of covering rough set and TOPSIS. This model has stronger capability than intuitionistic fuzzy set and Pythagorean fuzzy set to manage the uncertainty. Furthermore an example is given to demonstrate how the proposed method helps us in decision making problems. [ABSTRACT FROM AUTHOR]

Pythagorean cubic fuzzy set, is an extension of the interval valued Pythagorean fuzzy set which relax the condition of the square sum of its membership and non-membership degree is less than or equal to one to supremum square sum of its membership and non-membership functions is less than one. Based on this information and by combining the idea of the confidence levels of each Pythagorean cubic fuzzy number, the proposed study investigated a new averaging and geometric operators, namely confidence Pythagorean cubic fuzzy weighted and geometric operators along with their order are presented. Some of their desired properties related to the new defined operators have been investigated. Under the given information, a multi criteria decision making process has been proposed with a numerical example for showing the effectiveness and validity of it. [ABSTRACT FROM AUTHOR]

Hesitant Pythagorean fuzzy set (HPFS) is a generalization of fuzzy set (FS). By integrating HPFS with rough set (RS), the notion of single granulation hesitant Pythagorean fuzzy rough sets (SGHPFRSs) is firstly put forward. Then, in view of the multigranulation framework, two types of multigranulation hesitant Pythagorean fuzzy rough sets (MGHPFRSs) are proposed, which are called the optimistic multigranulation hesitant Pythagorean fuzzy rough sets (OMGHPFRSs) and pessimistic multigranulation hesitant Pythagorean fuzzy rough sets (PMGHPFRSs). The connections between serial hesitant Pythagorean fuzzy relations and hesitant Pythagorean fuzzy approximation operators are established. The relationship between the MGHPFRSs and SGHPFRSs will be explored. Moreover, the relationship between the OMGHPFRSs and PMGHPFRSs will be established subsequently. Ultimately, an illustrative example will be given to expound the availability about the MGHPFRSs in multi-attribute decision making. [ABSTRACT FROM AUTHOR]

The recently proposed q-rung orthopair fuzzy set (q-ROFS) is an effective tool for handling fuzziness and uncertainty in the process of multi-attribute group decision making (MAGDM). Considering that in most MAGDM problems, not only decision makers' quantitative evaluation information but also their qualitative assessment opinions should be taken into account, this paper proposes a concept of q-rung orthopair uncertain linguistic sets (q-ROULSs) by combining q-ROFSs with uncertain linguistic variables (ULVs). The proposed q-ROULSs parallel to intuitionistic uncertain linguistic sets (IULSs) and Pythagorean uncertain linguistic sets (PULSs), but provide more freedom for decision makers to express their evaluation information. Afterwards, we propose operational laws and a comparison law for q-ROULSs. To effectively aggregate q-rung orthopair uncertain linguistic information, we extend the Muirhead mean (MM) operator to q-ROULSs and propose a family of q-rung orthopair uncertain linguistic Muirhead mean operators. The prominent advantage of the proposed operators is that the interrelationship among any numbers of input arguments can be captured. In addition, some desirable properties and special cases of the proposed operators are studied. Moreover, we propose a new method to MAGDM problems based on the proposed operators within q-rung orthopair uncertain linguistic context. Finally, we conduct a numerical experiment to demonstrate the validity and superiorities of the proposed method. [ABSTRACT FROM AUTHOR]

The recently proposed partitioned Bonferroni mean (PBM) can effectively deal with situations in which attributes are partitioned into several parts, and attributes in the same part are dependent, whereas the attributes in different parts are independent. The power average (PA) operator has the capacity of reducing the negative effects of decision makers' unreasonable assessments on the decision result. To take advantages of PBM and PA, we propose a family of Pythagorean fuzzy aggregation operators based on the interaction operational laws of Pythagorean fuzzy numbers (PFNs), such as the Pythagorean fuzzy interaction power PBM (PFIPPBM), the Pythagorean fuzzy interaction power partitioned geometric Bonferroni mean (PFIPPGBM), and the weighted forms of PFIPPBM and PFIPPGBM. The proposed operators can not only handle the situations where attributes are partitioned into several parts and attributes in the same part are interrelated, but also reduce the bad influence of unreasonable assessments on the decision result, and simultaneously consider the interactions between membership and non-membership degrees. Based on the proposed operators, a novel approach to multiple attribute group decision making (MAGDM) is proposed and a numerical instance as well as comparative analysis is conducted to demonstrate the validity and superiorities of the proposed approach. [ABSTRACT FROM AUTHOR]

The Heronian mean and generalized Heronian mean provide two aggregation operators which consider the inter-dependent phenomena among the aggregated arguments. In this paper, the Heronian mean and generalized Heronian mean under Pythagorean fuzzy environments is studied. First, the generalized Pythagorean fuzzy Heronian mean (GPFHM) operator and generalized Pythagorean fuzzy weighted Heronian mean (GPFWHM) operator are proposed and some of their desirable properties and special cases are investigated in detail. Furthermore, we propose the Pythagorean fuzzy geometric Heronian mean (PFGHM) operator and Pythagorean fuzzy weighted geometric Heronian mean (PFWGHM) operator. Then, two approaches to multiple attribute decision making based on GPFWHM operator and PFWGHM operator are proposed. Finally, a practical example for supplier selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we utilize Hamacher operations and power aggregation operators to develop some Pythagorean fuzzy Hamacher power aggregation operators: Pythagorean fuzzy Hamacher power average (PFHPA) operator, Pythagorean fuzzy Hamacher power geometric (PFHPG) operator, Pythagorean fuzzy Hamacher power weighted average (PFHPWA) operator, Pythagorean fuzzy Hamacher power weighted geometric (PFHPWG) operator, Pythagorean fuzzy Hamacher power ordered weighted average (PFHPOWA) operator, Pythagorean fuzzy Hamacher power ordered weighted geometric (PFHPOWG) operator, Pythagorean fuzzy Hamacher power hybrid average (PFHPHA) operator and Pythagorean fuzzy Hamacher power hybrid geometric (PFHPHG) operator. The prominent characteristic of these proposed operators are studied. Then, we utilize these operators to develop some approaches to solve the multiple attribute decision making problems with Pythagorean fuzzy numbers (PFNs). Finally, a practical example is given to verify the developed approach and deliver a comparative analysis. [ABSTRACT FROM AUTHOR]

In this paper, we initiate some new operators for Pythagorean fuzzy set and discuss their properties in detail. Then, a new score function of Pythagorean fuzzy number (PFN) is proposed for solving the failure problems when comparing two PFNs. Later, we present an algorithm for solving multi-criteria decision making (MCDM) problem based on Weighted Distance Based Approximation (WDBA). Finally, the effectiveness and feasibility of approach is demonstrated by some numerical examples. The salient features of the proposed method, compared to the existing Pythagorean fuzzy decision making methods, are (1) it can derive a ranking without the complex process; (2) it can obtain the optimal alternative without counterintuitive phenomena; (3) it has a great power in distinguishing the optimal alternative. [ABSTRACT FROM AUTHOR]

Since Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets are better tools to deal with fuzziness and vagueness. Therefore, in this paper, we present the notion of Pythagorean cubic fuzzy sets in which the membership degree and non-membership degree are cubic fuzzy numbers which hold the conditions that the square sum of its membership degree is less than or equal to 〈 1 ˜ , 1 〉. We define some basic operators and to compare two Pythagorean cubic fuzzy numbers we develop score and accuracy function. We also define the distance between two Pythagorean cubic fuzzy numbers. Based on the defined operators, we propose Pythagorean cubic fuzzy weighted averaging (PCFWA), Pythagorean cubic fuzzy weighted geometric (PCFWG), Pythagorean cubic fuzzy ordered weighted averaging (PCFOWA) and Pythagorean cubic fuzzy ordered weighted geometric (PCFOWG) operators. We discuss some of its operational laws of the established operators and suggest a multi criteria decision-making (MCDM) approach based on the developed operators. Moreover, the methods and operators proposed in this paper are providing more general, more accurate and precise results as compared to the existing methods because these methods and operators are the generalization of their existing methods. Furthermore, the method for multi-criteria decision-making problems based on these proposed operators was developed, and the operational processes were also illustrated in detail. Finally, an illustrative example is given to show the decision-making steps in detail of these proposed methods and operators to show the validity, practicality and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we extend MM operator and dual MM (DMM) operator to process the interval-valued Pythagorean fuzzy numbers (IVPFNs) and then to solve the MADM problems. Firstly, we develop some interval-valued Pythagorean fuzzy Muirhead mean operators by extending MM and DMM operators to IVPFNs. Then, we prove some properties and discuss some special cases with respect to the parameter vector. Moreover, we present some new methods to deal with MADM problems with the IVPFNs based on the proposed MM and DMM operators. Finally, we verify the validity and reliability of our methods by using an application example for green supplier selections, and analyse the advantages of our methods by comparing it with other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiple attribute decision making problems with Pythagorean fuzzy information. Then, we first introduce some operations on Pythagorean fuzzy sets, and further develop the induced Pythagorean fuzzy ordered weighted geometric (IPFOWG) operator. We also establish some desirable properties of this operator, such as commutativity, idempotency and monotonicity. Then, we apply the induced Pythagorean fuzzy ordered weighted geometric (IPFOWG) operator to solve the Pythagorean fuzzy multiple attribute decision making problems. Finally, an illustrative example for evaluating the operation performance of high-tech zone technology business incubator network is given to verify the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]

Pythagorean Hesitant fuzzy set (PHFS) which permits the membership degree and non-membership degree of an element to a set represented by several possible values is deliberated as a powerful tool to express uncertain information in the process of multi-attribute decision making (MADM) problems. In this paper, we propose a novel approach based on TOPSIS method and the maximizing deviation method for solving MADM problems where the evaluation information provided by the decision makers (DMs) is expressed in form of Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. To determine the attribute weight we develop an optimization model based on maximizing deviation method. Finally we provide a practical decision-making problem to demonstrate the implementation process of the proposed method. [ABSTRACT FROM AUTHOR]

Recently, the TODIM method has been used to solve multiple attribute decision making (MADM) problems. The interval-valued Pythagorean fuzzy sets (IVPFSs) are useful tools to depict uncertainty of the MADM. In this paper, we will extend the TODIM method to the MADM with the interval-valued Pythagorean fuzzy numbers (IVPFNs). Firstly, the definition, comparison and distance of IVPFNs are briefly introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, the extended classical TODIM method is proposed to deal with MADM problems with the IVPFNs, and its significant characteristic is that it can fully consider the decision makers' bounded rationality which is a real action in decision making. Furthermore, we develop the concept of the q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs) and extend the TODIM method to q-RIVOFNs. Finally, a numerical example is proposed and a comparative analysis is given. [ABSTRACT FROM AUTHOR]

Retail store selection is an important decision for both customers and retailers because it is directly linked to customer satisfaction and profitability of retailers. Because of the competitive market conditions, retailers severely try to find out what they should do to be preferred by potential customers, and consequently to grow their sales. In this context, the criteria influencing customers' store selection decision have to be analyzed. Although the relationship between customer preferences and retail store attributes has been widely studied through exploratory studies, a comprehensive framework using a multi-criteria decision making method under uncertainty to provide an overall assessment for retail stores has not yet been proposed. Pythagorean fuzzy sets are quite capable of representing uncertainty and vagueness in a decision making process by providing a larger domain to experts in expressing their opinions. Therefore, in this study, a novel interval-valued Pythagorean fuzzy WASPAS method is developed to evaluate the performance of retail stores. The obtained results are compared with crisp WASPAS and interval-valued intuitionistic WASPAS methods and it is revealed that the proposed method provides reliable and informative outputs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a novel correlation-based compromise approach for addressing multiple criteria decision analysis problems under complex uncertainty based on Pythagorean fuzzy sets. Based on the concepts of information energy and correlations for Pythagorean fuzzy characteristics, this paper proposes useful correlation-based compromise indices and investigates their desirable properties. These compromise indices can be employed to underlie anchored judgments and to reflect a certain balance between the correlations with positive-ideal and negative-ideal points of reference. Moreover, they can fully take into consideration the amount of information associated with higher degrees of uncertainty and can effectively fuse subjective assessments conveyed by Pythagorean fuzziness. From two different perspectives of displaced and fixed ideals, this paper provides two algorithmic procedures of the proposed correlation-based compromise approach for conducting multiple criteria evaluation tasks within Pythagorean fuzzy environments. A real-world problem concerning a financing decision on working capital policies is investigated to show the feasibility and applicability of the developed techniques. The application results, along with a comparative analysis, demonstrate the practicality and effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this paper, we utilize Hamacher operations and Prioritized aggregation operators to develop some Pythagorean fuzzy Hamacher Prioritized aggregation operators: Pythagorean fuzzy Hamacher Prioritized average (PFHPA) operator, Pythagorean fuzzy Hamacher Prioritized geometric (PFHPG) operator, Pythagorean fuzzy Hamacher Prioritized weighted average (PFHPWA) operator and Pythagorean fuzzy Hamacher Prioritized weighted geometric (PFHPWG) 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 a comparative analysis is also delivered. [ABSTRACT FROM AUTHOR]

The sum of membership and non-membership degrees of the Pythagorean fuzzy set is greater than one with their square sum less than or equal to one. Thus, as an extension of intuitionistic fuzzy set, the Pythagorean fuzzy set is a powerful tool to describe fuzziness and uncertainty. The aim of this study is to introduce some new operators for aggregating Pythagorean fuzzy information and apply them to multi-attribute decision making. Considering the advantages of the power average operator and Muirhead mean, we introduce the power Muirhead mean operator and investigate it under Pythagorean fuzzy environment. Thus, some new Pythagorean fuzzy aggregation operators, such as the Pythagorean fuzzy power Muirhead mean and the weighted Pythagorean fuzzy power Muirhead mean are developed. The prominent advantage of these proposed operators is that they consider the relationships between fused data and the interrelationships between all aggregated values, thereby obtaining more information in the process of multi-attribute decision making. Furthermore, we introduce a novel approach to multi-attribute decision-making problems based on the proposed operators. Finally, we provide a numerical example to illustrate the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets, which is based on intuitionistic fuzzy sets (IFSs), is an important tool to solve problems and has attracted a large number of researchers in different fields. As we know, studies have focused on interval-valued Pythagorean fuzzy set and aggregated operators. However, few studies focus on point operators. This paper introduces and discusses what is the pythagorean fuzzy point operators, study their properties and relationships, which is seen as the extensions of intuitionistic fuzzy sets. The uncertainty regarding to Pythagorean fuzzy set could be decreased if we use the pythagorean fuzzy point operators. In the end, pythagorean fuzzy multi-attributes decision making based on analytic hierarchy procedure is put forward to cope with the complicated MADM (multi-attributes decision making) issues which can be very useful when we face the multi-level analysis. [ABSTRACT FROM AUTHOR]

Recently, the TODIM (an acronym in Portuguese for Interactive Multi-criteria Decision Making) approach, which can characterize the decision makers’ psychological behaviors under risk, has been introduced to handle multiple attribute decision making (MADM) problems. The Pythagorean 2-tuple linguistic sets (P2TLSs) are useful tools to depict uncertainty in MADM. This paper use TODIM method to solve MADM with the Pythagorean 2-tuple linguistic numbers (P2TLNs). Firstly, the definition, comparison and distance of P2TLNs are briefly presented, and classical TODIM for MADM are introduced. Then, the extended classical TODIM method is proposed to deal with MADM problems with P2TLNs, and its important characteristic is to fully consider the decision makers’ bounded rationality which is a real action in MADM. Furthermore, we extend TODIM method to Pythagorean interval 2-tuple linguistic sets (PI2TLSs). Finally, a numerical example is proposed and a comparative analysis is also given. [ABSTRACT FROM AUTHOR]

In this erratum, we point out that two theorems and six equations in a previous paper by G. Wei (2017) are incorrect by the Pythagorean fuzzy operations in detail, and present the modified theorems and equations. The original paper appeared as Wei, G. (2017). Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making. Journal of Intelligent & Fuzzy Systems, 33(4), 2119–2132. . [ABSTRACT FROM AUTHOR]

For multi-attribute group decision making (MAGDM) problem, there exist inter-dependent or interactive phenomena among criteria or preference of decision makers, therefor it is not suitable for us to aggregate them by conventional aggregation operators based on additive measures. In this paper based on fuzzy measures an interval-valued Pythagorean fuzzy Choquet integral geometric (IVPFCIG) operator is investigated for multiple criteria group decision making. First, some operational laws on interval-valued Pythagorean fuzzy numbers (IVPFNs) are introduced. Then an interval-valued Pythagorean fuzzy Choquet integral geometric (IVPFCIG) operator is proposed. Moreover, some of its properties are given and discussed in detail. Then Choquet integral-based distance between interval-valued Pythagorean fuzzy numbers is defined. Combining the IVPFCIG operator with Choquet integral-based distance, an extension of TOPSIS method is developed to deal with multi-attribute interval-valued Pythagorean fuzzy group decision making problems. Finally a numerical example is used to illustrate the developed procedures. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiple attribute decision making problems with interval-valued Pythagorean uncertain linguistic information. Then, we utilize arithmetic and geometric operations to develop some interval-valued Pythagorean uncertain linguistic aggregation operators: interval-valued Pythagorean uncertain linguistic weighted average (IVPULWA) operator, interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator, interval-valued Pythagorean uncertain linguistic ordered weighted average (IVPULOWA) operator, interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator, interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator and interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator, some interval-valued Pythagorean uncertain linguistic correlate aggregation operators, some interval-valued Pythagorean uncertain linguistic induced aggregation operators, some interval-valued Pythagorean uncertain linguistic induced correlate aggregation operators and some interval-valued Pythagorean uncertain linguistic power aggregating operators. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the interval-valued Pythagorean uncertain linguistic multiple attribute decision making problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]