Pythagorean Fuzzy Sets Research Articles

Advancing Disc-Based Pythagorean Fuzzy Sets with Distinct Radii

Advancing Disc-Based Pythagorean Fuzzy Sets with Distinct Radii

Pythagorean Fuzzy Soft Somewhat Continuous Functions

In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future-assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat metamorphism’s within the context of Pythagorean fuzzy soft somewhat continuous functions, offering new insights into their topological invariant. Through detailed analysis and examples, we demonstrate the applicability of these concepts in various scientific and engineering problems. This work provides a comprehensive framework for understanding and utilizing Pythagorean fuzzy soft sets in complex decision-making scenarios. Finally, we compare various relationships across some generalizations of Pythagorean fuzzy soft continuous functions.

Reliability analysis for patient safety in the healthcare sector using dual hesitant pythagorean fuzzy set

Reliability analysis for patient safety in the healthcare sector using dual hesitant pythagorean fuzzy set

Subgroups and Homomorphism Structures of Complex Pythagorean Fuzzy Sets

This research introduces the notion of complex Pythagorean fuzzy subgroup (CPFSG). Both complex fuzzy subgroup (CFSG) and complex intuitionistic fuzzy subgroup (CIFSG) have significance in assigning membership grades in the unit disk in the complex plane. CFSG has a limitation solved by CIFSG, while CIFSG deals with a limited range of values. The important novelty of the CPFSG lies in its ability to solve the above limitations simultaneously and gets a wider range of values to be engaged in CPFSG. This work has introduced and investigated CPFSG as a new algebraic structure via the conditions that the sum of the square membership and non-membership lies on the unit interval for both the amplitude term and phase term. The result as any CIFSG is CPFSG but the convers is not true has been proved. Complex Pythagorean fuzzy coset has been defined and complex Pythagorean fuzzy normal subgroup (CPFNSG) and their algebraic characteristic has been demonstrated. Homomorphism on the CPFSG is shown. Some results as the inverse image of CPFSG and CPFNSG under isomorphism function are also a CPFSG and CPFNSG, respectively.

Preventive Maintenance Strategy Prediction of the Firewater Systems Based on the Pythagorean Fuzzy Cost–Benefit–Safety Analysis and Fuzzy Dematel

The firewater system is a complex system associated with the safety process of Hydrogen storage tanks. Predicting preventive maintenance strategies is essential to ensure the long-term reliability of this system. Therefore, it is necessary to evaluate the multistate reliability of the firewater system in order to predict preventive maintenance strategies and provide safety measures. A polymorphic fuzzy fault tree analysis (PFFTA) for the risk analysis of complex systems has attracted much attention because of its powerful evaluation capability and its ability to analyze relationships among basic events. However, obtaining multistate failure probability (MFP) data for basic events in PFFTA has always been a major challenge. It is also difficult to quantify the minimum cut set (MCS) in PFFTA and determine the critical components for selecting a preventive maintenance strategy. In this study, we propose the Pythagorean fuzzy cost–benefit–safety analysis by using the PFFTA, an improved consistency aggregation method (I-CAM), and fuzzy Dematel for a predictive preventive maintenance strategy. In the proposed approach, the I-CAM method was used to collect and aggregate weights of experts’ opinions to evaluate the MFP of basic events in PFFTA. As a result, a triptych cost–benefit–safety analysis based on Pythagorean fuzzy sets (PFSs) and the sum-product method (SPM) was estimated to reduce expert subjectivity, support an improved cost-effectiveness index to rank critical components, and fuzzy Dematel to evaluate influence of proposed preventive maintenance actions. To clarify the effectiveness and feasibility of the proposed methodology, a case study of the firewater system related to the plant is located in SONELGAZ electricity power plant (OUMACHE Unit) was demonstrated. Both evaluations of the cost–benefit–safety analysis of the critical component were performed, and selected the influence of preventive maintenance strategy of the firewater system was predicted.

Research on optimization of mining methods for broken ore bodies based on interval-valued pythagorean fuzzy sets and TOPSIS-GRA.

Identifying the optimal mining methods plays a pivotal role in ensuring both economic efficiency and environmental sustainability. This study aims to propose a model that combines interval-valued Pythagorean fuzzy sets (IVPFS) and TOPSIS-GRA to select the optimal mining method for broken ore bodies. First, a multi-factor comprehensive evaluation system, including economic, safety, and technical aspects, was established. IVPFS was introduced to express the fuzzy information of the decision-making process within the evaluation system. Additionally, an objective method combining the principle of fuzzy entropy measurement with EWM was proposed to determine the weights of fuzzy information. This method distinguished the importance of decision-makers and indicators. Then, an integration of distance and similarity (TOPSIS-GRA) was employed for ranking alternative solutions to select the optimal one. This model was applied to the decision-making problem of mining methods for the broken and difficult-to-mine ore bodies in the Tanyaokou mining area. Initial fuzzy evaluation information was obtained by having decision-makers score the mining methods. Results showed that the comprehensive scores of four alternatives are 0.5172, 0.4683, 0.5192, and 0.5465, respectively. The optimal method was the point-pillar upward horizontal layered filling mining method. Finally, the sensitivity analysis confirmed the stability of the model. The comparative results under different fuzzy environments (PFS and TFS) demonstrated the strong capability of IVPFS in handling fuzzy information for optimizing mining methods.

A Distance-Based Approach to Fuzzy Cognitive Maps Using Pythagorean Fuzzy Sets

AbstractFuzzy Cognitive Maps (FCMs) have been attracting researchers from a wide application area due to being easy to apply and interpret. Since its proposal, the method has been improved to satisfy the diverse needs of practitioners such as solving different types of problems and representing particular types of uncertainty. The classical FCMs depend highly on the decision-maker judgments and the uncertainty inherent in the judgments deserves significant attention. Although there are several fuzzy extensions integrated into FCMs, the uncertainty caused by the lack of knowledge, the hesitancy of decision makers, and also the limited capacity of humans to deal with pre-defined rules should be considered. To address this issue, a new distance-based approach integrating Pythagorean Fuzzy Sets and FCMs is proposed. To the best of our knowledge, this is the first time this extension is integrated into FCMs. Besides allowing to represent the uncertainty until the end of the calculations, the new approach offers decision makers an easier and more flexible way to assess the strength of existing causal relationships. To provide a comparison between the proposed approach and the classical FCMs, two real-life applications are selected as case studies.

Enhanced Cross-Entropy Exponential TODIM Framework for Interval-Valued Pythagorean Fuzzy Information and Applications to Art Teaching Quality Evaluation

Art teachers in preschool education need to possess not only high professional competence but also a commitment to continuous learning to enhance their comprehensive teaching abilities. The art teaching quality evaluation in preschool education programs at vocational colleges is multiple-attribute group decision-making (MAGDM). Currently, the Exponential TODIM and cross-entropy approach was employed to address MAGDM. Interval-valued Pythagorean fuzzy sets (IVPFSs) serve as a strategy for characterizing uncertain data during the art teaching quality evaluation in preschool education programs at vocational colleges. In this study, the interval-valued Pythagorean fuzzy number cross-entropy Exponential TODIM (IVPFN-CE-ExpTODIM) approach is developed to bring up MAGDM problems under IVPFSs. To validate the proposed method, a numerical study is conducted on the art teaching quality evaluation in preschool education programs at vocational colleges.

Innovative player evaluation: Dual-possibility Pythagorean fuzzy hypersoft sets for accurate international football rankings

This study introduces an advanced approach for ranking international football players, addressing the inherent uncertainties in performance evaluations. By integrating dual possibility theory and Pythagorean fuzzy sets, the model accommodates varying degrees of ambiguity and imprecision in player attributes. Additionally, the use of hypersoft set theory enriches the analysis by capturing the multifaceted nature of player evaluations. The proposed aggregation operators refine the synthesis of diverse information sources, leading to a comprehensive and nuanced assessment. This research significantly enhances player evaluation methodologies, providing a more adaptable framework for a fair assessment of international football talent. A practical example illustrates the application of dual-possibility Pythagorean fuzzy hypersoft sets (DP-PFHSS). A numerical technique is proposed for solving multi-criteria decision-making (MCDM) challenges with known dual possibility information using the proposed aggregation operators. This decision-making algorithm effectively determines a football player's worth, contributing to the overall ranking and evaluation process. The approach aids in scouting and recruitment by facilitating talent identification and informed player signings. Graphical analysis, comparing existing and proposed methods using average and geometric operators, demonstrates the superiority of the proposed approach in the players evaluation, indicating that F1 is in the top ranking.

Various distance between generalized Diophantine fuzzy sets using multiple criteria decision making and their real life applications

This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies.

Interval-valued fermatean fuzzy Aczel-Alsina geometric aggregation operators and their applications to group decision-making

Developing new aggregation operators on various classes of fuzzy sets and their generalizations is important in modelling real-life decision-making problems. Interval-valued Fermatean fuzzy sets (IVFFs) generalize the idea of interval-valued Pythagorean fuzzy sets (IVPFS) play a crucial role in modelling problems involving inadequate information. Decision-making problems modelled using IVFFNs require different score functions and aggregation operators on the set of IVFFNs. This study mainly focuses on establishing a few interval-valued Fermatean fuzzy (IVFF) aggregation operators by integrating the Aczel-Alsina (AA) operations to deal with group decision-making (GDM) problems. In this work, first, we discuss various Aczel-Alsina-based IVFF operations such as AA sum, AA product, and AA scalar multiplication for proposing a few new aggregation operators for the IVFF environment based on the new IVFF operations. Secondly, we introduce a few operators, including the interval-valued Fermatean fuzzy Aczel-Alsina (IVFFAA) weighted geometric operator, the IVFFAA ordered weighted geometric (IVFFAAOWG) operator, and the IVFFAA hybrid geometric (IVFFAAHG) operator. Various important properties such as idempotency, boundness, and monotonicity have also been studied. Thirdly, we establish multi-criteria group decision-making (MCGDM) method for solving real-life decision-making problems. Fourthly, we solve a model GDM problem to show the applicability and efficacy of our proposed MCGDM method, which utilizes the IVFFAAWG operator. Further, a sensitivity analysis is performed to ensure better performance, and finally, a comparative study of our method is done by comparing our proposed MCGDM approach with different existing methods.

Exploring The Boundaries of Uncertainty: Interval Valued Pythagorean Neutrosophic Set and Their Properties

The interval-valued Pythagorean fuzzy set (IVPFS) presents a novel approach to tackling vagueness and uncertainty, while neutrosophic sets, a broader concept encompassing of fuzzy sets and intuitionistic fuzzy sets, are tailored to depict real-world data characterized by uncertainty, imprecision, inconsistency, and incompleteness. Additionally, the development of Interval Value Neutrosophic Sets (IVNS) enhances precision in handling problems involving a range of numbers within the real unit interval, rather than focusing solely on a single value. However, despite these advancements, there is a deficiency in research addressing practical implementation challenges, conducting comparative analyses with existing methods, and applying these concepts across various fields. This study aims to bridge this research gap by proposing a novel concept based on the Interval Valued Pythagorean Neutrosophic Set (IVPNS), which is a generalization of the IVPFS and INS. The development of IVPNS provides a more comprehensive framework for handling uncertainty, ambiguity, and incomplete information in various fields, leading to more robust decision-making processes, improved problem-solving capabilities, and better management of complex systems. Furthermore, this research introduces the algebraic operations for IVPNS, including addition, multiplication, scalar multiplication, and exponentiation and provides a comparative analysis with IVPFS and IVNS. The study incorporates illustrative numerical examples to demonstrate these operations in practice. Additionally, this study provides and rigorously proves the algebraic properties of IVPNS, specifically discussing their commutative and associative properties. This validation ensures compliance with established conditions for IVPNS, reinforcing their theoretical soundness and practical applicability.

Hamacher Aggregation Operators for Pythagorean Fuzzy Set and its Application in Multi-Attribute Decision-Making Problem

Pythagorean fuzzy set is a useful expansion of intuitionistic fuzzy set for dealing with ambiguities, which mostly occur in real-life problems. Hamacher t-norm also has important and compatible norms that incorporate a parameter that offers various options to decision-makers during the information fusion process, thereby enhancing their ability to model decision-making problems effectively compared to alternative methods. In this study, Hamacher operators are being used to introduce several Pythagorean fuzzy Hamacher interactive weighted averaging (PFHIWA), Pythagorean fuzzy Hamacher interactive ordered weighted averaging (PFHIOWA), Pythagorean fuzzy Hamacher interactive weighted geometric (PFHIWG), and Pythagorean fuzzy Hamacher interactive ordered weighted geometric (PFHIOWG) operators. The properties of these operators are examined in detail. The benefit of using progressive operators is that they deliver more understanding of the scenario to the decision-makers. Proposed operators are utilized to elaborate multi-attribute decision-making (MADM). By showing the sensitivity analysis, our proposed operator has high stability related to multi-attribute decision-making (MADM) under the Pythagorean fuzzy data set.

An integrated group decision support framework utilizing Pythagorean fuzzy DEMATEL–CoCoSo approach for medicine cold chain logistics provider selection

PurposeThe purpose of this study is to establish a synthetic group decision framework based on the Pythagorean fuzzy (PF) set to select the optimal medicine cold chain logistics provider (MCCLP). Fierce market competition makes enterprises must constantly improve every link in the process of enterprise sustainable development. The evaluation of MCCLP in pharmaceutical enterprises is an important link to enhance the comprehensive competitiveness. Because of the fuzziness of expert cognition and the complexity of the decision procedure, PF set can effectively handle the uncertainty and ambiguity in the process of multi-criteria group decision decision-making (MCGDM).Design/methodology/approachThis paper develops an integrated group decision framework through combining the Decision-Making Trial and Evaluation Laboratory (DEMATEL) technique and combined compromise solution (CoCoSo) approach to select a satisfactory MCCLP within PF circumstances. First, the PF set is used to process the ambiguity and uncertainty of the cognition ability of experts. Second, a novel PF knowledge measure is propounded to measure the vagueness of the PF set. Third, a comprehensive criterion weight determination technique is developed through aggregating subjective weights attained utilizing the PF DEMATEL approach and objective weight deduced by knowledge measure method. Furthermore, an integrated MCGDM approach based on synthetic weight and CoCoSo method is constructed.FindingsThe outcomes of sensibility analysis and comparison investigation show that the suggested decision framework can help decision experts to choose a satisfactory MCCLP scientifically and reasonably. Accordingly, the propounded comprehensive decision framework can be recommended to enterprises and organizations to assess the MCCLP for their improvement of core competitiveness.Originality/valueMCCLP selection is not only momentous for pharmaceutical enterprises to improve transportation quality and ensure medicine safety but also provides a strong guarantee for enterprises to improve their core competitiveness. Nevertheless, enterprises face certain challenges due to the uncertainty of the assessment environment as well as human cognition in the process of choosing a satisfactory MCCLP. PF set possesses a formidable capability to address the uncertainty and imprecision information in the process of MCGDM. Therefore, pharmaceutical enterprises can implement the proposed method to evaluate the suppliers to further improve the comprehensive profit of enterprises.

Evaluation of Spare Parts Support Capacity of Civil Aircrafts Based on Type-2 Hesitant Pythagorean Fuzzy Sets and Improved Technique for Order Preference by Similarity to Ideal Solution

To improve the spare parts support capacity of civil aircrafts and given the actual lack of evaluation methods at present, the evaluation problem of spare parts support capacity was solved in this study by proposing a multi-attribute decision method based on Type-2 hesitant Pythagorean fuzzy sets and an improved Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). First, the basic definitions of Type-2 hesitant Pythagorean fuzzy sets were given, which were further promoted to Type-n hesitant Pythagorean fuzzy sets, and the basic order relation criterion of Type-2 hesitant Pythagorean fuzzy sets was introduced. Second, a complete evaluation system for spare parts supply support capacity was established with the spare parts of civil aircrafts as the study objects, and each evaluation indicator was introduced in detail. Then, the spare parts support solutions were preferentially sorted using the correlation coefficient formula of Type-2 hesitant Pythagorean fuzzy sets and improved TOPSIS. Finally, the reliability and reasonability of the proposed method were verified through an example calculation and comparative analysis. The experimental results indicate that the proposed method can acquire the evaluation results of spare parts support capacity more scientifically and can be referenced by relevant studies.

Integral-Valued Pythagorean Fuzzy-Set-Based Dyna Q+ Framework for Task Scheduling in Cloud Computing.

Task scheduling is a critical challenge in cloud computing systems, greatly impacting their performance. Task scheduling is a nondeterministic polynomial time hard (NP-Hard) problem that complicates the search for nearly optimal solutions. Five major uncertainty parameters, i.e., security, traffic, workload, availability, and price, influence task scheduling decisions. The primary rationale for selecting these uncertainty parameters lies in the challenge of accurately measuring their values, as empirical estimations often diverge from the actual values. The integral-valued Pythagorean fuzzy set (IVPFS) is a promising mathematical framework to deal with parametric uncertainties. The Dyna Q+ algorithm is the updated form of the Dyna Q agent designed specifically for dynamic computing environments by providing bonus rewards to non-exploited states. In this paper, the Dyna Q+ agent is enriched with the IVPFS mathematical framework to make intelligent task scheduling decisions. The performance of the proposed IVPFS Dyna Q+ task scheduler is tested using the CloudSim 3.3 simulator. The execution time is reduced by 90%, the makespan time is also reduced by 90%, the operation cost is below 50%, and the resource utilization rate is improved by 95%, all of these parameters meeting the desired standards or expectations. The results are also further validated using an expected value analysis methodology that confirms the good performance of the task scheduler. A better balance between exploration and exploitation through rigorous action-based learning is achieved by the Dyna Q+ agent.

An integrated multi-attributive border approximation area comparison (MABAC) method for evaluating resilience and knowledge sharing of suppliers in pythagorean fuzzy environment

This paper aims to determine the resilience scores of suppliers by considering the importance of knowledge sharing and incorporating a set of criteria that affect the supply chain’s resilience, along with knowledge sharing. In order to achieve this goal, we propose a new multi-criteria decision-making (MCDM) method named MULTIMABAC to evaluate suppliers’ resilience. This method is based on the multi-attributive border approximation area comparison (MABAC) method. The new method consists of criteria weighting, expert weighting, and supplier evaluation. In the criteria weighting section, we extend the BWM (Best–Worst method) using Pythagorean fuzzy sets (PFSs). The weight of experts is determined by incorporating the average concept into the MABAC method. The resilience score of suppliers is calculated using an extended MABAC method with ideal positive and negative solutions. All indices are aggregated using the ordinal priority approach (OPA) to obtain a unique ranking. Additionally, we extend the MULTIMABAC method using PFSs to address uncertainty in supply chain decision-making. Results and sensitivity analysis demonstrate that the proposed method effectively facilitates knowledge sharing in selecting resilient suppliers. Therefore, organizations can use the proposed method to evaluate and choose resilient suppliers. Moreover, the MULTIMABAC method demonstrates robustness to variations in the weights of its constituent parameters.

Development of Aczel-Alsina t-norm based linear Diophantine fuzzy aggregation operators and their applications in multi-criteria decision-making with unknown weight information

Aczel-Alsina t-norm and t-conorm are intrinsically flexible and endow Aczel-Alsina aggregation operators with greater versatility and robustness in the aggregation process than operators rooted in other t-norms and t-conorm families. Moreover, the linear Diophantine fuzzy set (LD-FS) is one of the resilient extensions of the fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PyFSs), and q-rung orthopair fuzzy sets (q-ROFSs), which has acquired prominence in decision analysis due to its exceptional efficacy in resolving ambiguous data. Keeping in view the advantages of both LD-FSs and Aczel-Alsina aggregation operators, this article aims to establish Aczel-Alsina operation rules for LD-FSs, such as Aczel-Alsina sum, Aczel-Alsina product, Aczel-Alsina scalar multiplication, and Aczel-Alsina exponentiation. Based on these operation rules, we expose the linear Diophantine fuzzy Aczel-Alsina weighted average (LDFAAWA) operator, and linear Diophantine fuzzy Aczel-Alsina weighted geometric (LDFAAWG) operator and scrutinize their distinctive characteristics and results. Additionally, based on these aggregation operators (AOs), a multi-criteria decision-making (MCDM) approach is designed and tested with a practical case study related to forecasting weather under an LD-FS setting. The developed model undergoes a comparative analysis with several prevailing approaches to demonstrate the superiority and accuracy of the proposed model. Besides, the influence of the parameter Λ on the ranking order is successfully highlighted.

Novel similarity measures under complex pythagorean fuzzy soft matrices and their application in decision making problems

Complex fuzzy soft matrices play a crucial role in various applications, including decision-making, pattern recognition, signals processing, and image processing. The main objective of this study is to introduce the unique notions of complex Pythagorean fuzzy soft matrices (CPFSMs), which provide more flexibility and accuracy in modelling uncertainty. CPFSMs incorporate Pythagorean fuzzy soft matrices, allowing for more sophisticated uncertainty modeling. The key findings of CPFSMs, specific instances, and certain fundamental set-theoretic operations and principles were covered. A set of new distance metrics between two CPFSMs has been defined. In the context of complex Pythagorean fuzzy soft sets and complex Pythagorean fuzzy soft matrices, we created a CPFS decision-making technique. Moreover, the application’s numerical example and comparison analysis have been effectively demonstrated. Thus, by integrating the concepts of Pythagorean fuzzy sets, soft matrices, and complex numbers, CPFSMs provide a robust framework with membership and non-membership degrees for complex decision-making modeling and analyzing uncertain data.

EDAS method for circular pythagorean fuzzy with improved Dombi power aggregation operators and their application in policy analysis and decision support systems

A circular Pythagorean fuzzy set is a very dominant and reliable technique for coping with uncertain and vague information in genuine life problems because it contains the membership function, non-membership function, and radius between both grades with a condition that is the sum of the pair will be contained in the unit interval. The objective of this article is to define some improved Dombi operational laws based on circular improved Pythagorean fuzzy values. Moreover, we discuss some drawbacks of the existing Dombi aggregation operators based on circular improved Pythagorean fuzzy numbers. We also defined some improved Dombi averaging and geometric aggregation operators and their properties. Later on, a technique of evaluation based on distance from the average solution is stated by using the proposed operators. Additionally, we resolve the problem of policy analysis and decision support systems based on multi-attribute decision-making problems for initiated techniques to enhance the worth of the derived theory. Finally, we illustrate some examples for showing the supremacy and validity of the proposed theory with the help of comparative analysis between proposed ranking values and some existing ranking values.