Pythagorean Fuzzy Sets Research Articles

Belief structure-based Pythagorean fuzzy entropy and its application in multi-source information fusion

Non-standard fuzzy sets play a significant role in uncertainty modeling. In addition to membership and non-membership degree, how to handle the hesitant degree is the key issue in the uncertain information process. In this paper, we model the Pythagorean fuzzy set (PFS) under the belief structure and measure its uncertainty based on fractal-based belief (FB) entropy. A novel fuzzy entropy for PFS called belief structure-based Pythagorean fuzzy (BSPF) entropy is proposed, whose effectiveness and advantages are proven based on mathematical analysis and numerical examples. A comparative analysis between BSPF entropy and other methods shows that BSPF entropy can obtain more reasonable results. Besides, a BSPF entropy-based multi-criteria decision-making (MCDM) method and a classification method are designed to solve practical problems. The experimental results demonstrate the effectiveness of these two proposed methods in solving real-world problems of decision-making and classification.

A Novel Similarity Measure and Score Function of Pythagorean Fuzzy Sets and Their Application in Assignment Problem

A Novel Similarity Measure and Score Function of Pythagorean Fuzzy Sets and Their Application in Assignment Problem

Another approach to linear Diophantine fuzzy rough sets on two universes and its application towards decision-making problems

Theories of the rough set (RS) and the fuzzy set (FS) are constructed to accommodate the uncertainty in the data analysis. Linear Diophantine FS (LD-FS) as a novel approach to decision-making (DM), broadening the predominating theories of intuitionistic FS (IFS), Pythagorean FS (PFS), q-rung orthopair FS (q-ROFS) deals with uncertain and vague information by considering the control or reference parameters. Exploring RSs in the framework of LD-FS is a propitious direction in RS theory, where LD-FSs are approximated by Linear Diophantine fuzzy relation (LD-FR). The primary aim of this article is to develop a new linear Diophantine fuzzy RS (LDF-RS) model based on an LD-FR over dual universes. The notions of lower and upper approximations of an LD-FS are introduced by using an LD-FR, and several fundamental structural properties are explored. Moreover, a connection between LDF-RSs and linear Diophantine fuzzy topology (LDF-topology) is established. In addition, some similarity relations among LD-FSs based on their lower and upper approximations are studied. Finally, a DM approach is crafted for the ranking of alternatives using the notions of LDF-RS. Moreover, a numerical example is designed and compared with some existing techniques.

CODAS extension using novel decomposed Pythagorean fuzzy sets: Strategy selection for IOT based sustainable supply chain system

Decomposed fuzzy sets (DFSs), which have been recently introduced in the field of fuzzy sets are a fuzzy set extension that deals with the membership degree and the non-membership degree of the elements in the set from a functional and dysfunctional point of view in more detail. On the other hand, Pythagorean fuzzy sets (PFSs) are another fuzzy set extension reflecting the hesitancy of decision-makers in a wider area. The main target of this study is to develop a new fuzzy set extension, decomposed Pythagorean fuzzy sets (DPFSs) in order to utilize the properties of DFSs and PFSs together. First, the basic principles and mathematical operators of the DPFSs are developed with their proofs. Then, a new fuzzy multi-criteria decision-making (MCDM) method, DPF- Combinative Distance based Assessment (CODAS) is developed to demonstrate the feasibility and effectiveness of the DPF extension. The developed method is handled in the strategy selection problem for Internet of Things (IoT)-based sustainable supply chain systems. A comprehensive sensitivity analysis is performed on both the criteria weights and the decision makers' weights to verify the stability and effectiveness of the obtained results. In addition, a comparative analysis with decomposed fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Pythagorean fuzzy CODAS and Pythagorean fuzzy TOPSIS methods is presented to show the validity, superiority and advantages of the developed method.

Socio-political evaluation of renewable energy resources under uncertain environment

Countries are developing green transition policies to reduce their dependency on conventional resources and to eliminate environmental concerns while meeting increasing energy demand. Renewable energy, thus, has a crucial effect on accomplishing these policies. To ensure a rapid spread of renewable energy and develop resilient renewable energy policies concerning the evaluation of renewable energy resources, socio-political factors should be considered. However, evaluating renewable energy technologies regarding the socio-political aspect is clearly more difficult and uncertain than a study focused only on quantifiable objectives, but it is critical since real projects are affected by such aspects. Multi-criteria decision-making (MCDM) techniques are best for addressing such issues with conflicting objectives. In this study, an integrated AHP-TOPSIS method with Pythagorean fuzzy sets (PFSs) was used to create the socio-political index for evaluating renewable energy hybrid technologies and Battery Energy Storage Systems (BESSs) in Turkey. The results showed that while solar energy has the highest socio-political index, Run of River with BESSs alternative ranks as the lowest.

Evaluating the blockchain-based healthcare supply chain using interval-valued Pythagorean fuzzy entropy-based decision support system

In the current era, blockchain technology has emerged as a novel technique to maintain the operations of healthcare management systems. The global COVID-19 disease has led to increase in the utilization of technology for healthcare supply chain, patient data management, and claims settlement. Data management in healthcare industry is a complex procedure where multiple organizations offer appropriate supply chain facilities in everyday life. Inadequate data handling disrupts the supply chain, which has a longstanding impact on the healthcare region. Blockchain-based solutions are renowned for their ability to produce safe and traceable results and effective in the health sector for secure data retrieval and storage, resulting in more effectual product creation and tracking. These technologies can deliver data provenance, sponsors candid healthcare industry demands, and certifies the immutability of multi direction transactions. Due to involvement of multiple factors/criteria, the selection of an effective blockchain platform-based healthcare supply chain can be treated as a decision support system (DSS) problem. Thus, the objective of this paper is to present a novel interval-valued Pythagorean fuzzy DSS for evaluating the blockchain platforms for healthcare supply chain and selecting a suitable one with respect to multiple criteria and uncertainty. The proposed DSS framework is divided into three stages. First, some fairly aggregation operators are presented to combine the individual’s information and also discussed their desirable characteristics on interval-valued Pythagorean fuzzy sets (IVPFSs). Second, an integrated weighting tool is developed using entropy-based tool for objective weight of criteria and pivot pairwise relative criteria importance assessment (PIPRECIA) model for subjective weight of criteria with IVPFSs. For this purpose, new entropy is presented on IVPFSs. Third, the multi-attribute ideal-real comparative analysis (MAIRCA) model is developed with the combination of fairly aggregation operators, entropy-based weighting tool and PIPRECIA model with interval-valued Pythagorean fuzzy information. Further, the proposed DSS is used on a case study of blockchain platforms for healthcare supply chain evaluation, which confirms its applicability and usefulness. Sensitivity and comparative assessments are carried out to check the consistency, robustness and efficiency of the presented method.

Similarity measures of Pythagorean fuzzy sets based on Lp metric and its applications to multicriteria decision-making with Pythagorean VIKOR and clustering

Similarity measures of Pythagorean fuzzy sets based on Lp metric and its applications to multicriteria decision-making with Pythagorean VIKOR and clustering

Fermatean Fuzzy Fairly Aggregation Operators with Multi-Criteria Decision-Making

A Fermatean fuzzy set (FRFS) is the extension of a fuzzy set, an intuitionistic fuzzy set, and a Pythagorean fuzzy set, and is used in different fields. Unlike other fuzzy structures, the sum of cubes of membership grades in FRFSs approximates a unit interval, increasing uncertainty. In this study, we intend to provide unique operational rules and aggregation operators (AOs) inside a Fermatean fuzzy environment. To develop a fair remedy for the membership degree and non-membership degree features of “Fermatean fuzzy numbers (FRFNs)”, our solution introduces new neutral or fair operating principles, which include the concept of proportional distribution. Based on the suggested operating principles, we provide the “Fermatean fuzzy fairly weighted average operator and the Fermatean fuzzy fairly ordered weighted averaging operator”. Our suggested AOs provide more generalized, reliable, and exact data than previous techniques. Combining the recommended AOs with multiple decision-makers and partial weight information under FRFSs, we also devised a technique for “multi-criteria decision-making”. To illustrate the application of our novel method, we provide an example of the algorithm’s effectiveness in addressing decision-making challenges.

Interval-valued Pythagorean fuzzy operational competitiveness rating model for assessing the metaverse integration options of sharing economy in transportation sector

The rise of the sharing economy (SE) has enabled new forms of urban mobility and had a substantial impact on the transportation sector. The applications of SE transportation lessen vehicle ownership and single-occupancy vehicles, which contributes to the sustainability of a region. Metaverse is a digital ecosystem that integrates SE applications with transportation sector and improves the sustainability of SE applications. The aim of this work is to assess and prioritize the Metaverse integration options of sharing economy from multiple criteria and uncertainty perspectives. In this regard, the proposed study develops a four stage decision-making method on interval-valued Pythagorean fuzzy (IVPF) setting. First, new score function is proposed to compare the interval-valued Pythagorean fuzzy numbers (IVPFNs) and further used to derive the weights of decision experts in the proposed method. Second, some IVPF interaction aggregation operators are proposed to aggregate the individual’s information in group decision-making. Third, an integrated weighting process is presented to obtain the objective weights of criteria using cross entropy-based formula and subjective weights by relative closeness coefficient-based model with IVPF information. For this purpose, a cross entropy is introduced for interval-valued Pythagorean fuzzy sets. Fourth, a hybrid operational competitiveness ratings assessment (OCRA) model is developed to rank the alternatives with IVPF information. Moreover, a case study of Metaverse integration options of sharing economy in transportation sector is taken to express the practicability and effectiveness of the presented method. Sensitivity analysis with respect to various values of parameter is discussed to reveal the stability of outcomes. It is also compared with some extant models to validate the superiority of developed method.

Multiobjective Optimization for FJSP Under Immediate Predecessor Constraints Based OFA and Pythagorean Fuzzy Set

In the flexible job shop scheduling problem (FJSP), immediate predecessor operations have an explicit impact on the scheduling results, and are often ignored. To analyze the influence of immediate predecessor operations on machine tool manufacturing workshops, four forms of immediate predecessor operations with the consideration of uncertain transportation and preparation time are defined. Then, the four forms of immediate predecessor operations are integrated into the FJSP, and a multiobjective FJSP is constructed. An improved optimal foraging algorithm (OFA) and Pythagorean fuzzy set (PFS) are combined to establish a multiobjective optimization algorithm, named PFSOFA. This algorithm is then used to solve the multiobjective FJSP. In PFSOFA, the Pareto fronts are mapped into PFSs. The Pythagorean fuzzy numbers (PFNs) in a PFS are transformed into right triangles, and the distances between the PFNs and the reference PFNs are defined as the distances between their right triangle centroids. Then, all distances of a PFS are integrated by the distance prospect function to obtain a distance prospect value. This value is then used to lead the iteration of PFSOFA. Through extensive experiments, including a real factory application case, the performance of PFSOFA is verified to be better than four classical multiobjective optimization algorithms for solving the multiobjective FJSP constructed in this paper. And for the factory case, PFSOFA also obtained the better schemes.

An investment decision framework for offshore wind-solar-seawater pumped storage power project under interval-valued Pythagorean fuzzy environment

Offshore wind-solar-seawater pumped storage (wind-PV-SPS) power system will be a very competitive offshore new energy project in the future because it can realize the complementarities of wind and photovoltaic resources in the dimensions of time and space, and reduce the waste of resources caused by voltage instability. In order to promote the sustainable development of offshore wind-PV-SPS system, it is particularly important to construct a scientific and reasonable investment decision-making framework. Firstly, a comprehensive criteria system is established in this paper, which includes economy, resource, environment, supporting conditions and market. Secondly, the interval-valued Pythagorean fuzzy set (IVPFS) is introduced to describe the uncertainty and fuzziness of information in the process of investment decision, and the expert weight correction model is established based on the similarity of IVPFSs. At the same time, IVPF-DEMATEL method is utilized to determine the weight of criteria, which takes the interaction between criteria into account. In addition, considering the decision-maker's risk preference psychology, we introduce the GOWUA-HARA operator into the PROMETHEE II method, and extend it to the interval-valued Pythagorean fuzzy environment to choose the optimal investment alternative. Finally, the proposed framework is applied to a case study. Then sensitivity analysis and comparative analysis are carried out to verify its feasibility and robustness.

Pythagorean fuzzy information processing based on centroid distance measure and its applications

By using Pythagorean fuzzy set (PFS), people can quantitatively describe and process information with fuzziness, and then make reasonable and effective judgments or decisions. In this process, distance measure often serves as a useful and important tool. The purpose of this paper is to explore some information processing methods and techniques in Pythagorean fuzzy environment. Firstly, the research progress of PFS is reviewed, and the main research directions and some hot issues in relevant research fields are discussed. Secondly, by analyzing the information of hesitation degree of Pythagorean fuzzy number (PFN), definitions of hesitation factor and centroid distance measure based on the representation of centroid coordinates of the hesitation regions are proposed, and properties of hesitation factor and centroid distance measure are discussed. Finally, the novel clustering algorithm, ranking method and multi criteria decision making (MCDM) method are developed in Pythagorean fuzzy environment, which are based on the new centroid distance measure. The effectiveness of the proposed algorithm and methods are verified through numerical examples and comparative analysis. The research results indicate that the centroid distance measure proposed in this paper can effectively characterize the distance between PFNs, thus making the clustering algorithm of PFNs, the ranking method of PFNs, and the Pythagorean fuzzy MCDM method have obvious superiorities and advantages.

University’s recruitment process using Fermatean fuzzy Einstein prioritized aggregation operators

In comparison to intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS), the Fermatean Fuzzy Set (FFS) is more efficacious in dealing ambiguous and imprecise data when making decisions. In this paper, we propose unique operations on Fermatean fuzzy information based on prioritized attributes, as well as Einstein’s operations based on adjusting the priority of characteristics in the Fermatean fuzzy environment. We use Einstein’s operations with prioritized attributes to propose new operations on Fermatean fuzzy numbers (FFNs), and then introduce basic aspects of these operations. Motivated by Einstein operations on FFNs, we develop Fermatean fuzzy Einstein prioritized arithmetic and geometric aggregation operators (AOs). In the first place, the concepts of a Fermatean fuzzy Einstein prioritized average (FFEPA), Fermatean fuzzy Einstein prioritized weighted average (FFEPWA), and Fermatean fuzzy Einstein prioritized ordered weighted average (FFEPOWA)-operators are introduced. Then, Fermatean fuzzy Einstein prioritized geometric (FFEPG) operator, Fermatean fuzzy Einstein prioritized weighted geometric (FFEPWG) operator, Fermatean fuzzy Einstein prioritized ordered weighted geometric (FFEPOWG) operator, and Fermatean fuzzy Einstein hybrid geometric (FFEHG) operator are given. We also go through some of the key characteristics of these operators. Moreover, using these operators, we establish algorithm for addressing a multiple attribute decision-making issue using Fermatean fuzzy data and attribute prioritizing. The case of university faculty selection is taken as a scenario to analyze and demonstrate the applicability of our suggested model. In addition, a comparison of the proposed and current operators is conducted, and the impact of attribute priority on the ranking order of alternatives is explored.

MADM framework based on the triangular Pythagorean fuzzy sets and applications to college public English teaching quality evaluation

The so-called “college English” teaching quality evaluation is to provide a basic, comprehensive, and realistic evaluation of the relevant aspects and management of teaching implementation on the basis of following the general laws of higher education; It is a comprehensive inspection of “College English” teaching and an important means of quality monitoring and policy adjustment for “College English”. As mentioned earlier, teaching evaluation is a comprehensive evaluation of teaching. Therefore, our evaluation of the quality of university public education is actually an examination of our specific measures in evaluating teaching, teaching methods and methods, teaching literature, and other aspects. The college public English teaching quality evaluation is a classical multiple attribute decision making (MADM). In this paper, we define the triangular Pythagorean fuzzy sets (TPFSs) and investigate the MADM problems under TPFSs. Based on the traditional dual generalized weighted Bonferroni mean (DGWBM) operator and dual generalized weighted geometric Bonferroni mean (DGWGBM) operator, some triangular Pythagorean fuzzy operators are proposed: triangular Pythagorean fuzzy DGWBM (TPFDGWBM) operator and triangular Pythagorean fuzzy DGWGBM (TPFDGWGBM) operator. Accordingly, we have took advantage of these operators to develop some approaches to work out the triangular Pythagorean fuzzy MADM. Ultimately, a practical example for college public English teaching quality evaluation is took advantage of to validate the developed approach, and an influence analysis of the parameter on the final results is been presented to attest its availability and validity.

Novel Approach to Multi-Criteria Decision-Making Based on the n,mPR-Fuzzy Weighted Power Average Operator

A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the nth power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a nth power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPAmn) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPAmn operator with the square-root fuzzy weighted power average (SR-FWPA), the nth power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators.

New approach for quality function deployment based on social network analysis and interval 2-tuple Pythagorean fuzzy linguistic information

Quality function deployment (QFD) is a useful quality management tool for the transformation of customer requirements (CRs) into engineering characteristics (ECs) in product development. In the literature, a lot of QFD approaches have been proposed to overcome the deficiencies and enhance the performance of the traditional QFD method. However, there are several issues deserving further investigation for improving QFD: majority of the existing methods describe experts’ opinions with single linguistic terms directly, which may produce the loss of information; most of the improved QFD methods cannot take the consensus levels among experts into account. In this study, we develop a new integrated QFD approach based on the interval 2-tuple Pythagorean fuzzy linguistic sets (I2PFLSs), the social network consensus reaching (SNCR) model, and an extended combined compromise solution (CoCoSo) method. There are three main contributions of the presented study: First, the correlation assessments between CRs and ECs given by experts are handled with the I2PFLSs. Second, the SNCR with minimum adjustment distance is employed to coordinate experts’ conflict correlation assessments by considering their trust relationships. Third, an extended CoCoSo method is employed to determine the importance ranking of ECs within the interval 2-tuple Pythagorean fuzzy linguistic environment. Finally, we present an electric vehicle product development case to demonstrate the effectiveness and superiority of the proposed QFD approach. The results show that the new QFD can express experts’ fuzzy and uncertain linguistic assessment information flexibly, deal with experts’ consensus in correlation assessment process, and yield more precise importance ranking of ECs for product improvement.

A novel goal programming approach based on accuracy function of pythagorean fuzzy sets

A novel goal programming approach based on accuracy function of pythagorean fuzzy sets

A Novel Pythagorean Fuzzy Set–Based Risk-Ranking Method for Handling Human Cognitive Information in Risk-Assessment Problems

With the rapid evolution of the information age and the development of artificial intelligence, processing human cognitive information has become increasingly important. The risk-priority-number (RPN) approach is a natural language-processing method and is the most widely used risk-evaluation tool. However, the typical RPN approach cannot effectively process the various forms of human cognitive information or hesitant information provided by experts in risk assessments. In addition, it cannot process the relative-weight consideration of risk-assessment factors. In order to fully grasp the various forms of human cognitive information provided by experts during risk assessment, this paper proposes a novel Pythagorean fuzzy set–based (PFS) risk-ranking method. This method integrates the PFS and the combined compromise-solution (CoCoSo) method to handle human cognitive information in risk-assessment problems. In the numerical case study, this paper used a healthcare waste-hazards risk-assessment case to verify the validity and rationality of the proposed method for handling risk-assessment issues. The calculation results of the healthcare waste-hazards risk-assessment case are compared with the typical RPN approach, intuitionistic fuzzy set (IFS) method, PFS method, and the CoCoSo method. The numerical simulation verification results prove that the proposed method can comprehensively grasp various forms of cognitive information from experts and consider the relative weight of risk-assessment factors, providing more accurate and reasonable risk-assessment results.

Evaluating Supply Resilience Performance of an Automotive Industry during Operational Shocks: A Pythagorean Fuzzy AHP-VIKOR-Based Approach

An integral aspect of global businesses and economic activities is the supply chain networks. Importantly, the coronavirus (COVID-19) pandemic scenario has further shown that the outbreak of diseases can create a global network-scale disruption to supply chain or logistics, thereby damaging several aspects of economic activities and business life. Hence, this study aims to assess the resilient supplier selection (RSS) process in the wake of the COVID-19 outbreak. A two-stage hybrid decision model using Pythagorean fuzzy sets was proposed as a case study from the automotive industry to deal with RSS during the COVID-19 outbreak. In the first stage, significant criteria and their corresponding sub-criteria were determined through a vast review of the literature and nominal group technique, while the relative weights for RSS were obtained through the Pythagorean Fuzzy Analytic Hierarchy Process (PFAHP) method. In the second stage, nine suppliers were evaluated with Pythagorean Fuzzy VIKOR (PFVIKOR) method. The results of the hybrid approach revealed that flexibility is the most important criterion among resilience criteria that constitute the most significant dimensions for RSS. In many studies, strategic criteria such as quality, cost, and delivery are found to be the most important criteria in supplier selection, however, in the wake of the COVID-19 outbreak, the opinions of decision-makers were significantly changed as the present study reveals that flexibility is the most important criterion to improve the operations of the supply chain for RSS. Next to flexibility is process capabilities, while quality (Q), and cost (C) existed as the first and second in the category of influential criteria for strategic supplier selection criteria, respectively. The managerial and practical implication is that, in the wake of COVID-19 disruptions, suppliers need to be re-evaluated based on resilience-related indicators.

An extended TOPSIS and entropy measure based on Sugeno integral in Pythagorean fuzzy set setting

As an extension of the concepts of fuzzy set and intuitionistic fuzzy set, the concept of Pythagorean fuzzy set better models some real life problems. Distance, entropy, and similarity measures between Pythagorean fuzzy sets play important roles in decision making. In this paper, we give a new entropy measure for Pythagorean fuzzy sets via the Sugeno integral that uses fuzzy measures to model the interaction between criteria. Moreover, we provide a theoretical approach to construct a similarity measure based on entropies. Combining this theoretical approach with the proposed entropy, we define a distance measure that considers the interaction between criteria. Finally, using the proposed distance measure, we provide an extended Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) for multi-criteria decision making and apply the proposed technique to a real life problem from the literature. Finally, a comparative analysis is conducted to compare the results of this paper with those of previous studies in the literature.