Pythagorean Fuzzy Environment Research Articles

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.

Hellinger distance measures on Pythagorean fuzzy environment via their applications

Pythagorean fuzzy sets (PFSs) are a versatile tool for handling uncertain problems and have proven effective in practical applications. However, many existing Pythagorean fuzzy distance measures have counter-intuitive situations, making it challenging to measure the difference between PFSs accurately. To address this issue, we propose two distance measures for PFSs inspired by the Hellinger distance measure. We also explore the properties of the proposed measures and provide several comparative examples with existing measures for PFSs, illustrating their superior performance in processing fuzzy information from PFSs. Finally, we further develop a new decision-making method on top of the proposed measures and evaluate its performance in two applications.

River ecosystem health assessment in the Qinghai-Tibet Plateau: A novel hybrid method based on artificial intelligence and multi-source data fusion

River ecosystem health assessment (REHA), an effective approach for identifying river ecosystem health, is crucial for achieving sustainable river management and ensuring water security. However, existing REHA methods still fail to consider the cumulated influences of uncertain inputs, stochastic environment and limited rationality of decision makers on REHA. Additionally, current REHA studies have mainly concentrated on plain areas, while the Qinghai-Tibet Plateau (QTP) remains largely unknown. Developing REHA techniques for plateau rivers is an urgent matter, due to the heightened fragility and complexity of river ecosystems in the QTP. To accurately assess river ecosystem health in the QTP, this study proposed Pythagorean fuzzy cloud (PFC) via coupling the Pythagorean fuzzy sets and cloud model. A novel PFC-TODIM model was developed by extending TODIM (the acronym in Portuguese for interactive and multicriteria decision making) to the Pythagorean fuzzy environment. The hybrid decision making framework was then created to handle REHA with uncertain inputs and stochastic environment, and the Senge Tsangpo River (STR) served as a case study in the QTP. We developed the indicator system based on multi-source data fusion, and employed Bayesian model averaging (BMA) method to reveal the potential risks and driving factors of river ecosystem health. Results showed that the developed models considered the limited rationality of decision makers, effectively handled REHA with uncertainties, and avoided overestimating river health levels due to ignoring the randomness and fuzziness of REHA. In STR, health statuses exhibited marked spatial differences. Sampling sites of 9.091%, 77.273 % and 13.636 % were excellent, healthy and subhealthy, respectively. Our findings highlight that dams, urban development, fish release, and grazing have adverse impacts on STR health, and effective protection measures are required to minimize human interferences on ecologically fragile areas. These findings can improve our understanding of the human disturbances and natural factors that interfere with river ecosystem health in the QTP.

Clustering analysis for Pythagorean fuzzy sets and its application in multiple attribute decision making

Pythagorean fuzzy sets, as a generalization of intuitionistic fuzzy sets, have a wide range of applications in many fields including image recognition, data mining, decision making, etc. However, there is little research on clustering algorithms of Pythagorean fuzzy sets. In this paper, a novel clustering idea under Pythagorean fuzzy environment is presented. Firstly, the concept of feature vector of Pythagorean fuzzy number (PFN) is presented by taking into account five parameters of PFN, and some new methods to compute the similarity measure of PFNs by applying the feature vector are proposed. Furthermore, a fuzzy similarity matrix by utilizing similarity measure of PFNs is established. Later, the fuzzy similarity matrix is transformed into a fuzzy equivalent matrix which is utilized to establish a novel Pythagorean fuzzy clustering algorithm. Based on the proposed clustering algorithm, a novel multiple attribute decision making (MADM) method under Pythagorean fuzzy environment is presented. To illustrate the effectiveness and feasibility of the proposed technique, an application example is offered.

Contamination control in the oil and gas industries using industrial separation technologies: A sustainability and safety-oriented DSS under Pythagorean fuzzy environment

ABSTRACT The challenges related to the sustainability and safety policies to use the technologies for separating industrial contaminants in the oil and gas industry necessitate an investigation based on these policies. The criteria are in four economic, technical, social, and environmental aspects (as sustainability principles), and three applicability, functionality, and integrity aspects (as safety principles). The research aims at extending a novel decision support system (DSS) by integrating MEREC (method based on the removal effects of criteria) and MABAC (multi-attributive border approximation area comparison) methods under the Pythagorean fuzzy set. Its notable advantages are that renders a stronger expression of uncertainty and the decision-making process is more closely managed. Also, it allows the decision-maker (DM) to evaluate in a wider area and avoid data and conflict among the decision-making team. The industrial separation technologies prioritization is implemented with the DSS. These technologies have been taken into account for contamination control in the sector of Iran’s natural gas processing. The results of the prioritization are: High-efficiency liquid–gas coalescer technology, mesh pad technology, filter separator technology, cyclonic separator technology, vane pack technology, two-phase separator technology, and supersonic separator technology. The high-efficiency liquid–gas coalescer technology is selected as the preferable alternative.

A new CoCoSo ranking-based QFD approach in Pythagorean fuzzy environment and its application on evaluating design attributes of mobile medical App

Quality function deployment (QFD) is a customer-driven product development technique that converts customer requirements (CRs) into design attributes (DAs) of a product and service. Nevertheless, in real situations, the traditional QFD method has been found that possesses some deficiencies, such as the accuracy assessment of relationships between CRs and DAs, and the inter-relationships among DAs. To fill in the above gaps, this study develops a new QFD approach by a CoCoSo-based ranking method under Pythagorean fuzzy environment. To begin with, an extended Pythagorean fuzzy decision-making trial and evaluation laboratory (DEMATEL) method is proposed to identify the relationships within DAs. Second, the aggregation method of the weighted average method and objective penalty function are propounded to construct the programming models for calculating the importance of DAs under Pythagorean fuzzy setting. Third, a new CoCoSo-based ranking method for Pythagorean triangular fuzzy numbers (PTrFNs) is proposed to obtain the ranking of DAs. Lastly, a case regarding “Ping An Health” mobile medical App is carried out to verify the effectiveness and superiority of the proposed QFD approach. The results show that the top DA is perceptibility. Therefore, perceptibility should be focus on firstly in the “Ping An Health” App design, such as system fluency, interface comfort and network stability. Additionally, the results show that the new QFD can express experts’ hesitant assessment information, deal with the interrelations among DAs, and yield more precise rankings of DAs in QFD.

A similarity-based three-way multiattribute decision model under constrained Pythagorean fuzzy environment

A variety of fuzzy sets, especially Pythagorean fuzzy sets (PFSs), are commonly utilized to describe the ambiguity of information in three-way decision (3WD) models. PFSs can handle fuzzy information well but ignore the description of stochastic information. To address this issue, in this paper, we attempt to construct a novel three-way multiattribute decision model by combining 3WD with constrained Pythagorean fuzzy sets (CPFSs) which can measure the reliability of PFSs. Firstly, we introduce the concept of constrained Pythagorean fuzzy information systems (CPFISs). Comprehensive positive and negative similarities of objects are given in a CPFIS by introducing normalization, score function, comparison rules, and positive and negative ideal solutions. Secondly, by comprehensive positive and negative similarities, novel formulas for calculating comprehensive relative utility function values are obtained and an objective method for estimating conditional probabilities is presented in the absence of decision attributes. Based on the above knowledge, we establish a similarity-based three-way multiattribute decision model in a CPFIS. Furthermore, we conduct a case analysis of appendicitis to realize the application of our proposed model and several comparative analyses to demonstrate the effectiveness and superiority of our model. Finally, the good performance of the new model is verified through experimental analyses.

An Approach to Lagrange’s Theorem in Pythagorean Fuzzy Subgroups

The Pythagorean fuzzy environment is a modern way of depicting uncertainty. The concept of Pythagorean fuzzy semi-level subgroups of any group is described in this paper. The Pythagorean fuzzy order of an element in a Pythagorean fuzzy subgroup is introduced and established various algebraic attributes. The relation between the Pythagorean fuzzy order of an element of a group and the order of that group is established. The Pythagorean fuzzy normalizer and Pythagorean fuzzy centralizer of Pythagorean fuzzy subgroups are discussed. Further, the concept of Pythagorean fuzzy quotient group and the index of a Pythagorean fuzzy subgroup are defined. Finally, a framework is developed for proving Lagrange’s theorem in Pythagorean fuzzy subgroups.

Interval-valued Pythagorean fuzzy entropy and its application to multi-criterion group decision-making

<abstract><p>Entropy is an important tool of information measurement in the fuzzy set and its inference. The research on information measurement based on interval-valued Pythagorean fuzzy sets mostly involves the distance formula for interval-valued Pythagorean fuzzy numbers, but seldom involves the measurement of fuzziness. In view of this situation, we have aimed to propose new interval-valued Pythagorean fuzzy entropy and weighted exponential entropy schemes. Based on the interval-valued Pythagorean fuzzy weighted averaging operator, a strategy based on weighted exponential entropy is proposed to solve the multi-criteria group decision-making (MCGDM) problem in the interval-valued Pythagorean environment. Two examples illustrate that this paper provides a feasible new method to solve the MCGDM problem in an interval-valued Pythagorean fuzzy (IVPF) environment. Finally, by comparing with the existing methods, it is concluded that the entropy measure of IVPF schemes and the corresponding MCGDM method can select the optimal solution of the practical problem more precisely and accurately. Therefore, the comparative analysis shows that the proposed measurement method has the characteristics of flexibility and universality.</p></abstract>

Multi-criteria group decision making based on graph neural networks in Pythagorean fuzzy environment

Given that the majority of existing approaches for multi-criteria group decision making (MCGDM) rely solely on the preferences of decision makers (DMs) and fail to consider the various relationships between alternatives, this paper attempts to model the relevant relational structures using graphs and introduce the concept of graph neural networks (GNNs) in the context of group decision-making. By leveraging the powerful expressive capabilities of GNNs, the aim is to mine additional information pertinent to the decision-making process and screen out alternatives for the final decision. To begin, we provide a mapping of MCGDM to the graph domain and construct a corresponding relation graph among alternatives. Additionally, to deal with uncertain or vague information, we transform the group decision-making problem into a Pythagorean fuzzy environment and define a novel measure of entropy specifically designed for Pythagorean fuzzy sets (PFSs) in the entropy weight model to determine the weights of criteria. Simultaneously, we propose a new distance measure for PFSs, which is then applied to the extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method to rank alternatives. Furthermore, we develop a GNNs-based Pythagorean fuzzy MCGDM approach that incorporates the aforementioned techniques for group decision-making. Finally, to validate the effectiveness and superiority of this approach, we employ it to address a supplier selection issue. Compared with baseline group decision-making approaches, our approach can indeed capture the relationships among alternatives in complex group decision-making scenarios and outperforms the best-performing baseline by nearly 2.8% in terms of ranking accuracy.

An integrated linguistic Pythagorean fuzzy decision-making approach for risk analysis of offshore wind turbine

The evaluation of risk factors within the context of condition-based maintenance (CBM) for offshore wind turbines has been a challenging issue. In this paper, a novel risk analysis method that combines the fuzzy weighted with zero inconsistency (FWZIC) technique and the DEMATEL method under the linguistic Pythagorean fuzzy environment is proposed. Firstly, 75 risk factors linked to offshore wind turbines are identified by leveraging analysis and prior literature, which are modeled using linguistic Pythagorean fuzzy sets (LPFSs) to facilitate the complexity of expert opinions in the face of uncertainty. Then, the hybrid expert weight calculation method and the FWZIC-based criteria weight calculation method are introduced to determine the weights of the experts and the criteria while considering uncertainty. Next, the extended DEMATEL method under the linguistic Pythagorean fuzzy environment is adopted to determine the significance and influence of the pinpointed risk factors, and an illustrative cause–effect diagram is crafted to dissect the interconnections between these factors. Through this intricate analysis, the cause factors and effect factors specific to offshore wind turbines are identified, and the results obtained from the proposed method are validated through sensitivity analysis and comparative analysis. The results of this study could provide valuable recommendations and references for the implementation of CBM in offshore wind turbines.

A hybrid multi-criteria decision-making framework of EWM-BWM-TODIM based on Linguistic Pythagorean fuzzy environment

Linguistic Pythagorean fuzzy set (LPFS) combines Pythagorean fuzzy sets and linguistic term sets, which can effectively deal with fuzzy information in multi-criteria decision-making (MCDM). The entropy weight method (EWM) can reflect the objectivity of decision information, while the best-worst method (BWM) can reflect the subjectivity of decision-makers. The interactive multi-criteria decision-making (TODIM) method can describe the different preferences of decision-makers for gains and losses. In this paper, EWM, BWM, and TODIM are combined and applied to LPFS for the first time. First, we calculate the objective weight and subjective weight of each criterion through EWM and BWM and combine them to get the final weight to balance subjectivity and objectivity. Then, this paper selects the best scheme through TODIM sorting. In conclusion, the LPFS-EWM-BWM-TODIM model is established in this paper. Finally, the paper applies this model to the selection of corporate investment strategy and green mine, verifies the effectiveness of the method, and carries out comparative analysis and sensitivity analysis, proving the rationality and robustness of the model.

Group decision-making method based on advanced aggregation operators with entropy and divergence measures under 2-tuple linguistic Pythagorean fuzzy environment

2-tuple linguistic Pythagorean fuzzy values (2TLPFVs) are flexible and more efficient tools to represent fuzzy linguistic information in complex real-world situations than 2-tuple linguistic values (2TLVs) and 2-tuple linguistic intuitionistic fuzzy values (2TLIFVs). This paper introduces a new score value function for 2TLPFVs and discusses its major properties. Two new aggregation operators (AOs) are presented, namely, the advanced 2-tuple linguistic Pythagorean fuzzy weighted average (Adv-2TLPFWA) operator and the advanced 2-tuple linguistic Pythagorean fuzzy weighted geometric (Adv-2TLPFWG) operator, which aggregate 2-tuple linguistic Pythagorean fuzzy information based on novel operational laws of 2TLPFVs. The work also exhibits several properties of these AOs in detail. Furthermore, the article discusses entropy and divergence measures under the 2-tuple linguistic Pythagorean fuzzy information environment. Finally, a new multiple attribute group decision-making (MAGDM) method is constructed to solve group decision-making problems in the 2-tuple linguistic Pythagorean fuzzy context. A real-life group decision-making problem of selecting the best green supplier for fruits and vegetables is investigated to verify the applicability and validity of the proposed approach. The sensitivity investigation and comparative analysis are conducted to illustrate the advantages of the proposed method.

Multi-criteria group decision-making method based on total distance and BWM with spatial information in Hesitant Pythagorean fuzzy environment

The processing method of fuzzy information is a critical element in multi-criteria group decision-making (MCGDM). The hesitant Pythagorean fuzzy set (HPFS) has a higher capacity in express the uncertainty of human inherent preference. A composite weighted mathematical programming model with prospect theory and best-worst method (BWM) is proposed to solve the uncertainty of criterion weight acquisition and decision-makers (DMs) psychological behavior under the HPF environment. The decision-making process is as follows: Firstly, a novel spatial distance measurement method is designed which considers the extension space of HPFSs space by five parameters under the HPF environment. Secondly, the optimal criteria weights model minimizes the total distance between the alternatives and the HPF positive ideal solution (HPFPIS), as well as minimizes the consistency ratio of BWM. Thirdly, we propose the prospect decision matrix by the prospect theory and optimal weights, then use the ordered weighted average operator under the normal distribution to calculate the weight of DMs and rank the decision alternatives. Finally, an example is illustrated here, sensitivity and reliability, and comparative analysis are conducted to verify the effectiveness of the proposed method.

A MCDM-based framework for selection of photovoltaic cell technology using novel information measure under Pythagorean fuzzy environment

A MCDM-based framework for selection of photovoltaic cell technology using novel information measure under Pythagorean fuzzy environment

Multi-attribute group decision making method based on Pythagorean fuzzy Einstein interactive power averaging approach for sustainable cement industry

Pythagorean fuzzy numbers have been playing an essential role in decision-making problems. The existing operational rules (ORs) constructed based on Einstein t-norm and t-conorm operators in the Pythagorean fuzzy environment have some drawbacks. This type of t-norm and t-conorm operators need to consider the interaction between the membership degree and non-membership degree of Pythagorean fuzzy numbers and, consequently, produces some unreasonable results in some realistic decision-making problems. To eliminate such deficiencies, interactive operational rules have been formed, revising the existing operational rules. A new aggregation operator, namely, Pythagorean fuzzy Einstein interactive weighted averaging operator, has been presented in this article based on newly revised ORs, and some properties of this operator are discussed. At the same time to eliminate or reduce the effect of existing biases that occurred from the end of decision makers in the evaluation procedure of alternatives in multi-attribute group decision-making problem, we have presented the Pythagorean fuzzy Einstein interactive power weighted averaging operator. A novel multi-attribute group decision-making method for the best sustainable cement industry evaluation in India is presented based on Pythagorean fuzzy Einstein interactive power weighted averaging and Pythagorean fuzzy Einstein interactive weighted averaging operators. The validity of the method is tested by comparing some existing methods. At last, a comparative study with sensitivity analysis shows that the proposed method is superior to some considered existing methods.

Ranking approach based on compression transformation and distance factor in Pythagorean fuzzy environment with its application in multi-attribute group decision-making

Ranking approach based on compression transformation and distance factor in Pythagorean fuzzy environment with its application in multi-attribute group decision-making

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.

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.