Pythagorean Fuzzy Sets Research Articles

Hybrid MCDM method on pythagorean fuzzy set and its application

Here in this article, a hybrid MCDM method on the Pythagorean fuzzy-environment is presented. This method is based on the Pythagorean Fuzzy Method based on Removal Effects of Criterion (PF-MEREC) and Stepwise Weight Assessment Ratio Analysis (SWARA) approaches. Here, the objective and subjective weights are assessed by PF-MEREC, SWARA model and the preference order ranking of the various alternatives is done through Complex Proportional Assessment (COPRAS) framework on the Pythagorean fuzzy set (PFS). The proposed method is the hybrid model of MEREC, SWARA and COPRAS methods. Further, the proposed model is used to identify the best banking management software (BMS) so that the bank can choose the robust bank management software tool to enhance its efficiency and excellence. Thereafter, a comparative discussion and sensitivity analysis of the proposed model is done with the existing techniques to judge the reasonability and efficiency of the proposed model.

Hybridizations of Archimedean copula and generalized MSM operators and their applications in interactive decision-making with q-rung probabilistic dual hesitant fuzzy environment

The q-rung probabilistic dual hesitant fuzzy sets (qRPDHFSs), which outperform dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets, and probabilistic dual hesitant Pythagorean fuzzy sets, are used in this research to develop an interactive group decision-making approach. We first suggest the Archimedean Copula-based operations on q-rung probabilistic dual hesitant fuzzy (qRPDHF) components and investigate their key features before constructing the approach. We then create some new aggregation operators (AOs) in light of these operations, including the qRPDHF generalized Maclaurin symmetric mean (MSM) operator, qRPDHF geometric generalized MSM operator, qRPDHF weighted generalized MSM operator, and qRPDHF weighted generalized geometric generalized MSM operator. These aggregation operators are better than current operators on qRPDHF because they can take into account the interactions between a large number of criteria and probability distributions. The evaluation findings are distorted since the present methodologies do not take expert involvement into account in order to achieve the required consistency level. We employ the idea of interaction, consistency, resemblance, and consensus-building among the decision-makers in our method to get around this. We create an optimization model based on the cross-entropy of the qRPDHF components to estimate the weights of the criterion. We provide contextual research on the choice of open-source software LMS in order to demonstrate the relevance of the recommended AOs. Likewise, we ran a sensitivity test on the weights of the criterion to make sure that our model is consistent. The comparison investigation has demonstrated that the suggested approach can overcome the challenges of previous works.

Artificial Neural Network Dual Hesitant Fermatean Fuzzy Implementation in Transportation of COVID-19 Vaccine

This article, in order to address impreciseness, initiated the notion of dual hesitant fermatean fuzzy sets (DHFFSs), as a generalization of the combination of dual hesitant fuzzy set (DHFS), dual hesitant Pythagorean fuzzy set (DHPFS) and Fermatean fuzzy set (FFS). The authors defined the fundamental set of operations for DHFFS. Additionally, the authors have also proposed two ranking functions and an accuracy function for the ordering of this novel set. In order to facilitate the pragmatic implementation of DHFFS in optimization, the authors formulated three types of transportation problem with dual hesitant Fermatean fuzzy (DHFF) parameters. To optimize the DHFF-TP, an algorithm was proposed with the help of one of the proposed ranking functions. Artificial neural network is also applied to the transportation problems in DHFF environment. A numerical example based on the transportation of COVID-19 vaccine with DHFF cost has also been carried out to validate out to validate our technique. In this article the proposed fuzzy system is included into this model to control uncertainties in Time, Quality, Cost, Reliability, and Availability induced in the transportation interruptions. The paper illustrates implication of the proposed fuzzy based model using a simulation in the context of a transportation process. The proposed model seeks to satisfy customers' demands for high-quality products and services, on a timely manner, at lowest cost possible. These characteristics are essential during times of disturbance, such as the COVID-19 pandemic.

A hybrid group decision-making approach involving Pythagorean fuzzy uncertainty for green supplier selection

Green supplier selection is a significant issue in green supply chain management field and has drawn worldwide attention from academia, industry, and governments. Numerous methodological models have emerged for correctly choosing an appropriate green supplier, but some inadequate aspects of these models, such as the use of high-order fuzzy uncertainty, incompatible designed criteria, the weight methods used by decision experts, and consensus on distinct ranks, should be improved. This research propounds a hybrid approach that combines the analytic hierarchy process (AHP), technique for order preference by similarity to ideal solutions (TOPSIS), and linear assignment method (LAM) under Pythagorean fuzzy (PF) uncertainty to assess candidate suppliers. A new hierarchical evaluative criteria system with green factors that run throughout procurement, production, and transaction is built. The ordered weighted averaging operation predicated on a normal distribution is adopted to figure out the significance of the relevant decision experts, and the PF-AHP technique is exploited to recognize relevant criteria significance. To find the best green supplier, the PF-TOPSIS technique employs the most recent generalized distance measure and magnitudes. The LAM is exploited to arrive at a consensus on distinct ranks depending on differentiated distance parameter values in the distance formula for the PF sets, and then the optimal green supplier can be obtained. Additionally, the presented PF-AHP-TOPSIS-LAM approach was applied to an illustrative case, and the outcomes were shown to be more beneficial and efficacious than the solutions by other existing techniques. Hence, our presented hybrid approach provides a better methodological framework, establishes a critical foundation for artificial intelligence applications within a decision support field, and contributes to such competitiveness of microenterprises and the long-term development of society.

Emission strategy selection for the circular economy-based production investments with the enhanced decision support system

The purpose of this study is to identify appropriate strategies to minimize carbon emission problem with a novel fuzzy decision-making model. First, the missing expert decisions are imputed for selecting the emission strategies of circular economy-based production investments. Secondly, the emission strategy perspectives of circular economy-based production investments are weighted with multi stepwise weight assessment ratio analysis (M-SWARA) methodology based on bipolar q-rung orthopair fuzzy sets (q-ROFSs). Finally, the industry alternatives are ranked by bipolar q-ROFS the elimination and choice translating reality (ELECTRE). The main novelty of this study is to generate optimal strategies to reduce carbon emissions with a new fuzzy decision-making model. Furthermore, all calculations are also made by using intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) with the aim of making comparative evaluations. It is concluded that the results are quite similar for both weighting the perspectives and ranking the alternatives. Hence, it is understood that the findings of the proposed model are coherent and reliable. It is identified that the best emission strategy is the long-term growth with the global sustainability by the combination of the perspective 1 (globalization) and perspective 4 (environment). It is also determined that textile is the most critical industry to cope with the carbon emission problem. In this context, it is necessary to introduce some legal regulations to prevent the increased trade volume from causing carbon emissions. In this framework, using filters or carbon capture technology will contribute to the solution of this problem. Such applications will create extra costs for businesses. Therefore, it would not be right to leave such applications to the decision of the enterprises. Hence, these applications should be made compulsory by legal regulations.

Multi-attribute decision-making based on novel Fermatean fuzzy similarity measure and entropy measure.

To deal with situations involving uncertainty, Fermatean fuzzy sets are more effective than Pythagorean fuzzy sets, intuitionistic fuzzy sets, and fuzzy sets. Applications for fuzzy similarity measures can be found in a wide range of fields, including clustering analysis, classification issues, medical diagnosis, etc. The computation of the weights of the criteria in a multi-criteria decision-making problem heavily relies on fuzzy entropy measurements. In this paper, we employ t-conorms to suggest various Fermatean fuzzy similarity measures. We have also discussed all of their interesting characteristics. Using the suggested similarity measurements, we have created some new entropy measures for Fermatean fuzzy sets. By using numerical comparison and linguistic hedging, we have established the superiority of the suggested similarity metrics and entropy measures over the existing measures in the Fermatean fuzzy environment. The usefulness of the proposed Fermatean fuzzy similarity measurements is shown by pattern analysis. Last but not least, a novel multi-attribute decision-making approach is described that tackles a significant flaw in the order preference by similarity to the ideal solution, a conventional approach to decision-making, in a Fermatean fuzzy environment.

Some New Correlation Coefficient Measures Based on Fermatean Fuzzy Sets using Decision Making Approach in Pattern Analysis and Supplier Selection

Fermatean fuzzy set (FFS) is an effective tool to depict expert reasoning information in the decision‐making process than fuzzy sets (FS), intuitionistic fuzzy sets (IFS), and Pythagorean fuzzy sets (PFS). Keeping in mind the importance of correlation coefficient and application in medical diagnosis, decision making and pattern recognition, several studies on correlation coefficient measures have been proposed in the literature. As there does not exist any study concerning correlation coefficient measures for FFS, in this communication, we propose novel entropy-correlation measures for Fermatean fuzzy sets and applied it decision making problems of pattern analysis and multi-criteria decision making for supplier selection. With the help of proposed correlation coefficient, we establish some weighted measures for FFS. Using numerical computations, we determine the efficacy of the suggested measures over other measures. The aim of this study is to propose a novel and efficient methodology for evaluation of supplier’s selection with uncertain information. Finally, we establish the comparative study of our developed measures over the existing correlation coefficient measures. The analysis showed that the suggested methodology is reliable, flexible, and consistent with the existing techniques.

A design model of FBS based on interval-valued Pythagorean fuzzy sets

The FBS (Function-Behaviour-Structure) model is a research model that stimulates creative thinking of designers in the design process. In order to reduce the influence of user requirement ambiguity on design results in the product design process and improve the accuracy of user requirements in the function-behavior-structure (FBS) design model, this paper proposes an interval-valued Pythagorean fuzzy set-based FBS model integrating AHP and HOQ methods. Firstly, the design model will use IVPF-AHP method to study user requirements and use interval-valued Pythagorean linguistic terms to replace the traditional scoring method of AHP to get the weight of each user requirement. Secondly, the conversion between user requirements and functions will be realized by IVPF-HOQ method, which converts customer requirements into functional characteristics and calculates the weights of each functional characteristic. Finally, the design focus will be filtered according to the order of importance of the functional characteristics, which will be used as functions to guide the development of the FBS model. In this paper, the feasibility and effectiveness of the proposed method will be verified by an application example of a hand-held fluorescence spectrometer. The results show that the proposed FBS model can effectively reduce the subjectivity and ambiguity in the decision-making process, improve the accuracy and information richness of user requirements, and effectively highlight the focus of the design study. The innovation of the proposed method is to provide a more objective and accurate innovative design method for user requirements through the integration of AHP, HOQ and FBS to effectively explore and analyze user requirements. The use of IVPFS to deal with fuzzy information in the design process in a more flexible manner can reduce the ambiguity of requirements when user data is small, and effectively improve the limitations of the FBS design model which is more subjective.

A Proposal of Robust Condition Monitoring Scheme for Industrial Systems

The Industry 4.0 paradigm aims to obtain high levels of productivity and efficiency, more competitive final products and compliance with the demanding regulations related to industrial safety. To achieve these objectives, the industrial systems must be equipped with condition monitoring systems for early detection, isolation, and location of faults. The paper presents a proposal for a condition monitoring system characterized by its robustness in presence of noise and missing variables in the measurements. The proposal combines the use of simple and effective imputation algorithms with a fuzzy classification kernel algorithm based on the use of the non-standard Pythagorean fuzzy sets. The proposed scheme was validated using the known DAMADICS test problem with excellent results.

Geometric Aggregation Operators for Solving Multicriteria Group Decision-Making Problems Based on Complex Pythagorean Fuzzy Sets

The Complex Pythagorean fuzzy set (CPyFS) is an efficient tool to handle two-dimensional periodic uncertain information, which has various applications in fuzzy modeling and decision making. It is known that the aggregation operators influence decision-making processes. Algebraic aggregation operators are the important and widely used operators in decision making techniques that deal with uncertain problems. This paper investigates some complex Pythagorean fuzzy geometric aggregation operators, such as complex Pythagorean fuzzy weighted geometric (CPyFWG), complex Pythagorean fuzzy ordered weighted geometric (CPyFOWG), complex Pythagorean fuzzy hybrid geometric (CPyFHG), induced complex Pythagorean fuzzy ordered weighted geometric (I-CPyFOWG), and induced complex Pythagorean fuzzy hybrid geometric (I-CPyFHG), and their structure properties, such as idempotency, boundedness, and monotonicity. In addition, we compare the proposed model with their existing models, such as complex fuzzy set and complex intuitionistic fuzzy set. We analyze an example involving the selection of an acceptable location for hospitals in order to demonstrate the effectiveness, appropriateness, and efficiency of the novel aggregation operators.

A comparative study of evaluating and benchmarking sign language recognition system-based wearable sensory devices using a single fuzzy set

Recent research has focused on developing real-time sign language recognition systems (SLRSs) based on gesture recognition to classify hand motions into their equivalent meaning in spoken language, but no comprehensive system with all desirable features has been presented. The existence of different systems has hindered the process of selecting the most preferred system. Therefore, many researchers have compared and evaluated several recognition systems to identify the best one using multicriteria decision-making methods. These studies extended the fuzzy decision by opinion score method (FDOSM) using a single Likert scale under the Pythagorean fuzzy set (PFS) or one of its extensions. However, no comparative study has examined the influence of using multiple Likert scales with a single fuzzy set. Furthermore, the effect of employing multiple Likert scales on benchmarking results is a challenging task. Therefore, this paper examines the three Likert scales (five-, seven- and ten-point) under the same fuzzy environment. This paper extends FDOSM into PFSs based on the power Bonferroni mean (PBM) operator (named PFDOSM-PBM) to benchmark the real-time SLRS. The decision matrix is constructed based on 30 real-time SLRS-based wearable sensory devices and the 11 evaluation criteria. The results reveal that the five-point Likert scale is superior to other scales (i.e., seven- and ten-point) as it is flexible, easy to use and generates more accurate findings on the basis of uncertainty compared to other scales. Systematic ranking and comparative analysis are conducted to validate and evaluate the proposed method.

Development of feature extraction method based on interval-valued Pythagorean fuzzy decision theory

Feature extraction is an important step in image processing and the accuracy of the extraction affects the result of subsequent target detection. There are many feature extraction algorithms, but there are few theories and methods about the accuracy of the extraction. Moreover, most of the existing extraction methods are computationally complex, and they will expose deficiencies once they encounter a large number of standard cases. In order to fill this gap, this study proposes a new soft computing multi-criteria group decision-making (MCGDM) method based on interval-valued Pythagorean fuzzy set information to deal with situations where multiple criteria are involved in complex decisions. Thus, the power average operator is used to eliminate the effects of unreasonable input arguments on decision-making results, and the Maclaurin symmetric mean operator is used to comprehensively consider the mutual relationship between input arguments in this article. Combining with the concept of IVPFS, the interval-valued Pythagorean fuzzy power Maclaurin symmetric mean (IVPFPMSM) operator and weighted IVPFPMSM operator are proposed. Then, the TOPSIS method is used to determine the weight vector of decision-making set, and a MCGDM method is developed under the interval-valued Pythagorean fuzzy framework. Next, aiming at the defects of the existing evaluation criteria, we consider the different ranges of membership degree and non-membership degree to propose a new evaluation criterion of MCGDM method. Finally, the proposed method is applied to the optimal selection of characteristic interval of foreign fibers to verify its practicability and effectiveness. The proposed method has certain guiding significance for the extension of decision-making fuzzy operators and the exploration of feature extraction methods.

Investigating river health across mountain to urban transitions using Pythagorean fuzzy cloud technique under uncertain environment

River health assessment (RHA) is a robust methodology for ascertaining the health of riverine ecosystems, and offering solutions for river conservation and management. Currently, RHA studies have been conducted mainly in mountainous and urban areas, while investigations across mountain to urban transitions remain scarce and insufficient attention has been paid to RHA under uncertain environments. To systematically investigate the RHA across mountain to urban transitions, this study proposed Pythagorean fuzzy cloud (PFC) via integrating the cloud model and Pythagorean fuzzy sets. TOPSIS (Technique for order of preference by similarity to ideal solution) was expanded to the PFC environment to developed a novel PFC-TOPSIS model. The hybrid framework was then created to handle RHA with uncertainty, and the Pi River in China served as a case study. Results showed that the developed models broadened the capabilities of current techniques, and offered a more efficient way for dealing with RHA with uncertainty. In Pi River, the health status showed considerable spatial heterogeneity. Upper reaches were markedly healthier than the lower reaches. Evaluation indicators were at unsatisfactory health levels, with only 68.75% and 18.75% healthy attainment rates in the upper and lower reaches of Pi River, respectively. Meanwhile, 68.75% of the sampling sites were subhealthy and 31.25% were unhealthy. Our research emphasizes that urban development, agricultural practices and dam construction have affected river health in some areas, and appropriate measures are needed to reduce their impacts in Pi River.

An advanced similarity measure for Pythagorean fuzzy sets and its applications in transportation problem

An advanced similarity measure for Pythagorean fuzzy sets and its applications in transportation problem

Some applications in decision-making using cosine maps and the relevance of the Pythagorean fuzzy

The non-classical set, generally known as Pythagorean fuzzy (PF), is one of mathematics’ most essential and it is used in our approach, which is crucial and useful, because it can handle more uncertainty than intuitionistic fuzzy sets and others non-classical sets, so has a wider range of applications. In nature, the periodicity and symmetry are provided by the well-known cosine trigonometric function, meeting decision-makers’ expectations for multi-time process parameters. Introduce the new cosine trigonometric operational laws (CTOLs) with (PF) structure, keeping the peculiarities of the cosine function and the relevance of the Pythagorean fuzzy set (PFS) in mind. New cosine trigonometric Pythagorean fuzzy (CTPF) aggregation operators are created on the basis of these (CTOLs). The main focus of the research is on a decision-making algorithm for multi-attribute decision-making situations. It is based on a planned aggregating operation employing unknown weight information for the specified standards. In the end, to demonstrate the effectiveness, an example of internet finance soft power evaluation (IFSPE) is offered. Sensibility and comparison analyses are also used to evaluate the method’s stability and validity.

Algorithm for Energy Resource Selection Using Priority Degree-Based Aggregation Operators with Generalized Orthopair Fuzzy Information and Aczel–Alsina Aggregation Operators

Many aggregation operators are studied to deal with multi-criteria group decision-making problems. Whenever information has two aspects, intuitionistic fuzzy sets and Pythagorean fuzzy sets are employed to handle the information. However, q-rung orthopair fuzzy sets are more flexible and suitable because they cover information widely. The current paper primarily focuses on the multi-criteria group decision-making technique based on prioritization and two robust aggregation operators based on Aczel–Alsina t-norm and t-conorm. This paper suggests two new aggregation operators based on q-rung orthopair fuzzy information and Aczel–Alsina t-norm and t-conorm, respectively. Firstly, novel q-rung orthopair fuzzy prioritized Aczel–Alsina averaging and q-rung orthopair fuzzy prioritized Aczel–Alsina geometric operators are proposed, involving priority weights of the information. Several related results of the proposed aggregation operators are investigated to see their diversity. A multi-criteria group decision-making algorithm based on newly established aggregation operators is developed, and a comprehensive numerical example for the selection of the most suitable energy resource is carried out. The proposed aggregation operators are compared with other operators to see some advantages of the proposed work. The proposed aggregation operators have a wider range for handling information, with priority degrees, and are based on novel Aczel–Alsina t-norm and t-conorm.

Attribute reduction and information granulation in Pythagorean fuzzy formal contexts

Pythagorean fuzzy set theory is one of the significant tools to deal with real-life problems which are prone to imprecision, partial truth or uncertainty. A Pythagorean fuzzy set reports the degree of truthness as well as falsity of a statement in order to illustrate the imprecision of that statement. Motivated by the resourcefulness of this theory, this paper is aimed to explore formal concept analysis in Pythagorean fuzzy environment. It defines Pythagorean fuzzy formal concept and Pythagorean fuzzy concept lattice associated with a Pythagorean fuzzy formal context. The proposed Pythagorean fuzzy formal concept is a duple whose first argument is (crisp) extension and second argument is Pythagorean fuzzy intension. Moreover, the join of different combinations of object concepts of a Pythagorean fuzzy formal context returns different Pythagorean fuzzy formal concepts. We also studied the knowledge reduction in Pythagorean fuzzy formal contexts in association with Pythagorean fuzzy concept lattice. The key ideas of attribute reduct, core and discernibility matrix for a Pythagorean fuzzy formal context are illustrated. Further, we merged the notions of information granulation and attribute reduction with Pythagorean fuzzy formal context. Various results of formal concepts and concept lattices are extended in Pythagorean fuzzy set theory. At last, an algorithm is developed which discovers non-redundant Pythagorean fuzzy formal concepts for a given Pythagorean fuzzy formal context. Eventually, an example is presented which elaborates the procedure of this algorithm.

Additive manufacturing process selection for automotive industry using Pythagorean fuzzy CRITIC EDAS.

For many different types of businesses, additive manufacturing has great potential for new product and process development in many different types of businesses including automotive industry. On the other hand, there are a variety of additive manufacturing alternatives available today, each with its own unique characteristics, and selecting the most suitable one has become a necessity for relevant bodies. The evaluation of additive manufacturing alternatives can be viewed as an uncertain multi-criteria decision-making (MCDM) problem due to the potential number of criteria and candidates as well as the inherent subjectivity of various decision-experts engaging in the process. Pythagorean fuzzy sets are an extension of intuitionistic fuzzy sets that are effective in handling ambiguity and uncertainty in decision-making. This study offers an integrated fuzzy MCDM approach based on Pythagorean fuzzy sets for assessing additive manufacturing alternatives for the automotive industry. Objective significance levels of criteria are determined using the Criteria Importance Through Inter-criteria Correlation (CRITIC) technique, and additive manufacturing alternatives are prioritized using the Evaluation based on Distance from Average Solution (EDAS) method. A sensitivity analysis is performed to examine the variations against varying criterion and decision-maker weights. Moreover, a comparative analysis is conducted to validate the acquired findings.

Another view on tolerance based multigranulation q-rung orthopair fuzzy rough sets with applications

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.

New optimization technique for group decision analysis with complex Pythagorean fuzzy sets

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.