Pythagorean Fuzzy Sets Research Articles

Trisection-fusion and fusion-trisection methods of three-way conflict analysis with Pythagorean fuzzy information

A basic task of Pawlak conflict analysis is to cluster agents based on their attitudes towards various issues. When ratings of agents are imprecise and uncertain, for example, as represented by a Pythagorean fuzzy situation table, single-measure based methods fall short in ensuring the accuracy of agent clusters. Multi-measure based conflict analysis methods are more effective for generating higher-quality agent clusters. The objective of this paper is to propose two new multi-measure based methods under uncertainty represented by Pythagorean fuzzy sets. We introduce three distinct conflict measures regarding an issue by leveraging the maximum positive and negative agents. The first measure is based on support degrees, the second measure incorporates opposition degrees, and the third measure considers both support and opposition degrees. These measures trisect a set of agents into three disjoint coalitions concerning an individual issue. For multiple issues, we propose two models by combing the two fundamental tasks, namely, trisection and fusion. A trisection-fusion model amalgamates a family of trisections generated from multiple issues based on conflict measures regarding single issues. Matrix representations of trisections are provided, and trisection fusions are transformed into a series of matrix operations. A fusion-trisection model trisects the set of agents according to fused conflict measures on multiple issues. To demonstrate the value of the two models, we apply them in assisting decision-makers in Hunan Province, China, to adjust development plans. The experimental results show that multi-measure based models offer decision-makers more comprehensive guidance.

An investigation on the recurrent function of the Pythagorean fuzzy cellular topological dynamical system

This study is about Pythagorean fuzzy cellular topological dynamical system which is generated using Pythagorean fuzzy cellular space. A dynamical system receives input for a certain function and performs an iterative procedure for that same function. A continuous function can be employed in a topological dynamical system and the same function is iterated again and again. As it is an iterative process, there is a perspective that the views of every individual may be ambiguous or imprecise. To overcome this uncertainty, Pythagorean fuzzy sets are employed in dynamical system. Fixing a boundary on the Pythagorean fuzzy dynamical system culminates in a Pythagorean fuzzy cellular topological dynamical system. It is shown that the Pythagorean fuzzy cellular space is compact, normal and homeomorphic and it is called the Pythagorean fuzzy cellular topological dynamical system. The Pythagorean fuzzy sets in the Pythagorean fuzzy cellular topological dynamical system are iterated under the action of Pythagorean fuzzy cellular continuous map. Then, the Pythagorean fuzzy orbit* set is obtained. Additionally, it is discussed that the stipulated dynamical system is topologically transitive and different aspects of topological transitivity are investigated.

A multiple attribute group decision making model based on 2-tuple linguistic pythagorean fuzzy dombi aggregation operators for optimal selection of potential global suppliers

Multiple-Attribute Group Decision-Making (MAGDM) is a significant area of research in decision-making, and its principles and methodologies are widely implemented. A Pythagorean Fuzzy Set (PFS) is an extension of an Intuitionistic Fuzzy Set (IFS) that is highly valuable for representing uncertain information in real-world scenarios. The 2-Tuple Linguistic Pythagorean Fuzzy Number (2TLPFN) is a specific type of Pythagorean Fuzzy Number (PFN) that can be used to represent uncertainty in real-world decision making through the use of 2-Tuple Linguistic Terms (2TLTs). This paper focuses on the examination of Multiple Attribute Group Decision Making (MAGDM) using 2TLPFNs. Dombi's t-norm and t-conorm operations were commonly referred to as Dombi operations, which might have been greater degree of applicability if offered in a new form of flexibility within the general parameter. In this research, we implement Dombi operations to construct some 2-Tuple Linguistic Pythagorean Fuzzy (2TLPF) Dombi Aggregation operators. These operators include the 2TLPF Dombi Weighted Averaging (2TLPFDWA) operator, 2TLPF Dombi Ordered Weighted Averaging (2TLPFDOWA) operator, 2TLPF Dombi Weighted Geometric (2TLPFDWG) operator, and 2TLPF Dombi Ordered Weighted Geometric (2TLPFDOWA) operator. An analysis is conducted to examine the unique characteristics of these suggested operators. Subsequently, we leveraged the proposed operators to develop a model aimed at tackling the MAGDM problems in the 2TLPF environment. Eventually, a suitable instance has been demonstrated to validate the formation of the model as well as exhibit its implementation and resilience.

Pythagorean Fuzzy Set for Enhancement of Low Contrast Mammogram Images

ABSTRACTBreast masses are often one of the primary signs of breast cancer, and precise segmentation of these masses is essential for accurate diagnosis and treatment planning. Diagnosis may be complex depending on the size and visibility of the mass. When the mass is not visible clearly, precise segmentation becomes very difficult and in that case enhancement is essential. Inadequate compression, patient movement, or paddle/breast movement during the exposure process might cause hazy mammogram images. Without enhancement, accurate segmentation and detection cannot be done. As there exists uncertainties in different regions, reducing uncertainty is still a main problem and so fuzzy methods may deal these uncertainties in a better way. Though there are many fuzzy and advanced fuzzy methods, we consider Pythagorean fuzzy set as one of the fuzzy sets that may be powerful to deal with uncertainty. This research proposes a new Pythagorean fuzzy methodology for mammography image enhancement. The image is first transformed into a fuzzy image, and the nonmembership function is then calculated using a newly created Pythagorean fuzzy generator. Membership function of Pythagorean fuzzy image is computed from nonmembership function. The plot between the membership value and the hesitation degree is used to calculate a constant term in the membership function. Next, an enhanced image is obtained by applying fuzzy intensification operator to the Pythagorean fuzzy image. The proposed method is compared qualitatively and quantitatively with those of non‐fuzzy, intuitionistic fuzzy, Type 2 fuzzy, and Pythagorean fuzzy methods, it is found that the suggested method outperforms the other methods. To show the usefulness of the proposed enhanced method, segmentation is carried out on the enhanced images.

Optimizing cloud resource utilization in the digital economy: An integrated Pythagorean fuzzy-based decision-making approach

The digital economy has experienced an unheard-of surge in the twenty-first century, revolutionizing how businesses run, societies function, and people interact. The utilization of cutting-edge technologies to optimize resources and streamline operations has become crucial with the changing digital landscape. Cloud computing has become a potential enabler in this transformative era, providing a flexible and scalable solution that meets the needs of the rapidly growing digital ecosystem. This study aims a novel PF-DNMA-SWARA method that amalgamates Stepwise Weight Assessment Ratio Analysis (SWARA) and double normalization-based multiple aggregation (DNMA) constructed on Pythagorean Fuzzy Sets (PFSs) to identify potential barriers to efficient cloud resource utilization for large-scale implementation in the digital economy. The results of this study uncovered a compelling narrative that emphasized the developed approach’s unique advantages in terms of its capacity to function naturally within the context of PFSs. The method offered a more complex and finely tuned evaluation framework by utilizing the inherent flexibility and interpretability of PFSs. This factor is of utmost importance, especially in complex situations where there are many uncertainties and imprecisions, making it impossible for conventional methods to accurately capture the nuances of real-world resource utilization dynamics.

Ranking of sectors in organized industrial zones according to Natech risk criteria: An application for Gaziantep province in Türkiye

Technological accidents triggered by natural disasters are called Natech accidents. Natech accidents have the impact of increasing the negative effects of disasters. One of the places at risk for a Natech accident is the Organized Industrial Zones (OIZs), where many industrial establishments are located. Although businesses within OIZs are evaluated in terms of security, Natech risk assessment is often ignored. This situation causes Natech accidents to occur. For these reasons, the study focused on evaluating OIZs regarding Natech risk. Many criteria such as electrical resources, natural gas, distance between buildings and so forth are taken into account in Natech's risk assessment. However, it is challenging to make clear decisions and predictions about these criteria due to the nature of the disaster. There is inherently uncertainty and confusion during and after a disaster. For this reason, using fuzzy sets to evaluate criteria is more appropriate. Therefore, the study uses Pythagorean fuzzy sets, which have a wider fuzzy evaluation area. This study also includes the evaluation of alternative sectors in terms of Natech risk. For this purpose, Pythagorean Fuzzy Analytic Hierarchy Process (PFAHP) and Pythagorean Fuzzy the Technique for Order of Preference by Similarity to Ideal Solution (PFTOPSIS) methods are used. OIZ in Gaziantep has the largest area compared to those in other cities in Türkiye, is chosen as the application area of the study. As a result of the study, it is determined that the release of hazardous substances and flammable criteria is the most crucial criterion, and the most dangerous sector in terms of Natech risk is the chemical sector.

A comprehensive risk assessment framework for occupational health and safety in pharmaceutical warehouses using Pythagorean fuzzy Bayesian networks

Risk assessment plays a crucial role in managing occupational health and safety in various industries, including pharmaceutical warehouses. Bayesian Networks (BN) have been widely employed for risk assessment due to their ability to handle uncertainty and quantify risks. However, the traditional BN approach has limitations in dealing with ambiguity and continuous variables. To address this, the fuzzy BN technique, combining fuzzy logic with BN, has emerged as an effective method for risk assessment. In this study, a fuzzy BN model using Pythagorean fuzzy sets is proposed for risk assessment in a pharmaceutical warehouse. The model incorporates 24 identified risk factors, and survey data is used to determine the conditional probabilities of these factors. The novelty of the study lies in the application of Pythagorean fuzzy sets and the development of risk assessment criteria specifically for pharmaceutical warehouses. The results of a comprehensive literature review and the proposed methodology are presented. A real case analysis is conducted, followed by validation and sensitivity analysis. The results provide valuable insights into enhancing occupational health and safety practices in pharmaceutical warehouses. The study contributes to enhancing occupational health and safety practices in pharmaceutical warehouses and provides a framework for future research.

Evaluating performance of novel similarity measures of Pythagorean fuzzy sets and their applications in pattern recognition and medical diagnosis

Evaluating performance of novel similarity measures of Pythagorean fuzzy sets and their applications in pattern recognition and medical diagnosis

Effective multi-attribute group decision-making approach to study astronomy in the probabilistic linguistic q-rung orthopair fuzzy VIKOR framework

This study employs a novel fuzzy logic-based framework to address multi-attribute group decision-making problems commonly encountered in modern astronomy. Our approach utilizes the probabilistic linguistic q-rung orthopair fuzzy set (PLq-ROFS) to handle the inherent uncertainties associated with astronomical data. The PLq-ROFS offers significant advantages over existing fuzzy sets like probabilistic hesitant, linguistic intuitionistic, and linguistic Pythagorean fuzzy sets, which comprise both stochastic and non-stochastic uncertainties simultaneously. To aggregate the probabilistic linguistic decision information effectively, we propose two novel operators: the PLq-ROF weighted power average (PLq-ROFWPA) and the PLq-ROF weighted power geometric (PLq-ROFWPG). These operators form the foundation of a novel method within the PLq-ROF environment. Furthermore, this study integrates the PLq-ROF framework with the VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) model, a widely used decision-making (DM) tool known for its ability to balance group utility maximization with individual regret minimization. This integration leads to the PLq-ROF-VIKOR model, a novel approach for ranking alternative solutions based on the subjective preferences of decision-makers. The effectiveness of the proposed method is demonstrated through a real-world case study in astronomy, accompanied by both parameter and comparative analyses. These analyses highlight the efficiency and accuracy of the PLq-ROF-VIKOR model, ultimately leading to the conclusion that cosmology is the most optimal key finding in this case study.

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.

Complex Pythagorean Normal Interval-Valued Fuzzy Aggregation Operators for Solving Medical Diagnosis Problem

This paper presents a new methodology for solving multiple-attribute decision-making problems (MADMs) using a complex Pythagorean normal interval-valued fuzzy set (CPNIVFS), which is an extended concept of a complex Pythagorean fuzzy set. Four types of different aggregating operations (AOs), including CPNIVF weighted averaging (CPNIVFWA), CPNIVF weighted geometric (CPNIVFWG), generalized CPNIVFWA (CGPNIVFWA), and generalized CPNIVFWG (CGPNIVFWG), are discussed. The scoring function, accuracy function, and operational laws of the CPNIVFS are defined. Algebraic structures, such as associative, distributive, idempotent, bounded, commutativity, and monotonicity properties, are also shown to be satisfied by complex Pythagorean normal interval-valued fuzzy numbers. Furthermore, an algorithm is proposed to solve the MADM problems based on the defined AOs. The proposed approach is then used for a medical diagnosis problem about brain tumors because computer science and machine tool technology are among the most important components of brain tumor research. The five types of brain tumors diagnosed in these patients are gliomas, meningiomas, metastases, embryonal tumors, and ependymomas. Several types of treatments are available, which are often combined as part of an overall treatment plan. Brain tumors can be treated in various ways, including surgery, radiation therapy, chemotherapy, immunotherapy, and clinical trials. Based on the comparisons and options gathered, the most suitable treatment can be chosen. In this regard, it is evident that the value of the integer ⅁\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Game $$\\end{document} plays a significant role in determining the model. The candidate models under consideration can be validated by comparing them with the previously proposed ones. The proposed technique is compared with the existing method to demonstrate its superiority and validity, and the results conclude that the former is more reliable and effective than the latter. Finally, the criteria are evaluated by expert assessments to determine the most appropriate options.

A novel perspective on the selection of an effective approach to reduce road traffic accidents under Fermatean fuzzy settings.

Road traffic accidents (RTAs) pose a significant hazard to the security of the general public, especially in developing nations. A daily average of more than three thousand fatalities is recorded worldwide, rating it as the second most prevalent cause of death among people aged 5-29. Precise and reliable decisionmaking techniques are essential for identifying the most effective approach to mitigate road traffic incidents. This research endeavors to investigate this specific concern. The Fermatean fuzzy set (FFS) is a strong and efficient method for addressing ambiguity, particularly when the concept of Pythagorean fuzzy set fails to provide a solution. This research presents two innovative aggregation operators: the Fermatean fuzzy ordered weighted averaging (FFOWA) operator and the Fermatean fuzzy dynamic ordered weighted geometric (FFOWG) operator. The salient characteristics of these operators are discussed and important exceptional scenarios are thoroughly delineated. Furthermore, by implementing the suggested operators, we develop a systematic approach to handle multiple attribute decisionmaking (MADM) scenarios that involve Fermatean fuzzy (FF) data. In order to show the viability of the developed method, we provide a numerical illustration encompassing the determination of the most effective approach to alleviate road traffic accidents. Lastly, we conduct a comparative evaluation of the proposed approach in relation to a number of established methodologies.

Novel Distance Measures of q-Rung Orthopair Fuzzy Sets and Their Applications

The q-rung orthopair fuzzy sets (q-ROFSs), a novel concept for processing vague information, offer a more potent and all-encompassing method compared to traditional fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets. The inclusion of the parameter q allows for the q-rung orthopair fuzzy sets to capture a broader range of uncertainty of information. In this paper, we present two novel distance measures for q-ROFSs inspired by the Jensen–Shannon divergence, called DJS_2D and DJS_3D, and we analyze some properties they satisfy, such as non-degeneracy, symmetry, boundedness, and triangular inequality. Then, the normalized distance measures, called DJS_2D˜ and DJS_3D˜, are proposed and we verify their rationality through numerical experiments. Finally, we apply the proposed distance measures to practical scenarios, including pattern recognition and multicriteria decision-making, and the results demonstrate the effectiveness of the proposed distance measures.

The application of cosine similarity measures with Laplacian energy to q-rung orthopair fuzzy graphs in decision-making problems

Q-rung orthopair fuzzy sets (q-ROFS), which are better than the intuitionistic and Pythagorean fuzzy sets, are a significant tool for expressing ambiguous information. Their key feature is that their ability to represent a larger space of uncertain information is based on the fact that the product of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1. Under these circumstances, we train group decision-making problems in the study using q-rung orthopair fuzzy inclination associations. Through the calculation of the standard deviation of one separable q-rung orthopair inclination association to the others and the unclear evidence of q-rung orthopair fuzzy inclination connections, we propose a novel approach to estimate the qualified reputation weights of authority. The “internal” and “impartial” evidence of authority is taken into consideration by this new mindset. Subsequently, we included the weights of authorities into the q-rung orthopair fuzzy inclination relations and used a relative similarity approach to determine the relevance of replacements and the best substitutes. The planned techniques' usefulness and realism are demonstrated by the contrast analysis with additional methods through mathematical demonstrations, both of which show the fuzzy set’s membership degree and non-membership degree, respectively.

To Solve Pythagorean Fuzzy Decagonal Transportation Problem Using Bipartite Graph with Optimal Solution

Objectives: This study aims to solve the Pythagorean fuzzy transportation problem by means of providing a novel approach. Methods: To determine the optimum decision-making, we applied the MODI approach. To illustrate the proposed approach, we applied several types of numerical issues. An algorithm for solving the suggested technique is described. Findings: This study presents a bipartite graph-based strategy to the Pythagorean fuzzy transportation problem. Applying TORA statistical software, the data analysis discovered that the crisp model becomes linear when the membership function is applied. Novelty: This study has paved a way for better optimal results to obtain when the Pythagorean fuzzy set is converted using the score function instead of the standard techniques for Defuzzification. Keywords: Score Function, Pythagorean Fuzzy, Defuzzification, Pythagorean Fuzzy Transportation Problem, Decagonal Fuzzy Number, Ranking Method, Initial Basic Feasible Solution, and Optimal Solution

Study on the Method of Selecting Sustainable Food Suppliers Considering Interactive Factors

The existing sustainable supplier selection methods are not sufficient to deal with the problem of sustainable food supplier selection with the interaction of criteria under uncertainty. Therefore, this paper proposes a method of sustainable food supplier selection based on an extended decision model. Firstly, a processing method for supplier evaluation information is constructed using the Pythagorean fuzzy set has the function of processing complex uncertain information. Secondly, to obtain the objective weights of decision experts, a Pythagorean fuzzy weighted distance measure model is constructed, and an expert information fusion method based on a weighted power mean operator is proposed to construct the group decision matrix. Then, the decision experiment and evaluation experiment methods are integrated with the traditional MARCOS method, to construct a sustainable food supplier selection method considering the interaction of factors. This method can effectively deal with the complicated and uncertain problem of sustainable food supplier selection with interactive factors. Finally, the feasibility of the proposed method is verified by an example of sustainable food supplier selection. In addition, parameter sensitivity analysis and multi-method comparative analysis verified the rationality of the proposed selection method for sustainable food supplier selection.

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.

Power Dombi Aggregation Operators for Complex Pythagorean Fuzzy Sets and Their Applications in Green Supply Chain Management

Power Dombi Aggregation Operators for Complex Pythagorean Fuzzy Sets and Their Applications in Green Supply Chain Management

A matrix norm-based Pythagorean fuzzy metric and its application in MEREC-SWARA-VIKOR framework for solar panel selection

Due to the escalating electricity demand from industrialization and modernization, it is imperative to explore clean and sustainable energy sources. Solar energy has become a feasible solution for this surging demand. Solar panels, vital components of solar power systems, are crucial in converting sunlight into electricity. However, selecting the most suitable solar panel is a complex task involving subjective and measurable parameters involving uncertainties. To address the uncertainties and ambiguities in real-life problems, the theory of Pythagorean fuzzy sets with distance measures has emerged as a flexible and superior tool. Nevertheless, many existing distance functions often fail to meet necessary metric conditions, leading to inaccurate and unreasonable outcomes. These limitations are pointed out by many literature. In addition, we observe a serious drawback of parameter dependency of some existing measures. This paper proposes a novel metric for Pythagorean fuzzy sets using the matrix norm to overcome these drawbacks. Additionally, we discuss various mathematical properties and provide geometrical representations for the proposed distance measure. To demonstrate the superiority of the proposed measure, we conduct a comparative study against existing distance measures. Furthermore, we develop a Pythagorean fuzzy-method based on the removal effects of criteria-stepwise weight assessment ratio analysis-vlse kriterijumska optimizacija kompromisno resenje (PF-MEREC-SWARA-VIKOR) method for selecting solar panel systems. Our approach incorporates the MEREC method for objective criteria, utilizes the SWARA method to calculate subjective weights, and employs the VIKOR method in the Pythagorean fuzzy context to determine the preference order of alternatives. We conduct sensitivity analysis and comparative studies with existing developed methods to validate our method. Highlight the effectiveness and advantages of our novel approach in selecting solar panels. Finally, exploring the future research directions and challenges associated with the proposed methodology.