Search Results (37)

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending the idea of symmetry found within groups. In this study, we specifically focus on semigroups. The main objective of this research is to apply the notion of m-polar picture fuzzy sets (m-PPFSs), with m being a natural number, in investigations into semigroups, as this concept generalizes m-polar fuzzy sets (m-PFSs) and picture fuzzy sets (PFSs). This research introduces the concepts of m-polar picture fuzzy left ideals (m-PPFLs), m-polar picture fuzzy right ideals (m-PPFRs), m-polar picture fuzzy ideals (m-PPFIs), m-polar picture fuzzy bi-ideals (m-PPFBs), and m-polar picture fuzzy generalized bi-ideals (m-PPFGBs) in semigroups. This study examines the relationships between these concepts, showing that every m-PPFL (m-PPFR) in the semigroups is also an m-PPFB, and that every m-PPFB in the semigroups is an m-PPFGB. However, the opposite is not true. Additionally, we provide the characteristics of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. We further discuss the connections between the m-PPFLs (m-PPFIs) and the m-PPFBs within the framework of regular semigroups, and most importantly, we show that, if the semigroup is regular, then the m-PPFBs and m-PPFGBs are equal. Finally, we utilize the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to explore the classifications of regular semigroups. Full article

Reliable operation of marine equipment is crucial for ensuring vessel performance and safeguarding the safety of personnel and the marine environment. However, the complexity of evaluation criteria and the subjectivity inherent in expert judgments pose significant challenges for effective reliability assessment. To address these challenges, this study proposes an extended VIKOR method within a group decision-making (GDM) framework based on picture fuzzy numbers. The method first collects expert evaluations through questionnaires and voting to construct individual decision matrices, and then it applies a newly developed entropy-based approach to determine attribute weights, resulting in a group-weighted decision matrix. Subsequently, an extended VIKOR model is introduced, where the group utility measure is derived from one positive reference matrix and two negative reference matrices, while the group regret measure is based on two negative reference matrices. To improve assessment precision, this study also introduces a novel normalized projection measure to evaluate the closeness between decision matrices. Finally, two ranking strategies are developed, for static and dynamic environments, respectively. The proposed method is validated through a case study on marine equipment reliability assessment, confirming its effectiveness and feasibility. This study provides valuable insights for both theoretical research and practical applications in maritime engineering. Full article

This paper proposes the concept of a centroid for picture fuzzy numbers and particularly for triangular picture fuzzy numbers. The concept allows the implementation of a ranking function for the triangular picture fuzzy numbers, which has the advantage of reuniting the symmetry and asymmetry of the information. Then, empirical applications are considered for the picture fuzzy numbers. Specifically, multiple TPFNs are considered. The ranked, A comparison study is conducted for said ranked TPFNs relative to other methodologies in the specialized literature, illustrating that these methods exhibit limitations in specific scenarios. An additional compelling example is provided: before elections, opinion surveys are extensively utilised to assess voter intentions about candidates. The survey findings can be analysed through PFNs and the ranking mechanism proposed in this study. Another contribution of this paper is the development an algorithm meant to solve decision making problems in an uncertain environment. This is applied in the practical context of comparing the performance of several standards in two successive evaluations. Full article

The idea of strong and weak adjacencies between vertices has been generalized into fuzzy graphs and intuitionistic fuzzy graphs (IFGs), and it is an important part of making decisions. However, one or two membership degrees are not always sufficient for making decisions on real-world problems that need an answer of types “yes, neutral, and no”. Consequently, in previous work, we generalized the concept into picture fuzzy graphs (PFGs) where each element in the PFG has membership, neutral, and non-membership degrees. Moreover, we constructed the notion of the coloring of PFGs based on strong and weak adjacencies between vertices. In this paper, we investigate some properties of the chromatic number of PFGs based on the concept of strong and weak adjacencies between vertices. According to these properties, we construct an algorithm to find the chromatic number of PFGs. The algorithm is useful when we work with large PFGs. Further, we improve the method to implement the PFG’s coloring for determining traffic signal phasing at an intersection. A case study has also been carried to evaluate the method. Full article

Supplier selection as a multiattribute decision-making (MADM) problem has various inherent uncertainties due to a number of symmetrical variables. In order to handle such information-based uncertainties, rational models like intuitionistic fuzzy sets have already been introduced in the literature. However, a picture fuzzy set (PiFS) with four dimensions of positive, neutral, negative, and rejection is better at capturing and interpreting such kinds of ambiguous information. Additionally, fuzzy parameterization (FPara) is helpful for evaluating the degree of uncertainty in the parameters. This study aims to develop a fuzzy parameterized picture fuzzy soft set (FpPiFSS) by integrating the ideas of PiFS and FPara. This integration is more adaptable and practical since it helps decision makers manage approximation depending on their objectivity and parameterization uncertainty. With the assistance of instructive examples, some of the set-theoretic operations are examined. A decision support framework is constructed using matrix manipulation, preferential weighting, fuzzy parameterized grades based on Pythagorean means, and the approximations of decision makers. This framework proposes a reliable algorithm to evaluate four timber suppliers (initially scrutinized by perusal process) based on eight categorical parameters for real estate projects. In order to accomplish suppliers evaluation, crucial validation outcomes are taken into account, including delivery level, purchase cost, capacity, product quality, lead time, green degree, location, and flexibility. To assess the advantages, dependability, and flexibility of the recommended strategy, comparisons in terms of computation and structure are provided. Consequently, the results are found to be reliable, analog, and consistent despite the use of fuzzy parameterization and picture fuzzy setting. Full article

The selection of offshore wind farm site (OWFS) has important strategic significance for vigorously developing offshore new energy and is deemed as a complicated uncertain multicriteria decision-making (MCDM) process. To further promote offshore wind power energy planning and provide decision support, this paper proposes a hybrid picture fuzzy (PF) combined compromise solution (CoCoSo) technique for prioritization of OWFSs. To begin with, a fresh PF similarity measure is proffered to estimate the importance of experts. Next, the novel operational rules for PF numbers based upon the generalized Dombi norms are defined, and four novel generalized Dombi operators are propounded. Afterward, the PF preference selection index (PSI) method and PF stepwise weights assessment ratio analysis (SWARA) model are propounded to identify the objective and subjective weight of criteria, separately. In addition, the enhanced CoCoSo method is proffered via the similarity measure and new operators for ranking OWFSs with PF information. Lastly, the applicability and feasibility of the propounded PF-PSI-SWARA-CoCoSo method are adopted to ascertain the optimal OWFS. The comparison and sensibility investigations are also carried out to validate the robustness and superiority of our methodology. Results manifest that the developed methodology can offer powerful decision support for departments and managers to evaluate and choose the satisfying OWFSs. Full article

The development of information technology has had a significant impact on various areas of human activity, including medicine. It has led to the emergence of the phenomenon of Industry 4.0, which, in turn, led to the development of the concept of Medicine 4.0. Medicine 4.0, or smart medicine, can be considered as a structural association of such areas as AI-based medicine, telemedicine, and precision medicine. Each of these areas has its own characteristic data, along with the specifics of their processing and analysis. Nevertheless, at present, all these types of data must be processed simultaneously, in order to provide the most complete picture of the health of each individual patient. In this paper, after a brief analysis of the topic of medical data, a new classification method is proposed that allows the processing of the maximum number of data types. The specificity of this method is its use of a fuzzy classifier. The effectiveness of this method is confirmed by an analysis of the results from the classification of various types of data for medical applications and health problems. In this paper, as an illustration of the proposed method, a fuzzy decision tree has been used as the fuzzy classifier. The accuracy of the classification in terms of the proposed method, based on a fuzzy classifier, gives the best performance in comparison with crisp classifiers. Full article

The aim of this paper is to assess the extent to which the Sad-Kalan watershed in Iran participates in floods and rank the Sad-Kalan sub-watersheds in terms of flooding potential by utilizing multi-criteria decision-making approaches. We employed the entropy of a drainage network, stream power index (SPI), slope, topographic control index (TCI), and compactness coefficient (Cc) in this investigation. After forming a decision matrix with 25 possibilities (sub-watersheds) and 5 evaluation indices, we used four MCDM approaches, including the analytic hierarchy process (AHP), best–worst method (BWM), interval rough numbers AHP (IRNAHP), picture fuzzy with AHP (PF-AHP), and picture fuzzy with linear assignment model (PF-LAM, hereafter PICALAM) algorithms, to rank the sub-watersheds. The study results demonstrated that PICALAM exhibited superior performance compared to the other methods due to its consideration of both local and global weights for each criterion. Additionally, among the methods used (AHP, BWM, and IRNAHP) that showed similar performances in ranking the sub-watersheds, the BWM method proved to be more time-efficient in the ranking process. Full article

The notion of fuzzy graph (FG) is widely used in many problems arising from partial or incomplete descriptions of the real world and in particular from fields such as engineering, economics, computer science, social disciplines, or medical diagnostics, and has been used in many fields of pure mathematics as well as in several areas of applied sciences such as decision making, statistics and networking. In this paper we will deal with the graph of the picture fuzzy(symmetric) set using the notion of domination in picture fuzzy graph (PFG) as a generalization of both the concept of fuzzy graph domination and intuitionistic fuzzy graph (IFG) domination. The concepts of domination theory (DT) and double domination theory (DDT) of a PFG are introduced, studied and concretely applied to the real case of an election competition to determine the minimum number of citizens a politician should meet in person in order to win the election. The choice of fuzzification (symmetric) and defuzzification (anti-symmetric) methods depends on the specific application and the type of fuzzy sets being used, whether they are symmetric or anti-symmetric. There are various methods for each process, such as centroid, max-min, and weighted average methods for defuzzification. Finally, in the last section, drawing from the application example, the features and benefits of PFGs with respect to fuzzy graphs and intuitionistic fuzzy graphs are compared and discussed. Full article

The process of bundling and clustering hasno clear boundaries; hence, their analysis contains uncertainities. Thus, it is more suitable to deal withbundling and clusteringby usingfuzzy graphs. Since picture fuzzy sets (PFSs) are more accurate, compatible, and flexible compared to the other generalizations of fuzzy sets (FSs),hence, it would be more effective to present edge bundling and clustering usingpicture fuzzy line graphs (PFLGs). The aim of our study is to introduce the notions of picture fuzzy intersection graphs (PFIGs) and picture fuzzy line graphs (PFLGs). These concepts are the generalizations of fuzzy intersection graphs (FIGs) and fuzzy line graphs (FLGs), respectively. We begin our discussion by introducing some fresh and useful terminologies in the theory of fuzzy graphs such as fuzzy intersection number, picture fuzzy intersection number, etc., and we explore few interesting results related to them. Based on these concepts, first we introduce the notion of picture fuzzy intersection graphs (PFIGs) and discuss manyimportant characteristics of these graphs. Afterwards, we introduce the notion of picture fuzzy line graphs (PFLGs) and discuss their various properties. We also investigate some structural properties of our newly established fuzzy graphs usingweak isomorphism and isomorphism. Finally, we provide an outline of the applications of picture fuzzy line graphs (PFLGs) in terms of cluster-based picture fuzzy edge bundling (CBPFEB) and the picture fuzzy c-mean algorithm. Since asymmetrical clusters ensure that the databases remain identical across the clusters, our study is deeply related to symmety. Full article

A medical diagnosis is one the most efficient processes of determining a disease based on a person’s symptoms and signs. In recent days, due to the complexities of the same type of diseases, it is very difficult to diagnose a disease by using old methods and techniques. In this way, new and efficient medical diagnosis methods can help a lot in reaching an accurate conclusion, depending upon the timing and sequences of symptoms and medical history. The physician relies on other clues like medical tests and imaging tests. So, in this way, a list of possible diagnoses can be determined, which are referred to as different diagnoses. To handle these types of issues in this manuscript, additional information is identified, and possible disease is confirmed. Under the consideration of classical data, it is a very difficult task to deal with complex and asymmetric sorts of data. Fuzzy set theory has a wide range of applications, from engineering to the medical field. Different methods and techniques have been proposed to support the decision-making process in medical fields. Picture fuzzy soft sets are more generalized structures and efficient tools to formalize the information more decently and accurately. So, devoted from this notion, in this article based on picture fuzzy soft settings, we firstly have established some basic operational laws for picture fuzzy soft number; then based on these operational laws, we have developed some aggregation operators named as picture fuzzy soft prioritized average and geometric aggregation operators. In real-world problems, these operators can be useful in analyzing uncomfortable and asymmetric information. Furthermore, some basic properties of the introduced operators have been initiated and discussed briefly. Moreover, to show the effective use of this developed approach to medical diagnoses, we have proposed an algorithm, along with a descriptive example. Additionally, a comparative analysis of the proposed work shows the superiority and effectiveness of the introduced approach. Full article

The purpose of this paper is to introduce new methods to measure the indirect control power of firms in complex corporate shareholding structures using the concept of power indices from cooperative game theory. The proposed measures vary in desirable properties satisfied, as well as in the bargaining models of power indices used to construct them. Hence, they can be used to produce different pictures of the coalitional strength of firms in control of other firms in mutual shareholding networks with the presence of cycles. Precisely, in the framework of Karos and Peters from 2015, ten power indices substitute the original Shapley and Shubik power index in a modular fashion. In this way, we obtain a set of new measures called aggregated indices. The float shareholders typically hold less than 5 percent of the outstanding shares, which is an uncertain element of indirect control in complex shareholding structures. The fuzzy number seems appropriate to model these shareholders’ behavior. The novelty is that we model the behavior of float using Z-fuzzy numbers. The new methods are tested in an example. Full article

As an alternative for sustainable transportation and economic development, biofuels are being promoted as renewable and climate-friendly resources of energy which can help to reduce the consumption of fossil fuels, some pollutant emissions and mitigate the climate change impact from transport. With the successful development of the biofuel industry, the location selection for biofuel production plant is one of the major concerns for the governments and policymakers. Finding locations for the construction of new biofuel production plants includes several dimensions of sustainability, including economic, social and environmental; therefore, this selection process can be considered a complex multi-criteria decision-making problem with uncertainty. As an advanced version of fuzzy set, picture fuzzy set (PiFS) is one of the comprehensive tools to handle the uncertainty with the account of truth, abstinence and falsity membership degrees. Thus, this work proposes a new decision-making methodology based on the weighted aggregated sum product assessment (WASPAS) approach and similarity measure with picture fuzzy information. By using picture fuzzy numbers, the proposed methodology can effectively address the uncertain information and qualitative data that often occurs in practical applications. In this methodology, a picture fuzzy similarity measure-based weighting model is proposed to find the criteria weights under picture fuzzy environment. For this purpose, a new similarity measure is introduced to measure the degree of similarity between picture fuzzy numbers. Moreover, the rank of the options is determined based on an integrated WASPAS approach under a PiFS context. To illustrate the effectiveness of the proposed framework, a case study of biofuel production plant location selection is presented from the picture fuzzy perspective. Further, a comparison with existing methods is conducted to test the validity and applicability of the obtained results. The sensitivity analysis is performed with respect to different values of decision parameter, which proves the stability, robustness, and practicality of the proposed approach. The presented picture fuzzy WASPAS approach feasibly enables the policymakers to identify the most desirable location for a biofuel production plant by considering the social, environmental and economic aspects of sustainability. Full article

Blood vessels are harmed in diabetic retinopathy (DR), a condition that impairs vision. Using modern healthcare research and technology, artificial intelligence and processing units are used to aid in the diagnosis of this syndrome and the study of diagnostic procedures. The correct assessment of DR severity requires the segmentation of lesions from fundus pictures. The manual grading method becomes highly difficult and time-consuming due to the wide range of the morphologies, number, and sizes of lesions. For image segmentation, traditional fuzzy clustering techniques have two major drawbacks. First, fuzzy memberships based clustering are more susceptible to outliers. Second, because of the lack of local spatial information, these techniques often result in oversegmentation of images. In order to address these issues, this research study proposes an outlier-based skimpy regularization fuzzy clustering technique (OSR-FCA) for image segmentation. Clustering methods that use fuzzy membership with sparseness can be improved by incorporating a Gaussian metric regularisation into the objective function. The proposed study used the symmetry information contained in the image data to conduct the image segmentation using the fuzzy clustering technique while avoiding over segmenting relevant data. This resulted in a reduced proportion of noisy data and better clustering results. The classification was carried out by a deep learning technique called convolutional neural network (CNN). Two publicly available datasets were used for the validation process by using different metrics. The experimental results showed that the proposed segmentation technique achieved 97.16% and classification technique achieved 97.26% of accuracy on the MESSIDOR dataset. Full article

Vendor management systems (VMSs) are web-based software packages that can be used to manage businesses. The performance of the VMSs can be assessed using multi-attribute decision-making (MADM) techniques under uncertain situations. This article aims to analyze and assess the performance of VMSs using MADM techniques, especially when the uncertainty is of complex nature. To achieve the goals, we aim to explore Hany mean (HM) operators in the environment of complex picture fuzzy (CPF) sets (CPFSs). We introduce CPF Hamy mean (CPFHM) and CPF weighted HM (CPFWHM) operators. Moreover, the reliability of the newly proposed HM operators is examined by taking into account the properties of idempotency, monotonicity, and boundedness. A case study of VMS is briefly discussed, and a comprehensive numerical example is carried out to assess VMSs using the MADM technique based on CPFHM operators. The sensitivity analysis and comprehensive comparative analysis of the proposed work are discussed to point out the significance of the newly established results. Full article