Concept Of Fuzzy Sets Research Articles

Robust intuitionistic fuzzy clustering with bias field estimation for noisy image segmentation

The concept of intuitionistic fuzzy set has been found to be highly useful to handle vagueness in data. Based on intuitionistic fuzzy set theory, intuitionistic fuzzy clustering algorithms are proposed and play an important role in image segmentation. However, due to the influence of initialization and the presence of noise in the image, intuitionistic fuzzy clustering algorithm cannot acquire the satisfying performance when applied to segment images corrupted by noise. In order to solve above problems, a robust intuitionistic fuzzy clustering with bias field estimation (RIFCB) is proposed for noisy image segmentation in this paper. Firstly, a noise robust intuitionistic fuzzy set is constructed to represent the image by using the neighboring information of pixels. Then, initial cluster centers in RIFCB are adaptively determined by utilizing the frequency statistics of gray level in the image. In addition, in order to offset the information loss of the image when constructing the intuitionistic fuzzy set of the image, a new objective function incorporating a bias field is designed in RIFCB. Based on the new initialization strategy, the intuitionistic fuzzy set representation, and the incorporation of bias field, the proposed method preserves the image details and is insensitive to noise. Experimental results on some Berkeley images show that the proposed method achieves satisfactory segmentation results on images corrupted by different kinds of noise in contrast to conventional fuzzy clustering algorithms.

COVID-19 safe campus evaluation for universities by a hybrid interval type-2 fuzzy decision-making model.

The fight against the COVID-19 pandemic, which has affected the whole world in recent years and has had devastating effects on all segments of society, has been one of the most important priorities. The Turkish Standards Institution has determined a checklist to contribute to developing safe and clean environments in higher education institutions in Turkey and to follow-up on infection control measures. However, this study is only a checklist that makes it necessary for decision-makers to make a subjective evaluation during the evaluation process, while the need to develop a more effective, systematic framework that takes into account the importance levels of multiple criteria has emerged. Therefore, this study applies the best-worst method under interval type-2 fuzzy set concept (IT2F-BWM) to determine the importance levels of criteria affecting the “COVID-19 safe campus” evaluation of universities in the context of global pandemic. A three-level hierarchy consisting of three main criteria, 11 sub-criteria, and 58 sub-criteria has been created for this aim. Considering the hierarchy, the most important sub-criterion was determined as periodic disinfection. The high contribution of the interval-valued type-2 fuzzy sets in expressing the uncertainty in the decision-makers’ evaluations and the fact that BWM provides criterion weights with a mathematical optimization model that produces less pairwise comparisons and higher consistency are the main factors in choosing this approach. Simple additive weighting (SAW) has also been injected into the IT2F-BWM to determine the safety level of any university campus regarding COVID-19. Thus, decision-makers will be better prepared for the devastating effects of the pandemic by first improving the factors that are relatively important in the fight against the pandemic. In addition, a threshold value will be determined by considering all criteria, and it will prepare the ground for a road map for campuses. A case study is employed to apply the proposed model, and a comparison study is also presented with the Bayesian BWM to validate the results of the criteria weights.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11356-022-22796-1.

Fuzzy Option Pricing Based on Fuzzy Number Binary Tree Model

In this article, a new method of option pricing in the form of fuzzy number is established based on fuzzy number binary tree model. Firstly,by using the operation rules of fuzzy number addition and number multiplication in cut set form, the fuzzy number binary tree model is equivalently transformed into two families of classical binary tree models which change with the level value, and some properties of the two binary tree model families are obtained, which are related to solve the problem of fuzzy option pricing. Then, based on the results obtained by us, the concepts of fuzzy set option price and fuzzy number option price are defined, and a sufficient condition is obtained for fuzzy set option price to be fuzzy number option price in the fuzzy number binary tree model. And then, for the convenience of practical application, the concepts of fold line fuzzy set and fuzzy number option price are introduced, a sufficient condition for fold line fuzzy set option price to be fold line fuzzy number option price is obtained, and a specific solution method of the fold line set and the fold line fuzzy number option pricing are set up. At last, two specific examples are given to show how to solve the fold line set and the fold line fuzzy number option price.

Utilization of interval-valued intuitionistic fuzzy soft set in the health sector

The soft set theory plays a key role for dealing with uncertainty, fuzziness and vagueness. The concept of fuzzy soft set which can be seen as a new mathematical approach to vagueness is used in many applications including reliability evaluation, multi criteria decision making and medical diagnosis problems. Later, it is generalized to an interval-valued intuitionistic fuzzy soft set. In this attempt we present the definition and operations of an interval-valued intuitionistic fuzzy soft set. Furthermore, based on the analysis of several operations on interval-valued intuitionistic fuzzy soft set in the study, we provide some notions such as the restricted intersection and restricted union of two interval-valued intuitionistic fuzzy soft sets for selection of appropriate hospital for patient affected by specific dieses.

A new integrated methodology for constructing linguistic pythagorean fuzzy statements for decision making problems

As a combining concept of Pythagorean fuzzy sets (PFSs) and linguistic fuzzy sets (FSs), linguistic PFSs (LPFSs) has been suggested in the literature to deal with the uncertain and inconsistent information in multi-criteria decision making (MCDM) process. The LPFSs based procedure has been built by assuming that the experts make assessments suitable with PFS. It does not provide a mechanism to ensure the suitability of the assessments with theory of PFSs but there are other type of non-standard fuzzy sets such as Neutrosophic sets (NSs) used for modeling with inconsistent information. The main motivation of this study is to offer an assessment collection method to guarantee that the input statements will be Pythagorean fuzzy linguistic expressions. As a second motivation, it is aimed to extend the PFS method for the fuzzy modeling of the other type of decision-making problems apart from MCDM which do not require aggregation and comparison operations and continue with precise fuzzy modeling (PFM). The third motivation of this study is to offer enhancements on the LPFSs method to increase the sensitivity of the modeling while protecting the interpretability. For these purposes, a new methodology based on LPFSs has been proposed and applied on a decision-making problem in a comparative way.

Spatial fuzzy topological space

The concept of spatial fuzzy set is introduced in this article. We have established some fundamental conclusions on the spatial fuzzy set and the spatial fuzzy topological space. Because fuzziness is an internal feature of spatial objects, we used topological relations to build internal properties and relationships between them.

ON NEW INTUITIONISTIC FUZZY(\ep,η)-NEGATION AND (\ep,η)-IMPLICATION

The concept of an Intuitionistic Fuzzy Set (IFS) was introduced in 1983 in \cite{ITKR} and again there in it, the first definition of operation intuitionistic fuzzy negation over IFS was introduced. Later, in \cite{IFPC}, two forms of the operation intuitionistic fuzzy implication were introduced. As of the 2000s, more than 200 different implications and more than 50 different negations arise and in [5 -- 21, 23, 24], their basic properties were discussed. In \cite{e,ee}, two negations, namely $\ep$-negation and $(\ep,\eta)$-negation, were introduced and over their bases $\ep$- and $(\ep,\eta)$-implications were defined, respectively. It was shown that the $(\ep,\eta)$-negation and $(\ep,\eta)$-implication are extensions of the $\ep$-negation and $\ep$-implication, respectively.

Novel distance and entropy definitions for linear Diophantine fuzzy sets and an extension of TOPSIS (LDF‐TOPSIS)

AbstractThe literature of multiple attribute decision making (MADM) is fruitful since there are various and successful applications of different fuzzy set extensions such as intuitionistic, Pythagorean and q‐Rung orthopair fuzzy sets (IFS, PFS and q‐ROFS). Besides their powerful aspects, some definitional limitations are known. In order to eradicate these boundaries regarding the definitions of membership and non‐membership degrees, linear Diophantine fuzzy set (LDFS) concept has been recently emerged. By considering two parameters, LDFS extends the representation area of the previous fuzzy set definitions and provides more extensive human judgement coverage field. In this study, the first distance and entropy measures in the literature have been developed for LDFSs. Their axiomatic definitions are given, and the proofs are shown. Also, thanks to our extensive literature review, we became aware that there is no MADM extension dedicatedly proposed for LDFS. So, the first MADM method extension for LDFS environment has also been developed in this study. A very well‐known MADM approach, TOPSIS, has been extended into LDFS environment for the first time in the literature. The applicability is shown in a healthcare management decision problem and the validity is checked and approved by comparing the alternative rankings LDF‐TOPSIS and the aggregation operators that were obtained from the literature produced.

Analysis of Risks and Security of E-Commerce by Using the Novel Concepts of Complex Cubic Picture Fuzzy Information

Digital technology is so broad today as to encompass almost everything. It eases the difficulties of day-to day life with internet. E-commerce plays an important role in digital age of technology. Instead of many benefits of online system, it faces many types of threats. Sometime, choosing the best e-commerce security has a lot of uncertainties and ambiguities to reduce the effect of threats. To reduce these uncertainties, this article developed a new concept of complex cubic picture fuzzy set (CoCPFS) of fuzzy algebra that explains the positive effect, neutral effect, and negative effect of any object and additionally discussed the Cartesian product between two CoCPFSs, CoCPF relations, and their types. The relationship among the e-commerce securities and threats is investigated for the first time in the history of fuzzy algebra. These relations show the ways of disabling the effects of a threat by an effective security technique. Furthermore, the advantages and benefits of CoCPFS are explained through comparison tests with preexisting frameworks of fuzzy sets.

Fuzzy Logic for Educational Purposes

Fuzzy logic is a popular logical system used in many applications in real life. The aim of this article is to briefly explain the basic concepts of fuzzy sets and fuzzy logic. We define an architecture of fuzzy logic systems and their parts. Using an example of an air-conditioning system we show how formal definitions of fuzzy logic are represented in an implementation.

Anatomy and the type concept in biology show that ontologies must be adapted to the diagnostic needs of research

BackgroundIn times of exponential data growth in the life sciences, machine-supported approaches are becoming increasingly important and with them the need for FAIR (Findable, Accessible, Interoperable, Reusable) and eScience-compliant data and metadata standards. Ontologies, with their queryable knowledge resources, play an essential role in providing these standards. Unfortunately, biomedical ontologies only provide ontological definitions that answer What is it? questions, but no method-dependent empirical recognition criteria that answer How does it look? questions. Consequently, biomedical ontologies contain knowledge of the underlying ontological nature of structural kinds, but often lack sufficient diagnostic knowledge to unambiguously determine the reference of a term.ResultsWe argue that this is because ontology terms are usually textually defined and conceived as essentialistic classes, while recognition criteria often require perception-based definitions because perception-based contents more efficiently document and communicate spatial and temporal information—a picture is worth a thousand words. Therefore, diagnostic knowledge often must be conceived as cluster classes or fuzzy sets. Using several examples from anatomy, we point out the importance of diagnostic knowledge in anatomical research and discuss the role of cluster classes and fuzzy sets as concepts of grouping needed in anatomy ontologies in addition to essentialistic classes. In this context, we evaluate the role of the biological type concept and discuss its function as a general container concept for groupings not covered by the essentialistic class concept.ConclusionsWe conclude that many recognition criteria can be conceptualized as text-based cluster classes that use terms that are in turn based on perception-based fuzzy set concepts. Finally, we point out that only if biomedical ontologies model also relevant diagnostic knowledge in addition to ontological knowledge, they will fully realize their potential and contribute even more substantially to the establishment of FAIR and eScience-compliant data and metadata standards in the life sciences.

A note on “Pythagorean uncertain linguistic hesitant fuzzy weighted averaging operator and its application in financial group decision making”

Shakeel et al. (Soft Comput 24: 1585–1597, 2020) proposed the concept of a Pythagorean uncertain linguistic hesitant fuzzy set (PULHFS), some arithmetic operations of Pythagorean uncertain linguistic hesitant fuzzy sets (PULHFSs), an approach for comparing PULHFSs and a Pythagorean uncertain linguistic hesitant fuzzy weighted averaging (PULHFWA) operator as well as it extensions. Also, using the proposed comparing approach and the proposed aggregation operators, Shakeel et al. proposed a method for solving multi-attribute group decision making (MAGDM) problems. In future other researchers may use Shakeel et al.’s work in their research work. However, it is observed that the approach for comparing PULHFSs and aggregation operators, proposed by Shakeel et al. (Soft Comput 24: 1585–1597, 2020), are not appropriate. Hence, the method for solving MAGDM problems, proposed by Shakeel et al., is also not appropriate. The aim of this note is to make the researchers aware of the inappropriateness of Shakeel et al. work. Furthermore, point out that to resolve the inappropriateness of Shakeel et al. work (Soft Comput 24: 1585–1597, 2020) is a challenging open research problem.

A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases.

Fuzzy parameterized fuzzy hypersoft set (-set) is more flexible and reliable model as it is capable of tackling features such as the assortment of attributes into their relevant subattributes and the determination of vague nature of parameters and their subparametric-valued tuples by employing the concept of fuzzy parameterization and multiargument approximations, respectively. The existing literature on medical diagnosis paid no attention to such features. Riesz Summability (a classical concept of mathematical analysis) is meant to cope with the sequential nature of data. This study aims to integrate these features collectively by using the concepts of fuzzy parameterized fuzzy hypersoft set (-set) and Riesz Summability. After investigating some properties and aggregations of -set, two novel decision-support algorithms are proposed for medical diagnostic decision-making by using the aggregations of -set and Riesz mean technique. These algorithms are then validated using a case study based on real attributes and subattributes of the Cleveland dataset for heart-ailments-based diagnosis. The real values of attributes and subattributes are transformed into fuzzy values by using appropriate transformation criteria. It is proved that both algorithms yield the same and reliable results while considering hypersoft settings. In order to judge flexibility and reliability, the preferential aspects of the proposed study are assessed by its structural comparison with some related pre-developed structures. The proposed approach ensures that reliable results can be obtained by taking a smaller number of evaluating traits and their related subvalues-based tuples for the diagnosis of heart-related ailments.

Non-linear Neutrosophic Numbers and Its Application to Multiple Criteria Performance Assessment

The concept of fuzzy set has been extended by neutrosophic fuzzy sets to represent sets whose elements have different degrees of membership characterized by a truth-membership function, an indeterminacy-membership function and a falsity-membership function. It is usually assumed that these functions are linear, hence excluding the possibility of non-linearity in many decision-making situations. From an alternative definition of non-linear neutrosophic numbers, we develop the concepts of (alpha ,beta ,gamma )-cuts, possibility mean, variance, skewness and a new possibility score function. These concepts are useful to deal with multiple criteria decision making problems. We illustrate the practical use of these concepts by means of a real case study in supply chain risk management in the motor industry. Due to the fact that neutrosophic sets have been used in several areas of decision-making, finance and economics, we argue that our proposal contributes to enhance the application of neutrosophic numbers.

Linear Diophantine Fuzzy Set Theory Applied to BCK/BCI-Algebras

In this paper, we apply the concept of linear Diophantine fuzzy sets in BCK/BCI-algebras. In this respect, the notions of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are introduced and some vital properties are discussed. Additionally, characterizations of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are considered. Moreover, the associated results for linear Diophantine fuzzy subalgebras, linear Diophantine fuzzy ideals and linear Diophantine fuzzy commutative ideals are obtained.

On fuzzy γµ-open sets in generalized fuzzy topological spaces

In this paper, we explore the existence of operation approach on open sets in a generalized fuzzy topological space. We introduce the concept of fuzzy γµ-open set and study some basic properties of it. We obtain an interesting result that the intersection of two fuzzy γµ-open sets may not be a fuzzy γµ-open set, but if the operation is regular then the intersection becomes a fuzzy γµ-open set. We also initiate the notions of fuzzy minimal γµ-open set and fuzzy γµ-locally finite space and establish various results related to these.

A Fuzzy Imperfect Production Inventory Model Based on Fuzzy Differential and Fuzzy Integral Method

In the inventory theory, to treat the uncertainty, the fuzzy set concept is used in order to provide a feasible approach to deal with the uncertainty problem. In this research work, a fuzzy economic production quantity model with interactive fuzzy demands is proposed. In a production process, in the beginning, the system is assumed to be in a controlled state in which only perfect items are manufactured. Later, the manufacturing production process shifts to be an out-of-control-state system; producing both perfect and imperfect items simultaneously, this is considered as a fuzzy state. The defective production rate is also taken into account as a fuzzy state. Here, the selection process of produced items is realized during the production period. With the aim of studying the practical feasibility of the fuzzy economic production inventory model along with a sensitivity analysis of some parameters, different numerical examples are illustrated.

Methods for Multiple Attribute Group Decision Making Based on Picture Fuzzy Dombi Hamy Mean Operator

In this work, ambiguity and unclarity are coped with the effective tools of picture fuzzy sets (PFSs), especially where the conditions demand simulation of various dimensions for evaluation, for example decision making. PFS requires operators to measure the coordination of two PFSs. As far as this paper is concerned, we bring new operators to PFSs with an application, validating this as the generalization of the concept of Fuzzy Sets (FSs) and Intuitionistic Fuzzy Sets (IFSs). The hybrid structure of PFSs has been incorporated with other operators to develop picture fuzzy Dombi Hamy mean (PFDHM) operator, picture fuzzy weighted Dombi Hamy mean (PFWDHM) operator, picture fuzzy Dombi dual Hamy mean (PFDDHM) operator, and PF weighted Dombi dual Hamy mean (PFWDDHM) operator. Further, the properties such as Idempotency, Monotonicity, Boundedness, and Commutativity related to each proposed operator have been discussed. By using these operators the multiple attribute group decision-making (MAGDM) methods are proposed. Moreover, we have explained the application by providing an example of a car supplier. The results are concluded by selecting the best car on the basis of attributes such as quality, production, service efficiency, and risk factors using operators defined on PFSs. A comparative study is also conducted to study the significance of the developed work.

Decision-Making Framework Based on Multineutrosophic Soft Rough Sets

The uncertain data appear in majority of real-world phenomena regarding business, engineering, manufacturing, industries, medical, and other social fields. The neutrosophic sets (NSs), soft sets (SSs), and rough sets (RSs) are distinct mathematical models to deal with different types of uncertainties in data and information. Nowadays, several hybrid models are emerging as their combination or their further generalizations. In this paper, the concept of multineutrosophic fuzzy soft rough sets (MNFSRSs) by combining multineutrosophic fuzzy soft sets (MNSSs) with RSs is presented. Certain eminent features of the proposed structure are investigated. Moreover, a general decision-making (DM) algorithm is described for DM problems to indicate the effectiveness of the presented model.

Interval-Valued Intuitionistic Fuzzy Subalgebras/Ideals of Hilbert Algebras

In this paper, the concept of interval-valued intuitionistic fuzzy sets to subalgebras and ideals of Hilbert algebras is introduced. The inverse image of interval-valued intuitionistic fuzzy subalgebras and interval-valued intuitionistic fuzzy ideals of Hilbert algebras is studied and some related properties are investigated. Equivalence relations on interval-valued intuitionistic fuzzy ideals are discussed.