Bipolar Fuzzy Graphs

Fuzzy concepts are represented by fuzzy sets, so a fuzzy knowledge is defined to be a collection of fuzzy sets (concepts) which includes two constants, 1 (true) and 0 (false). Abstractly, a logical derivation is a finite series of logical operations acting on a given knowledge. In fuzzy world, logical operations are defined by mathematical operations. So fuzzy logical derivations are mathematical derivations, and vice versa In this paper, the closure of fuzzy reasoning is characterized by fuzzy rough sets, and by topologies (neighborhood systems). A new fuzzy concept F N is derivable from the old knowledge K iff the membership function of F N is continuous in K-topology, or equivalently, iff F N is a rough fuzzy set (concept) of I N D(K),or plainly, iff the membership function of F N is a “step” function in the sense that it takes a constant value in each equivalence class of I N D(K).A fuzzy set L partitions the universe into equivalence classes; each equivalence class consists of those elements which have the same degree of membership. The indiscernibility relation I N D(K)over K is the “intersection” of all the equivalence relations induced by the fuzzy sets in K. The membership function of a fuzzy set is a real valued function, so U can be given the minimal topology such that the membership function of each fuzzy set in K is a continuous function. Such topology is called K-topology.

In 1965, Zadeh [1] introduced fuzzy set as a generalization of crisp set by considering membership degrees of elements. That membership degree is represented gradually as a real number in an interval [0, 1]. Wang et al. [2] generalized the concept of fuzzy set, called dynamic fuzzy sets (DFS). In the DFS, each membership degree of an element is given by a membership function dealing with time variable. Thus, the membership degree of an element in a given dynamic fuzzy set might dynamically change according to the time’s variable. In 2002, Intan and Mukaidono [3] proposed an extended concept of fuzzy set, called knowledge-based fuzzy sets (KFS). In the KFS, the membership degree of an element given a fuzzy set is subjectively determined by a single knowledge. The membership degree of an element with respect to a given fuzzy set may be different provided by different knowledge of persons. This paper combines both concepts, DFS and KFS, called knowledge-based dynamic fuzzy sets (KDFS), by realizing that membership function of a given fuzzy set provided by a certain knowledge may be dynamically changed over time. Three kinds of summary fuzzy sets are proposed and discussed. Some basic operations of KDFS are defined. Their properties are verified and examined.

There are several kinds of fuzzy set extensions in the fuzzy set theory. Among them, this paper is concerned with interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets. In interval-valued fuzzy sets, membership degrees are represented by an interval value that reflects the uncertainty in assigning membership degrees. In intuitionistic fuzzy sets, membership degrees are described with a pair of a membership degree and a nonmembership degree. In bipolar-valued fuzzy sets, membership degrees are specified by the satisfaction degrees to a constraint and its counter-constraint. This paper investigates the similarities and differences among these fuzzy set representations.

Mathematical morphology on bipolar fuzzy sets: general algebraic framework

Fuzzy sets provide an effective method for dealing with uncertain and imprecise problems. For data of intermediate fuzzy distribution, membership degrees of objects whose attribute values are larger or smaller than the normal value would be the same and carried out the same decision. However, objects with different values mean that the information they contain is different for the decision-making problem. The decision process of calculating membership degrees in fuzzy set will lose the information of data itself. Therefore, bilateral fuzzy sets and their three-way decisions are proposed. First, the deviation degree is proposed in order to distinguish these objects. Compared with the membership degree, the deviation degree extends the mapping range from [0, 1] to [- 1, 1]. For six typical membership functions, their corresponding deviation functions are discussed and deduced. Second, the concept of bilateral fuzzy sets is proposed and the corresponding operation rules are analyzed and proved. Then, three-way decisions and approximations based on bilateral fuzzy sets are constructed. Next, for the optimization of threshold, principle of least cost is extended to the three-way decisions model based on bilateral fuzzy sets, and theoretical derivation is carried out. Finally, based on probability statistics, the principle based on confidence interval is proposed, which provides a new perspective for threshold calculation.

Outlier detection in neutrosophic sets by using rough entropy based weighted density method

In 1983 the Intuitionistic Fuzzy Sets (IFSs) were introduced as an extension of Zadeh's fuzzy sets. Fifteen years later the Bipolar Fuzzy Sets (BFSs) were introduced as a new extension of fuzzy sets. In the article it is proved that each BFS can be transformed injectively or bijectively to an IFSs.

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

Intuitionistic fuzzy probabilities are an extension of the concept of probabilities with application in several practical problem solving tasks. The former are probabilities represented through intuitionistic fuzzy numbers, to indicate the uncertainty of the membership and nonmembership degrees in the value assigned to probabilities. Moreover, a dual hesitant fuzzy set (DHFS) is an extension of an intuitionistic fuzzy set, and its membership degrees and nonmembership degrees are represented by two sets of possible values; this new theory of fuzzy sets is known today as dual hesitant fuzzy set theory. This work will extend the notion of dual hesitant fuzzy probabilities by representing probabilities through the dual hesitant fuzzy numbers, in the sense of Zhu et al., instead of intuitionistic fuzzy numbers. We also give the concept of dual hesitant fuzzy probability, based on which we provide some main results including the properties of dual hesitant fuzzy probability, dual hesitant fuzzy conditional probability, and dual hesitant fuzzy total probability.

The concept of bipolar fuzzy sets is an extension of fuzzy sets. Our aim is to define bipolar fuzzy rough topology by means of the lower and upper approximations on bipolar fuzzy sets, bipolar fuzzy rough image, bipolar fuzzy rough inverse image, bipolar fuzzy rough subspace and bipolar fuzzy rough continuous functions between these topologies. We construct some functions which are bipolar fuzzy rough continuous and further prove pasting lemma for bipolar fuzzy rough continuous mappings.

Multicriteria decision-making (MCDM) models are valuable for handling complex problems with various criteria and choices. Conventional MCDM lacks a generic paradigm for many real-world situations, like group choices and ambiguous preferences, and only considers crisp decision problems. Therefore, it makes sense to consider most MCDM problems in the actual world to be fuzzy MCDM problems. Spherical fuzzy sets extend the concept of fuzzy sets by considering uncertainty in three ways rather than fuzzy sets' membership degrees only. Effectively identifying, characterizing, and comparing the many variables that may affect a decision is the multi-attribute utility theory (MAUT) approach. This research proposes a novel and expanded MAUT approach that deals with rough approximations under a spherical fuzzy environment. The entropy weighting technique, a key component of our proposed method, assigns weights to criteria based on their relative importance, thereby providing a robust and objective basis for decision-making. This method is applied when selecting medication service robots (MSR) for COVID-19 patients. MSRs are increasingly being used in healthcare to reduce global fatality and spread rates, particularly for highly infectious diseases like COVID-19. However, selecting the most suitable robot from various options can be complex, necessitating mathematical tools for proper selection. The proposed method's effectiveness is not just a claim but is backed by sensitivity and comparative analyses, where we compared the results of our method with those of other established methods, demonstrating its superiority in terms of accuracy and efficiency.

Understanding bipolar information is crucial as it enables individuals to make informed decisions that consider both extremes of a spectrum, leading to more balanced and effective outcomes. Interval-valued bipolar fuzzy set (IVBFS) has already been introduced in the literature as a great decision-making tool that can capture interval-valued bipolar information to properly address uncertainty. In this article, we introduce a hybrid of Interval-valued bipolar fuzzy set (IVBFS) and bipolar hypersoft sets (BHSS) called interval-valued bipolar fuzzy hypersoft set [Formula: see text], which merges the capabilities of IVBFS and BHSS. The rationale behind the design of the presented data structure is to manipulate and process information in decision-making scenarios when the data is bipolar, has multiple attributes that need to be addressed up to a sub-attributive level to get a proper representation of the data provided, and needs to be presented in the form of intervals. In [Formula: see text], two hyper soft sets (HSSs) are used, one providing positive interval-valued membership information and the other providing negative interval-valued membership information. We outline the essential features and basic operations of [Formula: see text] in this paper, examining its commutative, associative, distributive, and De Morgan laws to ensure a comprehensive analysis. To demonstrate the significance of [Formula: see text], we develop a preferential decision support algorithm for selecting the best alternative in e-learning, such as identifying the most suitable instructional method, which can effectively be formulated as a Multi-Attribute Decision-Making (MADM) problem. This approach allows for the systematic evaluation of various alternatives based on multiple parameters and sub-parameters, enabling a rational and well-informed decision. This algorithm helps select the best alternative from a given set of options, leveraging the versatile nature of [Formula: see text]. The presented study conducts both computation-based and structural comparisons to evaluate the adaptability and reliability of the proposed framework.

In many domains of information processing, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools. Here we propose to extend these tools by defining algebraic relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators.

The 2022 International Conference on Image Processing and Intelligent Information Systems (IPIIS 2022) was successfully held online during October 22-23, 2022 in Zhengzhou, China. In the context of the conference, scholars and experts in the fields of image processing and intelligent information systems, involved both in experimental and theoretical research, were invited from various universities and institutions. The conference focused on the most recent advancements in topics such as 'Wireless Sensor Networks', 'Digital Signal Processing', 'Embedded Systems' and others.
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Human kind is exposed to large amounts of data. According to statistics, more than 80% of information received by humans comes from the visual system. Therefore, image information processing is not only an important research topic but also a challenging task. The unique information processing mechanism of the human visual system provides it with fast, accurate, and efficient image processing capabilities. At present, many advanced image analysis and processing techniques have been widely used in image communication, geographic information systems, medical image analysis, and virtual reality. However, there is still a large gap between these technologies and the human visual system. Therefore, building an image system research mechanism based on the biological vision system is an attractive but difficult target. Although it is a challenge, it can also be considered as an opportunity which utilizes biologically inspired ideas. Meanwhile, through the integration of neural biology, biological perception mechanisms, and computer science and mathematical science, related research can bridge biological vision and computer vision. Finally, the biologically inspired image analysis and processing system is expected to be built on the basis of further consideration of the learning mechanism of the human brain.