Search Results (24)

Multiple criteria decision-making (MCDM) is a significant area of decision-making theory that certainly warrants attention. It might be difficult to accurately convey the necessary decision facts when navigating decision-making problems since we frequently run into complicated issues and unpredictable situations. To address this, introducing the novel idea of the bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS), a hybrid structure of the bipolar fuzzy set (BFS) and the pqr-spherical fuzzy set (pqr-SFS), is the main goal of this work. The fundamental (set-theoretic and algebraic) operations on ${\mathrm{B}}_{\mathrm{pqr}}$SFSs are explained as well as their relations to several known models. A distance measure, such as Euclidean distance, among ${\mathrm{B}}_{\mathrm{pqr}}$SFNs, is provided. Afterward, we expand the fundamental aggregation operators to the pqr-spherical fuzzy (${\mathrm{B}}_{\mathrm{pqr}}$SF) environment by developing bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs. According to the aforementioned distance measure and operators, an MCDM approach is established consisting of two algorithms, namely, the TOPSIS method and the method using the proposed operators in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context. Moreover, a numerical example is provided in order to ensure that the presented model is applicable. By using the two algorithms, a comparative analysis of the proposed method with other existing ones is given in order to verify the feasibility of the suggested decision-making procedure. Full article

The bipolar fuzzy model is a rapidly evolving research area that provides a robust framework for addressing real-world problems, with wide-ranging applications in scientific and technical domains. Within this framework, bipolar fuzzy graphs play a significant role in decision-making and problem-solving, particularly through domination theory, which helps tackle practical challenges. This study explores various operations on product bipolar fuzzy graphs, including union (∪), join (+), intersection (∩), Cartesian product (×), composition (∘), and complement, leading to the generation of new graph structures. Several important results related to complete product bipolar fuzzy graphs under these operations are established. Additionally, we introduce key concepts such as dominating sets, minimal dominating sets, and the domination number $⋎\left(\mathcal{H}\right)$, supported by illustrative examples. This study further investigates the properties of domination in the context of these operations. To demonstrate practical applicability, we present a decision-making problem involving the optimization of bus routes and the strategic placement of bus stations using domination principles. This research contributes to the advancement of bipolar fuzzy graph theory and its practical applications in real-world scenarios. Full article

This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept’s distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network. Full article

The aim of this study is to provide neighborhood structures in bipolar fuzzy supra topological space and to show the applicability of bipolar fuzzy supra topology to the medical diagnosis problem. Firstly, we give some properties related to bipolar fuzzy points and their neighborhood structure in bipolar fuzzy supra topological spaces. Then, we consider another important structure, “quasi-coincident”, in the case of bipolar fuzzy points and bipolar fuzzy sets. Then, we introduce the corresponding neighborhood structure called “Q-neighborhood system” by using the quasi-coincident relations. Furthermore, we also investigate the characterization of bipolar fuzzy supra topological space in terms of quasi-neighborhoods. Finally, we present a new method to solve medical diagnosis problems by using the bipolar fuzzy score function. Full article

It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in mPFG involving m saturations for each element in the membership value array of a vertex. This explains $\alpha$-saturation and $\beta$-saturation. We investigate intriguing properties such as $\alpha$-vertex count and $\beta$-vertex count and establish upper bounds for particular instances of mPFGs. Using the concept of $\alpha$-saturation and $\alpha$-saturation, block and bridge of mPFG are characterized. To identify the $\alpha$-saturation and $\beta$-saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG is determined. The time complexity of these algorithms is $O\left(\right|V{|}^3)$, where $\left|V\right|$ is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point. Full article

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment. Full article

These days, multi-criteria decision-making (MCDM) approaches play a vital role in making decisions considering multiple criteria. Among these approaches, the picture fuzzy soft set model is emerging as a powerful mathematical tool for handling various kinds of uncertainties in complex real-life MCDM situations because it is a combination of two efficient mathematical tools, namely, picture fuzzy sets and soft sets. However, the picture fuzzy soft set model is deficient; that is, it fails to tackle information symmetrically in a bipolar soft environment. To overcome this difficulty, in this paper, a model named picture fuzzy bipolar soft sets (P${}_{\mathrm{R}}$FBSSs, for short) is proposed, which is a natural hybridization of two models, namely, picture fuzzy sets and bipolar soft sets. An example discussing the selection of students for a scholarship is added to illustrate the initiated model. Some novel properties of P${}_{\mathrm{R}}$FBSSs such as sub-set, super-set, equality, complement, relative null and absolute P${}_{\mathrm{R}}$FBSSs, extended intersection and union, and restricted intersection and union are investigated. Moreover, two fundamental operations of P${}_{\mathrm{R}}$FBSSs, namely, the AND and OR operations, are studied. Thereafter, some new results (De Morgan’s law, commutativity, associativity, and distributivity) related to these proposed notions are investigated and explained through corresponding numerical examples. An algorithm is developed to deal with uncertain information in the P${}_{\mathrm{R}}$FBSS environment. To show the efficacy and applicability of the initiated technique, a descriptive numerical example regarding the selection of the best graphic designer is explored under P${}_{\mathrm{R}}$FBSSs. In the end, concerning both qualitative and quantitative perspectives, a detailed comparative analysis of the initiated model with certain existing models is provided. Full article

In this study, firstly, we interpret the level set, support, kernel for bipolar complex fuzzy (BCF) set, bipolar complex characteristic function, and BCF point. Then, we interpret the BCF subgroup, BCF normal subgroup, BCF conjugate, normalizer for BCF subgroup, cosets, BCF abelian subgroup, and BCF factor group. Furthermore, we present the associated examples and theorems, and prove these associated theorems. After that, we interpret the image and pre-image of BCF subgroups under homomorphism and prove the related theorems. Full article

In today’s world, the countries that have easy access to energy resources are economically strong, and thus, maintaining a better geopolitical position is important. Petroleum products such as gas and oil are currently the leading energy resources. Due to their excessive worth, the petroleum industries face many risks and security threats. Observing the nature of such problems, it is asserted that the complex bipolar fuzzy information is a better choice for modeling them. Keeping the said problem in mind, this article introduces the novel structure of complex bipolar fuzzy relation (CBFR), which is basically used to find out the relationships between complex bipolar fuzzy sets (CBFSs). Similarly, the types of CBFRs are also defined, which is helpful during the process of analyzing and interpreting the problem. Moreover, some useful results and interesting properties of the proposed structures are deliberated. Further, a new modeling technique based on the proposed structures is initiated, which is used to investigate the security risks to petroleum industries. Furthermore, a detailed comparative analysis proves the advantages and supremacy of CBFRs over other structures. Therefore, the results achieved by the proposed methods are substantially reliable, practical and complete. Full article

This article introduces the notion of bipolar complex fuzzy soft set as a generalization of bipolar complex fuzzy set and soft set. Furthermore, this article contains elementary operations for bipolar complex fuzzy soft sets such as complement, union, intersection, extended intersection, and related properties. The OR and AND operations for bipolar complex fuzzy soft set are also initiated in this study. Moreover, this study contains the decision-making algorithm and real-life examples to display the success and usability of bipolar complex fuzzy soft sets. Finally, the comparative study of initiated notions with some prevailing ideas are also interpreted in this study. Full article

The bipolar fuzzy (BF) set is an extension of the fuzzy set used to solve the uncertainty of having two poles, positive and negative. The rough set is a useful mathematical technique to handle vagueness and impreciseness. The major objective of this paper is to analyze the notion of approximation of BF ideals of semirings by combining the theories of the rough and BF sets. Then, the idea of rough approximation of BF subsemirings (ideals, bi-ideals and interior ideals) of semirings is developed. In addition, semirings are characterized by upper and lower rough approximations using BF ideals. Further, it is seen that congruence relations (CRs) and complete congruence relations (CCRs) play fundamental roles for rough approximations of bipolar fuzzy ideals. Therefore, their associated properties are investigated by means of CRs and CCRs. Full article

Many papers on fuzzy risk analysis calculate the similarity between fuzzy numbers. Usually, they use symmetric and reflexive similarity measures between parameters of fuzzy sets or “centers of gravity” of generalized fuzzy numbers represented by real numbers. This paper studies bipolar similarity functions (fuzzy relations) defined on a domain with involutive (negation) operation. The bipolarity property reflects a structure of the domain with involutive operation, and bipolar similarity functions are more suitable for calculating a similarity between elements of such domain. On the set of real numbers, similarity measures should take into account symmetry between positive and negative numbers given by involutive negation of numbers. Another reason to consider bipolar similarity functions is that these functions define measures of correlation (association) between elements of the domain. The paper gives a short introduction to the theory of correlation functions defined on sets with an involutive operation. It shows that the dissimilarity function generating Pearson’s correlation coefficient is bipolar. Further, it proposes new normalized similarity and dissimilarity functions on the set of real numbers. It shows that non-bipolar similarity functions have drawbacks in comparison with bipolar similarity functions. For this reason, bipolar similarity measures can be recommended for use in fuzzy risk analysis. Finally, the correlation functions between numbers corresponding to bipolar similarity functions are proposed. Full article

The central objective of the proposed work in this research is to introduce the innovative concept of an m-polar fuzzy set (m-PFS) in semigroups, that is, the expansion of bipolar fuzzy set (BFS). Our main focus in this study is the generalization of some important results of BFSs to the results of m-PFSs. This paper provides some important results related to m-polar fuzzy subsemigroups (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs), m-polar fuzzy quasi-ideals (m-PFQIs) and m-polar fuzzy interior ideals (m-PFIIs) in semigroups. This research paper shows that every m-PFBI of semigroups is the m-PFGBI of semigroups, but the converse may not be true. Furthermore this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semigroups by the properties of m-PFIs and m-PFBIs. Full article

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define $(\alpha ,\beta )$-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. Full article

Research on the protein folding problem differentiates the protein folding process with respect to the duration of this process. The current structure encoded in sequence dogma seems to be clearly justified, especially in the case of proteins referred to as fast-folding, ultra-fast-folding or downhill. In the present work, an attempt to determine the characteristics of this group of proteins using fuzzy oil drop model is undertaken. According to the fuzzy oil drop model, a protein is a specific micelle composed of bi-polar molecules such as amino acids. Protein folding is regarded as a spherical micelle formation process. The presence of covalent peptide bonds between amino acids eliminates the possibility of free mutual arrangement of neighbors. An example would be the construction of co-micelles composed of more than one type of bipolar molecules. In the case of fast folding proteins, the amino acid sequence represents the optimal bipolarity system to generate a spherical micelle. In order to achieve the native form, it is enough to have an external force field provided by the water environment which directs the folding process towards the generation of a centric hydrophobic core. The influence of the external field can be expressed using the 3D Gaussian function which is a mathematical model of the folding process orientation towards the concentration of hydrophobic residues in the center with polar residues exposed on the surface. The set of proteins under study reveals a hydrophobicity distribution compatible with a 3D Gaussian distribution, taken as representing an idealized micelle-like distribution. The structure of the present hydrophobic core is also discussed in relation to the distribution of hydrophobic residues in a partially unfolded form. Full article