Search Results (21)

This study investigates linear Diophantine fuzzy structures within the framework of Sheffer stroke BCK-algebras (SBCK-algebras). We introduce and characterize linear Diophantine fuzzy SBCK-subalgebras and linear Diophantine fuzzy SBCK-ideals, establishing fundamental connections between these fuzzy structures and their corresponding crisp subalgebras and ideals. In particular, we prove that the level sets of linear Diophantine fuzzy SBCK-subalgebras form SBCK-subalgebras, and, conversely, every SBCK-subalgebra gives rise to such a fuzzy structure. Additionally, we show that every linear Diophantine fuzzy SBCK-ideal induces a linear Diophantine fuzzy SBCK-subalgebra; however, the converse does not necessarily hold. Several structural properties, homomorphic images, and intersections of such fuzzy ideals are also examined. These results demonstrate how linear Diophantine logic naturally integrates with Sheffer stroke BCK-algebras and enriches their algebraic behavior. Full article

The symmetrical linear Diophantine fuzzy Hamacher aggregation operators play a fundamental role in many decision-making applications. The selection of a cyber security system is of paramount importance for maintaining digital assets. It necessitates a comprehensive review of threat landscapes, vulnerability assessments, and the specific needs of the organization in order to ensure the implementation of effective security measures. Smart grid (SG) technology uses modern communication and monitoring technologies to enhance the management and regulation of electricity production and transmission. However, greater dependence on technology and connection creates new vulnerabilities, exposing SG communication networks to large-scale attacks. Unlike previous surveys, which often give broad overviews of SG design, our research goes a step further, giving a full architectural layout that includes major SG components and communication linkages. This in-depth review improves comprehension of possible cyber threats and allows SGs to analyze cyber risks more systematically. To determine the best cybersecurity strategies, this study introduces a multi-criteria group decision-making (MCGDM) approach using the linear Diophantine fuzzy Hamacher prioritized aggregation operator (LDFHPAO). In real-world applications, aggregation operators (AOs) are essential for information fusion. This research presents innovative prioritized AOs designed to address MCGDM problems in uncertain environments. We developed the LDF Hamacher prioritized weighted average (LDFHPWA) and LDF Hamacher prioritized weighted geometric (LDFHPWG) operators, which address the shortcomings of traditional operators and provide a more robust modeling approach for MCGDM challenges. This study also outlines key characteristics of these new prioritized AOs. An MCGDM approach incorporating these operators is proposed and demonstrated to be effective through an example that compares and selects the optimal cybersecurity. Full article

The theory of spherical linear Diophantine fuzzy sets (SLDFS) boasts several advantages over existing fuzzy set (FS) theories such as Picture fuzzy sets (PFS), spherical fuzzy sets (SFS), and T-spherical fuzzy sets (T-SFS). Notably, SLDFS offers a significantly larger portrayal space for acceptable triplets, enabling it to encompass a wider range of ambiguous and uncertain knowledge data sets. This paper delves into the regularity of spherical linear Diophantine fuzzy graphs (SLDFGs), establishing their fundamental concepts. We provide a geometrical interpretation of SLDFGs within a spherical context and define the operations of complement, union, and join, accompanied by illustrative examples. Additionally, we introduce the novel concept of a spherical linear Diophantine isomorphic fuzzy graph and showcase its application through a social network scenario. Furthermore, we explore how this amplified depiction space can be utilized for the study of various graph theoretical topics. Full article

The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree $\alpha$ of $\alpha$ level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree $\alpha$, we can solve DM from the set of $\alpha$-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on $\alpha$-cut analysis bestowing the degree $\alpha$ given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example. Full article

The concept of linear Diophantine fuzzy set (LDFS) theory with its control parameters is a strong model for machine learning and data-driven multi-criteria decision making (MCDM). The sine-trigonometric function (STF) has two significant features, periodicity and symmetry about the origin that are very useful tools for information analysis. Keeping in view the characteristics of both STF and LDFS theory, this article introduces the sine-trigonometric operations for linear Diophantine fuzzy numbers (LDFNs). These operational laws lay a foundation for developing new linear Diophantine fuzzy sine-trigonometric aggregation operators (LDFSTAOs). The integration of Industry 4.0 technology into healthcare has the potential to revolutionize patient care. One of the most challenging tasks is the selection of efficient suppliers for the healthcare supply chain (HSC). The traditional suppliers are not efficient in accordance with Industry 4.0, with particular uncertainties. A new MCDM framework is presented based on LDFSTAOs to examine the HSC performance in industry 4.0. A credibility test, sensitivity analysis and comparative analysis are performed to express the novelty, reliability, and efficiency of the proposed methodology. Full article

The notion of a linear diophantine fuzzy set as a generalization of a fuzzy set is a mathematical approach that deals with vagueness in decision-making problems. The use of reference parameters associated with validity and non-validity functions in linear diophantine fuzzy sets makes it more applicable to model vagueness in many real-life problems. On the other hand, subspaces of vector spaces are of great importance in many fields of science. The aim of this paper is to combine the two notions. In this regard, we consider the linear diophantine fuzzification of a vector space by introducing and studying the linear diophantine fuzzy subspaces of a vector space. First, we studied the behaviors of linear diophantine fuzzy subspaces of a vector space under a linear diophantine fuzzy set. Second, and by means of the level sets, we found a relationship between the linear diophantine fuzzy subspaces of a vector space and the subspaces of a vector space. Finally, we discuss the linear diophantine fuzzy subspaces of a quotient vector space. Full article

Intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and q-rung orthopair fuzzy sets (q-ROFS) are among those concepts which are widely used in real-world applications. However, these theories have their own limitations in terms of membership and non-membership functions, as they cannot be obtained from the whole unit plane. To overcome these restrictions, we developed the concept of a complex linear Diophantine fuzzy set (CLDFS) by generalizing the notion of a linear Diophantine fuzzy set (LDFS). This concept can be applied to real-world decision-making problems involving complex uncertain information. The main motivation behind this paper is to study the applications of CLDFS in a non-associative algebraic structure (AG-groupoid), which has received less attention as compared to associative structures. We characterize a strongly regular AG-groupoid in terms of newly developed CLDF-score left (right) ideals and CLDF-score $(0,2)$-ideals. Finally, we construct a novel approach to decision-making problems based on the proposed CLDF-score ideals, and some practical examples from civil engineering are considered to demonstrate the flexibility and clarity of the initiated CLDF-score ideals. Full article

Nowadays, there is an ever-increasing diversity of materials available, each with its own set of features, capabilities, benefits, and drawbacks. There is no single definitive criteria for selecting the perfect biomedical material; designers and engineers must consider a vast array of distinct biomedical material selection qualities. The goal of this study is to establish fairly operational rules and aggregation operators (AOs) in a linear Diophantine fuzzy context. To achieve this goal, we devised innovative operational principles that make use of the notion of proportional distribution to provide an equitable or fair aggregate for linear Diophantine fuzzy numbers (LDFNs). Furthermore, a multi-criteria decision-making (MCDM) approach is built by combining recommended fairly AOs with evaluations from multiple decision-makers (DMs) and partial weight information under the linear Diophantine fuzzy paradigm. The weights of the criterion are determined using incomplete data with the help of a linear programming model. The enhanced technique might be used in the selection of compounds in a variety of applications, including biomedical programmes where the chemicals used in prostheses must have qualities similar to those of human tissues. The approach presented for the femoral component of the hip joint prosthesis may be used by orthopaedists and practitioners who will choose bio-materials. This is due to the fact that biomedical materials are employed in many sections of the human body for various functions. 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

The advantages of the intuitionistic fuzzy set, Pythagorean fuzzy set, and q-rung orthopair fuzzy set are all carried over into the linear Diophantine fuzzy set by extending the restrictions on the grades. Linear Diophantine fuzzy sets offer a wide range of practical applications because the reference parameters allow evaluation andto express their judgments about membership and nonmembership degrees in a variety of ways. Linguistic-valued information cannot be described by linear Diophantine fuzzy numbers since precise numbers are used in linear Diophantine fuzzy systems. In this paper, we first present the novel idea of a linguistic linear Diophantine fuzzy set, which is the hybrid structure of the linear Diophantine fuzzy set and the linguistic term set. Furthermore, some basic operational rules with novel distance measures, namely, Hamming, Euclidean, and Chebyshev distance measures, are established. Based on the newly defined concept of distance measure, an extended TOPSIS technique is presented to tackle the linguistic uncertainty in real-world decision support problems. A numerical example is illustrated to support the applicability of the proposed methodology and to analyze symmetry of the optimal decision. A comparison analysis is constructed to show the symmetry, validity, and effectiveness of the proposed method over the existing decision support techniques. Full article

In this paper, we focus on several ideas associated with linear Diophantine fuzzy soft sets (LDFSSs) along with its algebraic structure. We provide operations on LDFSSs and their specific features, elaborating them with real-world examples and statistical depictions to construct an inflow of linguistic variables based on linear Diophantine fuzzy soft (LDFSS) information. We offer a study of LDFSSs to the multi-criteria decision-making (MCDM) process of university determination, together with new algorithms and flowcharts. We construct LDFSS-TOPSIS, LDFSS-VIKOR and the LDFSS-AO techniques as robust extensions of TOPSIS (a technique for order preferences through the ideal solution), VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) and AO (aggregation operator). We use the LDFSS-TOPSIS, LDFSS-VIKOR and LDFSS-AO techniques to solve a real-world agricultural problem. Moreover, we present a small-sized robotic agri-farming to support the proposed technique. A comparison analysis is also performed to examine the symmetry of optimal decision and to analyze the efficiency of the suggested algorithms. Full article

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. Full article

We introduce the notion of the interval-valued linear Diophantine fuzzy set, which is a generalized fuzzy model for providing more accurate information, particularly in emergency decision-making, with the help of intervals of membership grades and non-membership grades, as well as reference parameters that provide freedom to the decision makers to analyze multiple objects and alternatives in the universe. The accuracy of interval-valued linear Diophantine fuzzy numbers is analyzed using Frank operations. We first extend the Frank t-conorm and t-norm (FT${}_c$T${}_n$) to interval-valued linear Diophantine fuzzy information and then offer new operations such as the Frank product, Frank sum, Frank exponentiation, and Frank scalar multiplication. Based on these operations, we develop novel interval-valued linear Diophantine fuzzy aggregation operators (AOs), including the “interval-valued linear Diophantine fuzzy Frank weighted averaging operator and the interval-valued linear Diophantine fuzzy Frank weighted geometric operator”. We also demonstrate various features of these AOs and examine the interactions between the proposed AOs. FT${}_c$T${}_n$s offer two significant advantages. Firstly, they function in the same way as algebraic, Einstein, and Hamacher t-conorms and t-norms. Secondly, they have an additional parameter that results in a more dynamic and reliable aggregation process, making them more effective than other general t-conorm and t-norm approaches. Furthermore, we use these operators to design a method for dealing with multi-criteria decision-making with IVLDFNs. Finally, a numerical case study of the novel carnivorous issue is shown as an application for emergency decision-making based on the proposed AOs. The purpose of this numerical example is to demonstrate the practicality and viability of the provided AOs. Full article

In this article, a new hybrid model named linear Diophantine fuzzy rough set (LDFRS) is proposed to magnify the notion of rough set (RS) and linear Diophantine fuzzy set (LDFS). Concerning the proposed model of LDFRS, it is more efficient to discuss the fuzziness and roughness in terms of linear Diophantine fuzzy approximation spaces (LDFA spaces); it plays a vital role in information analysis, data analysis, and computational intelligence. The concept of $(<\mathfrak{p},{\mathfrak{p}}^{\prime }>,<\mathfrak{q},{\mathfrak{q}}^{\prime }>)$-indiscernibility of a linear Diophantine fuzzy relation (LDF relation) is used for the construction of an LDFRS. Certain properties of LDFA spaces are explored and related results are developed. Moreover, a decision-making technique is developed for modeling uncertainties in decision-making (DM) problems and a practical application of fuzziness and roughness of the proposed model is established for medical diagnosis. Full article

It is quite beneficial for every company to have a strong decision-making technique at their disposal. Experts and managers involved in decision-making strategies would particularly benefit from such a technique in order to have a crucial impact on the strategy of their company. This paper considers the interval-valued linear Diophantine fuzzy (IV-LDF) sets and uses their algebraic laws. Furthermore, by using the Muirhead mean (MM) operator and IV-LDF data, the IV-LDF power MM (IV-LDFPMM) and the IV-LDF weighted power MM (IV-LDFWPMM) operators are developed, and some special properties and results demonstrated. The decision-making technique relies on objective data that can be observed. Based on the multi-attribute decision-making (MADM) technique, which is the beneficial part of the decision-making strategy, examples are given to illustrate the development. To demonstrate the advantages of the developed tools, a comparative analysis and geometrical interpretations are also provided. Full article