This work presents a modified Pythagorean fuzzy similarity operator and utilizes its potential in the analysis of questionnaire. Similarity operator is a formidable methodology for decision-making under uncertain domains. Pythagorean fuzzy set is an extended form of intuitionistic fuzzy set with a better accuracy in complex real-world applications. Lots of discussions bordering on the uses of Pythagorean fuzzy sets have been explored based on Pythagorean fuzzy similarity operators. Among the extant Pythagorean fuzzy similarity operators, the work of Zhang et al. is significant but it contains some flaws which need to be corrected/modified to enhance reliable interpretation. To this end, this work explicates the Zhang et al.’s techniques of Pythagorean fuzzy similarity operator by pinpointing their drawbacks to develop an enhanced Pythagorean fuzzy similarity operator, which appropriately satisfies the similarity conditions and yields consistent results in comparison to the Zhang et al.’s techniques. Succinctly speaking, the aim of the work is to correct the flaws in Zhang et al.’s techniques via modifications. To theoretically validate the enhanced Pythagorean fuzzy similarity operator, we discuss it properties and find out that the similarity conditions are well satisfied. In addition, the enhanced PFSO and the Zhang et al.’s PFSOs are compared in the context of precision, and it is verified that the enhanced Pythagorean fuzzy similarity operator can successfully measure the similarity between vastly related but inconsistent PFSs and as well yields a very reasonable results. Furthermore, the enhanced Pythagorean fuzzy similarity operator is applied to the analysis of questionnaire on virtual library to ascertain the extent of awareness and effects of virtual library on students’ academic performance via real data collected from fieldwork. Finally, it is certified that the enhanced Pythagorean fuzzy similarity operator can handle diverse everyday problems more precisely than the Zhang et al.’s Pythagorean fuzzy similarity operators.
Graph structure (GS) is an advancement of the graph concept which effectively represents intricate situations with various connections, frequently used in computer science and mathematics to illustrate relationships among objects and extensively researched in fuzzy sets (FS), intuitionistic fuzzy set (IFS), pythagorean fuzzy set (PFS) and q-rung orthopair fuzzy set (q-ROFS). Meanwhile, a linear Diophantine fuzzy set (LDFS) is a remarkable extension of the existing notions of a FS, IFS, PFS and q-ROFS by comporting reference parameters that removed all the limitations related to membership degree (MD) and non-membership degree (NMD). According to the best of our knowledge, there is a lack of elegantly proposed GS extension for LDFSs in the current literature. As a result, this research focuses on introducing first linear Diophantine fuzzy graph structure (LDFGS) concept which extends the existing notions of GS in various contexts of FSs. Several key concepts in LDFGSs are presented, such as ˘ρi-edge, ˘ρi-path, strength of ˘ρi-path, ˘ρi-strength of connectedness, ˘ρi-degree of a vertex, vertex degree, total ρ˘i-degree of a vertex, and total vertex degree in an LDFGS. In addition, we introduce the ˘ρi-size, size, and order of an LDFGS. Moreover, this article presents the ideas of the maximal product of two LDFGSs, strong LDFGS, degree and ˘ρi-degree of the maximal product, ˘ρi-regular and regular LDFGSs, along with examples for clarification. Certain significant results related to the proposed concepts also demonstrated with explanatory examples such as the maximal product of two strong LDFGSs is also a strong LDFGS, the maximal product of two connected LDFGSs is also a connected LDFGS but the maximal product of two regular LDFGS may not be a regular LDGS. Moreover, many interesting and alternative formulas for calculating ˘ρi-degrees of an LDFGS in various situations are proved with examples. LDFGSs are highly beneficial for solving numerous combinatorial problems involving multiple relations, and they surpass existing concepts of GSs within the FS context due to their flexibility in selecting MD and NMD alongside their reference parameters.
A hybrid method for the numerical solution of the system of delayed linear fuzzy mixed VolterraFredholm integral equations (FMDVFIES) is introduced. Using the hybrid of Bernstein polynomials and blockpulse functions (HBBFs), an approximate solution for the equations system is provided. Firstly, the HBBFs and their operational matrices are introduced, and some of their characteristics are described. Then by applying the operational matrices on FMDVFIES convert it to the algebraic equations system. The numerical solution is obtained by solving this algebraic system. Then the convergence is investigated and some numerical examples are presented to show the effectiveness of the method.
Professor John N. Mordeson is a distinguished mathematician and educator who has made significant contributions to the field of fuzzy logic and its applications. He is currently a Professor Emeritus of Mathematics at Creighton University. Dr. Mordeson earned his B.S., M.S., and Ph.D. degrees from Iowa State University. Throughout his career, he has authored twenty books and more than two hundred journal articles on fuzzy science, making remarkable advancements in the field. He also serves on the editorial boards of numerous academic journals, continuing to make valuable contributions to fuzzy science.
Using the concept of fuzzy points, the notion of fuzzy filters in pre-ordered residuated systems is introduced, and their relevant properties are investigated and analyzed. Characterization of fuzzy filters are displayed. Fuzzy filters are formed using filters. The concepts of positive set, $\in_{t}$-set, (extended) $q_{t}$-set are defined and the conditions under which they become filters are explored. The concept of fuzzy filter with thresholds is introduced and related properties are investigated.
In this work, we define a new sequence denominated by fuzzy Leonardo numbers. Some algebraic properties of this new sequence are studied and several identities are established. Moreover, the relations between the fuzzy Fibonacci and fuzzy Lucas numbers are explored, and several results are given. In addition, some sums involving fuzzy Leonardo numbers are provided.
The focus of this work is to study sequences of interactive fuzzy numbers. The interactivity relation is associated with the concept of joint possibility distribution. In this case, the type of interactivity studied is linked to a family of joint possibility distributions (Jγ), in which the parameter γ intrinsically models levels of interactivity between the fuzzy numbers involved. Each element of the sequence of interactive fuzzy numbers is obtained through a discrete equation, and the arithmetic operations present in the equation are extended to this type of fuzzy number. Some simulations are performed to illustrate the behavior of the sequences, called interactive, and to compare them with the sequences obtained by other fuzzy arithmetic operations.
In this paper, we continue the investigation started in [1]. We obtain new results derived from novel concepts developed in analogy with others already established, e.g., the fact that leftoids (X, ∗) for φ are super-transitive if and only if φ(φ(x)) = φ(x) for all x ∈ X. In addition we apply fuzzy subsets in this context and we derive a number of results as consequences.
The n,m‐rung orthopair fuzzy set theory is a robust model for managing uncertainty, particularly in multi‐attribute decision‐making. Meanwhile, the hesitant fuzzy model is a well‐established tool in decision‐making processes. Recognizing the similarities between these models, we propose a new framework called "c,d‐rung orthopair hesitant fuzzy sets," which integrates both approaches. We examine key operations such as union, intersection, complement, subset, and equality, and introduce aggregation operators like the c,d‐RHFPA, c,d‐RHFWA, c,d‐RHFPG, and c,d‐RHFWPG operators. Additionally, an algorithm for multi‐attribute decision‐making is developed, which is applied to determine optimal business strategies for sustainable supply chain management. A comparative analysis with existing methods demonstrates the model's effectiveness, offering insights into its strengths and limitations. This paper introduces a novel approach to decision‐making, outlining its real‐world application and future research directions
Alzheimers disease is an unpredictable and progressive neurodegenerative disorder that initially affects memory thinking and behavior. Some key features of Alzheimers disease are memory loss, cognitive decline, behavioral changes, disorientation, and physical symptoms. In this article, we design the procedure of a multiattributive border approximation area comparison deep learning algorithm for the diagnosis of Alzheimers Disease. For this, first, we goal to design the model of complex propositional linear Diophantine fuzzy information with their basic operational laws. In addition, we analyze the model of complex propositional linear Diophantine fuzzy power average operator, complex propositional linear Diophantine fuzzy weighted power average operator, complex propositional linear Diophantine fuzzy power geometric operator, complex propositional linear Diophantine fuzzy weighted power geometric operator, and also initiate their major properties. Additionally, the key role of this paper is to arrange relevant from different sources for diagnosing Alzheimers disease under the consideration of the designed technique. Finally, we compare both (proposed and existing) ranking information to address the supremacy and strength of the designed models.
Atanassov's intuitionistic fuzzy set is more adept at representing and managing uncertainty. Within intuitionistic fuzzy set theory, intuitionistic fuzzy measure is a significant field of study. In order to address decision making, we present a novel similarity metric between intuitionistic fuzzy sets in this study. First, based on the minimum and maximum levels of similarity, we suggest a new similarity metric between intuitionistic fuzzy values. It is capable of overcoming the limitations of current approaches to gauging the degree of resemblance between fuzzy intuitionistic sets. It is also possible to show some aspects of the suggested similarity measure between intuitionistic fuzzy sets by taking into account the modal operators and their different extensions. Finally, we apply the proposed similarity measure between intuitionistic fuzzy sets to deal with a real life problem. The suggested action can provide a precise outcome. The application section examines a real-world issue of choosing the best course of action among n options based on m criteria. A fictitious case study is created along with the method's algorithm.
In [1], states are ranked with respect to the best states to work. In [2], states are ranked with respect to the peace and security for women. We determine the fuzzy similarity measure of these to rankings. We find the similarity to be high for one of the measures and very high for the other. We then break the United States into regions and determine the fuzzy similarity measure of these two rankings for each region. The fuzzy similarity here is medium for one measure and high for the other. Similarity plays a role in many fields. There exists many special definitions of similarity which have been used in different areas. We choose to use fuzzy similarity measures which seem appropriate in rankings. In fact, we develop some new measures.
The credibility theory was introduced by B. Liu as a new way to describe the fuzzy uncertainty. The credibility measure is the fundamental notion of the credibility theory. Recently, L.Yang and K. Iwamura extended the credibility measure by defining the parametric measure m_λ (λ is a real parameter in the interval [0,1] and for λ= 1/2 we obtain as a particular case the notion of credibility measure). By using the m_λ-measure, we studied in this paper a risk neutral multi-item inventory problem. Our construction generalizes the credibilistic inventory model developed by Y. Li and Y. Liu in 2019. In our model, the components of demand vector are fuzzy variables and the maximization problem is formulated by using the notion of m_λ-expected value. We shall prove a general formula for the solution of optimization problem, from which we obtained effective formulas for computing the optimal solutions in the particular cases where the demands are trapezoidal and triangular fuzzy numbers. For λ=1/2 we obtain as a particular case the computation formulas of the optimal solutions of the credibilistic inventory problem of Li and Liu. These computation formulas are applied for some m_λ-models obtained from numerical data.
This study introduces a novel framework, Generalized Interval-Valued Neutrosophic Rough Soft Sets (GIVNRS sets), designed to improve handling uncertainty, imprecision, and vagueness in complex decision-making scenarios. By integrating soft, rough, and generalized interval-valued neutrosophic set theories, the framework offers a robust methodology for addressing indeterminacy and incomplete data. The theoretical foundation of GIVNRS sets is built upon fundamental operations, including intersection, union, complement, and novel aggregation union operators tailored for multi-criteria decision-making (MCDM) applications. The practical applicability of the framework is demonstrated through a water quality assessment, where it successfully classifies river segments based on key water quality parameters such as pH, Dissolved Oxygen (DO), and Biochemical Oxygen Demand (BOD). The case study results show that the pollution scores for the river segments were computed, classifying the segments such as “Good,” “Moderate,” and “Poor,” with corresponding pollution levels. These findings highlight the framework’s ability to manage incomplete and inconsistent data, providing a reliable and comprehensive water quality evaluation. Compared to traditional models, the GIVNRS set approach offers enhanced flexibility, stability, and adaptability. This study not only contributes to the theoretical development of neutrosophic, soft, and rough set theories but also establishes GIVNRS sets as a powerful tool for water quality decision-making. Future research will explore further advancements in the application and computational efficiency of this framework.
This article focuses on evaluating the success or failure of kidney transplantation using Shannon entropy, fuzzy sets, and Scaf. The data for Scaf references used in this study for both healthy individuals and kidney transplant recipients have been collected from the relevant literature. For both groups, Scaf's Shannon entropy values have been calculated using an appropriate probability density function and formulation, and sequences have been generated for CAF and Scr biomarkers from entropy values, with findings interpreted. These sequences are called healing sequences. A case study demonstrating whether the transplant procedure was successful or unsuccessful was presented using sequences that we refer to as healing sequences. In this context, the utilization of mathematical tools such as fuzzy sets, Shannon entropy, and reference intervals becomes evident. These tools provide a systematic and quantitative approach to assessing the outcomes of kidney transplantation. By leveraging the principles of Shannon entropy, we gain insights into the degree of unpredictability and fuzziness associated with biomarker values, which can be indicative of the transplant's success. Furthermore, the concept of healing sequences provides a valuable framework for tracking the progression of patients post-transplantation. By monitoring changes in CAF and Scr biomarkers over time, healthcare professionals can make informed decisions and interventions to ensure the well-being of kidney transplant recipients.
As the digital landscape continues to evolve, the selection of an appropriate online shop-ping platform has become increasingly crucial for both consumers and businesses. This paper introduces a novel approach that combines the Fermatean fuzzy set theory with the triangular divergence distance measure in Compromise Ranking of Alternatives from Distance to Ideal Solution (CRADIS) method to streamline the decision-making process in online platform selection. Through a comprehensive example, we illustrate the application of this approach in evaluating and ranking four distinct online shopping latforms based on multiple criteria. Through this integrated approach, decision-makers can gain valuable insights into the rela-tive merits of each online shopping platform, allowing them to make informed choices aligned with their preferences and requirements. Furthermore, by accommodating uncertainty and imprecision, the Fermatean fuzzy set theory enhances the robustness of the decision-making process, minimizing the risk of making suboptimal decisions. Overall, this paper demon-strates the practical applicability of Fermatean fuzzy set theory in decision support systems for online platform selection. To demonstrate the proposed method’s applicability, we have compared the results with existing Multi-attribute decision making (MADM) methods. To establish its stability, we conducted a sensitivity analysis. By leveraging the CRADIS method alongside Fermatean fuzzy set theory, decision-makers can navigate the complex landscape of online shopping platforms with greater confidence and efficiency, ultimately leading to more satisfactory outcomes for both consumers and businessesalike.
This paper aims at solving a class of linear LR complex fuzzy matrix equations $A\widetilde{X}B=\widetilde{C}$ using a matrix approach. By using the basic operation of LR fuzzy number matrix, the original complex fuzzy matrix equation is transformed into a clear matrix equation group. Two new and simplified models for calculating fuzzy solutions are designed in detail, and sufficient conditions for strong fuzzy solutions are analyzed. Finally, two examples are given to illustrate the feasibility and effectiveness of the proposed method. Now that the complex fuzzy numbers can describe uncertain factors more vivid and reasonable than the real fuzzy numbers sometimes and the wide application of matrix equations under uncertain conditions, our research work enriches the fuzzy linear systems theory.
In this paper, the calculation methods of the real eigenvalues and LR fuzzy eigenvectors of clear real symmetry matrices are deeply considered. The original fuzzy feature problem is extended by using the arithmetic algorithm of LR fuzzy numbers into a simple feature problem with a high-order clear real symmetry matrix. We discuss two cases: (a) λ is a non-negative unknown eigenvalue; (b) λ is a negative unknown eigenvalue. We established two computational models and proposed an algorithm for finding the fuzzy eigenvectors of the true symmetry matrix. Some numerical examples are used to illustrate our proposed method.
In this study, we present the ideas of logical entropy and logical conditional entropy for partitions in interval-valued intuitionistic fuzzy sets, and we establish their fundamental properties. First, we establish the definitions of logical entropy and logical conditional entropy, demonstrating their key characteristics and relationships. We then define logical mutual information and explore its properties, providing a comprehensive understanding of its behavior within the context of interval-valued intuitionistic fuzzy sets. Additionally, we propose the concept of logical divergence of states defined on interval-valued intuitionistic fuzzy sets and examine its properties in detail, including its application and implications for understanding state transitions within these fuzzy sets. Finally, we extend our study to dynamical systems, introducing the logical entropy of such systems when modeled with intervalvalued intuitionistic fuzzy sets. We present several results related to this extension, highlighting the applicability and relevance of logical entropy in analyzing and understanding the behavior of dynamical systems. Overall, this paper offers a thorough exploration of logical entropy, mutual information, and divergence within the framework of interval-valued intuitionistic fuzzy sets, providing new insights and potential applications in various fields.
In this article, we introduce a new concept of fuzzy measurement, the space of fuzzy measurable functions and fuzzy integral, which has a dynamic position and is different from previous approaches. With this concept, we create a new version of measurement theory and fuzzy integral. The main goal of this paper is to define the fuzzy integral in the fuzzy size space. First, we introduce fuzzy measurable functions and $L^{+}$ essential and related concepts in fuzzy space. In the continuation of the work, with the help of fuzzy measurable functions, we define the fuzzy integral in the fuzzy measurement space and examine the theorems related to it and the relationship between them in the fuzzy measurement space. The next step is to establish one of the fundamental convergence theorems with the uniform convergence theorem in the fuzzy measurement space and prove it. Finally, we prove Fatou's lemma as an application of the theorems raised in the fuzzy measurement space.
