Research on Fuzzy Logic and Mathematics with Applications, Third Edition

Preface

The notion of the non-rigid fuzzy set was introduced by Lotfi A. Zadeh [1] in 1965 as a generalization of the ordinary sets. In effect, the theory of fuzzy sets sparked some new computing methods which have applications in decision making, identification, optimization, information, control, etc. To be more specific, Zadeh is also the founder of fuzzy logic [2,3,4]. Since the advent of the notion of the fuzzy set, Zadeh, his colleagues and followers have used this important and interesting set and established a great deal of important and interesting research in fuzzy logic, fuzzy topology, fuzzy arithmetic, etc. Indeed, the theory of fuzzy mathematics stems from the notion of graded membership which is allegedly motivated by the processes of human perception and cognition. It is well-known that the fuzzy set theory and fuzzy logic are used in different parts of engineering such as control systems, robotics, automation, consumer electronics, image processing, etc. This Special Issue deals with fuzzy logic and mathematics with applications in decision making, graph theory, soft set theory, fuzzy semantics, fuzzy and neutrosophic topologies, decision-making, theory of fuzzy graphic binary metroid and theory of neutrosophic rhotrices.

Twelve research articles appear in the Special Issue “Research on Fuzzy Logic and Mathematics with Applications, Third Edition” of the MDPI Symmetry journal. In what follows we will give a brief account of their results and findings.

In Contribution 1, “A Novel IVBPRT-ELECTRE III Algorithm Based on Bidirectional Projection and Its Application”, the authors Juxiang Wang, Min Xu, Yanjun Wang and Ziqi Zhu present an extension of bidirectional projection to probabilistic linguistic term sets in order to preserve the completeness of information to the level it is possible. In fact, the authors extend, in a probabilistic linguistic environment, the large-scale group decision-making problem to limited interval values. Moreover, they proposed a new group decision-making method named IVBPRT-ELECTRE III algorithm (ELECTRE III based on bidirectional projection and regret theory under limited interval-valued probabilistic linguistic term set.

In Contribution 2, “The Neutrosophization of $\delta$-Separation Axioms”, the authors Ahu Açikgöz, Ferhat Esenbel, Abdulhamit Maman and Seher Zorlu offer the new notion of neutrosophic $\delta$-open sets. They also propose the interesting and useful notions of neutrosophic $\delta$-separation axioms, which stem from the concept of neutrosophic $\delta$-open sets. Moreover, they study the interplay between these separation properties and their characteristics within subspaces.

In Contribution 3, “A New Graph Vulnerability Parameter: Fuzzy Node Integrity”, the authors Ferhan Nihan Murater and Goksen Bacak-Turan introduce the useful parameter of fuzzy node integrity, which takes into account the number of disrupted elements and the strength of residual connections. They demonstrate the applicability of this parameter, by deriving general formulas for different types of symmetric and asymmetric fuzzy graph structures, including cycle graphs, wheel graphs, and star graphs. Their findings are important with respect to both theoretical research and practical improvements in transportation, disaster management, and communication networks.

In Contribution 4, “Some Properties of Boolean-like Laws in Fuzzy Logic”, the focus of the authors, Sevilay Demir Sağlam and Gül Karadeniz Gözeri, is on the relationships between fuzzy logic and classical logic properties using fuzzy t-norms, t-conorms, and fuzzy implications. Indeed, the purpose is to extend the Boolean laws in classical logic to fuzzy logic. They establish the necessary and sufficient conditions for validating the generalizations of the proposed properties from classical to fuzzy logic. They also give some examples to demonstrate practical applicability and advantages of their approach.

In Contribution 5, “Characterization of Degree Energies and Bounds in Spectral Fuzzy Graphs”, the authors Ruiqi Cai, Buvaneswari Rangasamy, Senbaga Priya Karuppusamy and Aysha Khan focus largely on the degree energy of fuzzy graphs to determine fundamental spectral bounds and characterize adjacency structures. In effect, they deduce the upper bounds on the sum of squared degree eigenvalues based on vertex degree distributions and establish constraints by utilizing the characteristic polynomial of the maximum degree matrix. Moreover, they showed that the average degree energy of a connected fuzzy graph remains strictly positive. They applied their proposed framework to protein–protein interaction networks to exhibit the critical proteins and to enhance the network resilience analysis.

In Contribution 6, “Some Characterizations of k-Fuzzy $\gamma$-Open Sets and Fuzzy $\gamma$-Continuity with Further Selected Topic”, Fahad Alsharari, Hind Y. Saleh and Islam M. Taha introduced the notion of k-fuzzy $\gamma$-open (k-$\mathcal{F}$-$\gamma$-open) sets as a new generalized class of fuzzy open ($\mathcal{F}$-open) sets on fuzzy topological spaces in the sense of Šostak. By utilizing this notion, they offered several properties of new mappings, separation axioms and covering properties.

In Contribution 7, “Distance Measures of $(m,a,n)$-Fuzzy Neutrosophic Sets and Their Applications in Decision Making”, Samajh Singh Thakur pointed out that an $(m,a,n)$-fuzzy neutrosophic set is more flexible and efficient than the existing extensions of neutrosophic sets when discussing the distance between multiple objects. In this regard, he not only gives ten distance measures for comparing $(m,a,n)$-fuzzy neutrosophic sets but also apply them in pattern classifications and multi-criteria decisions. Moreover, he provides some numerical examples to show the practical use and scientific applications of these distance measures by taking into account the classifying materials and investment problems. By utilizing the comparative analysis and graphical interpretations, he demonstrates the advantage and effectiveness of the proposed measures.

In Contribution 8, “Fuzzy Graphic Binary Matroid Approach to Power Grid Communication Network Analysis”, the authors Jing Li, Buvaneswari Rangasamy, Saranya Shanmugavel and Aysha Khan give the conditions for fuzzy graphic binary matroids, and also propose a systematic method for analyzing the reliability of the power grid network. This framework ensures the identification of critical circuits and enhances the network performance and stability. The investigation presented in this paper exhibits that matroid-based analysis offers a structured approach to network reliability assessment.

In Contribution 9, “Belief Entropy-Based MAGDM Algorithm Under Double Hierarchy Quantum-like Bayesian Networks and Its Application to Wastewater Reuse”, Juxiang Wang, Yaping Li, Xin Wang and Janjun Wang applied a multi-attribute quantum group decision-making method based on PLTS (probabilistic linguistic term set) and QLBN (quantum-like Bayesian network) in order to select the optimal wastewater reuse alternative. Among others, they proposed a new method for solving the quantum interference term. They are able to explain the uncertain belief state of the DM (decision maker) by applying quantum probability theory to solve the multi-attribute group decision-making problem. They also provide a new approach for multi-attribute group decision-making. The model is applied to the case of selecting the wastewater reuse alternative in Hefei City, and the comparative analysis demonstrates that the method is effective in simulating the disagreement, contradiction, and unpredictability inherent in the human decision-making process.

In Contribution 10, “m-Polar Picture Fuzzy Bi-Ideals and Their Applications in Semigroups”, the authors Warud Nakkhasen, Atthchai Chada and Teerapan Jodnok, utilizing the concept of m-PPFSs (m-polar picture fuzzy sets) for the study of semigroups, are able to introduce and explore fundamental algebraic structures such as m-PPFSubs (m-polar picture fuzzy subsemigroups), m-PPFLs (m-polar picture fuzzy left ideals), m-PPFRs (m-polar picture fuzzy right ideals), m-PPFIs (m-polar picture fuzzy ideals), m-PPFBs (m-polar picture fuzzy bi-ideals), and m-PPFGBs (m-polar picture fuzzy generalized bi-ideals) in semigroups. In this regard, they obtained several characterizations and fundamental properties of such sets. Furthermore, they studied the relationships among various m-PPFIs in regular semigroups. They also characterized the regular semigroups by their m-PPFSubs, m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs.

In Contribution 11, “Double-Framed Bipolar Fuzzy Soft Sets and Algorithmic Approaches with Symmetry for Multi-Criteria Decision-Making Under Uncertainty”, Shadya M. Mershkhan and Baravan A. Asaad define and analyze the fundamental concepts and operations of double-framed bipolar fuzzy soft sets (DFBFSSs). They also provide numerical examples in order to show the practicality and applicability of DFBFSSs in complex decision-making and information modeling scenarios. Indeed, the authors in this chapter establish a formally symmetric extension of bipolar fuzzy soft set theory, introducing a nontrivial automorphism-invariant structure that is not captured by existing fuzzy or bipolar models.

In Contribution 12, “On the Symmetric Structure of Neutrosophic Rhotrices: Index Characterization, Invertibility, and Decision-Making Applications”, Paulraj Gnanachandra, Saeid Jafari, Ananda Priya Baskar and Cheirmaraj Arunthathi investigated the structure of invertible refined neutrosophic rhotrices, thereby extending the classical theory of rhotrices into the domain of refined neutrosophics. Moreover, they not only analyze the index structures of rhotrices of small orders, which establishes the general characterization of the rhotrix index set and reveals an inherent symmetry of the rhotrix structure under the interchange of indices, but also propose a rhotrix-based representation of binary relations, which shows the broader applicability of refined neutrosophic rhotrices in modeling relational and algebraic structures. To demonstrate the practical relevance of the developed framework, the authors apply a refined neutrosophic decision-making model to the evaluation of selected women empowerment schemes. In effect, the proposed model provides a systematic mechanism for multi-criteria assessment and ranking of policy interventions.