Martin Kalina; Juraj Kalafut
Volume 23, Issue 5 , September and October 2026, Pages 1-7
Abstract
The paper deals with smooth uninorms and smooth nullnorms on finite lattices. These operations were first introduced on the unit interval and later generalized to bounded lattices. One of the most important and most studied properties is continuity. However, when working on finite lattices (finite structures, ...
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The paper deals with smooth uninorms and smooth nullnorms on finite lattices. These operations were first introduced on the unit interval and later generalized to bounded lattices. One of the most important and most studied properties is continuity. However, when working on finite lattices (finite structures, in general), the topological continuity is not very meaningful. For this reason, Mayor and Torrens (2005) introduced several notions that replace continuity. Smoothness and divisibility are among them. We study smooth nullnorms on finite lattices looking for conditions under which, when the nullnorm is defined on a lattice L, it is possible to extend the smooth nullnorm to a smooth uninorm defined on the horizontal sum of L and a finite chain. We provide necessary and sufficient conditions for such an extension to yield a smooth uninorm. Finally, we give a necessary and sufficient condition under which a smooth uninorm is also a divisible uninorm.
Optimization
Majid Darehmiraki; Madineh Farnam
Volume 23, Issue 5 , September and October 2026, Pages 9-36
Abstract
This paper introduces a Hybrid Neuro-Fuzzy-Evolutionary (HNFE) framework for multi-objective robust optimization under granular uncertainty. The framework achieves a novel synthesis of three computational paradigms: granular calculus for handling fuzzy uncertainty, evolutionary algorithms for global ...
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This paper introduces a Hybrid Neuro-Fuzzy-Evolutionary (HNFE) framework for multi-objective robust optimization under granular uncertainty. The framework achieves a novel synthesis of three computational paradigms: granular calculus for handling fuzzy uncertainty, evolutionary algorithms for global exploration, and recurrent neural networks for local refinement. Unlike existing methods that rely on a single paradigm, our tripartite architecture enables simultaneous handling of multiple uncertainty levels while maintaining both global exploration capabilities and local convergence properties.The core theoretical contribution is the formulation of Granular Karush-Kuhn-Tucker (KKT) conditions, which provide a foundational optimality framework for fuzzy multi-objective optimization through α-cut decomposition and weighted aggregation. The HNFE framework operationalizes this through three components: a granular reformulation module that transforms fuzzy problems into crisp multi-objective formulations; an enhanced evolutionary engine with granular non-dominated sorting; and a neuro-dynamic refinement system based on recurrent neural networks that ensures local convergence to granularly optimal solutions.Comprehensive theoretical analysis establishes the framework's global convergence to Pareto-optimal solutions under mild regularity conditions. The framework's robustness across varying uncertainty levels and problem complexities, coupled with its adaptive parameter control, makes it suitable for real-world applications in finance, manufacturing, and complex systems engineering. Extensive numerical experiments demonstrate superior performance in hypervolume, generational distance, and solution diversity compared to state-of-the-art methods, with statistical significance tests confirming these advantages.This work represents a significant advance in fuzzy optimization by providing a unified framework that seamlessly integrates global exploration, local refinement, and rigorous uncertainty handling, establishing new standards for computational intelligence under uncertainty.
Fuzzy Math
Ping Yu; Waqar Ul Hassan; Abeeha Mishal; Aseel Smerat; Azmat Ullah Khan Niazi; Naveed Iqbal
Volume 23, Issue 5 , September and October 2026, Pages 37-66
Abstract
This paper addresses the consensus problem in multi-robot systems subject to time delays, unknown nonlinearities, and cyber-physical attacks. To effectively handle uncertainties and mitigate the adverse effects of malicious disruptions, a Type-3 fuzzy logic system is introduced to construct a robust ...
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This paper addresses the consensus problem in multi-robot systems subject to time delays, unknown nonlinearities, and cyber-physical attacks. To effectively handle uncertainties and mitigate the adverse effects of malicious disruptions, a Type-3 fuzzy logic system is introduced to construct a robust hybrid control framework. The considered network is exposed to sensor attacks, actuator attacks, and Byzantine disruptions, where particular emphasis is placed on mitigating sensor and actuator attacks.The proposed controller is designed to achieve two main objectives: minimizing the consensus error among robots and maintaining balanced update behavior during communication processes. Further more, a constrained control strategy based on Model Predictive Control (MPC) is incorporated to optimize resource utilization and enhance system security. By embedding security constraints into the MPC formulation, the proposed approach effectively reduces the influence of sensor attacks on the networked robots. To validate the effectiveness of the developed framework, the Duffing-Holmes chaotic system is employed as a benchmark model, demonstrating improved performance under communication delays and cyber-physical disturbances. In addition, Lyapunov-Krasovskii functional analysis combined with Linear Matrix Inequalities (LMIs) is utilized to establish the convergence and stability conditions of the multi-robot network. Finally, extensive simulation results are provided to verify the theoretical findings and demonstrate the robustness and effectiveness of the proposed strategy under attack scenarios and time-delay conditions.
Fuzzy Math
Meenakshi A; Dhanushiya S; H. Rashmanlou; B. Pourhassan; F. Mofidnakhaei
Volume 23, Issue 5 , September and October 2026, Pages 67-86
Abstract
In the modern digital era, the level of security is much dependent on the factors of reliability and robustness. Domination theory provides a powerful mathematical tool for the purpose of controlling and monitoring access among nodes in a fuzzy network. Network models in the form of fuzzy graphs have ...
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In the modern digital era, the level of security is much dependent on the factors of reliability and robustness. Domination theory provides a powerful mathematical tool for the purpose of controlling and monitoring access among nodes in a fuzzy network. Network models in the form of fuzzy graphs have become important in recent times in view of providing significant development related to lightweight cryptographic systems as they can handle issues related to uncertainty and partial membership effectively. Polynomials are fundamental in cryptography because they are the basis of many algebraic systems that are used in encryption and, most importantly, decryption techniques. Polynomials are employed explicitly or implicitly in all cryptosystems, creating a solid platform upon which communication occurs. This study aims to develop a fuzzy graph network, describing an encrypted version of secret information. Mathematical modeling of the fuzzy graph network is derived based on a multi-prime RSA system. Considering the given text, it creates a polynomial as part of the main key of a cryptosystem. Algorithms for encryption and decryption are developed and explained with the aim of showing the effectiveness of the proposed method. Also, the security level of the developed fuzzy graph network is improved with the incorporation of the domination parameters and the polynomial transformations.
Fuzzy modelling
Long Kim Ngo; Binh Van Ngo
Volume 23, Issue 5 , September and October 2026, Pages 87-105
Abstract
This study develops a decomposed fuzzy controller for a self-balancing two-wheel robot, with the learning rates of the fuzzy modules optimized by a modified Jaya algorithm. The controller is structured into two interacting parts-one regulating the body tilt and the other compensating for wheel dynamics—to ...
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This study develops a decomposed fuzzy controller for a self-balancing two-wheel robot, with the learning rates of the fuzzy modules optimized by a modified Jaya algorithm. The controller is structured into two interacting parts-one regulating the body tilt and the other compensating for wheel dynamics—to improve the handling of nonlinearities and reduce the computational load during real-time operation. The modified Jaya algorithm is incorporated to refine the adaptation process, enabling faster convergence and more reliable parameter updates under varying operating conditions. The proposed method is assessed through both simulation and hardware experiments. The experimental platform consists of an Arduino Mega 2560 board, an L298N motor driver, and an MPU6050 sensor, which provides real-time attitude measurements for feedback control. Results show that the optimized fuzzy controller stabilizes the robot more quickly and maintains balance more effectively in the presence of disturbances compared with a conventional fuzzy design without optimization. These outcomes confirm the suitability of the modified Jaya–based optimization in enhancing the performance of decomposed fuzzy control for self-balancing robotic systems.
M. B. Dowlatshahi; S. Beiranvand
Volume 23, Issue 5 , September and October 2026, Pages 107-127
Abstract
Large Language Models (LLMs) have demonstrated remarkable capabilities in natural language understanding and generation; however, they continue to suffer from critical limitations, including predictive uncertainty, factual hallucination, and lack of interpretability. Fuzzy logic, with its inherent capacity ...
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Large Language Models (LLMs) have demonstrated remarkable capabilities in natural language understanding and generation; however, they continue to suffer from critical limitations, including predictive uncertainty, factual hallucination, and lack of interpretability. Fuzzy logic, with its inherent capacity to model graded membership and reason under imprecision, has emerged as a promising complementary paradigm to mitigate these shortcomings. Despite the rapidly growing body of literature, a systematic review that comprehensively synthesizes integration methodologies, empirical gains, and deployment challenges in fuzzy-LLM systems remains absent. This study conducts a systematic literature review of 70 peer-reviewed studies published between November 2022 and June 2026, organized around five research questions: (1) theoretical frameworks for fuzzy-LLM integration, (2) improvements in reasoning performance, (3) allucination mitigation mechanisms, (4) fuzzy evaluation metrics, and (5) practical implementation obstacles. Our analysis reveals that fuzzy rule-based systems dominate the integration landscape, comprising 40 studies (57.1%), whereas fuzzy attention mechanisms remain critically underexplored, with only 3 studies (4.3%). Uncertainty reasoning constitutes the predominant application domain, addressed in 54 studies (77.1%), while causal reasoning remains entirely uninvestigated. Empirical findings indicate that fuzzy mechanisms yield measurable accuracy improvements between 1.5% and 3.0%, alongside substantial hallucination reduction. Among mitigation strategies, fuzzy refinement emerges as both the most frequently adopted and effective approach, reported in 19 studies (27.1%). Nevertheless, large-scale deployment continues to face persistent barriers, particularly computational overhead and rule explosion, partially mitigated through Type-2 fuzzy systems and differentiable fuzzy layers. This review identifies fuzzy attention, standardized benchmarks, and causal reasoning as pivotal directions for future inquiry, offering a foundational reference for researchers and practitioners developing reliable, interpretable, and uncertainty-aware LLM-based systems.
Fuzzy Math
Hadi Yavari; Seyed Naser Hosseini
Volume 23, Issue 5 , September and October 2026, Pages 129-145
Abstract
In this paper, we investigate the existence of colimits in the category of O-structural spaces, denoted by OSS. We show that OSS admits all small colimits. Special types of colimits, such as coproducts, coequalizers, and pushouts, are studied in detail. Based on these results, we explicitly describe ...
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In this paper, we investigate the existence of colimits in the category of O-structural spaces, denoted by OSS. We show that OSS admits all small colimits. Special types of colimits, such as coproducts, coequalizers, and pushouts, are studied in detail. Based on these results, we explicitly describe the O-structures of these colimits in several topological structures that are O-structural spaces, including the categories of fuzzy topological spaces in the sense of Chang, intuitionistic fuzzy topological spaces in the sense of Coker, fuzzy topological spaces in the sense of Chakraborty and Ahsanullah, soft and fuzzy soft topological spaces, neutrosophic topological spaces, hypersoft topological spaces and various types of topological molecular lattice structures. In particular, we show how these colimits can be recovered via the underlying O-structural framework. The results shed light on the internal structure of these categories and offer categorical tools for further analysis of generalized topological structures.
Fuzzy Math
M. Shamsizadeh; Kh. Abolpor; H. Bordbar
Volume 23, Issue 5 , September and October 2026, Pages 147-156
Abstract
In this paper, we introduce the concept of vector simulation for vector general fuzzy automata (VGFAs). An algorithm is developed to decide vector simulation between two VGFAs in polynomial time. We prove that the union of vector simulations is again a vector simulation. For any VGFA, we construct its ...
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In this paper, we introduce the concept of vector simulation for vector general fuzzy automata (VGFAs). An algorithm is developed to decide vector simulation between two VGFAs in polynomial time. We prove that the union of vector simulations is again a vector simulation. For any VGFA, we construct its quotient automaton and show that the identity relation is the only vector simulation on the quotient, which implies its minimality.
Fuzzy modelling
L. Jing; H. Zhang
Volume 23, Issue 5 , September and October 2026, Pages 157-183
Abstract
This research presents the Tail-Risk-Aware Fuzzy Distributional Reinforcement Learning (T-FDRL) framework, an innovative method for robust portfolio management in dynamic, non-stationary financial markets. By combining Fuzzy Markov Decision Processes, distributional reinforcement learning, and Wasserstein-based ...
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This research presents the Tail-Risk-Aware Fuzzy Distributional Reinforcement Learning (T-FDRL) framework, an innovative method for robust portfolio management in dynamic, non-stationary financial markets. By combining Fuzzy Markov Decision Processes, distributional reinforcement learning, and Wasserstein-based robust optimization, T-FDRL tackles uncertainty in market signals, heavy-tailed return distributions, and shifts in market dynamics. It employs fuzzy sets to model vague market data, captures the complete return distribution, and optimizes portfolio allocations to maximize returns while minimizing tail risks using Conditional Value at Risk (CVaR). Tested on historical data from Apple, Amazon, Google, and Microsoft stocks spanning 2016 to 2025, T-FDRL delivers a cumulative return of 1544.21%, an annualized return of 21.22%, and a Sharpe Ratio of 1.60, surpassing benchmarks like Deep Q-Networks (272.38%), Fuzzy Reinforcement Learning (1120.22%), Value-Based DQN (1067.86%), and Robust Markov Decision Processes (652.63%). With low volatility (13.23%) and a CVaR 5% of -1.64%, T-FDRL excels in risk management, especially during turbulent periods such as 2020 and 2022. This framework provides a resilient, adaptive approach to portfolio optimization, offering valuable insights for investors and policymakers navigating complex financial landscapes.
Fuzzy Scheduling
B. Goyal; D. Gupta; R. Rani
Volume 23, Issue 5 , September and October 2026, Pages 185-203
Abstract
This study addresses a two-machine flow shop scheduling problem with uncertain processing durations represented as trapezoidal fuzzy numbers, which are converted to crisp values using Yager’s ranking index. Job-block formation is introduced to reduce sequencing complexity and suppress waiting on ...
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This study addresses a two-machine flow shop scheduling problem with uncertain processing durations represented as trapezoidal fuzzy numbers, which are converted to crisp values using Yager’s ranking index. Job-block formation is introduced to reduce sequencing complexity and suppress waiting on the second machine, and new theorems establish how block aggregation influences waiting-time behavior. A structured heuristic generates the sequence with minimum total waiting time of jobs.Extensive computational experiments on specially structured instances demonstrate consistent improvements in waiting time. Statistical analysis, including paired t-tests and Wilcoxon signed-rank tests, confirms that the proposed heuristic achieves significant reductions compared to classical rules and recent heuristics across multiple job sizes (p < 0.01), supporting the superiority of the method. The results highlight the practical effectiveness of integrating fuzzy modelling, crisp conversion, and block-based sequencing, providing a theoretically grounded and empirically validated approach for modern production environments with imprecise processing times.
