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27 pages, 2227 KB  
Article
Application of a Reinforcement Learning-Based Improved Genetic Algorithm in Flexible Job-Shop Scheduling Problems
by Guoli Zhao, Jiansha Lu, Gangqiang Liu, Weini Weng and Ning Wang
Mathematics 2026, 14(2), 307; https://doi.org/10.3390/math14020307 - 15 Jan 2026
Abstract
This paper addresses the limitations of genetic algorithms in solving the Flexible Job-Shop Scheduling Problem (FJSP) including slow convergence, susceptibility to local optima, and sensitivity to parameter settings. The paper proposes an Improved Genetic Algorithm based on Reinforcement Learning (IGARL). First, a hybrid [...] Read more.
This paper addresses the limitations of genetic algorithms in solving the Flexible Job-Shop Scheduling Problem (FJSP) including slow convergence, susceptibility to local optima, and sensitivity to parameter settings. The paper proposes an Improved Genetic Algorithm based on Reinforcement Learning (IGARL). First, a hybrid population selection mechanism that combines the Queen Bee Mating Flight (QBMF) strategy with the Tournament Selection (TS) method is introduced. This mechanism significantly accelerates convergence by optimizing the population structure. Second, a dynamic population update strategy based on tunnel vision, termed the Solution Space Diversity Awakening (SSDA) strategy, is developed. When the population becomes trapped in local optima, this strategy intelligently triggers random perturbations and introduces high-potential individuals to enhance the algorithm’s ability to escape local optima and promote population diversity. Third, a novel multi-Q-table reinforcement learning framework is embedded within the iterative process to dynamically adjust key genetic algorithm parameters (such as selection, mutation, and crossover rates) and enable multi-dimensional performance evaluation, thereby effectively guiding the search toward better solutions. Experimental results demonstrate that the IGARL algorithm achieves a 10% to 60% improvement in convergence speed on Brandimarte benchmark instances, with solution quality significantly surpassing that of the basic genetic algorithm. Moreover, the fluctuation of the average optimal solution remains within 20%, indicating strong stability and robustness. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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19 pages, 627 KB  
Article
Stress-Testing Slovenian SME Resilience: A Scenario Model Calibrated on South African Evidence
by Klavdij Logožar and Carin Loubser-Strydom
Sustainability 2026, 18(2), 828; https://doi.org/10.3390/su18020828 - 14 Jan 2026
Abstract
Small and medium-sized enterprises (SMEs) play a central role in employment and regional economic development, yet they are highly vulnerable to shocks such as pandemics, energy price spikes, and supply chain disruptions. Scenario modelling, stress testing, and digital twins are used to assess [...] Read more.
Small and medium-sized enterprises (SMEs) play a central role in employment and regional economic development, yet they are highly vulnerable to shocks such as pandemics, energy price spikes, and supply chain disruptions. Scenario modelling, stress testing, and digital twins are used to assess resilience, yet most applications focus on large firms in single-country settings. This article develops a model to stress test the resilience of Slovenian SMEs, calibrated with parameters and mechanisms derived from South African SME resilience studies. A system dynamics model with stocks for cash, inventory, and productive capacity is specified and subjected to demand, supply, financial, and compound shock scenarios, with and without resilience measures such as liquidity buffers, customer and supplier diversification, and basic digital planning capabilities. Results indicate non-linear tipping points where small reductions in liquidity sharply increase the likelihood of distress, and show that combinations of liquidity, diversification, and collaborative supply chain practices reduce the depth and duration of output losses. The study demonstrates how evidence from an African context can inform resilience strategies in a small European economy and provides a transparent, portable modelling architecture that can be adapted to other settings. Implications are discussed for SME managers and for policies supporting sustainable, resilient enterprise ecosystems. Full article
(This article belongs to the Special Issue Advancing Innovation and Sustainability in SMEs and Entrepreneurship)
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23 pages, 1377 KB  
Review
Immunomodulatory Effects of Lidocaine: Mechanisms of Actions and Therapeutic Applications
by Jianwei Wu, Quanfu Chen, Zhiling He, Bin Yang, Zhenhua Dai and Feifei Qiu
Pharmaceuticals 2026, 19(1), 134; https://doi.org/10.3390/ph19010134 - 12 Jan 2026
Viewed by 128
Abstract
Lidocaine, an amide-type regional anesthetic, has been an important medication in the field of anesthesia since its clinical approval. Recently, lidocaine has emerged as a powerful immunomodulatory agent beyond its classical anesthetic properties. This review has summarized the recent basic and clinical studies [...] Read more.
Lidocaine, an amide-type regional anesthetic, has been an important medication in the field of anesthesia since its clinical approval. Recently, lidocaine has emerged as a powerful immunomodulatory agent beyond its classical anesthetic properties. This review has summarized the recent basic and clinical studies with sufficient evidence on the multifaceted effects of lidocaine on both innate and adaptive immune cells, including macrophages, neutrophils, eosinophils, basophils, natural killer (NK) cells, mast cells, dendritic cells (DCs), monocytes, and T lymphocytes. We have also detailed how lidocaine affects critical cellular processes, such as cellular polarization, cytokine production, phagocytosis, and apoptosis, through multiple signaling pathways, including NF-κB, TLR4/p38 MAPK, voltage-sensitive sodium channels, HIF1α, TGF-β/Smad3, AMPK-SOCS3, TBK1-IRF7, and G protein-coupled receptors. These immunoregulatory effects of lidocaine are dependent on its concentration, duration of action, and the microenvironment. The immunomodulatory actions of lidocaine may contribute to its potential therapeutic value in various settings of diseases, such as cancer, sepsis, acute lung injury, asthma, organ transplantation, ischemia–reperfusion injury (IRI), and diabetes. We propose that lidocaine can be repurposed as an immunomodulator for treating immune-mediated inflammatory diseases. However, future research should define optimal dosing strategies, validate its mechanisms of action in clinical trials, and explore its novel clinical applications as a complementary immunotherapy. Full article
(This article belongs to the Section Pharmacology)
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18 pages, 8082 KB  
Article
Application of Attention Mechanism Models in the Identification of Oil–Water Two-Phase Flow Patterns
by Qiang Chen, Haimin Guo, Xiaodong Wang, Yuqing Guo, Jie Liu, Ao Li, Yongtuo Sun and Dudu Wang
Processes 2026, 14(2), 265; https://doi.org/10.3390/pr14020265 - 12 Jan 2026
Viewed by 161
Abstract
Accurate identification of oil–water two-phase flow patterns is essential for ensuring the safety and operational efficiency of oil and gas extraction systems. While traditional methods using empirical models and sensor technologies have provided basic insights, they often struggle to capture the nonlinear features [...] Read more.
Accurate identification of oil–water two-phase flow patterns is essential for ensuring the safety and operational efficiency of oil and gas extraction systems. While traditional methods using empirical models and sensor technologies have provided basic insights, they often struggle to capture the nonlinear features of complex operational conditions. To address the challenge of data scarcity commonly found in experimental settings, this study employs a data augmentation strategy that combines the Synthetic Minority Over-sampling Technique (SMOTE) with Gaussian noise injection, effectively expanding the feature space from 60 original experimental nodes. Next, a physics-constrained attention mechanism model was developed that incorporates a physical constraint matrix to effectively mask irrelevant feature interactions. Experimental results show that while the standard attention model (83.88%) and the baseline BP neural network (84.25%) have limitations in generalizing to complex regimes, the proposed physics-constrained model achieves a peak test accuracy of 96.62%. Importantly, the model demonstrates exceptional robustness in identifying complex transition regions—specifically Dispersed Oil-in-Water (DO/W) flows—where it improved recall rates by about 24.6% compared to baselines. Additionally, visualization of attention scores confirms that the distribution of attention weights aligns closely with fluid-dynamic mechanisms—favoring inclination for stratified flows and flow rate for turbulence-dominated dispersions—thus validating the model’s interpretability. This research offers a novel, interpretable approach for modeling dynamic feature interactions in multiphase flows and provides valuable insights for intelligent oilfield development. Full article
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16 pages, 336 KB  
Article
An Exact Algorithm for Counting the Number of Independent Sets of a Graph
by Guillermo De Ita Luna, J. Raymundo Marcial-Romero, Pedro Bello López and Meliza Contreras González
Mathematics 2026, 14(2), 275; https://doi.org/10.3390/math14020275 - 12 Jan 2026
Viewed by 82
Abstract
For a graph G of a degree greater than or equal to 3, counting the number of independent sets (denoted as i(G)) is a classical #P-complete problem. Here, we establish a new worst-case upper bound time complexity for computing [...] Read more.
For a graph G of a degree greater than or equal to 3, counting the number of independent sets (denoted as i(G)) is a classical #P-complete problem. Here, we establish a new worst-case upper bound time complexity for computing i(G) for any non-constraint undirected graph. Our proposal applies the vertex division rule i(G)=i(G{x})+i(GN[x]) over a vertex x which satisfies some conditions, and considers cactus and outerplanar graphs as basic subgraphs.Our algorithm establishes a leading worst-case upper bound of O*(1.2321n), where n is the number of vertices in the graph and O* omits polynomial terms in n. Full article
(This article belongs to the Special Issue Computational Algorithms and Models for Graph Problems)
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18 pages, 4452 KB  
Article
Structural Basis of Chemokine CXCL8 Monomer and Dimer Binding to Chondroitin Sulfate: Insights into Specificity and Plasticity
by Bryon P. Mahler, Balaji Nagarajan, Nehru Viji Sankaranarayanan, Prem Raj B. Joseph, Umesh R. Desai and Krishna Rajarathnam
Biomolecules 2026, 16(1), 124; https://doi.org/10.3390/biom16010124 - 12 Jan 2026
Viewed by 131
Abstract
Chemokines play a central role in orchestrating neutrophil recruitment from the bloodstream and determining their effector functions at sites of infection. Chemokine activity is determined by three key properties: reversible monomer–dimer equilibrium, binding to glycosaminoglycans (GAGs), and signaling through the GPCR class of [...] Read more.
Chemokines play a central role in orchestrating neutrophil recruitment from the bloodstream and determining their effector functions at sites of infection. Chemokine activity is determined by three key properties: reversible monomer–dimer equilibrium, binding to glycosaminoglycans (GAGs), and signaling through the GPCR class of receptors CXCR1 and CXCR2. In this study, we investigated the structural basis of CXCL8 monomer and dimer binding to GAG chondroitin sulfate (CS) using nuclear magnetic resonance (NMR) spectroscopy, docking, and molecular dynamics (MD) measurements. Our studies reveal that both the monomer and dimer use essentially the same set of basic residues for binding, that the interface is extensive, that the dimer is the high-affinity CS ligand, and that the CS-binding residues form a contiguous surface within a monomer. Several of these residues also participate in receptor interactions, suggesting that CS-bound CXCL8 is likely impaired in its ability to bind receptors. Notably, we observe that the same basic residues are involved in binding CS and heparin/heparan sulfate, even though these GAGs differ in backbone structures and sulfation patterns. We conclude that the strategic distribution and topology of basic residues on the CXCL8 scaffold enable engagement with diverse GAG structures, which likely allows fine-tuning receptor signaling to regulate neutrophil trafficking and effector functions. Full article
(This article belongs to the Special Issue The Role of Glycosaminoglycans and Proteoglycans in Human Disease)
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46 pages, 1489 KB  
Article
Active Inference and Functional Parametrisation: Differential Flatness and Smooth Random Realisation
by Hugues Mounier, Thomas Parr and Karl Friston
Entropy 2026, 28(1), 87; https://doi.org/10.3390/e28010087 - 11 Jan 2026
Viewed by 90
Abstract
This paper is a first attempt to marry constructive nonlinear control theory techniques with active inference. Specifically, we are interested in the relationship between differential flatness and the design of generative models for use in control settings. We place specific emphasis on the [...] Read more.
This paper is a first attempt to marry constructive nonlinear control theory techniques with active inference. Specifically, we are interested in the relationship between differential flatness and the design of generative models for use in control settings. We place specific emphasis on the pathwise properties of differentially flat systems that inherit from their definition in terms of successive temporal derivatives and relate this to the use of generalised coordinates of motion in formulating continuous-time generative models in active inference. To illustrate the basic concepts, we appeal to the example of oculomotor control. Full article
(This article belongs to the Section Complexity)
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30 pages, 6245 KB  
Article
Learning to Engineer: Integrating Robotics-Centred Project-Based Learning in Early Undergraduate Education
by Pg Emeroylariffion Abas
Educ. Sci. 2026, 16(1), 105; https://doi.org/10.3390/educsci16010105 - 10 Jan 2026
Viewed by 200
Abstract
Engineering programmes have been giving more weight to experiential learning, largely because many students still find it difficult to see how classroom theory connects to the work that engineers handle on the ground. With this in mind, a robotics-centred Project-based Learning (PBL) module [...] Read more.
Engineering programmes have been giving more weight to experiential learning, largely because many students still find it difficult to see how classroom theory connects to the work that engineers handle on the ground. With this in mind, a robotics-centred Project-based Learning (PBL) module was introduced to first-year general engineering students as part of the faculty’s engineering spine. The module asks students to design, build, and program small autonomous robots capable of navigating and competing in a set arena. Even a simple task of this kind draws together multiple strands of engineering. Students shift between sketching mechanical layouts, wiring basic circuits, writing code, testing prototypes, and negotiating the usual challenges that arise when several people share responsibility for the same piece of hardware. To explore how students learned through the module, a mixed-methods evaluation was carried out using survey responses alongside reflective pieces written by the students themselves. Certain patterns appeared repeatedly. Many students felt that their technical skills had grown, particularly in breaking down a messy problem into smaller, more workable components. Teamwork also surfaced as a prominent theme. Groups often had to sort out issues such as a robot veering off course due to a misaligned sensor or a block of code producing unpredictable behaviour. These issues were undoubtedly challenging for the students, but they also had a certain pedagogical flavour, with many students describing them as a source of frustration as well as a learning opportunity. Later iterations of the module may benefit from more targeted support at key stages. Despite the many challenges, robotics has been shown to be an attractive way for students to step into engineering practice. The project helped them build technical capability, but it also encouraged habits that matter just as much in real work, such as planning, communicating clearly, and returning to a problem until it behaves as expected. Taken together, the experience offers useful guidance for curriculum designers seeking to create early learning environments that feel authentic and manageable and for motivating students who are just beginning their engineering journey. Full article
(This article belongs to the Special Issue Engineering Education: Innovation Through Integration)
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33 pages, 1529 KB  
Article
An SQL Query Description Problem with AI Assistance for an SQL Programming Learning Assistant System
by Ni Wayan Wardani, Nobuo Funabiki, Htoo Htoo Sandi Kyaw, Zihao Zhu, I Nyoman Darma Kotama, Putu Sugiartawan and I Nyoman Agus Suarya Putra
Information 2026, 17(1), 65; https://doi.org/10.3390/info17010065 - 9 Jan 2026
Viewed by 224
Abstract
Today, relational databases are widely used in information systems. SQL (structured query language) is taught extensively in universities and professional schools across the globe as a programming language for its data management and accesses. Previously, we have studied a web-based programming learning assistant [...] Read more.
Today, relational databases are widely used in information systems. SQL (structured query language) is taught extensively in universities and professional schools across the globe as a programming language for its data management and accesses. Previously, we have studied a web-based programming learning assistant system (PLAS) to help novice students learn popular programming languages by themselves through solving various types of exercises. For SQL programming, we have implemented the grammar-concept understanding problem (GUP) and the comment insertion problem (CIP) for its initial studies. In this paper, we propose an SQL Query Description Problem (SDP) as a new exercise type for describing the SQL query to a specified request in a MySQL database system. To reduce teachers’ preparation workloads, we integrate a generative AI-assisted SQL query generator to automatically generate a new SDP instance with a given dataset. An SDP instance consists of a table, a set of questions and corresponding queries. Answer correctness is determined by enhanced string matching against an answer module that includes multiple semantically equivalent canonical queries. For evaluation, we generated 11 SDP instances on basic topics using the generator, where we found that Gemini 3.0 Pro exhibited higher pedagogical consistency compared to ChatGPT-5.0, achieving perfect scores in Sensibleness, Topicality, and Readiness metrics. Then, we assigned the generated instances to 32 undergraduate students at the Indonesian Institute of Business and Technology (INSTIKI). The results showed an average correct answer rate of 95.2% and a mean SUS score of 78, which demonstrates strong initial student performance and system acceptance. Full article
(This article belongs to the Special Issue Generative AI Transformations in Industrial and Societal Applications)
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14 pages, 1382 KB  
Article
Synthesis and Properties of Polyarylene Ether Nitrile and Polyphenylene Sulfone Copolymers
by Azamat Zhansitov, Kamila Shakhmurzova, Zhanna Kurdanova, Azamat Slonov, Ilya Borisov, Elena Rzhevskaya, Ismel Musov, Artur Baykaziev and Svetlana Khashirova
Polymers 2026, 18(2), 159; https://doi.org/10.3390/polym18020159 - 7 Jan 2026
Viewed by 188
Abstract
Copolymers of polyphenylene sulfone and polyarylene ether nitrile were synthesized using nucleophilic polycondensation. 2,6-difluorobenzonitrile (DFBN), 4,4′-dihydroxybiphenyl, and 4,4′-dichlorodiphenyl sulfone were used as monomers. The structure of the obtained copolymers was confirmed by means of IR spectroscopy, and their solubility in various solvents was [...] Read more.
Copolymers of polyphenylene sulfone and polyarylene ether nitrile were synthesized using nucleophilic polycondensation. 2,6-difluorobenzonitrile (DFBN), 4,4′-dihydroxybiphenyl, and 4,4′-dichlorodiphenyl sulfone were used as monomers. The structure of the obtained copolymers was confirmed by means of IR spectroscopy, and their solubility in various solvents was studied. Thermal properties were studied using differential scanning calorimetry (DSC) and thermogravimetric analysis, as well as a set of basic mechanical properties. It was found that both thermal stability and glass transition temperature are virtually independent of the copolymer composition, while samples with a DFBN monomer content of more than 75% exhibit a melting peak in the region of 357 °C on the DSC curves, indicating an increase in the degree of crystallinity, accompanied by a deterioration in the solubility of these polymers. With increasing DFBN content, a uniform increase in elastic modulus is observed, and both bending and tensile strength increase significantly. However, the introduction of DFBN segments into the polyphenylene sulfone structure leads to a decrease in impact strength. Full article
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51 pages, 4344 KB  
Review
Mechanistic Pathways and Product Selectivity in Pyrolysis of PE, PP and PVC: A Foundation for Applied Chemistry in Europe
by Tim Tetičkovič, Dušan Klinar, Klavdija Rižnar and Darja Pečar
Molecules 2026, 31(2), 202; https://doi.org/10.3390/molecules31020202 - 6 Jan 2026
Viewed by 444
Abstract
Plastic streams dominated by polyethylene (PE) including PE HD/MD (High Density/Medium Density) and PE LD/LLD (Low Density/Linear Low Density), polypropylene (PP), and polyvinyl chloride (PVC) across Europe demand a design framework that links synthesis with end of life reactivity, supporting circular economic goals [...] Read more.
Plastic streams dominated by polyethylene (PE) including PE HD/MD (High Density/Medium Density) and PE LD/LLD (Low Density/Linear Low Density), polypropylene (PP), and polyvinyl chloride (PVC) across Europe demand a design framework that links synthesis with end of life reactivity, supporting circular economic goals and European Union waste management targets. This work integrates polymerization derived chain architecture and depolymerization mechanisms to guide selective valorization of commercial plastic wastes in the European context. Catalytic topologies such as Bronsted or Lewis acidity, framework aluminum siting, micro and mesoporosity, initiators, and strategies for process termination are evaluated under relevant variables including temperature, heating rate, vapor residence time, and pressure as encountered in industrial practice throughout Europe. The analysis demonstrates that polymer chain architecture constrains reaction pathways and attainable product profiles, while additives, catalyst residues, and contaminants in real waste streams can shift radical populations and observed selectivity under otherwise similar operating windows. For example, strong Bronsted acidity and shape selective micropores favor the formation of C2 to C4 olefins and Benzene, Toluene, and Xylene (BTX) aromatics, while weaker acidity and hierarchical porosity help preserve chain length, resulting in paraffinic oils and waxes. Increasing mesopore content shortens contact times and limits undesired secondary cracking. The use of suitable initiators lowers the energy threshold and broadens processing options, whereas diffusion management and surface passivation help reduce catalyst deactivation. In the case of PVC, continuous hydrogen chloride removal and the use of basic or redox co catalysts or ionic liquids reduce the dehydrochlorination temperature and improve fraction purity. Staged dechlorination followed by subsequent residue cracking is essential to obtain high quality output and prevent the release of harmful by products within European Union approved processes. Framing process design as a sequence that connects chain architecture, degradation chemistry, and operating windows supports mechanistically informed selection of catalysts, severity, and residence time, while recognizing that reported selectivity varies strongly with reactor configuration and feed heterogeneity and that focused comparative studies are required to validate quantitative structure to selectivity links. In European post consumer sorting chains, PS and PC are frequently handled as separate fractions or appear in residues with distinct processing routes, therefore they are not included in the polymer set analyzed here. Polystyrene and polycarbonate are outside the scope of this review because they are commonly handled as separate fractions and are typically optimized toward different product slates than the gas, oil, and wax focused pathways emphasized here. Full article
(This article belongs to the Special Issue Applied Chemistry in Europe, 2nd Edition)
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38 pages, 7206 KB  
Article
Children’s Well-Being of Physical Activity Space Design in Primary School Campus from the Perspective of Basic Psychological Needs
by Qi Song, Yixin Liu, Yihao Zhang, Min Huang, Bingjie Sun and Yuting Li
Buildings 2026, 16(1), 222; https://doi.org/10.3390/buildings16010222 - 4 Jan 2026
Viewed by 280
Abstract
The issue of children’s mental health is becoming increasingly prominent worldwide. Research indicates that insufficient physical activity constitutes a significant risk factor for various health issues. As primary school campuses serve as the primary setting for children’s physical activities, their spatial design currently [...] Read more.
The issue of children’s mental health is becoming increasingly prominent worldwide. Research indicates that insufficient physical activity constitutes a significant risk factor for various health issues. As primary school campuses serve as the primary setting for children’s physical activities, their spatial design currently fails to adequately support children’s psychological needs, thereby hindering improvements in mental well-being. This study, grounded in the theory of basic psychological needs, employs a combined approach of quantitative bibliometric analysis and qualitative content analysis to systematically investigate the relationship between children’s psychological needs and the design of physical activity spaces within school environments. The study identified six major research clusters: children’s health, self-determination theory, evaluation, physical education, user-centred design, and physical health. These reveal the multidimensional relationship between spatial design and children’s well-being. The key findings suggest that optimising facility diversity, designing progressively challenging activity zones, and implementing function-oriented colour coding can fulfil children’s fundamental psychological needs for autonomy, competence, and relatedness. These measures significantly enhance their mental well-being and overall happiness. This study extends the application of basic psychological needs theory to the field of campus spatial design, providing practical guidance for designers, educators, and policymakers. It aims to collectively advance the continuous optimisation of school sports facilities, thereby creating more favourable conditions for children’s healthy development. Full article
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27 pages, 382 KB  
Article
Single Machine Scheduling Problems: Standard Settings and Properties, Polynomially Solvable Cases, Complexity and Approximability
by Nodari Vakhania, Frank Werner and Kevin Johedan Ramírez-Fuentes
Algorithms 2026, 19(1), 38; https://doi.org/10.3390/a19010038 - 4 Jan 2026
Viewed by 157
Abstract
Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems have appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often [...] Read more.
Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems have appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often particular subjects like special objective functions were discussed or more general scheduling problems were surveyed, in which a substantial part was devoted to single machine problems. In this paper, we focus on standard settings, basic structural properties of these settings, polynomial algorithms and complexity and approximation issues, which have not been reviewed so far, and suggest some future work in this area. Full article
(This article belongs to the Special Issue 2024 and 2025 Selected Papers from Algorithms Editorial Board Members)
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39 pages, 995 KB  
Article
Multi-Granulation Variable Precision Fuzzy Rough Set Based on Generalized Fuzzy Remote Neighborhood Systems and the MADM Application Design of a Novel VIKOR Method
by Xinyu Mei and Yaoliang Xu
Symmetry 2026, 18(1), 84; https://doi.org/10.3390/sym18010084 - 3 Jan 2026
Viewed by 184
Abstract
Variable precision fuzzy rough sets (VPFRSs) and multi-granulation fuzzy rough sets (MGFRSs) are both significant extensions of rough sets. However, existing variable precision models generally lack the inclusion property, which poses potential risks in applications. Meanwhile, multi-granulation models tend to emphasize either optimistic [...] Read more.
Variable precision fuzzy rough sets (VPFRSs) and multi-granulation fuzzy rough sets (MGFRSs) are both significant extensions of rough sets. However, existing variable precision models generally lack the inclusion property, which poses potential risks in applications. Meanwhile, multi-granulation models tend to emphasize either optimistic or pessimistic scenarios but overlook compromise situations. A generalized fuzzy remote neighborhood system is a symmetric union-fuzzified form of the neighborhood system, which can extend the fuzzy rough set model to a more general framework. Moreover, semi-grouping functions eliminate the left-continuity required for grouping functions and the associativity in t-conorms, making them more suitable for information aggregation. Therefore, to overcome the limitations of existing models, we propose an optimistic (OP), pessimistic (PE), and compromise (CO) variable precision fuzzy rough set (OPCAPFRS) based on generalized fuzzy remote neighborhood systems. The semi-grouping function and its residual minus are employed in the OPCAPFRS. We discuss the basic properties of the OPCAPFRS and prove that it satisfies the generalized inclusion property (GIP). This partially addresses the issue that a VPFRS cannot fulfill the inclusion property. A novel methodology for addressing multi-attribute decision-making (MADM) problems is developed through the fusion of the proposed OPCAPFRS framework and the VIKOR technique. The proposed method is applied to the problem of selecting an optimal CPU. Subsequently, comparative experiments and a parameter analysis are conducted to validate the effectiveness and stability of the proposed method. Finally, three sets of experiments are performed to verify the reliability and robustness of the new approach. It should be noted that the new method performed ranking on a dataset containing nearly ten thousand samples, obtaining both the optimal solution and a complete ranking, thereby validating its scalability. Full article
(This article belongs to the Special Issue Symmetry and Fuzzy Set)
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18 pages, 345 KB  
Article
Generalized Interval-Valued Convexity in Fractal Geometry
by Muhammad Zakria Javed, Muhammad Uzair Awan, Dafang Zhao, Awais Gul Khan and Lorentz Jäntschi
AppliedMath 2026, 6(1), 5; https://doi.org/10.3390/appliedmath6010005 - 3 Jan 2026
Viewed by 121
Abstract
The main goal of this study is to explain the idea of generalized interval-valued (I.V) convexity on a fractal set. We first define the basic operations for a generalized interval of Rs with 0<s1 [...] Read more.
The main goal of this study is to explain the idea of generalized interval-valued (I.V) convexity on a fractal set. We first define the basic operations for a generalized interval of Rs with 0<s1. Then, we expand the idea of (I.V) Riemann integration to (I.V) local fractal integration, which sets the stage for further research. This is followed by the proof of new Jensen, Hermite, Hadamard, Pachpatte, and Fejer inequalities that are (I.V) and have to do with the generalized class of (I.V) convexity defined over the fractal domain. We furnish validation through visual and comparative approaches. Our outcomes are the refinement of many existing results, indicating that they are fruitful. In fractal settings, this is the first paper to work on (I.V) convexity and some set-valued versions of Hermite–Hadamard-type containments. Full article
(This article belongs to the Special Issue Advances in Intelligent Control for Solving Optimization Problems)
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