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Volume 13
Issue 19
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Mathematics, Volume 13, Issue 19 (October-1 2025) – 184 articles

Cover Story (view full-size image): We study the regularity properties of the unique solution of a generalized mean-field G-SDE. More precisely, we consider a generalized mean-field G-SDE with a square-integrable random initial condition, establish its first- and second-order Fréchet differentiability in the stochastic initial condition, and specify the G-SDEs of the respective Fréchet derivatives. The first- and second-order Fréchet derivatives are obtained for locally Lipschitz coefficients admitting locally Lipschitz first- and second-order Fréchet derivatives respectively. Our approach heavily relies on the Grönwall inequality, which leverages the Lipschitz continuity of the coefficients. View this paper
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24 pages, 6122 KB  
Article
A Minimal CA-Based Model Capturing Evolutionarily Relevant Features of Biological Development
by Miguel Brun-Usan, Javier de Juan García and Roberto Latorre
Mathematics 2025, 13(19), 3238; https://doi.org/10.3390/math13193238 - 9 Oct 2025
Viewed by 168
Abstract
Understanding how complex biological forms emerge and evolve remains a central question in evolutionary and developmental biology. To explore this complexity, we introduce a minimal two-dimensional, cellular automaton (CA)-based model that captures key features of biological development—such as spatial growth, self-organization, and differentiation—while [...] Read more.
Understanding how complex biological forms emerge and evolve remains a central question in evolutionary and developmental biology. To explore this complexity, we introduce a minimal two-dimensional, cellular automaton (CA)-based model that captures key features of biological development—such as spatial growth, self-organization, and differentiation—while remaining computationally tractable and evolvable. Unlike most abstract genotype–phenotype mapping models, our approach generates emergent morphological complexity through spatially explicit rule-based interactions governed by a simple genetic vector, resulting in self-organized patterns reminiscent of biological morphogenesis. Using simulations, we show that, as observed in empirical studies, the resulting phenotypic distribution is highly skewed: simple forms are common, while complex ones are rare. The model exhibits a strongly non-linear genotype-to-phenotype mapping in such a way that small genetic changes can lead to disproportionately large morphological shifts. Notably, transitions toward complexity are less frequent than regressions to simplicity, reflecting evolutionary asymmetries observed in natural systems. We further demonstrate that, by allowing for mutations in the generative rules, our model is capable of adaptive evolution and even reproducing generic features of tumoral growth. These findings suggest that even minimal developmental rules can give rise to rich, hierarchical patterns and complex evolutionary dynamics, positioning our CA-based model as a powerful tool for investigating how developmental constraints and biases shape morphological evolution. Full article
(This article belongs to the Special Issue Trends and Prospects of Numerical Modelling in Bioengineering)
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22 pages, 1975 KB  
Article
Analysis of Blockchain Adoption in Environmental Monitoring Based on Evolutionary Game
by Lili Zhang, Shuolei Hu, Lei Qiao and Kai Zhong
Mathematics 2025, 13(19), 3237; https://doi.org/10.3390/math13193237 - 9 Oct 2025
Viewed by 109
Abstract
Environmental monitoring is the basis of environmental protection. China’s existing environmental monitoring system has been relatively perfect, but there are still data fraud and other illegal issues. Blockchain technology can well meet the requirements of environmental monitoring, but there are many obstacles in [...] Read more.
Environmental monitoring is the basis of environmental protection. China’s existing environmental monitoring system has been relatively perfect, but there are still data fraud and other illegal issues. Blockchain technology can well meet the requirements of environmental monitoring, but there are many obstacles in its adoption process, so this paper combines the characteristics of blockchain technology to integrate the two stakeholders of government and polluting enterprises into a unified model and introduces parameters related to smart contracts and corruption. The dynamic evolutionary game theory, combined with numerical simulation, is used to explore the behavioral decision-making characteristics and change rules of relevant stakeholders. The results show that there are stable conditions for the three strategies. Compared with the development cost of blockchain, the management cost of blockchain has a greater impact on the strategy choice of polluting enterprises because the income of polluting enterprises adopting blockchain technology can greatly affect the strategy choice of polluting enterprises, and there is a positive correlation between the income and the willingness of polluting enterprises to choose blockchain technology; only the construction cost of blockchain will cause fluctuations in the government’s strategy choice, and other factors will not have a greater impact on the government’s choice. This study provides a useful reference for promoting the adoption of blockchain technology in the field of environmental protection. Full article
(This article belongs to the Special Issue Mathematical Modelling and Optimization for Sustainable Operations Management, 2nd Edition)
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13 pages, 272 KB  
Article
On Smooth Solution to Three-Dimensional Incompressible Navier–Stokes Equations Based on Numerical Solutions by Finite Element Approximation
by Fengnan Liu, Junpeng Cao and Ziqiu Zhang
Mathematics 2025, 13(19), 3236; https://doi.org/10.3390/math13193236 - 9 Oct 2025
Viewed by 144
Abstract
In this paper, we develop a fully discrete finite element scheme, based on a second-order backward differentiation formula (BDF2), for numerically solving the three-dimensional incompressible Navier–Stokes equations. Under the assumption that the fully discrete solution remains bounded in a certain norm, we establish [...] Read more.
In this paper, we develop a fully discrete finite element scheme, based on a second-order backward differentiation formula (BDF2), for numerically solving the three-dimensional incompressible Navier–Stokes equations. Under the assumption that the fully discrete solution remains bounded in a certain norm, we establish that any smooth initial data necessarily gives rise to a unique strong solution that remains smooth. Moreover, we demonstrate that the fully discrete numerical solution converges strongly to this exact solution as the temporal and spatial discretization parameters approach zero. Full article
16 pages, 342 KB  
Article
Almost Nonlinear Contractions of Pant Type Employing Locally Finitely Transitive Relations with an Application to Nonlinear Integral Equations
by Faizan Ahmad Khan, Abdulrahman F. Aljohani, Adel Alatawi, Fahad M. Alamrani, Mohammed Zayed Alruwaytie and Esmail Alshaban
Mathematics 2025, 13(19), 3235; https://doi.org/10.3390/math13193235 - 9 Oct 2025
Viewed by 105
Abstract
In this research, a few metrical fixed-point outcomes consisting of an almost nonlinear Pant-type contraction employing a locally finitely transitive relation have been established. The findings of our research extrapolate, unify, develop, and improve a number of previously mentioned results. In the present [...] Read more.
In this research, a few metrical fixed-point outcomes consisting of an almost nonlinear Pant-type contraction employing a locally finitely transitive relation have been established. The findings of our research extrapolate, unify, develop, and improve a number of previously mentioned results. In the present investigation, we formulate a fixed-point finding for almost nonlinear Pant-type contractions in abstract metric space. To assist our study, we formulate numerous examples to illustrate our outcomes. Using our findings, we describe the existence and uniqueness of solutions to a nonlinear Fredholm integral equation. Full article
(This article belongs to the Special Issue Advanced Theories and Novel Methods for Nonlinear Analysis, Optimization and Applications)
16 pages, 3838 KB  
Article
Metric Morphological Interpretation of 3D Structures by Gray–Scott Model Simulation Utilising 2D Multifractal Analysis
by Akira Takahara and Yoshihiro Sato
Mathematics 2025, 13(19), 3234; https://doi.org/10.3390/math13193234 - 9 Oct 2025
Viewed by 154
Abstract
Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure [...] Read more.
Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure of three-dimensional slices. The proposed methods were verified using Monte Carlo and Gray–Scott simulations; the pixel-existence probability (PEP)-averaging method, which averages the pixel-existence probability in the slice direction, was confirmed to be the most suitable for analysing three-dimensional structures in two dimensions. This method enables a stable quantitative evaluation, regardless of the direction from which the three-dimensional structure is observed. Full article
(This article belongs to the Special Issue Advances in Fractal Geometry and Applications)
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26 pages, 1783 KB  
Article
Adaptive Tree-Structured MTS with Multi-Class Mahalanobis Space for High-Performance Multi-Class Classification
by Yefang Sun, Yvlei Chen and Yang Xu
Mathematics 2025, 13(19), 3233; https://doi.org/10.3390/math13193233 - 9 Oct 2025
Viewed by 124
Abstract
The traditional Mahalanobis–Taguchi System (MTS) employs two main strategies for multi-class classification: the partial binary tree MTS (PBT-MTS) and the multi-tree MTS (MT-MTS). The PBT-MTS relies on a fixed binary tree structure, resulting in limited model flexibility, while the MT-MTS suffers from its [...] Read more.
The traditional Mahalanobis–Taguchi System (MTS) employs two main strategies for multi-class classification: the partial binary tree MTS (PBT-MTS) and the multi-tree MTS (MT-MTS). The PBT-MTS relies on a fixed binary tree structure, resulting in limited model flexibility, while the MT-MTS suffers from its dependence on pre-defined category partitioning. Both methods exhibit constraints in adaptability and classification efficiency within complex data environments. To overcome these limitations, this paper proposes an innovative Adaptive Tree-structured Mahalanobis–Taguchi System (ATMTS). Its core breakthrough lies in the ability to autonomously construct an optimal multi-layer classification tree structure. Unlike conventional PBT-MTS, which establishes a Mahalanobis Space (MS) containing only a single category per node, ATMTS dynamically generates the MS that incorporates multiple categories, substantially enhancing discriminative power and structural adaptability. Furthermore, compared to MT-MTS, which depends on prior label information, ATMTS operates without predefined categorical assumptions, uncovering discriminative relationships solely through data-driven learning. This enables broader applicability and stronger generalization capability. By introducing a unified multi-objective joint optimization model, our method simultaneously optimizes structure construction, feature selection, and threshold determination, effectively overcoming the drawbacks of conventional phased optimization approaches. Experimental results demonstrate that ATMTS outperforms PBT-MTS, MT-MTS, and other mainstream classification methods across multiple benchmark datasets, achieving significant improvements in the accuracy and robustness of multi-class classification tasks. Full article
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16 pages, 843 KB  
Article
Mathematical Modeling and Intensive Simulations Assess Chances for Recovery of the Collapsed Azov Pikeperch Population
by Yuri V. Tyutyunov and Inna Senina
Mathematics 2025, 13(19), 3232; https://doi.org/10.3390/math13193232 - 9 Oct 2025
Viewed by 168
Abstract
The main objective of the study is to evaluate the recovery potential of the collapsed semi-anadromous pikeperch population (Sander lucioperca L.) in the Azov Sea during 2021–2030. We use a Ricker-based age-structured model that accounts for the effects of salinity and temperature [...] Read more.
The main objective of the study is to evaluate the recovery potential of the collapsed semi-anadromous pikeperch population (Sander lucioperca L.) in the Azov Sea during 2021–2030. We use a Ricker-based age-structured model that accounts for the effects of salinity and temperature on reproduction. In earlier work, the model predicted and explained the pikeperch stock collapse as the consequence of salinity and temperature exceeding the species’ tolerance limits. To assess the probability of stock recovery, we conducted a long-term retrospective validation and ran Monte Carlo projections under alternative climate scenarios with supplemental management actions. The results confirm that the dynamics of the pikeperch population in the Azov Sea are essentially environment-driven and negatively impacted by the large positive anomalies in both water temperature and salinity. Simulations suggest that either a substantial and persistent artificial restocking of juvenile recruits, or mostly unlikely scenarios of simultaneous reduction in salinity and temperature combined with additional restocking can provide conditions for the stock restoration within the decade considered. Based on these projections, we recommend a suite of urgent restoration measures to create the conditions required for future stock recovery. Full article
(This article belongs to the Special Issue Models in Population Dynamics, Ecology and Evolution)
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24 pages, 2777 KB  
Article
LightSeek-YOLO: A Lightweight Architecture for Real-Time Trapped Victim Detection in Disaster Scenarios
by Xiaowen Tian, Yubi Zheng, Liangqing Huang, Rengui Bi, Yu Chen, Shiqi Wang and Wenkang Su
Mathematics 2025, 13(19), 3231; https://doi.org/10.3390/math13193231 - 9 Oct 2025
Viewed by 328
Abstract
Rapid and accurate detection of trapped victims is vital in disaster rescue operations, yet most existing object detection methods cannot simultaneously deliver high accuracy and fast inference under resource-constrained conditions. To address this limitation, we propose the LightSeek-YOLO, a lightweight, real-time victim detection [...] Read more.
Rapid and accurate detection of trapped victims is vital in disaster rescue operations, yet most existing object detection methods cannot simultaneously deliver high accuracy and fast inference under resource-constrained conditions. To address this limitation, we propose the LightSeek-YOLO, a lightweight, real-time victim detection framework for disaster scenarios built upon YOLOv11. Our LightSeek-YOLO integrates three core innovations. First, it employs HGNetV2 as the backbone, whose HGStem and HGBlock modules leverage depthwise separable convolutions to markedly reduce computational cost while preserving feature extraction. Secondly, it introduces Seek-DS (Seek-DownSampling), a dual-branch downsampling module that preserves key feature extrema through a MaxPool branch while capturing spatial patterns via a progressive convolution branch, thereby effectively mitigating background interference. Third, it incorporates Seek-DH (Seek Detection Head), a lightweight detection head that processes features through a unified pipeline, enhancing scale adaptability while reducing parameter redundancy. Evaluated on the common C2A disaster dataset, LightSeek-YOLO achieves 0.478 AP@small for small-object detection, demonstrating strong robustness in challenging conditions such as rubble and smoke. Moreover, on the COCO, it reaches 0.473 mAP@[0.5:0.95], matching YOLOv8n while achieving superior computational efficiency through 38.2% parameter reduction and 39.5% FLOP reduction, and achieving 571.72 FPS on desktop hardware, with computational efficiency improvements suggesting potential for edge deployment pending validation. Full article
(This article belongs to the Special Issue Machine Learning Applications in Image Processing and Computer Vision)
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25 pages, 66105 KB  
Article
Toward Real-Time Scalable Rigid-Body Simulation Using GPU-Optimized Collision Detection and Response
by Nak-Jun Sung and Min Hong
Mathematics 2025, 13(19), 3230; https://doi.org/10.3390/math13193230 - 9 Oct 2025
Viewed by 315
Abstract
We propose a GPU-parallelized collision-detection and response framework for rigid-body dynamics, designed to efficiently handle densely populated 3D simulations in real time. The method combines explicit Euler time integration with a hierarchical Octree–AABB collision-detection scheme, enabling early pruning and localized refinement of contact [...] Read more.
We propose a GPU-parallelized collision-detection and response framework for rigid-body dynamics, designed to efficiently handle densely populated 3D simulations in real time. The method combines explicit Euler time integration with a hierarchical Octree–AABB collision-detection scheme, enabling early pruning and localized refinement of contact checks. To resolve collisions, we employ a two-step response algorithm that integrates non-penetration correction and impulse-based velocity updates, stabilized through smoothing, clamping, and bias mechanisms. The framework is fully implemented within Unity3D using compute shaders and optimized GPU kernels. Experiments across multiple mesh models and increasing object counts demonstrate that the proposed hierarchical configuration significantly improves scalability and frame stability compared to conventional flat AABB methods. In particular, a two-level hierarchy achieves the best trade-off between spatial resolution and computational cost, maintaining interactive frame rates (≥30 fps) under high-density scenarios. These results suggest the practical applicability of our method to real-time simulation systems involving complex collision dynamics. Full article
(This article belongs to the Topic Extended Reality: Models and Applications)
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31 pages, 2358 KB  
Article
Semi-Supervised Bayesian GANs with Log-Signatures for Uncertainty-Aware Credit Card Fraud Detection
by David Hirnschall
Mathematics 2025, 13(19), 3229; https://doi.org/10.3390/math13193229 - 9 Oct 2025
Viewed by 194
Abstract
We present a novel deep generative semi-supervised framework for credit card fraud detection, formulated as a time series classification task. As financial transaction data streams grow in scale and complexity, traditional methods often require large labeled datasets and struggle with time series of [...] Read more.
We present a novel deep generative semi-supervised framework for credit card fraud detection, formulated as a time series classification task. As financial transaction data streams grow in scale and complexity, traditional methods often require large labeled datasets and struggle with time series of irregular sampling frequencies and varying sequence lengths. To address these challenges, we extend conditional Generative Adversarial Networks (GANs) for targeted data augmentation, integrate Bayesian inference to obtain predictive distributions and quantify uncertainty, and leverage log-signatures for robust feature encoding of transaction histories. We propose a composite Wasserstein distance-based loss to align generated and real unlabeled samples while simultaneously maximizing classification accuracy on labeled data. Our approach is evaluated on the BankSim dataset, a widely used simulator for credit card transaction data, under varying proportions of labeled samples, demonstrating consistent improvements over benchmarks in both global statistical and domain-specific metrics. These findings highlight the effectiveness of GAN-driven semi-supervised learning with log-signatures for irregularly sampled time series and emphasize the importance of uncertainty-aware predictions. Full article
(This article belongs to the Special Issue Artificial Intelligence Techniques in the Financial Services Industry)
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19 pages, 19843 KB  
Article
Distinguishing Human- and AI-Generated Image Descriptions Using CLIP Similarity and Transformer-Based Classification
by Daniela Onita, Matei-Vasile Căpîlnaș and Adriana Baciu (Birlutiu)
Mathematics 2025, 13(19), 3228; https://doi.org/10.3390/math13193228 - 9 Oct 2025
Viewed by 279
Abstract
Recent advances in vision-language models such as BLIP-2 have made AI-generated image descriptions increasingly fluent and difficult to distinguish from human-authored texts. This paper investigates whether such differences can still be reliably detected by introducing a novel bilingual dataset of English and Romanian [...] Read more.
Recent advances in vision-language models such as BLIP-2 have made AI-generated image descriptions increasingly fluent and difficult to distinguish from human-authored texts. This paper investigates whether such differences can still be reliably detected by introducing a novel bilingual dataset of English and Romanian captions. The English subset was derived from the T4SA dataset, while AI-generated captions were produced with BLIP-2 and translated into Romanian using MarianMT; human-written Romanian captions were collected via manual annotation. We analyze the problem from two perspectives: (i) semantic alignment, using CLIP similarity, and (ii) supervised classification with both traditional and transformer-based models. Our results show that BERT achieves over 95% cross-validation accuracy (F1 = 0.95, ROC AUC = 0.99) in distinguishing AI from human texts, while simpler classifiers such as Logistic Regression also reach competitive scores (F1 ≈ 0.88). Beyond classification, semantic and linguistic analyses reveal systematic cross-lingual differences: English captions are significantly longer and more verbose, whereas Romanian texts—often more concise—exhibit higher alignment with visual content. Romanian was chosen as a representative low-resource language, where studying such differences provides insights into multilingual AI detection and challenges in vision-language modeling. These findings emphasize the novelty of our contribution: a publicly available bilingual dataset and the first systematic comparison of human vs. AI-generated captions in both high- and low-resource languages. Full article
(This article belongs to the Special Issue Advances in Machine Learning and Deep Learning: Innovations and Applications)
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16 pages, 324 KB  
Article
Doubly Robust Estimation of the Finite Population Distribution Function Using Nonprobability Samples
by Soonpil Kwon, Dongmin Jang and Kyu-Seong Kim
Mathematics 2025, 13(19), 3227; https://doi.org/10.3390/math13193227 - 8 Oct 2025
Viewed by 223
Abstract
The growing use of nonprobability samples in survey statistics has motivated research on methodological adjustments that address the selection bias inherent in such samples. Most studies, however, have concentrated on the estimation of the population mean. In this paper, we extend our focus [...] Read more.
The growing use of nonprobability samples in survey statistics has motivated research on methodological adjustments that address the selection bias inherent in such samples. Most studies, however, have concentrated on the estimation of the population mean. In this paper, we extend our focus to the finite population distribution function and quantiles, which are fundamental to distributional analysis and inequality measurement. Within a data integration framework that combines probability and nonprobability samples, we propose two estimators, a regression estimator and a doubly robust estimator, and discuss their asymptotic properties. Furthermore, we derive quantile estimators and construct Woodruff confidence intervals using a bootstrap method. Simulation results based on both a synthetic population and the 2023 Korean Survey of Household Finances and Living Conditions demonstrate that the proposed estimators perform stably across scenarios, supporting their applicability to the production of policy-relevant indicators. Full article
(This article belongs to the Special Issue Uncertainty Quantification Techniques in Statistics, Machine Learning and FinTech: 2nd Edition)
25 pages, 5983 KB  
Article
Theoretical Modeling of Light-Fueled Self-Harvesting in Piezoelectric Beams Actuated by Liquid Crystal Elastomer Fibers
by Lin Zhou, Haiming Chen, Wu Bao, Xuehui Chen, Ting Gao and Dali Ge
Mathematics 2025, 13(19), 3226; https://doi.org/10.3390/math13193226 - 8 Oct 2025
Viewed by 136
Abstract
Traditional energy harvesting systems, such as photovoltaics and wind power, often rely on external environmental conditions and are typically associated with contact-based vibration wear and bulky structures. This study introduces light-fueled self-vibration to propose a self-harvesting system, consisting of liquid crystal elastomer fibers, [...] Read more.
Traditional energy harvesting systems, such as photovoltaics and wind power, often rely on external environmental conditions and are typically associated with contact-based vibration wear and bulky structures. This study introduces light-fueled self-vibration to propose a self-harvesting system, consisting of liquid crystal elastomer fibers, two resistors, and two piezoelectric cantilever beams arranged symmetrically. Based on the photothermal temperature evolution, we derive the governing equations of the liquid crystal elastomer fiber–piezoelectric beam system. Two distinct states, namely a self-harvesting state and a static state, are revealed through numerical simulations. The self-oscillation results from light-induced cyclic contraction of the liquid crystal elastomer fibers, driving beam bending, stress generation in the piezoelectric layer, and voltage output. Additionally, the effects of various system parameters on amplitude, frequency, voltage, and power are analyzed in detail. Unlike traditional vibration energy harvesters, this light-fueled self-harvesting system features a compact structure, flexible installation, and ensures continuous and stable energy output. Furthermore, by coupling the light-responsive LCE fibers with piezoelectric transduction, the system provides a non-contact actuation mechanism that enhances durability and broadens potential application scenarios. Full article
(This article belongs to the Special Issue Mathematical Models in Mechanics and Engineering)
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35 pages, 454 KB  
Article
Two Versions of Dunkl Linear Canonical Wavelet Transforms and Applications
by Saifallah Ghobber and Hatem Mejjaoli
Mathematics 2025, 13(19), 3225; https://doi.org/10.3390/math13193225 - 8 Oct 2025
Viewed by 139
Abstract
Among the class of generalized Fourier transformations, the linear canonical transform is of crucial importance, mainly due to its higher degrees of freedom compared to the conventional Fourier and fractional Fourier transforms. In this paper, we will introduce and study two versions of [...] Read more.
Among the class of generalized Fourier transformations, the linear canonical transform is of crucial importance, mainly due to its higher degrees of freedom compared to the conventional Fourier and fractional Fourier transforms. In this paper, we will introduce and study two versions of wavelet transforms associated with the linear canonical Dunkl transform. More precisely, we investigate some applications for Dunkl linear canonical wavelet transforms. Next we will introduce and develop the harmonic analysis associated with the Dunkl linear canonical wavelet packets transform. We introduce and study three types of wavelet packets along with their associated wavelet transforms. For each of these transforms, we establish a Plancherel and a reconstruction formula, and we analyze the associated scale-discrete scaling functions. Full article
(This article belongs to the Section E: Applied Mathematics)
29 pages, 2037 KB  
Article
An Evolutionary Game Approach to Enhancing Semiconductor Supply Chain Security in China: Collaborative Governance and Policy Optimization
by Ye Yuan, Jingtao Zhao, Jiacheng Liu and Jiang Yu
Mathematics 2025, 13(19), 3224; https://doi.org/10.3390/math13193224 - 8 Oct 2025
Viewed by 287
Abstract
In response to the changing international landscape and the risks associated with China’s supply chain security, conducting policy simulations on semiconductor supply chain security helps clarify the industry’s policies and governance strategies for semiconductor supply chain security in China. It also enables a [...] Read more.
In response to the changing international landscape and the risks associated with China’s supply chain security, conducting policy simulations on semiconductor supply chain security helps clarify the industry’s policies and governance strategies for semiconductor supply chain security in China. It also enables a better understanding of the current state and focus areas of China’s semiconductor supply chain security, which is of great significance for improving the security levels of semiconductor supply chains across provinces and cities and for constructing a secure, efficient, and autonomous semiconductor supply chain system. Firstly, this paper reviews the current research on semiconductor supply chains, supply chain security, and industrial policies. Secondly, based on the industrial policies for semiconductor supply chain security, an evolutionary game model is constructed, involving government departments, chain owner enterprises, and upstream and downstream small and medium-sized enterprises (SMEs) within the supply chain. Finally, the MATLAB R2016b system simulation method is employed to conduct a policy simulation analysis of China’s semiconductor supply chain security and further analyze the industrial policies related to semiconductor supply chain security. The results show that: (1) Supply chain security depends on multi-agent collaborative governance, with government leadership, and chain owner enterprises driving innovation in SMEs, improving digitalization levels, and ensuring supply chain autonomy and control. (2) Increasing government management revenue, raising the responsibility costs for chain owner enterprises, and reducing the innovation costs for SMEs can accelerate the achievement of the ideal governance state. Lastly, policy recommendations are proposed to build an autonomous and controllable supply chain system. Full article
(This article belongs to the Section D: Statistics and Operational Research)
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17 pages, 905 KB  
Article
The Simplest 2D Quantum Walk Detects Chaoticity
by César Alonso-Lobo, Gabriel G. Carlo and Florentino Borondo
Mathematics 2025, 13(19), 3223; https://doi.org/10.3390/math13193223 - 8 Oct 2025
Viewed by 200
Abstract
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely [...] Read more.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ≃0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)≃1150 compared to the rectangular billiard (regular) case, where the average PR≃1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian. Full article
(This article belongs to the Section C2: Dynamical Systems)
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29 pages, 3821 KB  
Article
Mathematical Framework for Digital Risk Twins in Safety-Critical Systems
by Igor Kabashkin
Mathematics 2025, 13(19), 3222; https://doi.org/10.3390/math13193222 - 8 Oct 2025
Viewed by 229
Abstract
This paper introduces a formal mathematical framework for Digital Risk Twins (DRTs) as an extension of traditional digital twin (DT) architectures, explicitly tailored to the needs of safety-critical systems. While conventional DTs enable real-time monitoring and simulation of physical assets, they often lack [...] Read more.
This paper introduces a formal mathematical framework for Digital Risk Twins (DRTs) as an extension of traditional digital twin (DT) architectures, explicitly tailored to the needs of safety-critical systems. While conventional DTs enable real-time monitoring and simulation of physical assets, they often lack structured mechanisms to model stochastic failure processes; evaluate dynamic risk; or support resilient, risk-aware decision-making. The proposed DRT framework addresses these limitations by embedding probabilistic hazard modeling, reliability theory, and coherent risk measures into a modular and mathematically interpretable structure. The DT to DRT transformation is formalized as a composition of operators that project system trajectories onto risk-relevant features, compute failure intensities, and evaluate risk metrics under uncertainty. The framework supports layered integration of simulation, feature extraction, hazard dynamics, and decision-oriented evaluation, providing traceability, scalability, and explainability. Its utility is demonstrated through a case study involving an aircraft brake system, showcasing early warning detection, inspection schedule optimization, and visual risk interpretation. The results confirm that the DRT enables modular, explainable, and domain-agnostic integration of reliability logic into digital twin systems, enhancing their value in safety-critical applications. Full article
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12 pages, 250 KB  
Article
On Extended Perron Complements of Nonnegative Irreducible γ-Diagonally and Product γ-Diagonally Dominant Matrices
by Qin Zhong and Jie Wu
Mathematics 2025, 13(19), 3221; https://doi.org/10.3390/math13193221 - 8 Oct 2025
Viewed by 165
Abstract
This study explores the γ-diagonal dominance and product γ-diagonal dominance properties of extended Perron complements. We prove that nonnegative irreducible matrices that are (strictly) γ-diagonally dominant maintain their nonnegativity, irreducibility, and (strictly) γ-diagonal dominance after applying the extended Perron complement operation. Similarly, (strictly) [...] Read more.
This study explores the γ-diagonal dominance and product γ-diagonal dominance properties of extended Perron complements. We prove that nonnegative irreducible matrices that are (strictly) γ-diagonally dominant maintain their nonnegativity, irreducibility, and (strictly) γ-diagonal dominance after applying the extended Perron complement operation. Similarly, (strictly) product γ-diagonally dominant nonnegative irreducible matrices also retain these properties under the same operation. Two numerical examples illustrate and validate these theoretical results. Full article
30 pages, 1549 KB  
Article
Satellite Constellation Multi-Target Robust Observation Method Based on Hypergraph Algebraic Connectivity and Observation Precision Theory
by Jie Cao, Xiaogang Pan, Yuanyuan Jiao, Bowen Sun and Yangyang Lu
Mathematics 2025, 13(19), 3220; https://doi.org/10.3390/math13193220 - 8 Oct 2025
Viewed by 240
Abstract
A multi-target robust observation method for satellite constellations based on hypergraph algebraic connectivity and observation precision theory is proposed to address the challenges posed by the surge in space targets and system failures. First, a precision metric framework is constructed based on nonlinear [...] Read more.
A multi-target robust observation method for satellite constellations based on hypergraph algebraic connectivity and observation precision theory is proposed to address the challenges posed by the surge in space targets and system failures. First, a precision metric framework is constructed based on nonlinear batch least squares estimation theory, deriving the theoretical precision covariance through cumulative observation matrices to provide a theoretical foundation for tracking accuracy evaluation. Second, multi-satellite collaborative observation is modeled as an edge-dependent vertex-weighted hypergraph, enhancing system robustness by maximizing algebraic connectivity. A constrained simulated annealing (CSA) algorithm is designed, employing a precision-guided perturbation strategy to efficiently solve the optimization problem. Simulation experiments are conducted using 24 Walker constellation satellites tracking 50 targets, comparing the proposed method with greedy algorithm, CBBA, and CSA-bipartite Graph methods across three scenarios: baseline, maneuvering, and failure. Results demonstrate that the CSA-hypergraph method achieves 0.089 km steady-state precision in the baseline scenario, representing a 41.4% improvement over traditional methods; in maneuvering scenarios, detection delay is reduced by 34.3% and re-achievement time is decreased by 47.4%; with a 30% satellite failure rate, performance degradation is only 9.8%, significantly outperforming other methods. Full article
(This article belongs to the Section E: Applied Mathematics)
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19 pages, 826 KB  
Article
Minimum-Cost Shortest-Path Interdiction Problem Involving Upgrading Edges on Trees with Weighted l∞ Norm
by Qiao Zhang and Xiao Li
Mathematics 2025, 13(19), 3219; https://doi.org/10.3390/math13193219 - 7 Oct 2025
Viewed by 172
Abstract
Network interdiction problems involving edge deletion on shortest paths have wide applications. However, in many practical scenarios, the complete removal of edges is infeasible. The minimum-cost shortest-path interdiction problem for trees with the weighted l∞ norm (MCSPIT∞) is studied in [...] Read more.
Network interdiction problems involving edge deletion on shortest paths have wide applications. However, in many practical scenarios, the complete removal of edges is infeasible. The minimum-cost shortest-path interdiction problem for trees with the weighted l∞ norm (MCSPIT∞) is studied in this paper. The goal is to upgrade selected edges at minimum total cost such that the shortest root–leaf distance is bounded below by a given value. We designed an O(nlogn) algorithm based on greedy techniques combined with a binary search method to solve this problem efficiently. We then extended the framework to the minimum-cost shortest-path double interdiction problem for trees with the weighted l∞ norm, which imposes an additional requirement that the sum of root–leaf distances exceed a given threshold. Building upon the solution to (MCSPIT∞), we developed an equally efficient O(nlogn) algorithm for this variant. Finally, numerical experiments are presented to demonstrate both the effectiveness and practical performance of the proposed algorithms. Full article
(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)
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17 pages, 2245 KB  
Article
Complex Variable Approach for Thermoelastic Boundary Value Problem Using Rational Mapping Techniques
by Mai Taha, Mohamed A. Abdou, Amnah E. Shammaky, Abeer A. Al-Dohiman and Eslam M. Youssef
Mathematics 2025, 13(19), 3218; https://doi.org/10.3390/math13193218 - 7 Oct 2025
Viewed by 198
Abstract
This article presents a novel approach to looking at steady-state thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using a complex variable approach and rational conformal mappings. The physical domain with a non-circular opening is mapped conformally to the unit [...] Read more.
This article presents a novel approach to looking at steady-state thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using a complex variable approach and rational conformal mappings. The physical domain with a non-circular opening is mapped conformally to the unit disk. A thermoelastic potential combines the temperature distribution, which is determined by the Laplace equation with Neumann boundary conditions. Gaursat functions, which are shown as truncated power series, show the complicated stress and displacement fields. They are found by putting boundary constraints at certain collocation points. This procedure presents us with a linear system that can be solved using the least squares method. The method is applied in an annular shape that is exposed to a radial temperature gradient. This experiment shows how changes at the boundary affect the distribution of stress. According to numerical simulations, stress distributions are more uniform when boundaries are smoother, but stress concentrations increase with the size of geometric disturbances. The suggested approach remarkably captures the way geometry and thermal effects interact in two-dimensional thermoelasticity. It is a reliable tool for researching intricate, heated elastic domains. Full article
(This article belongs to the Section C4: Complex Analysis)
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27 pages, 2189 KB  
Article
Miss-Triggered Content Cache Replacement Under Partial Observability: Transformer-Decoder Q-Learning
by Hakho Kim, Teh-Jen Sun and Eui-Nam Huh
Mathematics 2025, 13(19), 3217; https://doi.org/10.3390/math13193217 - 7 Oct 2025
Viewed by 154
Abstract
Content delivery networks (CDNs) face steadily rising, uneven demand, straining heuristic cache replacement. Reinforcement learning (RL) is promising, but most work assumes a fully observable Markov Decision Process (MDP), unrealistic under delayed, partial, and noisy signals. We model cache replacement as a Partially [...] Read more.
Content delivery networks (CDNs) face steadily rising, uneven demand, straining heuristic cache replacement. Reinforcement learning (RL) is promising, but most work assumes a fully observable Markov Decision Process (MDP), unrealistic under delayed, partial, and noisy signals. We model cache replacement as a Partially Observable MDP (POMDP) and present the Miss-Triggered Cache Transformer (MTCT), a Transformer-decoder Q-learning agent that encodes recent histories with self-attention. MTCT invokes its policy only on cache misses to align compute with informative events and uses a delayed-hit reward to propagate information from hits. A compact, rank-based action set (12 actions by default) captures popularity–recency trade-offs with complexity independent of cache capacity. We evaluate MTCT on a real trace (MovieLens) and two synthetic workloads (Mandelbrot–Zipf, Pareto) against Adaptive Replacement Cache (ARC), Windowed TinyLFU (W-TinyLFU), classical heuristics, and Double Deep Q-Network (DDQN). MTCT achieves the best or statistically comparable cache-hit rates on most cache sizes; e.g., on MovieLens at M=600, it reaches 0.4703 (DDQN 0.4436, ARC 0.4513). Miss-triggered inference also lowers mean wall-clock time per episode; Transformer inference is well suited to modern hardware acceleration. Ablations support CL=50 and show that finer action grids improve stability and final accuracy. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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37 pages, 9471 KB  
Article
Mathematical Approach Integrating Surrogate Models in Heuristic Optimization for Gabion Retaining Wall Design
by Esra Uray and Zong Woo Geem
Mathematics 2025, 13(19), 3216; https://doi.org/10.3390/math13193216 - 7 Oct 2025
Viewed by 163
Abstract
This study focuses on the mathematical method developed by integrating the surrogate model as constraints for wall stability into the heuristic optimization algorithm to gain the optimum cost and CO2 emission value of the gabion retaining wall (GRW). This study also includes [...] Read more.
This study focuses on the mathematical method developed by integrating the surrogate model as constraints for wall stability into the heuristic optimization algorithm to gain the optimum cost and CO2 emission value of the gabion retaining wall (GRW). This study also includes the comparison of optimum GRW results with optimum cantilever retaining wall (CRW) designs for different design cases. The Harmony Search Algorithm (HSA), which efficiently explores the design space and robustly reaches the optimum result in solving optimization problems, was used as the heuristic optimization algorithm. The primary construction scenario was considered as an optimization problem, which involved excavating the slope, constructing the wall, and compacting the backfill soil to minimize the cost and CO⁢2 emissions for separate objective functions of GRW and CRW designs. Comparative results show that GRWs outperform CRWs in terms of sustainability and cost-efficiency, achieving 55% lower cost and 78% lower CO⁢2 emissions on average, while the HSA–surrogate model provides a fast and accurate solution for geotechnical design problems. The surrogate models for sliding, overturning, and slope stability safety factors of GRW exhibited exceptional accuracy, characterized by minimal error values (MSE, RMSE, MAE, MAPE) and robust determination coefficients (R2≈0.99), hence affirming their dependability in safety factor assessment. By integrating the surrogate model based on the statistical method into the optimization algorithm, a quick examination of the wall’s stability was performed, reducing the required computational power. Full article
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15 pages, 378 KB  
Article
Nonlinear Transmission Line: Shock Waves and the Simple Wave Approximation
by Eugene Kogan
Mathematics 2025, 13(19), 3215; https://doi.org/10.3390/math13193215 - 7 Oct 2025
Viewed by 125
Abstract
The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper we additionally include linear ohmic resistors. Thus, the dissipation being taken into account leads to the existence of shocks—the travelling waves [...] Read more.
The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper we additionally include linear ohmic resistors. Thus, the dissipation being taken into account leads to the existence of shocks—the travelling waves with different asymptotically constant values of the voltage (the capacitor charge) and the current before and after the front of the wave. For the particular values of ohmic resistances (corresponding to strong dissipation) we obtain the analytic solution for the profile of a shock wave. Both the charge on a capacitor and current through the inductor are obtained as the functions of the time and space coordinate. In the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper we consider the nonlinear lossless transmission line. We formulate a simple wave approximation for such transmission line, which decouples left/right-going waves. The approximation can also be used for the lossy transmission line, which is considered in the first part of the paper, to describe the formation of the shock wave (but, of course, not the shock wave itself). Full article
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13 pages, 314 KB  
Article
Thermodynamic Hamiltonian and Entropy Production
by Umberto Lucia and Giulia Grisolia
Mathematics 2025, 13(19), 3214; https://doi.org/10.3390/math13193214 - 7 Oct 2025
Viewed by 209
Abstract
The variational method holds considerable significance within mathematical and theoretical physics. Its importance stems from its capacity to characterise natural systems through physical quantities, irrespective of the chosen frame of reference. This characteristic makes it a powerful tool for understanding the behaviour of [...] Read more.
The variational method holds considerable significance within mathematical and theoretical physics. Its importance stems from its capacity to characterise natural systems through physical quantities, irrespective of the chosen frame of reference. This characteristic makes it a powerful tool for understanding the behaviour of diverse physical phenomena. A global and statistical approach originating from the principles of non-equilibrium thermodynamics has been developed. This approach culminates in the principle of maximum entropy generation, specifically tailored for open systems. The principle itself arises as a direct consequence of applying the Lagrangian approach to open systems. The work focuses on a generalised method for deriving the thermodynamic Hamiltonian. This Hamiltonian is essential to the dynamical analysis of open systems, allowing for a detailed examination of their time evolution. The analysis suggests that irreversibility appears to be a fundamental process related to the evolution of states within open systems. Full article
21 pages, 1094 KB  
Article
Dynamic Equivalence of Active Distribution Network: Multiscale and Multimodal Fusion Deep Learning Method with Automatic Parameter Tuning
by Wenhao Wang, Zhaoxi Liu, Fengzhe Dai and Huan Quan
Mathematics 2025, 13(19), 3213; https://doi.org/10.3390/math13193213 - 7 Oct 2025
Viewed by 265
Abstract
Dynamic equivalence of active distribution networks (ADNs) is emerging as one of the most important issues for the backbone network security analysis due to high penetration of distributed generations (DGs) and electricity vehicles (EVs). The multiscale and multimodal fusion deep learning (MMFDL) method [...] Read more.
Dynamic equivalence of active distribution networks (ADNs) is emerging as one of the most important issues for the backbone network security analysis due to high penetration of distributed generations (DGs) and electricity vehicles (EVs). The multiscale and multimodal fusion deep learning (MMFDL) method proposed in this paper contains two modalities, one of which is a CNN + attention module to simulate Newton Raphson power flow calculation (NRPFC) for the important feature extraction of a power system caused by disturbance, which is motivated by the similarities between NRPFC and convolution network computation. The other is a long short-term memory (LSTM) + fully connected (FC) module for load modeling based on the fact that LSTM + FC can represent a load′s differential algebraic equations (DAEs). Moreover, to better capture the relationship between voltage and power, the multiscale fusion method is used to aggregate load modeling models with different voltage input sizes and combined with CNN + attention, merging as MMFDL to represent the dynamic behaviors of ADNs. Then, the Kepler optimization algorithm (KOA) is applied to automatically tune the adjustable parameters of MMFLD (called KOA-MMFDL), especially the LSTM and FC hidden layer number, as they are important for load modeling and there is no human knowledge to set these parameters. The performance of the proposed method was evaluated by employing different electric power systems and various disturbance scenarios. The error analysis shows that the proposed method can accurately represent the dynamic response of ADNs. In addition, comparative experiments verified that the proposed method is more robust and generalizable than other advanced non-mechanism methods. Full article
(This article belongs to the Section C2: Dynamical Systems)
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25 pages, 2551 KB  
Article
Deep-Reinforcement-Learning-Based Sliding Mode Control for Optimized Energy Management in DC Microgrids
by Monia Charfeddine, Mongi Ben Moussa and Khalil Jouili
Mathematics 2025, 13(19), 3212; https://doi.org/10.3390/math13193212 - 7 Oct 2025
Viewed by 278
Abstract
A hybrid control architecture is proposed for enhancing the stability and energy management of DC microgrids (DCMGs) integrating photovoltaic generation, batteries, and supercapacitors. The approach combines nonlinear Sliding Mode Control (SMC) for fast and robust DC bus voltage regulation with a Deep Q-Learning [...] Read more.
A hybrid control architecture is proposed for enhancing the stability and energy management of DC microgrids (DCMGs) integrating photovoltaic generation, batteries, and supercapacitors. The approach combines nonlinear Sliding Mode Control (SMC) for fast and robust DC bus voltage regulation with a Deep Q-Learning (DQL) agent that learns optimal high-level policies for charging, discharging, and load management. This dual-layer design leverages the real-time precision of SMC and the adaptive decision-making capability of DQL to achieve dynamic power sharing and balanced state-of-charge levels across storage units, thereby reducing asymmetric wear. Simulation results under variable operating scenarios showed that the proposed method significantly improvedvoltage stability, loweredthe occurrence of deep battery discharges, and decreased load shedding compared to conventional fuzzy-logic-based energymanagement, highlighting its effectiveness and resilience in the presence of renewable generation variability and fluctuating load demands. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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28 pages, 567 KB  
Article
Fine-Tune LLMs for PLC Code Security: An Information-Theoretic Analysis
by Ping Chen, Xiaojing Liu and Yi Wang
Mathematics 2025, 13(19), 3211; https://doi.org/10.3390/math13193211 - 7 Oct 2025
Viewed by 342
Abstract
Programmable Logic Controllers (PLCs), widely used in industrial automation, are often programmed in IEC 61131-3 Structured Text (ST), which is prone to subtle logic vulnerabilities. Traditional tools like static analysis and fuzzing struggle with the complexity and domain-specific semantics of ST. This work [...] Read more.
Programmable Logic Controllers (PLCs), widely used in industrial automation, are often programmed in IEC 61131-3 Structured Text (ST), which is prone to subtle logic vulnerabilities. Traditional tools like static analysis and fuzzing struggle with the complexity and domain-specific semantics of ST. This work explores Large Language Models (LLMs) for PLC vulnerability detection, supported by both theoretical insights and empirical validation. Theoretically, we prove that control flow features carry the most vulnerability-relevant information, establish a feature informativeness hierarchy, and derive sample complexity bounds. We also propose an optimal synthetic data mixing strategy to improve learning with limited supervision. Empirically, we build a dataset combining real-world and synthetic ST code with five vulnerability types. We fine-tune open-source LLMs (CodeLlama, Qwen2.5-Coder, Starcoder2) using LoRA, demonstrating significant gains in binary and multi-class classification. The results confirm our theoretical predictions and highlight the promise of LLMs for PLC security. Our work provides a principled and practical foundation for LLM-based analysis of cyber-physical systems, emphasizing the role of domain knowledge, efficient adaptation, and formal guarantees. Full article
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15 pages, 332 KB  
Article
Multiple Positive Solutions of Nabla Fractional Equations with Summation Boundaries
by Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda
Mathematics 2025, 13(19), 3210; https://doi.org/10.3390/math13193210 - 7 Oct 2025
Viewed by 220
Abstract
The current work studies difference problems including two different nabla operators coupled with general summation boundary conditions that depend on a parameter. After we deduce the Green’s function, we obtain an interval of the parameter, where it is strictly positive. Then, we establish [...] Read more.
The current work studies difference problems including two different nabla operators coupled with general summation boundary conditions that depend on a parameter. After we deduce the Green’s function, we obtain an interval of the parameter, where it is strictly positive. Then, we establish a lower and upper bound of the related Green’s function and we impose suitable conditions of the nonlinear part, under which, using the classical Guo–Krasnoselskii fixed point theorem, we deduce the existence of at least one positive solution of the studied equation. After that, we impose more restricted conditions on the right-hand side and we obtain the existence of n positive solutions again using fixed point theory, which is the main novelty of this research. Finally, we give particular examples as an application of our theoretical findings. Full article
(This article belongs to the Special Issue Fractional Calculus: Advances and Applications)
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20 pages, 3266 KB  
Article
A Simulated Annealing Approach for the Homogeneous Capacitated Vehicle Routing Problem
by Dalia Vanessa Arce-Ortega, Federico Alonso-Pecina, Marco Antonio Cruz-Chávez and Jesús del Carmen Peralta-Abarca
Mathematics 2025, 13(19), 3209; https://doi.org/10.3390/math13193209 - 7 Oct 2025
Viewed by 288
Abstract
This study addresses the Capacitated Vehicle Routing Problem (CVRP) known to be NP-hard. In this problem, a set of customers with varying demands is considered. To solve the problem, routes were generated for several vehicles with identical capacity, which were responsible for delivering [...] Read more.
This study addresses the Capacitated Vehicle Routing Problem (CVRP) known to be NP-hard. In this problem, a set of customers with varying demands is considered. To solve the problem, routes were generated for several vehicles with identical capacity, which were responsible for delivering products to a set of geographically dispersed customers. The purpose of the problem is to minimize the total cost of all routes. This problem was solved by applying the metaheuristic Simulated Annealing (SA) and incorporating four different neighborhoods to improve the initial solution generated randomly. In the SA, a set of cooling factors is used. The best solution obtained by SA is refined by the use of Hill Climbing using a double neighborhood. The algorithm was tested with instances from the literature in order to measure its effectiveness in solution quality and execution time. We tested the approach with 106 instances from the literature and obtained the optimum in 93 instances. The average time in most instances was less than five minutes. Delivery companies can benefit from this approach. They only need to identify the depot, the clients, and the distance between locations, and this approach can be used with relative ease. Full article
(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)
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