The Fuzzy Job-Shop Scheduling Problem (FJSSP) extends the classical Job-Shop Scheduling Problem (JSSP), a well-known NP-hard combinatorial optimization task, by representing uncertainty in processing times, due dates, and related parameters via fuzzy sets. This paper presents a comprehensive literature review of the FJSSP, updating and extending the seminal survey published in 2014. Covering the period from 2014 to 2025, the review classifies FJSSP formulations based on fuzzy representations, including triangular and interval models, Intuitionistic Fuzzy Sets (IFS), and Interval Type-2 Fuzzy Sets (IT2FS). It summarizes constraint variants and objective functions, spanning completion-time criteria, energy and emission costs, robustness, and quality metrics. The review surveys exact, heuristic, and metaheuristic solvers alongside emerging learning-based methods, specifically Deep Reinforcement Learning (DRL) on graph representations. Furthermore, it analyzes benchmark datasets and fuzzification methods, including recent generative protocols for dynamic disruptions. Finally, the paper identifies open gaps, highlighting the need for hybrid fuzzy–stochastic uncertainty models, the standardization of IT2FS, the integration of structural fuzziness for problem definitions, and tighter coupling with Digital Twins (DT) in service-oriented scheduling.
At this work modern approaches are being explored to optimization of computational resources in multi-cloud architectures using intelligent orchestration systems. The main challenges which associated with managing heterogeneous cloud environments are considered and also the benefits of applying AI/ML-methods to automate workload placement and scaling. Was suggested a formal mathematical model of the optimization problem taking into account resource costs, network delays, SLA requirements and migration costs. It was shown that using of predictive models and RL-agents allows to increase the efficiency of resource allocation, reduce financial costs and ensure resilience to load changes. This work contains of analysis modern research, practical recommendations, and outlines directions for further scientific developments in the field of multi-cloud infrastructures.
The paper considers the application of global thresholding methods for lung region segmentation in chest X-ray images, in particular the Otsu, Kapur, and Balanced Histogram Thresholding (BHT) methods. A comparative analysis of the these methods is performed. The effectiveness of the approaches is evaluated using the Dice coefficient, Recall, and the over-segmentation index (OSI). Additionally, the impact of image preprocessing using Contrast Limited Adaptive Histogram Equalization (CLAHE) on segmentation quality is analyzed. The obtained results show that the use of CLAHE significantly improves segmentation performance for the Otsu and Kapur methods. The BHT method provides the highest segmentation completeness, but is characterized by an increased level of over-segmentation. It is demonstrated that the considered methods can be applied for initial lung region extraction.
The purpose of the article is to develop a graph neural network–based method for detecting coordination structures in network data based on graph topology analysis and machine learning techniques. Research methodology. The study is based on representing network data as an interaction graph, where vertices correspond to objects and edges represent their relationships. Structural representations of graph vertices are obtained using a GraphSAGE [4] graph neural network architecture, which aggregates information from the local neighborhood. The subsequent identification of coordination structures is performed using the density-based clustering algorithm DBSCAN [3] without an a priori specification of the number of clusters. Research results. The proposed method was evaluated on a synthetic network graph that reproduces key properties of real-world systems. Experimental results demonstrate high effectiveness of the approach: the average values of precision, recall, and F1-score for detecting coordination structures were 0.91, 0.98, and 0.94, respectively. The obtained results confirm the ability of the method to reliably identify dense coordination groups even in the absence of explicit individual anomalies. Practical significance. The proposed graph neural network–based method is universal, does not require fully labeled data, and can be integrated into large-scale network data analysis systems for detecting coordination structures in applied monitoring and information analytics tasks.
The theory of fixed points of operators and the corresponding algorithms are a powerful tool for studying nonlinear phenomena. The article considers the problems of finding fixed points of Fejer (quasi-nonexpansive) operators acting in Hilbert space, and variational inequalities on the set of fixed points. Strong convergence of the Halpern algorithm with averaging (the Halpern Suzuki algorithm) for finding fixed points of Fejer (quasi-nonexpansive) operators is proved.
This article focuses on solving the linear-quadratic (LQ) optimal control problem for a parabolic partial differential equation (PDE) operating under parametric uncertainty. To manage this uncertainty, we model the unknown parameter via a probability distribution and minimize the expected value of the cost functional. Applying a Dynamic Programming approach to this system yields an exact optimal state-feedback control law, which is governed by a infinite-dimensional Integro-Differential Riccati Equation (IDRE). Because solving this equation directly is computationally prohibitive, we employ a Spectral Galerkin method combined with a polynomial chaos expansion to approximate the stochastic parameter space. This mathematical transformation reduces the complex, stochastic IDRE into a standard Matrix Riccati Differential Equation (MRDE). By reducing the stochastic PDE control problem to an MRDE that can be solved in advance, our framework avoids the severe computational bottlenecks typically associated with uncertain environments. This provides a highly efficient baseline for robust controller design and future reinforcement learning implementations.
The aim of the article is to derive suficient conditions for stability and convergence of solutions to systems of differential equations that model neurodynamic processes, in particular the learning and operation dynamics of Hopfield neural networks. Research methodology. The analysis of asymptotic stability and convergence is based on Lyapunov's second method. Appropriate Lyapunov functions are constructed for the considered models, estimates for the derivative along trajectories are obtained, and conditions are formulated under which trajectories approach stationary regimes. Results of the research. The proposed approach extends the classical stability theory, which typically focuses on the stability of an individual trajectory (most often an equilibrium point), to the case of neurodynamic processes. Suficient conditions for asymptotic stability of equilibrium states and for convergence of solutions to these equilibria are derived for systems of difierential equations describing neurodynamic dynamics. For Hopfield network models, criteria are established that guarantee the decrease of a corresponding Lyapunov functional and, consequently, the convergence of learning and operating processes to stationary regimes. The obtained conditions are presented in a form suitable for verification in terms of model parameters and properties of the interconnection matrix. Practical significance. The derived stability and convergence criteria can be used in the design and tuning of Hopfield neural networks and related neurodynamic models to ensure guaranteed convergence of the learning process and stability of operating modes, as well as for parameter selection and validation in applied problems of information processing and optimization.
The aim of the article is to construct and analyze a statistical estimator for the real-valued impulse response function (IRF) of a time-invariant continuous linear system, and to establish its properties of asymptotic unbiasedness and consistency. The estimation of the response function is performed using a sample inputoutput cross-correlogram approach. The input signal is modeled as a stationary zero-mean Gaussian stochastic process, represented as a trimmed Fourier series using a trigonometric orthonormal basis. The study employs functional analysis and probability theory to derive upper bounds for the mathematical expectation and variance of the proposed estimator. The study establishes that the proposed integral cross-correlogram estimator is asymptotically unbiased as the cutoff level N approaches infinity. Furthermore, by evaluating the upper bounds of the bias and variance, it is proven that the estimator is consistent in the mean square sense as both the cutoff level N and the averaging interval length T approach infinity. The proposed approach offers a rigorous mathematical framework for system identification applicable to signal processing, automatic control, econometrics, and oceanology. The use of a Fourier series representation makes this method particularly effective for analyzing linear systems driven by periodic or quasi-periodic input signals.
In this paper, a new method for solving systems of nonlinear equations is introduced, obtained by combining Shamanskii’s method with line search. Local applicability and global convergence of the new method are proven. Its efficiency is demonstrated on a ten-dimensional test problem.
Most mathematical optimization models of the applied problems are multimodal. Many methods and algorithms have been developed and are being developed for their solution. To verify the effectiveness of such methods, many test functions and problems have been developed. But most of such test functions are simple. They are symmetric, of low dimension, and have known solutions. This complicates the verification of the effectiveness of existing and new methods of global optimization. The paper proposes modifications of known test functions that satisfy the efficiency conditions. The minima of these functions are found by the method of exact quadratic regularization. The results obtained are significantly better than the solutions obtained by other methods.
Automatic assembly of apictorial two-dimensional jigsaw puzzles is a typical problem of determining the correspondence between segments of two different curves (curve matching). Historically, this has been one of the earliest problems in such an important branch of computer science as pattern recognition. Curve matching is used in computer vision; reconstruction of destroyed paper documents or banknotes; archaeological restoration of mosaics, ornaments, pottery; interpretation of medical diagnostics results; and so on. A significant problem in these studies is the ordering of contour points, which are usually discrete and inaccurately determined by technical means due to the object’s volume, lighting and color peculiarities, and environmental contamination characteristics. All this necessitates smoothing and interpolation of the obtained points. However, smoothing also leads to the loss of useful information. The problem is further complicated by the fact that matching criteria are based on comparing the curvatures of adjacent lines, while the effect of measurement errors and their smoothing on calculated curvatures is insufficiently studied in the literature. The article considers a jigsaw puzzle containing 60 elements. Sides of all pieces are digitized using a medium-resolution camera. A conditional boundary is determined between the pixels of a piece and the general background. Typically, the constructed puzzle contour contains between 2000 and 3500 pixels. Then, a continuous curve is built using an original beam-based corotational spline, which treats the contour as a flexible beam, while the measurement points act as spring supports, with their stiffness controlling the degree of smoothing. A feature of the implemented method is that during calculation, the initially noisy boundary points may change their computed positions, which leads to the need for their renumbering. To address this, the concept of a projection of a measured point onto the calculated contour is introduced, defined as the nearest point, and renumbering is performed according to these projection sequences. To determine the corner points of a puzzle element, the points of maximum curvature are first found, where a so-called imaginary point is introduced. Unlike real points, here the condition of angular continuity of the contour is not fixed, and an angle jump occurs, the magnitude of which is obtained by minimizing the integral of the square of curvature in the vicinity of this vertex. The corner points divide the puzzle into four different contour segments. The correspondence between contours of different puzzle pieces (i.e., automatic assembly of the whole puzzle) is determined by minimizing the integral of the square of the difference of curvatures between two adjacent contour segments. Practical calculations and automatic puzzle assembly confirmed the effectiveness of the proposed method.
The purpose of this paper is to formulate and implement a generalized approach to the evolutionary reconstruction of stability boundaries of dynamical systems in the parameter space, based on experimental observations or numerical simulation results. The problem of stability boundary recovery is formulated as an inverse optimization problem, in which observed time series are transformed into stability indicators such as variance, autocorrelation, or generalized spectral characteristics. To approximate the bifurcation surface Γ(t), a parametric model g(λ, θ) is employed, whose parameters are adaptively updated over time using recursive or filtering algorithms. The proposed method enables the reconstruction of the dynamic stability boundary without prior knowledge of the governing equations of the system. Through examples involving a linear stochastic system, the Van der Pol oscillator, and an SIS-type model, the approach demonstrates its capability to accurately identify critical parameters and transition zones even in the presence of noise. The proposed method is applicable for the analysis and monitoring of complex technical, biological, social, and economic systems in which stability can only be indirectly assessed through experimental observations.
The aim of the article is to develop a methodology for classifying social network accounts into «bot», «non-bot», and «suspicious» categories using Multi-Criteria Decision-Making methods (MCDM). Research methodology. The study employs a hybrid MCDM approach, combining the Analytic Hierarchy Process (AHP) and entropy method to determine feature weights, and the TOPSIS method for final classification. The criteria integrate behavioral, structural, attributive, and content-based features. Results of the research. The proposed model was tested on a synthetic dataset of 100 accounts. It demonstrated high effectiveness, achieving 90% classification accuracy, with a precision of 0.85 and a recall of 0.89. The results confirm the model’s ability to reliably detect bots while minimizing false classifications of genuine users. Practical significance. The developed methodology provides a transparent, explainable, and adaptable tool for bot detection that can be integrated into social network monitoring systems, digital security tools, and information analytics platforms without the need for complete model retraining.
At the paper a linear regression model whose function has the form f(x) = ax + b, where a and b are unknown parameters, is studied. Approximate values (observations) of functions f(x) are registered at equidistant points of a line segment. It is also assumed that the covariance matrix of deviations is a tridiagonal bisymmetric matrix. In the theorem proved in the paper, necessary and sufficient condition for the elements of such matrix is found, which ensures the equality of LS and Aitken estimations both parameters of this model simultaneously. All elements are expressed in terms of two, which correspond to the variances of random deviations at the first two observation points. A sufficient condition for the connection between these two elements of the matrix to be positive definite was also found.
This paper addresses the design of a convolutional neural network architecture for processing chest X-ray images using pattern recognition methods in the context of classification into the following classes: COVID-19 viral pneumonia, non-COVID pneumonia, and absence of disease. The development of a convolutional neural network architecture is a key component of technologies for timely and accurate diagnosis of lung diseases. In this work, a CNN architecture consisting of five convolutional layers separated by pooling layers is proposed. The network was trained using a batch size of 32 and the Adam optimization algorithm, achieving an overall classification accuracy of 94%.
Cyber-physical systems generate multidimensional time series describing the state of the system. When the state of the system changes, it is necessary to detect the transition point in the time series. The article describes a new nonparametric method for detecting the transition point in multidimensional time series generated by components of cyber-system components, using the principal component analysis (PCA) as a dimensionality reduction method, and this dimensionality reduction is accompanied by the application of Petunin statistics to one-dimensional data sets. Numerical and quasi-real experiments demonstrate the high accuracy and stability of the proposed algorithm over a wide range of distributions and hypothetical examples of cyber-physical systems. The accuracy is measured by the number of steps after the transition point when it was detected. There is also a comparison with the already known methods — the Wilcoxon test and the KolmogorovSmirnov consistency test. Accuracy up to 20 steps from the transition point was achieved, and in most cases even less — no more than 10 steps. This method provides a clear and human-understandable interpretation of algorithms and their results.
The aim of the article is to construct and analyze a direct numerical method for solving systems of linear equations which are formed during numerical simulations of mass transfer process on graph. Research methodology. Proposed modification of Thomas method is based on the recursive removal of paths and cycles from the graph using the Thomas method and the cyclic Thomas method, respectively. Analysis of obtained numerical method is based on proving the main characteristics of numerical methods, such as correctness, stability, and asymptotic estimation of evaluating time. Results of the research. A direct numerical method for solving the system of linear equations on graph based on Thomas method and cyclic Thomas method is constructed. The correctness of the proposed modification is proven. A stability result for this numerical method is obtained. Asymptotic estimates of the execution time and the amount of additional memory depending on the number of graph vertices are obtained. Practical significance. Computational experiments indicate the superiority of the proposed algorithm over iterative numerical methods, so its application will positively affect the efficiency of numerical modeling of the mass transfer process on graphs.
We consider the equilibrium programming problems in 2-uniformly convex and uniformly smooth Banach spaces. We obtained a theorem about weak convergence of the two-stage proximal algorithm for pseudo-monotone equilibrium programming problems in 2-uniformly convex and uniformly smooth Banach spaces. We proposed an adaptive two-stage proximal algorithm for equilibrium programming problems. The parameter update rule does not use the values of the Lipschitz constants of the bifunction. In contrast to the rules of the linear search type, it does not require calculations of the bifunction values at additional points. For pseudo-monotone bifunctions of the Lipschitz type, we prove the theorem on weak convergence of the sequences generated by the algorithm.
This paper focuses on the robust version of the classical algorithmic problem of finding the best time to buy and sell stock. We consider its variants with different transaction limits, as well as other modifications like transaction fee or cooldown. We reduce each classical problem to its robust version, thereby obtaining lower bounds on the time complexity of all potential solutions. We extensively test all developed methods on random and adversarial data to ensure correctness and evaluate performance. We propose efficient methods for the robust counterparts of almost all problems. We also discuss suboptimal polynomial method based on dynamic programming techniques for the limited number of transactions. Developed methods are valuable in the practical applications of stock trading to obtain a minimum regret solution and evaluate the regret of an existing solution. We expect these applications of dynamic programming techniques to robust optimization problems to be relatively easy to extend and generalize for similar issues in combinatorial optimization.
This paper explores the operational principles of large language models (LLMs), focusing in particular on the mechanism of next-token generation within the process of autoregressive modeling. It outlines the theoretical foundations of neural language models, the transformer architecture with its self-attention mechanism, and the roles of tokenization and embedding in forming the input representation of text. The study analyzes the main methods for selecting the next token (greedy decoding, top-k sampling, top-p sampling, temperature), their impact on the stochasticity of results, and the trade-off between coherence and creativity. It also examines context length limitations, sources of training data, and challenges related to interpretability and the likelihood of «hallucinations». The article provides a comprehensive overview of the architectural and algorithmic foundations behind text generation in LLMs.
