Search Results (157)

This paper addresses a human resource allocation problem with spatial-temporal constraints (HRAP-SC) in the parallel assembly of complex products, such as satellites and aircraft. It involves coordinating a limited pool of multi-skilled workers across geographically distributed workstations, subject to rigorous constraints including team collaboration requirements, operation priorities, technological tail times (e.g., curing), and strict 8 h workdays. Existing exact approaches typically fail to converge due to the combinatorial explosion arising from the strong coupling of shared resources across workstations, while meta-heuristic methods often suffer from performance instability caused by hyper-parameter sensitivity. To overcome these limitations, we propose a pattern-based decomposition algorithm (PDA), a novel parameter-free exact solution framework. By exploiting the inherent symmetry of identical jobs and parallel workstations, PDA defines a set of canonical patterns to drastically reduce the search space. It employs an efficient traversal mechanism reinforced by rigorous mathematical bounds and pruning rules to eliminate unpromising solutions. Computational experiments demonstrate that PDA significantly outperforms state-of-the-art Mixed-Integer Programming (MIP) and Constraint Programming (CP) solvers. Unlike standard solvers, which frequently time out (3600 s), PDA strictly evaluates only a single pattern when proving optimality, and robustly scales to large industrial instances (e.g., six jobs comprising 78 operations) to provide high-quality schedules. By successfully solving complex scheduling problems that remain intractable for monolithic solvers, PDA provides a robust and automated decision-support tool for production management in complex manufacturing systems. Full article

This paper presents a mathematical reinterpretation of virtual multi-vector synthesis defined over finite vector sets. Unlike conventional approaches that treat multi-vector synthesis as an algorithmic technique, the proposed framework characterizes it as a structured problem combining convex geometry, combinatorial selection, and probabilistic averaging. First, it is shown that the set of all realizable virtual vectors coincides with the convex hull of a finite vector set, providing a geometric interpretation of the synthesis process. Based on this observation, a subset-based formulation is introduced, in which virtual vectors are constructed as averages over selected subsets. This formulation allows the synthesis problem to be interpreted as a combinatorial selection problem. Under a uniform subset selection model, closed-form expressions for the expectation and variance of the synthesized vectors are derived. In particular, it is demonstrated that the approximation behavior can be interpreted through the variance structure of subset-averaged vectors, and that increasing the subset size leads to a systematic reduction in variance. Furthermore, the trade-off between approximation accuracy and combinatorial complexity is analyzed, and the existence of an optimal subset size is established. The proposed framework provides a theoretical foundation for understanding multi-vector synthesis as a structured mathematical process, and offers a general perspective applicable to a wide class of approximation problems over finite vector sets. Full article

The main objective of this study was to reduce the dwell time of wagons at stations and to improve the efficiency of shunting locomotive utilization. This is a combinatorial problem, since an increase in the number of loading and unloading fronts leads to a sharp growth in the number of feasible service variants. During the research, a mathematical model describing the servicing process of industrial sidings was developed. This study addressed the problem of determining the optimal sequence of wagon deliveries and the optimal distribution of workload among shunting locomotives. For conditions under which two or more shunting locomotives are used, an optimization method based on the indicator of wagon-hour reduction (σ) was proposed for allocating loading and unloading fronts. Using combinatorial properties, it was shown that many possible allocation variants are symmetric, which allowed for the development of a mathematical solution that simplifies the search for an optimal solution. Computational results demonstrated that, at the hypothetical railway station “N-1”, applying the optimal service sequence reduces wagon dwell time by 21% compared with an arbitrary sequence. At the hypothetical station “N-2”, distributing wagon groups between two shunting locomotives improves the efficiency of the servicing process by 26% compared with using a single locomotive. The results based on real data from the “B-2” railway station show that the proposed method provides an improvement of approximately 31.3% compared to the current operational practice, while Smith’s rule achieves an improvement of 14.9%. Based on the proposed model and algorithm, a software tool was developed to automatically determine servicing sequences for loading and unloading fronts, analyze alternatives, and evaluate shunting locomotive efficiency. Full article

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models. Full article

This work presents a classical theoretical framework in which combinatorial optimization emerges from the nonlinear relaxation of coupled real-valued phase fields governed by a global Lyapunov energy functional. Each computational element (CF-bit) evolves in a bistable periodic potential while pairwise interactions encode problem-specific couplings, enabling gradient-descent minimization of QUBO and Ising objective functions. The key contribution is an explicit global energy functional from which all dynamics are derived, guaranteeing monotonic energy descent under damping. This distinguishes the approach from several existing oscillator-based Ising architectures where the governing dynamics contain non-gradient terms and an explicit global Lyapunov functional has not been derived in their standard formulations. Numerical simulations on instances up to 20 bits demonstrate deterministic phase-locking convergence, with optional transient noise improving the exploration of rugged landscapes. While limited in scale and not overcoming NP-hardness, this work provides a conceptual framework showing how discrete optimization can emerge from continuous classical dynamics with a mathematically transparent energy structure. Full article

This paper develops a rigorous mathematical and computational framework for four-dimensional chess defined on the discrete hypercubic lattice $\{1,\dots , 8{\}}^4$. We formalize piece movement using displacement sets in ${\mathbb{Z}}^4$, define adjacency via the Chebyshev metric, and analyze the resulting move graphs for rooks, bishops, knights, queens, and kings. We establish exact mobility formulas, parity invariants, and connectivity properties, consolidating known product-graph results for rooks and kings while introducing a boundary-sensitive analysis of the four-dimensional knight verified by exhaustive enumeration. The mathematical framework is complemented by a fully implemented 4D chess engine and interactive visualization environment rendering all 64 (z,w)-slices of the hypercube simultaneously. The system supports full move legality, generalized special rules, multi-king checkmate detection, and reproducible state enumeration. Performance measurements and exploratory branching-factor estimates are obtained through reproducible random playouts using the publicly available implementation. We contextualize this ruleset within existing work on move graphs on ${\mathbb{Z}}_n^m$, higher-dimensional leapers, spectral properties of grid graphs, toroidal analogs, and multidimensional visualization. Exploratory qualitative feedback (N = 18) is included to examine whether the visualization design is interpretable and navigable in practice, providing feasibility-oriented observations on how slice-based 4D projection and layered board rendering are perceived by non-expert users in an exploratory context. Together, the mathematical results, implemented engine, and visualization form a coherent foundation for the study of strategy, complexity, and human interaction in four-dimensional game systems. The framework provides a basis for future investigations into spectral analysis of move graphs, symmetry-aware search, hierarchical planning, and educational applications in high-dimensional geometry. Full article

The Graph Burning Problem (GBP) is a combinatorial optimization problem that has gained relevance as a tool for quantifying a graph’s vulnerability to contagion. Although it is based on a very simple propagation model, its decision version is NP-complete and its optimization version is NP-hard. This paper introduces novel mathematical programs for the GBP. Among the introduced programs are a Mixed-Integer Linear Program (MILP), a Constraint Satisfaction Problem (CSP), two Integer Linear Programs (ILPs), and two Quadratic Unconstrained Binary Optimization (QUBO) problems. Most optimization solvers can handle these, with QUBO problems being of capital interest in quantum computing. Nonetheless, the primary objective of this paper is not to solve instances of the GBP, but rather to deepen our understanding of it by identifying and examining what we believe to be its simplest mathematical formulations, that is, models that use as few variables and constraints as possible (compact mathematical programs). We believe that this collection of programs can provide ideas for modeling variants and related problems. As a marginal result, one of the proposed ILPs, equipped with a row generation technique, allowed a commercial solver to find optimal solutions for some of the largest and most challenging instances for the GBP. Full article

This study proposes an end-to-end mathematical framework to automatically transform warehouse layout images into optimization-ready route matrices. The objective is to convert visual spatial information into a discrete, graph-based representation suitable for combinatorial route optimization. The problem is formulated as a mapping from continuous image space to a structured grid representation, integrating image segmentation, graph construction, and Traveling Salesman Problem (TSP)-based routing. Synthetic warehouse layouts were generated to create labeled training data, and a U-Net convolutional neural network was trained to perform multi-class segmentation of warehouse elements. The predicted grid representation was then converted into a graph structure, where feasible cells define vertices and adjacency defines edges. Shortest path distances were computed using Breadth-First Search, and the resulting distance matrix was used to solve a TSP instance. The segmentation model achieved approximately 98% training accuracy and 95–97% validation accuracy. The generated route matrices enabled successful construction of feasible and optimal round-trip routes in all tested scenarios. The proposed framework demonstrates that warehouse layouts can be automatically transformed into discrete mathematical representations suitable for logistics optimization, reducing manual preprocessing and enabling scalable integration into digital logistics systems. Full article

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—${\Gamma }_E$, ${\Gamma }_0$, ${\Gamma }_{\mathrm{Cay}}$, ${\Gamma }_{\mathcal{K}}$, and $\Gamma \left(G_k\right)$—each encoding a distinct algebraic facet of $S{B}_i(\bowtie )B$. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and $\mathcal{D}$-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative $S{B}_i(\bowtie )B$, $({\Gamma }_E,{\Gamma }_{\mathcal{K}})$ uniquely determines $\mathcal{J}$-order, $\mathcal{D}$-classes, and $\mathcal{H}$-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

In the field of structural health monitoring systems, sensors serve as the fundamental components for assessing infrastructure integrity. The rationality of their spatial configuration significantly influences the precision of structural performance assessment, the efficacy of damage detection algorithms, and the operational reliability of the system throughout its designated lifecycle. A robust optimization methodology for the placement of multi-type sensors is proposed in this study, explicitly formulated to mitigate the negative impact of sensor malfunctions during long-term operation. First, a rigorous evaluation framework for sensor placement schemes is established based on Bayesian inference and the minimization of information entropy, thereby quantifying the uncertainty inherent in parameter identification. Then, a probabilistic model of sensor failure is developed utilizing the Weibull distribution to capture time-dependent reliability characteristics, combined with a modified information entropy calculation method that mathematically assimilates these failure probabilities into the optimization objective. Finally, a heuristic search strategy is employed to achieve the robust optimal placement of multi-type sensors, efficiently navigating the complex combinatorial search space. In contrast to deterministic information entropy (DIE) methodologies, which assume ideal sensor functionality, the robust information entropy (RIE) approach comprehensively accounts for the stochastic nature of sensor failures and their impact on the information content of the monitoring network, thereby significantly augmenting the robustness and redundancy of the sensor configuration. Validations utilizing a numerical frame structure and a finite element bridge model demonstrate that the RIE method effectively integrates the sensor failure probability model to yield robust optimal placement schemes, minimizing the risk of information loss and ensuring reliable structural health monitoring throughout the engineering lifecycle. Full article

Reconstructing a genome from collections of short DNA fragments is a fundamental problem in modern sequencing. Although genome assembly algorithms are widely used in practice, the mathematical conditions that allow exact reconstruction are not always clear. This study develops a graph-theoretic framework for genome reconstruction using De Bruijn graphs and Eulerian paths in an idealized, error-free setting. Each k-mer is represented as a directed edge connecting its $(k-1)$-length prefix and suffix. The resulting overlap graph is constructed using a balanced search tree and traversed with a stack-based Eulerian algorithm. Numerical experiments over a broad range of genome lengths and fragment lengths reveal a sharp transition in reconstruction accuracy. This transition is explained by a probabilistic model for prefix collisions in the directed graph. The theoretical predictions agree with simulation results and provide conditions on the fragment length required for reliable reconstruction. These results show that the difficulty of genome assembly is governed primarily by the combinatorial structure of the underlying graph rather than by algorithmic heuristics. Full article

Static honeypot deployment and one-shot attack-path analysis often become ineffective against adaptive adversaries because fixed decoy layouts are easy to fingerprint and risk estimates quickly go stale. This paper presents a unified, mathematically grounded TrapManager framework that couples graph representation learning with budget-constrained combinatorial optimization for dynamic cyber deception. We model attacker progression on vulnerability-based attack graphs and learn context-aware node embeddings using a Graph Attention Network (GAT) that fuses vulnerability-driven risk signals (e.g., CVSS-derived node scores) with structural features. The learned representations are used to estimate edge plausibility and rank candidate source–target routes at the path level. Given limited resources, we formulate pointTrap placement as a Mixed-Integer Programming (MIP) problem that maximizes the expected interception of high-risk paths while penalizing deployment cost under explicit budget constraints, including mandatory coverage of the top-ranked critical paths. To enable online adaptiveness, a pointTrap-triggered, event-driven feedback mechanism locally amplifies risk around alerted regions, updates path weights without retraining the GAT, and re-solves the MIP for rapid redeployment. Experiments on MulVAL-generated benchmark attack graphs and cross-domain transfer settings demonstrate fast convergence, strong discrimination between attack and non-attack edges, and early interception within a small number of hops even with minimal decoy budgets. Overall, the proposed framework provides a scalable and resource-efficient approach to closed-loop attack-path defense by integrating attention-based learning and integer optimization. Full article

Digital signal processing plays a central role in intelligent railway communications under high-mobility, strong-multipath, and time-varying-channel conditions. This review surveys representative techniques for multi-carrier modulation, precoding, index modulation, and chaos-inspired physical layer security and highlights their mathematical foundations. Core themes include transform-domain representations typified by time–frequency analysis, linear-algebraic formulations of precoding and equalization, combinatorial structures underlying index mapping and spectral efficiency gains, and nonlinear dynamical systems theory of chaotic encryption. The methods are compared in terms of bit error performance, peak-to-average power ratio, spectral efficiency, computational complexity, and information security, with emphasis on railway-specific deployment constraints. The synergistic application of these methods with intelligent railway transportation systems is expected to enhance the overall performance of railway transportation systems in terms of transmission efficiency, reliability, and security. It provides critical technological support for the efficient and secure operation of next-generation intelligent transportation systems. Full article

This Editorial introduces the Symmetry Special Issue “Theory and Applications of Special Functions II” and summarizes the nine contributions collected therein. The papers span the analytic continuation of multivariate hypergeometric functions; stability theory for differential equations via integral transforms; numerical schemes for multi-space fractional partial differential equations based on nonstandard finite differences and orthogonal polynomials; applications of the Lambert W function to viscoelastic creep modeling; algebraic constructions of new Hermite-type polynomial families via the monomiality principle; higher-level generalizations of poly-Cauchy numbers; Bell-polynomial expansions for Laplace transforms of higher-order nested functions; and two complementary studies on the physical implementation and algebraic description of Gaussian quantum states. Beyond the contributions of the Special Issue, we highlight methodological connections—continued fractions and complex analysis, transform techniques, special polynomials, and combinatorial sequences—and emphasize the unifying role of symmetry across mathematical structures and applications. Full article