Metaheuristic Optimization

In many real-world settings, problem instances that need to be solved are quite similar, and knowledge from previous optimization runs can potentially be utilized. We explore this for the Traveling Salesperson problem with time windows (TSPTW), which often arises in settings where the travel-time matrix is fixed but time-window constraints change across related tasks. Existing TSPTW studies, however, have not systematically compared solving such task sequences independently with sequential transfer from previously solved tasks. We address this gap using a multi-task benchmark in which each base instance is expanded into five related tasks under two environments: partial time-window expansion and swap-additive time reassignment. We compare a standard from-scratch protocol with an iterative protocol that initializes each task from the best tour of the previous task, using the popular local search approaches LNS, VNS, and LKH-3 under a common penalized-score objective. Our experimental results show that the iterative protocol is consistently superior in the progressive-relaxation setting and generally competitive under swap-additive changes, with improvements increasing on more difficult instances.

We present AgentGA, a framework that evolves autonomous code-generation runs by optimizing the agent seed: the task prompt plus optional parent archives that initialize a fresh workspace. The outer loop searches over these reusable starting conditions rather than editing code directly. Each generation launches a fresh autonomous run in an isolated workspace, while selected parent archives provide inherited artifacts that descendants can inspect and reuse. AgentGA couples a population-level genetic algorithm with long-horizon agents; selection uses deterministic 1:1 elite tournaments and operator allocation is adapted online with a modified Hedge controller. We instantiate the approach for tabular AutoML on the 16-competition Weco-Kaggle Lite benchmark. Across the full benchmark, AgentGA averages 71.90% Exceeds % of Human versus 51.38% for the AIDE reference, winning 15/16 competitions. Within AgentGA runs, descendants conditioned on inherited parent archives win 51.9% of 1,680 parent-child tournaments versus 8.6% for de novo proposals. These results support agent-seed optimization as a practical design choice for autonomous code-search systems.

The travelling thief problem (TTP) is a well-known multi-component optimisation problem that captures the interdependence between two components: the tour across cities and the packing of items. The packing while travelling problem (PWT) is an NP-hard subproblem of TTP where the packing of items should be optimised for a given fixed tour. In many solvers, the packing component is often addressed using greedy heuristics. Here, the use of suitable greedy functions is essential for the success of greedy algorithms. In this paper, we introduce new reward functions tailored to the PWT and extend them to a hyper-heuristic framework to achieve further advantage. Furthermore, we investigate the chance constrained PWT for greedy approaches and adopt the newly introduced reward functions for stochastic weights. The experimental results clearly demonstrate the benefit of the tailored heuristics over the standard heuristics in both deterministic and stochastic constraints.

The rapid growth of nature-inspired metaheuristics has exposed a persistent gap between metaphorical novelty and genuine algorithmic advancement. Motivated by the biophysics of chromatin loop extrusion -- a well-characterized genome-folding process driven by SMC motor complexes and conditional barriers -- we introduce the Loop-Extrusion Linkage (LEL) operator, a structure-learning wrapper that combines online variable-interaction estimation, spectral seriation via the Fiedler vector, and adaptive interval-based subspace search. LEL constructs a sparse interaction graph from successful optimization steps, derives a heuristic one-dimensional variable ordering, and generates overlapping evaluation subsets through stochastic interval growth modulated by learned boundary-crossing probabilities. We evaluate LEL on six synthetic diagnostic functions at d=96 designed to probe specific structural hypotheses -- contiguous blocks, permuted blocks, overlapping windows, banded chains, separable controls, and dense rotated couplings -- across 10^4 and 5 x 10^4 evaluation budgets with 15 independent seeds. Results are assessed via the Wilcoxon signed-rank test with Holm-Bonferroni correction and Vargha-Delaney A12 effect sizes. At 10^4 evaluations, Full LEL achieves the best median log-gap on 3 of 6 functions significantly outperforming all ablations and jSO on the structured tasks. At 5 x 10^4 evaluations, simpler ablations and baselines often surpass the full method, indicating that the adaptive barrier mechanism may over-constrain late-stage search on uniformly partitioned landscapes. The strongest supported finding is that learned spectral ordering consistently improves over graph-only grouping and random variable ordering, suggesting that interaction-graph seriation is the most valuable component of the proposed framework.

Metaheuristic algorithms such as Particle Swarm Optimization (PSO) and Evolutionary Algorithms (EA) excel at exploring solution spaces but lack mechanisms to accumulate and reuse procedural knowledge from successful search trajectories. This paper proposes Associative Constructive Evolution (ACE), a framework that enhances metaheuristics through learned generative guidance. ACE introduces a Generative Construction Automaton (GCA) -- a probabilistic model over operation sequences -- coupled with the base metaheuristic in a synergistic loop: the metaheuristic explores and provides trajectory samples, while the GCA consolidates successful patterns and guides future exploration. Three mechanisms realize this cooperation: Hebbian weight consolidation that strengthens associations between co-successful operations, guided sampling that biases search toward learned high-quality regions, and symbolic abstraction that extracts frequent patterns into reusable macro-operations. Experiments integrating ACE with EA and PSO on molecular design and maze navigation demonstrate consistent improvements. ACE-PSO achieves a 27.5% increase in success rate while reducing convergence time by 49.6%. In molecular design, ACE-EA improves fitness by 10.1% with 126 chemically interpretable macro-operations automatically discovered.

The domain of metaheuristic optimization has become vibrant due to a flood of new algorithms using a new nature-inspired metaphor but lacking clear methodological novelty. The Criticism behind the development of these algorithms has reached such an extent that the critics started to assert that all novel algorithms are only copies of already developed ones. In this study, we try to show that the situation is not so black and white. Therefore, we define a strong equivalence theorem for estimating the similarity between two nature-inspired metaheuristics, according to which two algorithms are equivalent if, and only if, the cosine similarity of their phenotypic and genotypic feature vectors, characterizing their behavior by searching for the optimal solutions, is above some threshold. On the theorem basis, a framework is developed for identifying the equivalence between nature-inspired metaheuristics. Extensive experimental work using the framework has shown that searching for conditions to achieve the high similarity of the more well-known nature-inspired metaheuristics is hard, or even not possible to achieve, in the limited computational environments.

Dogfight is a tactical behavior of cooperation between fighters. Inspired by this, this paper proposes a novel metaphor-free metaheuristic algorithm called Dogfight Search (DoS). Unlike traditional algorithms, DoS draws algorithmic framework from the inspiration, but its search mechanism is constructed based on the displacement integration equations in kinematics. Through experimental validation on CEC2017 and CEC2022 benchmark test functions, 10 real-world constrained optimization problems and mountainous terrain path planning tasks, DoS significantly outperforms 7 advanced competitors in overall performance and ranks first in the Friedman ranking. Furthermore, this paper compares the performance of DoS with 3 SOTA algorithms on the CEC2017 and CEC2022 benchmark test functions. The results show that DoS continues to maintain its lead, demonstrating strong competitiveness. The source code of DoS is available at https://ww2.mathworks.cn/matlabcentral/fileexchange/183519-dogfight-search.

In dynamic flexible job shops, order arrivals, machine breakdowns, and processing-time deviations continually reshape the scheduling state and the priority trade-offs behind dispatching decisions. Dispatching rules are well suited to this setting because they are fast, interpretable, and easy to deploy, and recent LLM-assisted automatic heuristic design further expands their expressiveness by evolving composite priority functions. The key challenge is to make these evolved rule behaviors state-adaptive without losing the rapid response needed for online rescheduling. This paper proposes a dynamic self-evolutionary framework DSevolve, which separates offline rule-library construction from online state-conditioned rule selection. Offline, an LLM-guided quality-diversity search combines multi-persona seeding, a MAP-Elites behavioral archive, and behavior-guided variation to evolve a library of complementary rules rather than a single elite, and event-level simulation then trains a neural selector to rank the rules by state. Online, after each rescheduling event, a neural selector maps a 22-dimensional state fingerprint to rule scores and dispatches the top-ranked rule within about a second, meeting the response-time requirement after each disruption. Experiments on dynamic instances derived from standard flexible job shop benchmarks show that DSevolve achieves lower mean makespan than individual LLM-evolved rules, classical dispatching rules, and learning-based baselines under a one-active-rule deployment protocol. Trained only on small instances, the selector transfers zero-shot to substantially larger dynamic shops. These results show that state-conditioned selection preserves the speed and interpretability of dispatching rules while improving adaptability.

We introduce the Free-Market Algorithm (FMA), a novel metaheuristic inspired by free-market economics. Unlike Genetic Algorithms, Particle Swarm Optimization, and Simulated Annealing -- which require prescribed fitness functions and fixed search spaces -- FMA uses distributed supply-and-demand dynamics where fitness is emergent, the search space is open-ended, and solutions take the form of hierarchical pathway networks. Autonomous agents discover rules, trade goods, open and close firms, and compete for demand with no centralized controller. FMA operates through a three-layer architecture: a universal market mechanism (supply, demand, competition, selection), pluggable domain-specific behavioral rules, and domain-specific observation. The market mechanism is identical across applications; only the behavioral rules change. Validated in two unrelated domains. In prebiotic chemistry, starting from 900 bare atoms (C, H, O, N), FMA discovers all 12 feasible amino acid formulas, all 5 nucleobases, the formose sugar chain, and Krebs cycle intermediates in under 5 minutes on a laptop -- with up to 240 independent synthesis routes per product. In macroeconomic forecasting, reading a single input-output table with zero estimated parameters, FMA achieves Mean Absolute Error of 0.42 percentage points for non-crisis GDP prediction, comparable to professional forecasters, portable to 33 countries. Assembly Theory alignment shows that FMA provides the first explicit, tunable mechanism for the selection signatures described by Sharma et al. (Nature, 2023). The event-driven assembly dynamics resonate with foundational programs in physics -- causal set theory, relational quantum mechanics, constructor theory -- suggesting that Darwinian market dynamics may reflect a deeper organizational principle that lead to the unfolding of Nature itself.

Several real-world optimization problems involve mixed-variable search spaces, where continuous, ordinal, and categorical decision variables coexist. However, most population-based metaheuristic algorithms are designed for either continuous or discrete optimization problems and do not naturally handle heterogeneous variable types. In this paper, we propose an adaptation of the Firefly Algorithm for mixed-variable optimization problems (FAmv). The proposed method relies on a modified distance-based attractiveness mechanism that integrates continuous and discrete components within a unified formulation. This mixed-distance approach enables a more appropriate modeling of heterogeneous search spaces while maintaining a balance between exploration and exploitation. The proposed method is evaluated on the CEC2013 mixed-variable benchmark, which includes unimodal, multimodal, and composition functions. The results show that FAmv achieves competitive, and often superior, performance compared with state-of-the-art mixed-variable optimization algorithms. In addition, experiments on engineering design problems further highlight the robustness and practical applicability of the proposed approach. These results indicate that incorporating appropriate distance formulations into the Firefly Algorithm provides an effective strategy for solving complex mixed-variable optimization problems.

This paper addresses the Service Network Design (SND) problem for a logistics service provider (LSP) operating in a multimodal freight transport network, considering uncertain travel times and limited truck fleet availability. A two-stage optimization approach is proposed, which combines metaheuristics, simulation and machine learning components. This solution framework integrates tactical decisions, such as transport request acceptance and capacity booking for scheduled services, with operational decisions, including dynamic truck allocation, routing, and re-planning in response to disruptions. A simulated annealing (SA) metaheuristic is employed to solve the tactical problem, supported by an adaptive surrogate model trained using a discrete-event simulation model that captures operational complexities and cascading effects of uncertain travel times. The performance of the proposed method is evaluated using benchmark instances. First, the SA is tested on a deterministic version of the problem and compared to state-of-the-art results, demonstrating it can improve the solution quality and significantly reduce the computational time. Then, the proposed SA is applied to the more complex stochastic problem. Compared to a benchmark algorithm that executes a full simulation for each solution evaluation, the learning-based SA generates high quality solutions while significantly reducing computational effort, achieving only a 5% difference in objective function value while cutting computation time by up to 20 times. These results demonstrate the strong performance of the proposed algorithm in solving complex versions of the SND. Moreover, they highlight the effectiveness of integrating diverse modeling and optimization techniques, and the potential of such approaches to efficiently address freight transport planning challenges.

Unmanned aerial vehicles (UAVs) have been widely used in urban missions, and proper planning of UAV paths can improve mission efficiency while reducing the risk of potential third-party impact. Existing work has considered all efficiency and safety objectives for a single decision-maker (DM) and regarded this as a multiobjective optimization problem (MOP). However, there is usually not a single DM but two DMs, i.e., an efficiency DM and a safety DM, and the DMs are only concerned with their respective objectives. The final decision is made based on the solutions of both DMs. In this paper, for the first time, biparty multiobjective UAV path planning (BPMO-UAVPP) problems involving both efficiency and safety departments are modeled. The existing multiobjective immune algorithm with nondominated neighbor-based selection (NNIA), the hybrid evolutionary framework for the multiobjective immune algorithm (HEIA), and the adaptive immune-inspired multiobjective algorithm (AIMA) are modified for solving the BPMO-UAVPP problem, and then biparty multiobjective optimization algorithms, including the BPNNIA, BPHEIA, and BPAIMA, are proposed and comprehensively compared with traditional multiobjective evolutionary algorithms and typical multiparty multiobjective evolutionary algorithms (i.e., OptMPNDS and OptMPNDS2). The experimental results show that BPAIMA performs better than ordinary multiobjective evolutionary algorithms such as NSGA-II and multiparty multiobjective evolutionary algorithms such as OptMPNDS, OptMPNDS2, BPNNIA and BPHEIA.

Selecting the most relevant or informative features is a key issue in actual machine learning problems. Since an exhaustive search is not feasible even for a moderate number of features, an intelligent search strategy must be employed for finding an optimal subset, which implies considering how features interact with each other in promoting class separability. Balancing feature subset size and classification accuracy constitutes a multi-objective optimization challenge. Here we propose MOELIGA, a multi-objective genetic algorithm incorporating an evolutionary local improvement strategy that evolves subordinate populations to refine feature subsets. MOELIGA employs a crowding-based fitness sharing mechanism and a sigmoid transformation to enhance diversity and guide compactness, alongside a geometry-based objective promoting classifier independence. Experimental evaluation on 14 diverse datasets demonstrates MOELIGA's ability to identify smaller feature subsets with superior or comparable classification performance relative to 11 state-of-the-art methods. These findings suggest MOELIGA effectively addresses the accuracy-dimensionality trade-off, offering a robust and adaptable approach for multi-objective feature selection in complex, high-dimensional scenarios.

Combinatorial optimization problems arise in logistics, scheduling, and resource allocation, yet existing approaches face a fundamental trade-off among generality, performance, and usability. We present cuGenOpt, a GPU-accelerated general-purpose metaheuristic framework that addresses all three dimensions simultaneously. At the engine level, cuGenOpt adopts a "one block evolves one solution" CUDA architecture with a unified encoding abstraction (permutation, binary, integer), a two-level adaptive operator selection mechanism, and hardware-aware resource management. At the extensibility level, a user-defined operator registration interface allows domain experts to inject problem-specific CUDA search operators. At the usability level, a JIT compilation pipeline exposes the framework as a pure-Python API, and an LLM-based modeling assistant converts natural-language problem descriptions into executable solver code. Experiments across five thematic suites on three GPU architectures (T4, V100, A800) show that cuGenOpt outperforms general MIP solvers by orders of magnitude, achieves competitive quality against specialized solvers on instances up to n=150, and attains 4.73% gap on TSP-442 within 30s. Twelve problem types spanning five encoding variants are solved to optimality. Framework-level optimizations cumulatively reduce pcb442 gap from 36% to 4.73% and boost VRPTW throughput by 75-81%. Code: https://github.com/L-yang-yang/cugenopt

This paper presents the constrained Hybrid Metaheuristic (cHM) algorithm as a general framework for continuous optimisation. Unlike many existing metaheuristics that are tailored to specific function classes or problem domains, cHM is designed to operate across a broad spectrum of objective functions, including those with unknown, heterogeneous, or complex properties such as non-convexity, non-separability, and varying smoothness. We provide a formal description of the algorithm, highlighting its modular structure and two-phase operation, which facilitates dynamic adaptation to the problem's characteristics. A key feature of cHM is its ability to harness synergy between both candidate solutions and component metaheuristic strategies. This property allows the algorithm to apply the most appropriate search behaviour at each stage of the optimisation process, thereby improving convergence and robustness. Our extensive experimental evaluation on 28 benchmark functions demonstrates that cHM consistently matches or outperforms traditional metaheuristics in terms of solution quality and convergence speed. In addition, a practical application of the algorithm is demonstrated for a feature selection problem in the context of data classification. The results underscore its potential as a versatile and effective black-box optimiser suitable for both theoretical research and practical applications.

The Capacitated Vehicle Routing Problem (CVRP), a fundamental combinatorial optimization challenge, focuses on optimizing fleet operations under vehicle capacity constraints. While extensively studied in operational research, the NP-hard nature of CVRP continues to pose significant computational challenges, particularly for large-scale instances. This study presents AILS-AHD (Adaptive Iterated Local Search with Automatic Heuristic Design), a novel approach that leverages Large Language Models (LLMs) to revolutionize CVRP solving. Our methodology integrates an evolutionary search framework with LLMs to dynamically generate and optimize ruin heuristics within the AILS method. Additionally, we introduce an LLM-based acceleration mechanism to enhance computational efficiency. Comprehensive experimental evaluations against state-of-the-art solvers, including AILS-II and HGS, demonstrate the superior performance of AILS-AHD across both moderate and large-scale instances. Notably, our approach establishes new best-known solutions for 8 out of 10 instances in the CVRPLib large-scale benchmark, underscoring the potential of LLM-driven heuristic design in advancing the field of vehicle routing optimization.

Mixed-Integer Programs (MIPs) are NP-hard optimization models that arise in a broad range of decision-making applications, including finance, logistics, energy systems, and network design. Although modern commercial solvers have achieved remarkable progress and perform effectively on many small- and medium-sized instances, their performance often degrades when confronted with large-cale or highly constrained formulations. This paper explores the use of the Random-Key Optimizer (RKO) framework as a flexible, metaheuristic alternative for computing high-quality solutions to MIPs through the design of problem-specific decoders. The proposed approach separates the search process from feasibility enforcement by operating in a continuous random-key space while mapping candidate solutions to feasible integer solutions via efficient decoding procedures. We evaluate the methodology on two representative and structurally distinct benchmark problems: the mean-variance Markowitz portfolio optimization problem with buy-in and cardinality constraints, and the Time-Dependent Traveling Salesman Problem. For each formulation, tailored decoders are developed to reduce the effective search space, promote feasibility, and accelerate convergence. Computational experiments demonstrate that RKO consistently produces competitive, and in several cases superior, solutions compared to a state-of-the-art commercial MIP solver, both in terms of solution quality and computational time. These results highlight the potential of RKO as a scalable and versatile heuristic framework for tackling challenging large-scale MIPs.

The optimization of complex medical appointment scheduling remains a significant operational challenge in multi-center healthcare environments, where clinical safety protocols and patient logistics must be reconciled. This study proposes and evaluates a Genetic Algorithm (GA) framework designed to automate the scheduling of multiple medical acts while adhering to rigorous inter-procedural incompatibility rules. Using a synthetic dataset encompassing 50 medical acts across four healthcare facilities, we compared two GA variants, Pre-Ordered and Unordered, against deterministic First-Come, First-Served (FCFS) and Random Choice baselines. Our results demonstrate that the GA framework achieved a 100% constraint fulfillment rate, effectively resolving temporal overlaps and clinical incompatibilities that the FCFS baseline failed to address in 60% and 40% of cases, respectively. Furthermore, the GA variants demonstrated statistically significant improvements (p < 0.001) in patient-centric metrics, achieving an Idle Time Ratio (ITR) frequently below 0.4 and reducing inter-healthcenter trips. While the GA (Ordered) variant provided a superior initial search locus, both evolutionary models converged to comparable global optima by the 100th generation. These findings suggest that transitioning from manual, human-mediated scheduling to an automated metaheuristic approach enhances clinical integrity, reduces administrative overhead, and significantly improves the patient experience by minimizing wait times and logistical burdens.

Multi-depot vehicle routing problems (MDVRPs) are prevalent in a variety of practical applications. However, they are computationally challenging to solve due to their inherent complexity. This paper proposes an effective hybrid algorithm for a class of MDVRPs. The algorithm integrates a learning-driven, diversity-controlled route-exchange crossover and a multi-depot-supported feasible-and-infeasible search framework guided by a multi-penalty evaluation function. Two dedicated depot-related local search operators are incorporated to further strengthen the search capability in multi-depot settings. To improve computational efficiency and scalability, an enhanced version of the algorithm is developed that uses a tensor-based GPU acceleration combined with a novel multi-move update strategy. Extensive computational experiments on benchmark instances of three MDVRP variants show that the proposed algorithms are highly competitive with state-of-the-art methods, especially for large-scale instances.

LLM-guided evolutionary computation, most notably AlphaEvolve, has been remarkably successful in discovering novel mathematical constructions by solving challenging optimization problems. The standard approach is to evolve a monolithic program that directly outputs a candidate solution. We present ImprovEvolve, an algorithmic alternative that drastically reduces cognitive load on the LLM. Instead of prompting the model for an end-to-end optimizer, we evolve a program with three specialized operators of initialization, local improvement, and perturbation. We then approach the optimum by iteratively applying local improvements and intensity-scheduled perturbations, effectively driving a basin-hopping search with LLM-evolved subroutines. For hexagon in hexagon packing, ImprovEvolve discovers new state-of-the-art packings of 11, 12, 15, and 16 hexagons, and additionally for 14, 17, and 23 hexagons after minimal expert tuning of the generated code. For the second autocorrelation inequality, the evolved and human-scaled program pushes the lower bound from 0.96102 to 0.96258. For spherical codes, the ImprovEvolve program lowers the best-known maximum cosine for the majority of 90 randomly chosen diverse state-of-the-art spherical codes, achieving relative improvements of up to 2.4%.

Over the past two decades, research in evolutionary multi-objective optimization has predominantly focused on continuous domains, with comparatively limited attention given to multi-objective combinatorial optimization problems (MOCOPs). Combinatorial problems differ significantly from continuous ones in terms of problem structure and landscape. Recent studies have shown that on MOCOPs multi-objective evolutionary algorithms (MOEAs) can even be outperformed by simple randomised local search. Starting with a randomly sampled solution in search space, randomised local search iteratively draws a random solution (from an archive) to perform local variation within its neighbourhood. However, in most existing methods, the local variation relies on a fixed neighbourhood, which limits exploration and makes the search easy to get trapped in local optima. In this paper, we present a simple yet effective local search method, called variable stepsize randomized local search (VS-RLS), which adjusts the stepsize during the search. VS-RLS transitions gradually from a broad, exploratory search in the early phases to a more focused, fine-grained search as the search progresses. We demonstrate the effectiveness and generalizability of VS-RLS through extensive evaluations against local search and MOEAs methods on diverse MOCOPs.

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.

This paper presents a large-scale budget-aware benchmark of multi-objective soft-computing metaheuristics for a bi-objective redundancy allocation problem in repairable k-out-of-n systems. The problem combines cost minimization and steady-state availability maximization under weight constraints, with binary subsystem-level decisions determining both the number of redundant components and the redundancy strategy. Four strategies are considered: cold standby, warm standby, hot standby, and a mixed active-warm standby strategy. Subsystem availability is evaluated using continuous-time Markov chains, while the optimization task is treated as a constrained mixed-integer multi-objective problem. The study compares 65 metaheuristic algorithms from multiple algorithmic families under two initialization settings, with and without Scaled Binomial Initialization (SBI), across six case studies of increasing structural and dimensional complexity and four weight limits per case. Performance is assessed using hypervolume, budget-dependent convergence profiles, and non-parametric statistical comparisons. The results show that algorithm rankings are strongly budget-dependent, indicating that a single final-budget ranking can be misleading. SBI provides a substantial early advantage and can change the relative performance of competing methods, especially for larger instances. The best-performing algorithms vary across budget regimes: NNIA-SBI and CMOPSO-SBI are competitive under tight budgets, whereas NSGA-II+ARSBX-SBI performs robustly for medium and large budgets. From a system design perspective, Pareto-optimal solutions are dominated by hot standby and mixed redundancy strategies, while cold and warm standby rarely appear. The benchmark highlights the importance of initialization, computational budget, and problem complexity when selecting soft-computing optimizers for practical redundancy allocation.

Designing high-performing heuristics for vehicle routing problems (VRPs) is a complex task that requires both intuition and deep domain knowledge. Large language model (LLM)-based code generation has recently shown promise across many domains, but it still falls short of producing heuristics that rival those crafted by human experts. In this paper, we propose VRPAgent, a framework for VRPs that integrates LLM-generated components into a metaheuristic and refines them through a novel genetic search. By using the LLM to generate problem-specific operators, embedded within a shared metaheuristic framework for VRPs, VRPAgent keeps tasks manageable, guarantees correctness, and still enables the discovery of novel and powerful strategies. Across multiple VRPs, including the capacitated VRP, the VRP with time windows, and the prize-collecting VRP, our method discovers heuristic operators that outperform handcrafted methods and recent learning-based approaches while requiring only a single CPU core. To our knowledge, VRPAgent is one of the first LLM-based methods to advance the state-of-the-art in VRPs, highlighting a promising future for automated heuristics discovery.

Swarm-based optimization algorithms have demonstrated remarkable success in solving complex problems, yet their widespread adoption remains limited due to poor transparency in how algorithmic components influence performance. This work presents a multi-faceted explainability framework for Particle Swarm Optimization (PSO) through two complementary perspectives: landscape-based and algorithmic explainability. From the landscape-based perspective, we develop a comprehensive characterization framework using Exploratory Landscape Analysis (ELA) to quantify problem difficulty, multimodality, and ruggedness, extracting ELA meta-features, dispersion measures, and information content statistics, while a machine learning approach employing Decision Tree and Random Forest classifiers enables prediction of optimal topology-specific hyperparameter configurations for unseen problems. From the algorithmic explainability perspective, we integrate IOHxplainer for temporal convergence profiling and Search Trajectory Networks (STN) for spatial navigation mapping, proposing three novel STN metrics-Connectivity Density, Fragmentation Score, and Search Efficiency-that enhance visual explainability by quantifying topology-specific search organization and transition effectiveness. Through systematic experimentation across 24 benchmark functions in multiple dimensions with Star, Ring, and Von Neumann topologies, we establish practical guidelines for topology selection and parameter configuration. Our findings uncover the black-box nature of PSO, providing greater transparency and interpretability to swarm intelligence systems. The source code is available at https://github.com/GitNitin02/ioh_pso.