The flexible job shop scheduling problem (FJSP) traditionally emphasizes machine flexibility, while worker heterogeneity is often simplified or neglected. In practical manufacturing systems, worker skill levels affect both processing efficiency and task feasibility, leading to the skilled-worker constrained flexible job shop scheduling problem (SWC-FJSP). This article investigates the SWC-FJSP with multi-level worker skills and skill-dependent processing efficiency, aiming to minimize the makespan. A mixed-integer linear programming model is first formulated to characterize the coupled operation–machine–worker assignment decisions. Then, a branching deep Q-network enhanced memetic algorithm, termed MABDQN, is developed as a learning-assisted evolutionary framework. MABDQN integrates an adaptive hybrid initialization strategy to generate high-quality and diverse populations, a BDQN-driven operator orchestration mechanism to select operator category and execution intensity according to the evolving search state, and a multi-stage adaptive restart strategy guided by a stage adaptive reward function to alleviate premature convergence. Extensive experiments on a newly constructed test set with different instance scales show that MABDQN outperforms several state-of-the-art algorithms in solution quality and robustness. Ablation studies further verify the contribution of the main components and demonstrate the effectiveness of state conditioned operator orchestration for complex SWC-FJSP instances.

Efficient transient scheduling of cluster tools is critical to stable fab-level operations in high-mix, low-volume semiconductor manufacturing. This study investigates the transient scheduling problem of dual-armed cluster tools (DACTs) processing two wafer types concurrently. From an engineering-management perspective, the problem is important because wafer release and robot sequencing decisions affect tool utilization, completion-time predictability, and fab-level responsiveness. A mixed-integer programming model is developed to jointly optimize wafer release order and robot task sequence with the objective of minimizing total completion time. To address the computational intractability of medium- and large-scale instances, a two-combined swap sequence (2-CSS) is proposed. Numerical results show that 2-CSS substantially reduces computation time while incurring only a 5.31% average increase in total completion time. The results also show that DACTs provide substantial throughput benefits over single-armed configurations, with improvements of up to 9.51%. Robustness analysis further shows that the observed dual-arm advantage persists across varying robot loading/unloading times and wafer residency time constraint-to-processing time ratios, while 2-CSS maintains small optimality gaps as the loading/unloading time and the processing time of a selected step vary, even when the system bottleneck shifts between processing steps. These findings can help fab managers evaluate transient schedules more quickly and coordinate tool operations more effectively in high-mix production.

With the increasing complexity of modern manufacturing processes, the demand for multi-shop joint scheduling has become increasingly important. The cascaded flowshop joint scheduling problem (CFJSP) has gained widespread attention due to its alignment with practical production needs. However, related research on this problem remains limited. Therefore, this paper proposes an elite-preserving iterated greedy with Q-learning (EIGQ) algorithm for solving the CFJSP, with the objective of minimizing the makespan. We introduce Q-learning into the perturbation mechanisms of the two phases, enabling the algorithm to adaptively select perturbation operators based on search status and historical performance. Four neighborhood operators are designed to balance global exploration and local exploitation. In addition, a strategy combining elite preservation and dynamic population injection is proposed to enhance the exploration of the solution space. Finally, the comprehensive experiments validate the effectiveness of the key components of the EIGQ algorithm and demonstrate its significant superiority over six state-of-the-art methods based on the results of 1200 test instances.

In recent years, the distributed permutation flowshop scheduling problem (DPFSP) has attracted increasing attention. However, due to machine upgrades or replacements, the processing time of the same stage for the same job may vary across different factories, leading to the DPFSP problem with heterogeneous characteristics. The release time is an important production constraint in real manufacturing, which reflects the completion status of the preparation work before production. Although the total weighted completion time is a significant objective, it has been less studied in the Distributed Heterogeneous Permutation Flowshop scheduling problem with Release Time (DHPFRT). Therefore, the DHPFRT problem is studied with the aim of minimizing the total weighted completion time. For solving the DHPFRT problem, a mathematical model is constructed and heuristic algorithms based on multi-stage or adaptive strategies are proposed. The proposed heuristic algorithms mainly include heuristic algorithms based on rules, two-stage heuristic algorithms with adaptability, and three-stage heuristic algorithms with adaptability. The proposed algorithms can obtain good solutions in a reasonable amount of time, which are very valuable for an actual production scheduling problem. Comprehensive experimental results demonstrate that the proposed algorithms exhibit significant advantages for solving the DHPFRT problem with the total weighted completion time criterion.

Networked cloud manufacturing requires coordinated scheduling of production and communication resources. However, operation assignment and network transmission are often modeled as weakly coupled decisions, making cyberphysical interactions difficult to capture. This paper investigates a networked dual-resource cloud manufacturing scheduling problem (N-DRCSP), where operation sequencing, processing node selection, machine and worker allocation, and transmission under network constraints are jointly optimized. A multiobjective mixed-integer linear programming model is formulated under a hierarchical hybrid control architecture to minimize total weighted tardiness and network load risk. To solve this large and strongly coupled problem, a bi-population heterogeneous graph attention network assisted non-dominated sorting genetic algorithm II, termed BP-HAN-NSGAII, is developed. Heterogeneous graph attention encodes operation-machine relations determined by worker availability, and the associated network states, while the bi-population mechanism balances exploitation and exploration. An online reverse teaching strategy further refines the learned search policy through Pareto guided policy distillation. Experiments on benchmark instances of different scales show that BP-HAN-NSGAII obtains better hypervolume and inverted generational distance values than the compared algorithms. Ablation studies, Pareto front comparisons, and statistical tests indicate that the heterogeneous representation, bipopulation cooperation, and reverse teaching mechanism improve convergence and diversity in complex networked dual-resource scheduling.

Iron and steel industry is a significant basic industry, where steelmaking–refining–continuous casting (SRCC) is a bottleneck. Efficient SRCC schedules can enhance the iron and steel production productivity greatly. SRCC scheduling problems are important and challenging industrial scheduling problems, as well as well-known nondeterministic polynomial time-hard problems. In the realistic SRCC process, the last production stage’s processing times are controllable, and the corresponding scheduling problems are named SRCC scheduling problems with controllable processing times (CPTs). To deal with the SRCC scheduling problems with CPTs efficiently, a new neighborhood structure (named restricted multiswap) and a new decoding method based on domain knowledge, and then a hybrid evolutionary algorithm (HEA) are proposed. The proposed HEA integrates several distinguished features: the aforementioned neighborhood structure and decoding method, local search, biased probability-guided crossover, and two diversification strategies including perturbation and mutation. The HEA achieves the lowest average relative percentage increase (RPI) in comparison with seven state-of-the-art scheduling algorithms, with the average RPI of 0.02

Energy conservation and carbon reduction are essential for achieving carbon peaking and neutrality, as well as advancing the green transformation of the economy and society. This research examines the energy-efficient distributed blocked flowshop scheduling problem with heterogeneous factories (EEDBFSP-HF), which aims to simultaneously minimize makespan and total energy consumption (TEC). To tackle the issue, a hybrid multi-objective variable-scale iterated greedy algorithm (HMOVIG) is developed. The algorithm incorporates several customized strategies to enhance solution quality and optimization efficiency. An elite selection strategy is employed to reserve high-quality individuals from the initial population. To promote population diversity and prevent premature convergence, a factory-aware crossover operator and a dynamic destruction intensity strategy are designed. Additionally, a problem-specific energy-saving heuristic is applied to lower TEC. A speedup-based deep local search is further integrated to refine solutions. Numerical experiments validate that HMOVIG consistently surpasses state-of-the-art algorithms in terms of scheduling effectiveness and solution quality.

The cascaded flow-shop joint scheduling problem (CFJSP) in printed circuit board manufacturing presents considerable challenges due to processing time uncertainty and inter-phase dependencies. Traditional approaches typically rely on deterministic assumptions or worst-case analysis, which limit their ability to accommodate real-world variability. This study reformulates the uncertain CFJSP as a many-objective optimization problem, treating each scenario-specific makespan as an independent objective. Each scenario represents a distinct realization of processing time variability, introducing implicit complex constraints that must be simultaneously satisfied to ensure robust and feasible scheduling. To address this, a multi-population co-evolutionary greedy algorithm is developed, incorporating fitness-aware offspring generation and constraint-preserving operators to maintain solution feasibility across all scenarios. A knowledge-guided interaction mechanism facilitates inter-population learning, improving convergence and maintaining diversity. Additionally, two greedy-based refinement mechanisms are introduced to intensify the search in promising regions, and a hybrid mutation operator is employed to strategically perturb elite solutions, preventing premature convergence and promoting global exploration. Extensive experiments on benchmark instances show that the proposed method significantly outperforms five state-of-the-art many-objective evolutionary algorithms in terms of convergence, diversity, and scheduling robustness, demonstrating its effectiveness and practical value for solving complex constrained scheduling problems under uncertainty.

Integrated coal mine energy systems (ICMES) generate streaming constraints where the number of active constraints fluctuates over time due to equipment switching, safety driven operations, and maintenance events. These dynamics cause abrupt contraction, expansion, or fragmentation of the feasible region. To address this challenge, we propose a federated learning (FL) variational autoencoder (VAE) evolutionary algorithm (FVE). Each constraint is mapped to an FL client so that local VAEs can learn heterogeneous constraint specific feasible subspaces as clients observe differently structured constraint data. Federated aggregation fuses these local latent models into a global generative model that adapts quickly and robustly to constraint inflow and outflow. An adaptive population correction mechanism repairs infeasible individuals, and enhanced dynamic dynamic nondominated sorting genetic algorithm-II tracks pareto front evolution under structural shifts. Comparative experiments on benchmark functions and an ICMES scheduling case demonstrate that FVE achieves faster feasibility restoration, improved convergence, and higher diversity than state-of-the-art methods. These results confirm the practicality of FVE for real-time industrial optimization under streaming constraints.

The manufacturing of complex products, prevalent in sectors such as electronics and automotive assembly, involves heterogeneous units which are connected in series and operate with interdependent scheduling decisions. This shift toward highly integrated production frameworks has rendered traditional single flowshop scheduling models inadequate. These models fail to meet the demands for global optimization across interconnected stages, making the coordinated scheduling of such multi-workshop systems a critical yet under-addressed challenge. To address this gap, we study the cascaded flowshop joint scheduling problem (CFJSP), which integrates a distributed permutation flowshop with a hybrid flowshop. Our proposed Adaptive Population-Based Iterated Greedy (APIG) algorithm begins with a collaborative initialization mechanism that blends diverse solution generation strategies. During the construction phase, an experience-driven skipping mechanism learns to evaluate operators adaptively. It intelligently prioritizes high-performance operations with a probabilistic set, effectively directing computational budget towards regions with higher payoff potential. To exploit neighborhood complementarity, the local search phase employs a hybrid strategy that alternates between insertion and swap operations. The efficacy of APIG is computationally confirmed by a 46.40

Advances in multi-robot technology have accelerated the development of smart agriculture, enabling tasks to be executed collaboratively with higher efficiency. In heterogeneous agricultural robots collaborative operation scheduling, fuzzy time window and matching constraints significantly increase the problem complexity. This paper proposes a multi-objective heterogeneous agricultural robot operation scheduling model with fuzzy service time window and matching constraints (MHROS_FT&M), aiming to optimize the total operation cost and service level. Given the NP-hard property of MHROS_FT&M, the hierarchical learning large neighborhood search algorithm (HLLNS) is developed. HLLNS incorporates the hierarchical reinforcement learning to enhance adaptability, a dynamic programming-based approach to improve service levels, and a sub-problem collaboration and mutation strategy to escape local optimum. By employing automated algorithm design technique to optimize 12 key parameters, the automated HLLNS (AHLLNS) is realized. In practical smart-farming scenarios, AHLLNS supports the joint scheduling of heterogeneous robots such as spraying drones, weeding robots, and seeding drones under uncertain service times, and explicitly balances operation cost against farmer satisfaction. The obtained schedules reduce unnecessary travel and resource consumption while keeping service times within acceptable ranges for farmers. Through automatic parameter tuning and the use of problem-specific operators, AHLLNS effectively addresses fuzzy time windows and matching constraints, achieving better performance across different problem scales. Experimental comparisons with Gurobi and state-of-the-art algorithms demonstrate AHLLNS superior computational efficiency and solution quality, validating its effectiveness for MHROS_FT&M. Note to Practitioners- This paper is motivated by the practical problem of scheduling heterogeneous agricultural robots in modern smart farming. In real-world agricultural production, different tasks require specific types of robots, and ensuring robots serve at the right time is crucial. However, current research often assumes homogeneous robot fleets, overlooking the need for task-robot compatibility. To tackle this, we introduce AHLLNS, designed to optimize multi-objective heterogeneous agricultural robot scheduling. By integrating reinforcement learning and AAD technology, AHLLNS improves decision-making, minimizes operational costs, and enhances farmer satisfaction. However, the current model assumes ideal conditions without accounting for robot failures. Future work will integrate deep reinforcement learning for real-time adaptability, allowing the system to handle equipment malfunctions dynamically. Beyond agriculture, this approach could also be applied to other fields that involve task scheduling with multiple types of agents, such as logistics and manufacturing.

With the increasing deployment of intelligent and unmanned technologies in smart farming, energy-efficient multi-robot task allocation and scheduling has become a critical challenge. This paper investigates the energy-efficient multi-robot task allocation and scheduling problem (EE-MRTASP), aiming to simultaneously minimize task completion time and total energy consumption. A knowledge-driven collaborative discrete Jaya algorithm (KC-DJaya) is proposed, incorporating a collaborative initialization strategy for generating high-quality initial solutions, a discrete Jaya update mechanism, and five knowledge-driven neighborhood operators to enhance local refinement and global search balance. Extensive experiments are conducted on benchmark instances with task scales ranging from 30 to 100 and robot numbers ranging from 2 to 7. Compared with seven state-of-the-art optimization algorithms, KC-DJaya consistently achieves the lowest inverted generational distance (IGD) and the highest hypervolume (HV) across all tested scales. In large-scale cases, IGD is reduced by approximately 30%-50% relative to most comparison algorithms, while HV improves by approximately 10%- 25%. Statistical tests indicate significant performance differences in favor of KC-DJaya. Ablation studies show that removing any single neighborhood operator results in up to 40% IGD degradation and 25% HV reduction, validating their complementary contributions. These results demonstrate the robustness and scalability of KC-DJaya for large-scale EE-MRTASP.

The research addresses a hybrid flowshop scheduling problem incorporating worker competency constraints. Unlike most existing studies that assume workers can operate all machines, our work accounts for the absence of certain worker skills. The added constraints substantially increase the problem's complexity, rendering traditional algorithms inadequate for obtaining feasible solutions. Therefore, a mixed-integer programming model is formulated, and a variable representation cooperative co-evolutionary algorithm (VRCCEA) is designed to achieve makespan minimization. Based on the decomposition idea, we use two populations to address the multicoupled problem and implement a cooperative mechanism by introducing a solution archive to promote the coevolution of populations. Given the limitations of a single encoding-decoding strategy, a variable representation mechanism is provided to balance the exploration scale and search efficiency. To prevent the failures of worker assignment, we design a heuristic based on resource constraint matrix (RCM), which conducts a greedy search within the feasible region. For the problem-specific knowledge, a reduced insertion neighborhood and an accelerated evaluation strategy are proposed to swiftly identify the best neighborhood solution. Finally, analytical experiments show the practical value of the algorithmic components and demonstrate that VRCCEA significantly outperforms five advanced metaheuristics.

With growing environmental awareness and increasing energy demands, sustainable manufacturing has become a focal point in the industry. Meanwhile, globalization has propelled distributed manufacturing systems as a dominant trend. This paper tackles the energy-efficient distributed heterogeneous hybrid flow-shop scheduling problem (EDHHFSP), aiming to minimize both makespan and total energy consumption. We first formulate a mixed-integer linear programming (MILP) model to provide a benchmark for small instances. More importantly, we propose a novel end-to-end deep reinforcement learning framework based on a heterogeneous graph neural network, which models the scheduling problem as a distributed decision-making process. A key innovation lies in the design of an action space composed of "job-factory" and "operation-machine" pairs, enabling fine-grained, decentralized scheduling decisions. Our approach starts with a novel heterogeneous graph representation of scheduling states, capturing complex interactions among jobs, factories, and machines. A three-stage embedding mechanism is developed to encode real-time scheduling environments. The agent then learns a parameterized policy using the proximal policy optimization (PPO) algorithm, guided by a reward function that balances makespan and energy efficiency. Experimental results demonstrate that our method generalizes well across different problem scales and significantly outperforms traditional heuristics and learning-based baselines in terms of both scheduling quality and energy savings. Note to Practitioners-Hybrid flow-shop scheduling in heterogeneous production environments is a common challenge faced by equipment manufacturers, particularly in the electronics manufacturing industry. These manufacturers are distributed across various locations, equipped with devices of differing performance, which often have limited buffer capacities and require varying setup times. To address this challenge, a machine learning-based method is used to schedule daily production tasks. This approach designs a novel heterogeneous graph structure to dynamically capture the complex relationships among factories, operations, and machines during scheduling. It employs an HGNN with a three-stage embedding mechanism to efficiently extract state features, which are then fed into a decision network to enable agents to take optimal actions. The method, trained on small-scale instances and directly applied to large-scale ones, improves both the computational time and performance of traditional optimization algorithms. The proposed method exhibits self-learning and adaptive capabilities, with its effectiveness validated through experiments on 1,600 test instances. Its practical application has been demonstrated in production scenarios at a printed circuit board assembly company. In the future, this method can be applied to address more complex distributed production scenarios.

In the context of intelligent agriculture, efficient multi-robot task allocation is essential for improving productivity and reducing operational costs. This study addresses a cooperative task allocation and path planning problem for multiple picking robots, with the objectives of minimizing the maximum completion time and the total travel distance. To this end, a novel decomposition-based multi-objective iterated greedy algorithm (D-MOIGA) is proposed by integrating a multi-objective decomposition strategy with an iterated greedy framework. The original bi-objective problem is first transformed into a set of single-objective subproblems using the Tchebycheff decomposition method, thereby reducing the dimensionality. A subproblem-oriented initialization strategy based on weight-guided task clustering (WTCSPI) is then designed to generate high-quality initial solutions aligned with the optimization direction of each subproblem. During the search process, the D-MOIGA integrates strategies including objective-driven greedy reconstruction, redundant subpath elimination, subpath splitting, and inter-robot subpath exchange to guide solutions in subproblem-specific directions and effectively approximate the true Pareto front (PF). Furthermore, four neighborhood search operators are designed to refine solutions, and an accelerated evaluation strategy is introduced to reduce the computational cost of neighborhood operations. Experimental results on large-scale orchard scenarios demonstrate that the proposed D-MOIGA achieves superior performance and robustness compared to several state-of-the-art multi-objective optimization algorithms.

The Flexible Job Shop Scheduling problem with Automated Guided Vehicles (FJSP-AGVs) is commonly encountered in modern manufacturing systems, with complexity increasing exponentially as the problem scales, making traditional methods inadequate. So far, a limitation of meta-heuristic algorithms incorporating machine learning for solving FJSP-AGVs is that machine learning is primarily used for heuristic selection (e.g., dispatching rules or local search operators) rather than actively guiding solution evolution based on the diverse features of solutions, leading to incomplete exploration and premature convergence. To tackle these issues, this paper first presents a Q-Learning Region Selection Enhanced Quality-Diversity (Q-QD) algorithm, which integrates a pairwise selection scheme to identify potential collaborative solutions within the feature space for two practical Chinese manufacturing scenarios: a coal machinery structural parts production workshop and a machining workshop in Nanjing. Unlike traditional approaches that focus solely on optimizing the objective, Q-QD leverages the customized feature space incorporating behaviors such as job transportation times and machine idle times. By combining Q-learning-based region selection with heuristic rule-based local search for AGV transportation, the proposed method effectively discovers high-performing and diverse solutions. The results obtained from two real-world cases and 21 benchmark instances indicate that the proposed Q-QD algorithm surpasses five advanced counterparts, achieving a makespan reduction of 12.2-30.8% relative to the strongest baseline. These substantial performance gains highlight the robustness and efficiency of QD-based optimization for solving FJSP-AGVs and suggest a valuable avenue for future research in intelligent job shop scheduling.

This article investigates a dynamic energy-efficient cascaded dual-shop collaborative scheduling problem with mating operations (DECDS-M), an underexplored yet practically important domain. We first formulate the problem as a mixed-integer linear programming (MILP) model to capture its structural complexity. To address the scalability and inefficient training challenges of conventional reinforcement learning (RL) methods, we propose a deep multi-agent reinforcement learning-enhanced iterated greedy algorithm (DMRLIG). An evolutionary RL mechanism is designed, where an iterated greedy controller dynamically adjusts the reward function to balance conflicting scheduling objectives and improve agent learning efficiency. A hierarchical multi-action design enables decentralized handling of both routing and sequencing decisions across workshops. Furthermore, a heterogeneous graph neural network (HGNN) captures dynamic interactions among factories, machines, and operations, facilitating high-quality policy representation. A proximal policy optimization (PPO) structure with dual actor-critics is employed to ensure robust and intelligent hierarchical decision-making. Extensive comparisons with six composite priority rules, two deep RL methods, and one metaheuristic method demonstrate the superiority of DMRLIG in optimizing both total tardiness and energy consumption. Finally, validation in an electronic equipment manufacturing company highlights the practical value and industrial applicability of the proposed approach

This paper addresses a hybrid flow shop scheduling problem in a human–machine collaborative production environment, where setup tasks are executed by workers and their durations vary according to stage-specific efficiencies. The goal is the minimization of total tardiness. To tackle this issue, a two-phase evolutionary algorithm (TEA) is developed. The algorithm incorporates a problem-specific solution representation, a heuristic population initialization strategy, and coordinated global and local search with dedicated job and worker operators. Extensive experiments demonstrate that TEA achieves better performance than both a knowledge-based iterated greedy algorithm and a variable representation cooperative co-evolutionary algorithm, especially for problems involving a large number of jobs and multiple processing stages, validating its effectiveness for human–machine collaborative scheduling.

The rapid deployment of intelligent unmanned systems is accelerating the transformation of precision agriculture. However, achieving autonomous, energy-efficient, and workload-balanced collaboration among heterogeneous weeding robot fleets in large-scale and structurally complex farmlands remains challenging. This study addresses a multi-objective collaborative scheduling problem for agricultural weeding robot fleets, aiming to minimize total travel distance, total energy consumption, and workload imbalance among robots. The problem is NP-hard, and conventional evolutionary approaches often suffer from limited adaptability due to static operator usage in complex search environments. To overcome these limitations, a deep reinforcement learning-guided memetic algorithm (DRL-MA) is developed. A dueling double deep Q-network is embedded within a Paretobased memetic framework, where operator selection is formulated as a Markov decision process to enable adaptive exploration-exploitation balance. The framework further incorporates task-priority-based encoding, feasibility-preserving decoding, and a multi-stage hybrid heuristic initialization strategy to enhance convergence and solution diversity. Extensive experiments on benchmark smart-farm scenarios and a real-farm case study demonstrate that DRL-MA outperforms representative evolutionary, reinforcement-learning-assisted, and recent learning-based baselines. In the benchmark experiments, DRL-MA improves the mean hypervolume by approximately 11.7% and reduces the mean inverted generational distance by approximately 64.2% compared with the strongest competing algorithm. In the real-farm case study, DRL-MA contributes 78.31% of the globally non-dominated solutions, indicating stronger overall Pareto competitiveness. Moreover, runtime and scalability analyses show that the proposed method remains computationally feasible as the problem scale increases. These results indicate that DRL-MA provides an effective, scalable, and practically applicable scheduling approach for collaborative weeding robot fleets in smart farming.

The coordination of weeding robots in multi-farm systems involves inherent trade-offs between efficiency (e.g., minimizing makespan and the total travel distance), posing an NP-hard optimization challenge. To address this, we propose a multi-objective multi-farm discrete artificial bee colony algorithm (MOMFDABC), which innovatively adapts the classical DABC framework for Pareto-optimal solutions. Key methodological advances include: A dual-phase initialization heuristic combining a multi-objective multi-farm Nawaz-Enscore-Ham (MOMFNEH) rule and an insertion-based greedy strategy for task sequencing and robot allocation, ensuring diversified initial solutions; Twelve adaptive local search operators are dynamically applied according to different stages of the algorithm (employed bee, onlooker bee, and scout bee phases) to optimize the Pareto frontier; An elitism-guided population update that preserves non-dominated solutions during the employed-onlooker-scout bee cycles. Comprehensive experimental results confirm the superior performance of MOMFDABC when compared with four state-of-the-art algorithms (AMOEA, MOTLBO, MODABC, MDTABC). Key findings include: The proposed MOMFDABC achieves HV values 34 % higher than those obtained by MDTLBO. When compared with MODABC, MOMFDABC reduces IGD by 89.7 %. Additional tests under varying farm scales and robot fleet sizes further demonstrate the robustness of this algorithm.