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This paper addresses the flexible job-shop scheduling problem with multiple time factors—namely, transportation time and setup time—as well as job priorities (referred to as FJSP-JPC-TST). An optimization model is established with the objective of minimizing the completion time. Considering the characteristics of the FJSP-JPC-TST, we propose an improved whale optimization algorithm that incorporates multiple strategies. First, a two-layer encoding mechanism based on operations and machines is introduced. To prevent illegal solutions, a priority-based encoding repair mechanism is designed, along with an active scheduling decoding method that fully considers multiple time factors and job priorities. Subsequently, a multi-level sub-population optimization strategy, an adaptive inertia weight, and a cross-population differential evolution strategy are implemented to enhance the optimization efficiency of the algorithm. Finally, extensive simulation experiments demonstrate that the proposed algorithm offers significant advantages and exhibits high reliability in effectively solving such scheduling problems.

In today’s complex and unpredictable manufacturing environment, dynamic events such as new job arrivals and urgent insertions can occur at any time. Meanwhile, multiple conflicting objectives need to be optimized simultaneously in the flexible job-shop scheduling problem (FJSP). This necessitates real-time multi-objective FJSP scheduling methods that can balance time efficiency and solution quality. Therefore, this paper proposes a dynamic multi-objective FJSP method based on a dynamic dual-attention network (DDAN) and multi-agent reinforcement learning. The DDAN captures global feature representations of operations and machines and deeply explores their complex dependencies. By integrating dynamic attention coefficients with job urgency factors, the DDAN can respond in real-time to dynamic events and provide effective support for subsequent decision-making. Additionally, a multi-agent reinforcement learning framework is introduced to balance the conflicting objectives of makespan and average tardiness. The higher level agent is designed to optimize makespan, while the lower level agent focuses on minimizing average tardiness. Collaboration between the two agents is facilitated by a carefully designed state-sharing mechanism and distinct reward functions, enabling more flexible and efficient resolution of conflicts in multi-objective optimization. Extensive testing has demonstrated the exceptional performance of the proposed method, which consistently and rapidly converges to the optimal solution, outperforming traditional priority dispatching rules (PDRs), metaheuristic algorithms, and state-of-the-art reinforcement learning methods, particularly in handling large-scale test instances.

ABSTRACT Cloud‐edge collaborative manufacturing supports cloud‐based training and edge‐side execution, requiring production scheduling policies with both rapid response and strong generalization. For the flexible job shop scheduling problem, existing deep reinforcement learning (DRL) methods usually rely on priority dispatching rules which limit generalization performance. To address this issue, this paper proposes heterogeneous graph attention‐enhanced DRL‐based scheduling (HGA‐DS), a heterogeneous graph attention‐enhanced deep reinforcement learning framework, and adopts dynamic joint operation‐machine selection for scheduling decisions. Specifically, an extended heterogeneous graph with job nodes is introduced to improve state representation, an estimated maximum finish time based reward function is designed to provide more effective training feedback, and a lightweight state‐conditioned attention mechanism is developed for low‐overhead neighbourhood information aggregation. Experimental results on datasets of different scales show that HGA‐DS outperforms existing learning‐based methods and composite dispatching rules, while exhibiting strong generalization ability and real‐time decision‐making performance.

Energy-aware distributed manufacturing has become a key focus in modern production systems due to the growing demand for sustainable and efficient operations. This study investigates the energy-aware distributed flexible job shop scheduling problem with job priority, where multiple factories cooperate to process prioritized jobs under energy consumption considerations. Considering job priorities is essential for reflecting the practical importance and urgency of different customer orders, which directly affects scheduling fairness and production responsiveness. The proposed bi-objective model aims to simultaneously minimize total weighted tardiness and total energy consumption, accounting for both processing and idle power. To effectively solve this complex NP-hard problem, a knowledge-guided deep reinforcement learning approach is developed. Domain knowledge is integrated into a double deep Q-network to guide the adaptive selection of local search operators, while a co-evolutionary mechanism maintains global exploration and accelerates convergence. Extensive computational experiments are conducted on 24 benchmark instances, which are categorized into five groups according to factory scale, with the maximum problem size reaching 160 jobs × 6 machines × 5 factories, together with a real-world case study. Compared with four state-of-the-art multi-objective baseline algorithms (NSGA-II, MOPSO, MOEA/D, and SPEA2), the proposed D2QN-COEA demonstrates substantial performance advantages. On average, it achieves an HV improvement of 23.1% compared with the best-performing baseline on each instance, while GD and IGD are reduced by 70.8% and 63.7%, respectively. When averaged across all four baseline algorithms, D2QN-COEA yields improvements of 203.4% in HV, 83.9% in GD, 79.9% in IGD, and 70.8% in Spacing, confirming its superior convergence accuracy and solution diversity. The results confirm that embedding domain knowledge into deep reinforcement learning enhances optimization robustness and provides an intelligent solution for energy-efficient distributed scheduling in modern manufacturing systems.

As the focus on environmental sustainability sharpens, the significance of low-carbon manufacturing and energy conservation continues to rise. While traditional flexible job shop scheduling strategies are primarily concerned with minimizing completion times, they often overlook the energy consumption of machines. To address this gap, this paper introduces a novel solution utilizing deep reinforcement learning. The study begins by defining the Low-carbon Flexible Job Shop Scheduling problem (LC-FJSP) and constructing a disjunctive graph model. A sophisticated representation, based on the Markov Decision Process (MDP), incorporates a low-carbon graph attention network featuring multi-head attention modules and graph pooling techniques, aimed at boosting the model’s generalization capabilities. Additionally, Bayesian optimization is employed to enhance the solution refinement process, and the method is benchmarked against conventional models. The empirical results indicate that our algorithm markedly enhances scheduling efficiency by 5% to 12% and reduces carbon emissions by 3% to 8%. This work not only contributes new insights and methods to the realm of low-carbon manufacturing and green production but also underscores its considerable theoretical and practical implications.

Quality inspection is a crucial step in ensuring product conformity and avoiding rework waste, while job priority constraints are prevalent in the production of complex products with assembly structures. This paper presents a modeling and solution framework for the multi-objective flexible job shop scheduling problem that incorporates both quality inspection activities and job priority constraints. An optimization model is constructed with the objectives of minimizing the makespan, minimizing the total energy consumption, and maximizing the processing quality. To solve this model, an improved multi-objective evolutionary algorithm based on decomposition is developed, which integrates several well-established mechanisms into a unified framework. The algorithm integrates multi-product assembly structures via virtual nodes, employs a two-vector encoding scheme, and incorporates a product—group repair mechanism based on binary sorting tree to handle job priority constraints. To maintain diversity among non-dominated solutions, a niching-based elite archive strategy is adopted. Furthermore, a quality enhancement strategy and a memory vector-based local search mechanism are embedded to strengthen the algorithm’s search capability. Simulation results demonstrate that the proposed algorithm outperforms the compared algorithms in terms of both convergence and diversity.

The utilization of a specific priority rule in scheduling operations in flexible job shop systems strongly influences production goals. In a context of production control in real practice, production performance indicators are evaluated always en bloc. This paper addresses the multi-criteria evaluating five selected conflicting production objectives via scalar simulation-based optimization related to applied priority rule. It is connected to the discrete-event simulation model of a flexible job shop system with partially interchangeable workplaces, and it investigates the impact of three selected priority rules—FIFO (First In First Out), EDD (Earliest Due Date), and STR (Slack Time Remaining). In the definition of the multi-criteria objective function, two scalarization methods—Weighted Sum Method and Weighted Product Method—are employed in the optimization model. According to the observations, EDD and STR priority rules outperformed the FIFO rule regardless of the type of applied multi-criteria method for the investigated flexible job shop system. The results of the optimization experiments also indicate that the evaluation via applying multi-criteria optimization is relevant for identifying effective solutions in the design space when the specific priority rule is applied in the scheduling operations.

The dynamic flexible job-shop problem (DFJSP) is a realistic and challenging problem that many production plants face. As the product line becomes more complex, the machines may suddenly break down or resume service, so we need a dynamic scheduling framework to cope with the changing number of machines over time. This issue has been rarely addressed in the literature. In this paper, we propose an improved learning-to-dispatch (L2D) model to generate a reasonable and good schedule to minimize the makespan. We formulate a DFJSP as a disjunctive graph and use graph neural networks (GINs) to embed the disjunctive graph into states for the agent to learn. The use of GINs enables the model to handle the dynamic number of machines and to effectively generalize to large-scale instances. The learning agent is a multi-layer feedforward network trained with a reinforcement learning algorithm, called proximal policy optimization. We trained the model on small-sized problems and tested it on various-sized problems. The experimental results show that our model outperforms the existing best priority dispatching rule algorithms, such as shortest processing time, most work remaining, flow due date per most work remaining, and most operations remaining. The results verify that the model has a good generalization capability and, thus, demonstrate its effectiveness.

The manuscript presents the importance of integrating mathematical methods for the determination of due date optimization parameter for maturity optimization in evolutionary computation (EC) methods in multi-objective flexible job shop scheduling problem (FJSSP). The use of mathematical modelling methods of due date optimization with slack (SLK) for low and total work content (TWK) for medium and high dimensional problems was presented with the integration into the multi-objective heuristic Kalman algorithm (MOHKA). The multi-objective optimization results of makespan, machine utilization and due date scheduling with the MOHKA algorithm were compared with two comparative multi-objective algorithms. The high capability and dominance of the EC method results in scheduling jobs for FJSSP production was demonstrated by comparing the optimization results with the results of scheduling according to conventional priority rules. The obtained results of randomly generated datasets proved the high level of job scheduling importance with respect to the interdependence of the optimization parameters. The ability to apply the presented method to the real-world environment was demonstrated by using a real-world manufacturing system dataset applied in Simio simulation and scheduling software. The optimization results prove the importance of the due date optimization parameter in highly dynamic FJSSP when it comes to achieving low numbers of tardy jobs, short job tardiness and potentially lower tardy jobs costs in relation to short makespan of orders with highly utilized production capacities. The main findings prove that multi-objective optimization of FJSSP planning and scheduling, taking into account the optimization parameter due date, is the key to achieving a financially and timely sustainable production system that is competitive in the global market.

In this paper, a new mathematical programming model is proposed for a partial flexible job shop scheduling problem with an integrated solution approach. The purpose of this model is the assignment of production operations to machines with the goal of simultaneously minimizing operating costs and penalties. These penalties include delayed delivery, deviation from a fixed time point for preventive maintenance, and deviation from the priorities of each machine. Considering the priorities for machines in partial flexible job shop scheduling problems can be a contribution in closer to the reality of production systems. For validation and evaluation of the effectiveness of the model, several numerical examples are solved by using the Baron solver in GAMS. Sensitivity analysis is performed for the model parameters. The results further indicate the relationship between scheduling according to priorities of each machine and production scheduling.