Multi-target energy-saving distributed flow shop group scheduling method and system

Abstract

The invention discloses a multi-target energy-saving distributed flow shop group scheduling method and system, and relates to the technical field of intelligent production scheduling and optimizing.The method comprises the steps that operation parameters of a distributed flow shop group scheduling scene are obtained, constructing a multi-objective mixed integer linear programming model taking minimization of the maximum completion time and the total energy consumption as optimization objectives; generating an initial population containing multiple feasible solutions by adopting multiple initialization strategies, and dividing the initial population into two sub-populations by adopting a Pareto frontier-based classification mechanism; constructing a dual Q learning mechanism, establishing two independent Q tables to respectively optimize two targets, generating a candidate solution in each sub-population by adopting an adaptive iteration greedy algorithm, a Q learning algorithm and a hybrid algorithm of the adaptive iteration greedy algorithm and the Q learning algorithm, screening an optimal solution by adopting non-dominated sorting and crowding distance sorting to update the sub-population, and continuously iterating and updating until a termination condition is met; and combining the solutions of the two sub-populations to form a Pareto frontier solution set, and obtaining an optimal scheduling scheme with a better effect.

Description

Technical Field

[0001] This invention relates to the field of intelligent production scheduling and optimization technology, and in particular to a multi-objective energy-saving distributed assembly line workshop scheduling method and system. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] As the manufacturing industry shifts from traditional centralized production to distributed and intensive production, the Distributed Flowshop Group Scheduling Problem (DFGSP) has become crucial for improving production efficiency. In practical production, some workpieces share similar processing routes or equipment requirements. To fully utilize the similarity in workpiece processing, grouping technology is introduced to group similar workpieces. All workpieces in a group are processed continuously and uninterruptedly in the process plant. This introduction of grouping technology in DFSP forms the Distributed Flowshop Group Scheduling Problem (DFGSP), which is widely found in industries such as label printing, metal stamping, and printed circuit board manufacturing. DFGSP not only needs to consider decision factors such as plant allocation, intra-group sequencing, and inter-group sequencing, but also needs to address the complexity brought about by the sequence-dependent setup time (SDST) and machine speed selection caused by group transitions. That is, when switching between groups, cleaning or tool changing is required. Longer SDSTs will delay completion, increase waiting time, and extend the production cycle. Higher machine speeds can shorten processing time but will increase energy consumption. Therefore, under the requirements of low-carbon manufacturing, total energy consumption and maximum completion time have become equally important optimization objectives in DFGSP.

[0004] In existing technologies, optimization methods for DFGSP mainly include iterative greedy algorithms (IG) and reinforcement learning algorithms (RL). However, these methods have certain shortcomings: most existing research focuses only on single-objective optimization, and does not adequately consider the collaborative optimization of multiple objectives such as energy consumption and maximum completion time. Existing schemes for multi-objective collaborative optimization mostly use single-policy methods to solve the scheduling strategy. However, single-policy algorithms such as IG have fixed destructive strength, which can easily lead to premature convergence or insufficient search diversity. Reinforcement learning algorithms use a single Q-table to optimize multiple objectives, resulting in unstable learning and poor performance of the final scheduling scheme. If multi-policy algorithms are used, the lack of effective cooperation between different search strategies leads to limited information sharing in complex distributed environments, slow convergence speed, poor solution distribution, and the performance of the generated scheduling scheme is difficult to meet practical needs. Summary of the Invention

[0005] To address the issues of multi-objective optimization imbalance and poor scheduling performance in existing DFGSP optimization strategies, this invention provides a multi-objective energy-saving distributed flow shop group scheduling method and system. Targeting the multi-objective and multi-constraint characteristics of the Distributed Flow Shop Group Scheduling Problem (DFGSP), this invention combines an improved iterative greedy strategy with dual Q-learning, while simultaneously optimizing maximum completion time and total energy consumption. This achieves a balance between search diversity and convergence stability, effectively improving the performance of the generated scheduling scheme.

[0006] In a first aspect, the present invention provides a multi-objective energy-saving distributed flow workshop group scheduling method.

[0007] A multi-objective energy-saving distributed flow workshop group scheduling method includes: Obtain the operating parameters of the distributed assembly line workshop group scheduling scenario, and construct a DFGSP multi-objective mixed integer linear programming model with the optimization objectives of minimizing the maximum completion time and total energy consumption; Multiple initialization strategies are used to generate an initial population containing multiple feasible solutions. A classification mechanism based on the Pareto front is used to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. A dual Q-learning mechanism is constructed, two independent Q-tables are established to optimize two objectives respectively, and candidate solutions are generated in each subpopulation using an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two. Then, non-dominated sorting and crowding distance sorting are used to select the optimal solution to update the subpopulation. The process is iterated and updated until the termination condition is met. The solutions from the two subpopulations are merged to form the Pareto front solution set, which yields the optimal scheduling scheme.

[0008] A further technical solution is that the operating parameters include the number of factories, the number of machines configured in each factory, the number of workpiece groups, and the number of workpieces in each workpiece group; In the multi-objective mixed-integer linear programming model, the optimization objectives include maximum completion time and total energy consumption, and the total energy consumption includes processing energy consumption, preparation energy consumption, and idle energy consumption; the decision variables include the group sorting sequence assigned to each factory, the workpiece sorting sequence within each group, and the processing speed of each workpiece on the machine; the constraints include factory allocation constraints, group sorting constraints, machine processing sequence constraints, speed selection constraints, time connection constraints, and energy consumption calculation constraints.

[0009] A further technical solution involves the following process for population initialization and subpopulation division: Multiple initialization strategies are adopted to generate feasible solutions, focusing on minimizing the maximum completion time, minimizing the total energy consumption, and randomly initializing the initial population respectively. Then, the initial populations of the various types are merged. A Pareto front-based classification mechanism is adopted. The initial population is evaluated by non-dominated sorting and crowding distance sorting. The elite solution set of the first Pareto front is extracted. According to the preference of each elite solution on the two optimization objectives, the elite solution set is divided into two subpopulations. The remaining non-elite solutions are allocated by the direct objective bias method to balance the size of the two subpopulations.

[0010] A further technical solution employs an adaptive iterative greedy algorithm to generate candidate solutions, including: Based on the operating parameters in the distributed assembly line workshop group scheduling scenario, the average number of groups in each factory is calculated to determine the scale of the factory, and then the magnitude of the damage intensity coefficient d is dynamically adjusted according to the factory scale. In the factory corresponding to the current solution, randomly remove d workpiece groups and temporarily store them in a temporary set. Insert the temporarily stored d workpiece groups one by one into all feasible positions, select the insertion scheme that makes the two optimization objectives optimal, and reconstruct the group allocation and sorting. The factory with the longest completion time among all factories is identified as the critical factory. The work group with the most workpieces in the critical factory is identified. Half of the total number of workpieces in the current group are randomly removed and temporarily stored. After perturbing the order of the remaining workpieces in the group, the temporarily stored workpieces are inserted into the optimal position in the group, and the workpiece allocation and sorting in the group are reconstructed. Reconstruct to form a new solution, verify whether the new solution satisfies all scheduling constraints, and output candidate solutions.

[0011] A further technical solution employs the Q-learning algorithm to generate candidate solutions, including: Take any random solution in the population as the current solution and perform random perturbation. Calculate the change in completion time and energy consumption of the solution before and after the perturbation. Match these changes with the nine system states represented by two dimensions to determine the current state. Based on the pre-trained Q-table corresponding to the optimization objective, query the optimal action corresponding to the current state, and use the optimal action to perturb the current solution to generate a new solution; Verify whether the generated new solution satisfies the scheduling constraints, and output the solutions that satisfy the constraints as candidate solutions.

[0012] A further technical solution involves the pre-training process of the Q-table as follows: Take any random solution in the population as the current solution and perform random perturbation. Calculate the change in completion time and energy consumption of the solution before and after the perturbation, and match them with the nine system states represented by two dimensions to determine the current state. Record the current state-action pair, calculate the reward value of the action according to the reward formula to quantify the degree of target optimization, use the reward value to correct and update the Q value of the current state-action pair in the Q table, and calculate the estimation error of the Q value; Continuously iterate the training until the Q-value estimation error tends to stabilize and approach 0, complete the Q-Learning training, save the final Q table, and obtain the pre-trained Q table.

[0013] A further technical solution employs a hybrid algorithm combining adaptive iterative greedy algorithm and Q-learning algorithm to generate candidate solutions, including: Taking any random solution in the population as the current solution, an adaptive iterative greedy algorithm is used to optimize the current solution. Through the steps of determining the factory scale, determining the damage strength coefficient, group-level destruction and reconstruction, and workpiece-level optimization, an intermediate solution is generated. The Q-learning algorithm is used to optimize the intermediate solution. The changes in completion time and energy consumption between the intermediate solution and the current solution are calculated to determine the state of the intermediate solution. Then, the optimal action is queried based on the Q table of the corresponding objective to fine-tune the intermediate solution and generate a new solution. Verify whether the new solution satisfies the scheduling constraints, and output a candidate solution that combines global search and local optimization.

[0014] A further technical solution involves generating three candidate solutions for each individual in each subpopulation using an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two. These solutions form a candidate solution set with the parent individuals. Then, non-dominated sorting and crowding distance sorting are used to select the optimal solution to update the subpopulation.

[0015] Secondly, the present invention provides a multi-objective energy-saving distributed assembly line workshop scheduling system.

[0016] A multi-objective energy-saving distributed assembly line scheduling system includes: The data acquisition and model building module is used to acquire the operating parameters of the distributed assembly line workshop group scheduling scenario and build a DFGSP multi-objective mixed integer linear programming model with the optimization objective of minimizing the maximum completion time and total energy consumption. The population initialization module is used to generate an initial population containing multiple feasible solutions using various initialization strategies. It adopts a classification mechanism based on the Pareto front to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. The iterative optimization module is used to construct a dual Q-learning mechanism. It establishes two independent Q-tables to optimize two objectives respectively. In each subpopulation, it uses an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two to generate candidate solutions. Then, it uses non-dominated sorting and crowding distance sorting to select the optimal solution to update the subpopulation. It iterates and updates until the termination condition is met. The scheduling scheme generation module is used to merge the solutions of the two subpopulations to form a Pareto front solution set, thereby obtaining the optimal scheduling scheme.

[0017] Thirdly, the present invention also provides an electronic device, comprising: a memory for storing executable instructions; and a processor for implementing the above-described multi-objective energy-saving distributed assembly line group scheduling method when executing the executable instructions stored in the memory.

[0018] Fourthly, the present invention also provides a computer-readable storage medium storing executable instructions for causing a processor to execute the executable instructions to implement the above-described multi-objective energy-saving distributed assembly line group scheduling method.

[0019] Fifthly, the present invention also provides a computer program product comprising executable instructions stored in a computer-readable storage medium; wherein, when the processor of an electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the above-mentioned multi-objective energy-saving distributed assembly line group scheduling method is implemented.

[0020] The above one or more technical solutions have the following beneficial effects: 1. This invention proposes a multi-objective energy-saving distributed flow shop group scheduling method and system. Addressing the multi-objective and multi-constraint characteristics of the Distributed Flow Shop Group Scheduling Problem (DFGSP), a MILP model with a dual objective is constructed. Through a Pareto front classification mechanism, the population is divided into dedicated subpopulations, focusing on optimizing the maximum completion time and total energy consumption. An adaptive iterative greedy strategy and a dual Q-learning mechanism are combined to simultaneously optimize the maximum completion time and total energy consumption. Furthermore, multiple key attributes based on the characteristics of the DFGSP problem are designed to guide the optimization process. A balance between search diversity and convergence stability is achieved through an iterative greedy module that dynamically adjusts the destructive strength and a reinforcement learning module that collaborates with the dual Q-table.

[0021] 2. This invention introduces three initialization strategies to balance high-quality solutions with diversity, avoiding a single initial search direction. During the iterative optimization process, three algorithms are used to generate candidate solutions: an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two. The adaptive IG algorithm dynamically adjusts the destruction intensity according to the factory scale, i.e., fine-tuning for small-scale factories and deep adjustment for large-scale factories, which can balance global exploration and local development. The independent Q-table design of dual Q-learning avoids target conflicts caused by a single Q-table, and finally generates a uniformly distributed Pareto optimal solution set. The hybrid operator combines the global search of IG and the local precise optimization of Q-learning, which significantly reduces the risk of premature convergence, improves the convergence speed and solution quality of the algorithm, and solves the problems of low search efficiency and premature convergence in existing technologies.

[0022] 3. The dual Q-learning mechanism adopted in this invention configures independent Q-tables for two major objectives, combining 9 state features, 23 customized actions, and targeted reward functions, so that each Q-table focuses on learning its own objective. By dynamically updating the Q-value of the state-action pair, the learning stability and policy adaptability are improved, ensuring the continuous output of high-quality solutions in complex scheduling scenarios, avoiding the drawbacks of unstable learning, and enhancing adaptability.

[0023] 4. This invention takes into account practical constraints such as SDST, machine speed level, and distributed factory layout. It guides the optimization of action design through five key attributes based on problem characteristics. Its HV and IGD values ​​are superior to existing CPLEX solvers and existing algorithms such as tIGA and RLMA. It exhibits strong robustness and generalization ability under different factory, group, and workpiece scales, and can be directly applied to actual scheduling scenarios in industries such as label printing and metal stamping.

[0024] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0025] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0026] Figure 1 This is a flowchart of the multi-objective energy-saving distributed flow workshop group scheduling method in Embodiment 1 of the present invention; Figure 2 This is a flowchart of the classification process for initializing the population in Embodiment 1 of the present invention; Figure 3 This is a Gantt chart of the initial scheduling solution in Embodiment 1 of the present invention; where (a) is factory 1 and (b) is factory 2; Figure 4 This is a Gantt chart after action A1 is performed in Embodiment 1 of the present invention; where (a) represents factory 1 and (b) represents factory 2. Figure 5 This is a Gantt chart after action A2 is performed in Embodiment 1 of the present invention; where (a) represents factory 1 and (b) represents factory 2. Figure 6 This is a Gantt chart after action A3 is performed in Embodiment 1 of the present invention; where (a) represents factory 1 and (b) represents factory 2. Figure 7 This is a schematic diagram of the error trend in Table 1 of the DQ-IG training in Embodiment 1 of the present invention; Figure 8 This is a schematic diagram of the error trend in Q-table 2 during DQ-IG training in Embodiment 1 of the present invention; Figure 9 The standard values ​​of the weighted parameter w1 of the reward function in Table 1 of the present invention at different levels are shown in Embodiment 1; where (a) is the HV trend and (b) is the IGD trend. Figure 10 The standard values ​​of the weighted parameter w1 of the reward function in Table 2 of the present invention at different levels are shown in Embodiment 1 of the present invention; where (a) is the HV trend and (b) is the IGD trend. Figure 11 This is a schematic diagram comparing the results of the initialization strategy variance analysis in Embodiment 1 of the present invention; where (a) is the RPI value of HV and (b) is the RPI value of IGD. Figure 12 This is a schematic diagram comparing the variance analysis results after removing the Q-learning module in Embodiment 1 of the present invention; where (a) is the RPI value of HV and (b) is the RPI value of IGD. Figure 13 This is a schematic diagram comparing the variance analysis results after removing the IG module in Embodiment 1 of the present invention; where (a) is the RPI value of HV and (b) is the RPI value of IGD. Figure 14 This is a schematic diagram comparing the variance analysis results of random action selection in Embodiment 1 of the present invention; where (a) is the RPI value of HV and (b) is the RPI value of IGD. Figure 15 This is a schematic diagram comparing the variance analysis results of the effective algorithm in Embodiment 1 of the present invention; where (a) is the RPI value of HV and (b) is the RPI value of IGD. Detailed Implementation

[0027] It should be noted that the following detailed descriptions are exemplary and are intended only to describe specific embodiments and to provide further explanation of the invention, and are not intended to limit the scope of exemplary embodiments of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0028] Example 1 To address the problems of multi-objective optimization imbalance, low search efficiency, and unstable learning in existing scheduling methods for the Distributed Flow Shop Group Scheduling Problem (DFGSP), this embodiment proposes a multi-objective energy-saving distributed flow shop group scheduling method. Considering the multi-objective and multi-constraint characteristics of DFGSP, a multi-objective optimization algorithm (DQ-IG) is designed. This algorithm combines an improved iterative greedy strategy (i.e., an adaptive iterative greedy strategy) with a dual Q-learning mechanism, simultaneously optimizing maximum completion time and total energy consumption. Based on this, this embodiment constructs a mixed-integer linear programming model to verify the effectiveness of the DFGSP problem and designs five key attributes based on the problem characteristics to guide the optimization process. Then, the aforementioned multi-objective optimization algorithm (DQ-IG) is used, achieving a balance between search diversity and convergence stability through a dynamically adjusted iterative greedy operator with destructive strength and a reinforcement learning operator cooperating with dual Q-tables.

[0029] The multi-objective energy-saving distributed flow workshop group scheduling method proposed in this embodiment, such as Figure 1 As shown, the specific steps include: Step S1: Obtain the operating parameters of the distributed flow workshop group scheduling scenario, and construct a DFGSP multi-objective mixed integer linear programming model with the optimization objective of minimizing the maximum completion time and total energy consumption; Step S2: Generate an initial population containing multiple feasible solutions using multiple initialization strategies. Use a classification mechanism based on the Pareto front to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. Step S3: Construct a dual Q-learning mechanism, establish two independent Q-tables to optimize the two objectives respectively, and use an adaptive iterative greedy algorithm, a Q-learning algorithm and a hybrid algorithm of the two to generate candidate solutions in each subpopulation. Then, use non-dominated sorting and crowding distance sorting to select the optimal solution to update the subpopulation. Iterate and update continuously until the termination condition is met. Step S4: Merge the solutions of the two subpopulations to form the Pareto front solution set, and obtain the optimal scheduling scheme.

[0030] The following content provides a more detailed introduction to the multi-objective energy-saving distributed flow workshop group scheduling method proposed in this embodiment.

[0031] In step S1, energy efficiency indicators are integrated into the flow shop scheduling problem with preparation time, distribution environment and speed constraints, and a multi-objective mixed integer linear programming model is established.

[0032] In this embodiment, the Distributed Flowshop Group Scheduling Problem (DFGSP) is defined as follows: The system has... The same factory ( Each factory is equipped with The machine has 100 machines and adopts a production line layout; there are 100 machines. A group of workpieces, denoted as Among them, the first Workpiece group Include Each workpiece group must be assigned to a specific factory for processing, and all workpieces within the group must be processed in the same sequence on each machine, continuously without interruption; each machine can operate at several discrete processing speed levels, denoted as... Additionally, consider the workpiece. At the factory The The processing steps on the machine are as follows Its standard processing time is When the machine is at actual speed During processing, the actual processing time is: (1) Based on the above-mentioned operating parameters such as the number of factories, the number of machines configured in each factory, the number of workpiece groups, and the number of workpieces in each workpiece group, this embodiment constructs a mixed integer linear programming (MILP) model for this multi-objective scheduling problem, and uses the CPLEX solver to verify the correctness and effectiveness of the model.

[0033] The MILP model built in this embodiment has the objective function of minimizing the maximum completion time C. max Total Energy Consumption (TEC) can be expressed as: min{C max , TEC}(2) In a distributed manufacturing system, the total energy consumption (TEC) consists of three main components: processing energy consumption (PEC), preparation energy consumption (SEC), and idle energy consumption (IEC). PEC exhibits a quadratic relationship with processing speed, and its power-speed characteristic is defined as follows: (32) In this embodiment, the power consumption during machine processing varies dynamically, while the machine power remains fixed when the machine is idle or during group switching. Specific energy consumption information is shown in Table 1 below.

[0034] Table 1 Energy consumption per unit time of the machine at different speed levels

[0035] Furthermore, the scheduling problem needs to consider three decision levels: (1) the order (or sorting sequence) of groups assigned to each factory, (2) the sorting sequence of workpieces within each group, and (3) the processing speed of each workpiece on the machine. To ensure a complete and feasible representation of the solution, three two-dimensional encoding structures are adopted: 1) Factory level: F vectors, each vector encoding an ordered sequence of groups assigned to a specific factory; 2) Group level: G vectors, each vector representing the processing sequence of workpieces within a certain group; 3) Machine level: J vector, each vector represents the specified workpiece processing speed on all machines.

[0036] Based on the above decision-making level, the decision variables of the MILP model are defined. These decision variables are the solutions to be solved (i.e., scheduling schemes). Specifically, these decision variables (Binary Decision Variables) include: • If the workpiece group Assigned to the factory If the value is 1, then the value is 0; otherwise, the value is 0. • If the workpiece group At the factory China occupies the first For each position, set it to 1; otherwise, set it to 0. • If the workpiece It occupies the first position in its workpiece group For each position, set it to 1; otherwise, set it to 0. • If the workpiece At the factory If the processing is in progress, the value is 1; otherwise, it is 0. • If in the factory medium workpiece Arranged in the workpiece Previously, the value was 1; otherwise, it was 0. • If the workpiece In the machine Speed ​​levels are adopted. If the value is 1, then the value is 0; otherwise, the value is 0. • If the workpiece At the factory ,machine Above speed If processed, the value is 1; otherwise, it is 0. • If in factory f, group Occupy position r, and the next position r+1 is grouped. If occupied, the value is 1; otherwise, it is 0.

[0037] Furthermore, the constraints of the MILP model include factory allocation constraints, intra-group sorting constraints, machine processing sequence constraints, speed selection constraints, time coordination constraints, and energy consumption calculation constraints, specifically:

[0038] Among them, constraint (2) specifies that the optimization objective is to minimize Cmax and TEC simultaneously; constraint (3) ensures that the start time of each operation is non-negative; constraint (4) specifies that no preparation time is required when switching between workpieces within the same group; constraints (5) and (8) guarantee that each group assigned to a factory occupies only one position; constraint (6) ensures that each group is assigned to only one factory; constraint (7) ensures that each position in a factory is occupied by at most one group; constraint (9) ensures that all groups assigned to the same factory are arranged consecutively, thereby eliminating any position gaps; constraints (10) and (11) ensure that each workpiece occupies exactly one position within its group, and each position is filled by exactly one workpiece; constraints (12) and (13) determine that the factory assignment of each workpiece is determined by the assignment of its associated group; constraint (14) ensures that each operation is executed at a processing speed level; constraint (15) ensures that the same machine in the factory occupies only one position. The workpieces processed on the machine do not overlap in time, while taking into account the preparation time between different groups; constraint (16) ensures that the process of the same workpiece on different machines follows the processing sequence of the assembly line; constraints (17)-(19) ensure that the completion time of each process is not less than the sum of its start time and actual processing time; constraint (20) defines the completion time of each factory as the completion time of its last scheduled workpiece; constraint (21) ensures that the maximum completion time is not less than the completion time of any factory; constraints (22)-(24) guarantee that the completion time of each machine is not less than the completion time of any workpiece processed on that machine; constraints (25)-(27) define the total processing time, total preparation time and total idle time of all factories respectively; constraints (28)-(30) define the processing energy consumption, preparation energy consumption and idle energy consumption of all factories respectively; constraint (31) defines the TEC of the entire system.

[0039] In the above formula, Indicates the total number of workpieces; Indicates the number of factories; This indicates the number of machines in each factory; Indicates the processing speed level; Indicates the number of workpiece groups; Indicates the workpiece index. ; Indicates the factory index. ; Indicates machine index, ; Indicates speed level index, ; Indicates the workpiece group index. ; Represents a set of workpieces. ; Indicates workpiece The workpiece group to which it belongs; This represents the collection of machines in a factory. ; Indicates workpiece group The number of workpieces in the quantity satisfies ; Indicates workpiece In the machine Standard processing time; Indicates speed level The actual processing speed ( (Maximum speed); Indicates in the machine From the workpiece group Switch to Preparation time, and ; Indicates machine In speed rating Energy consumption per unit time during processing; Indicates machine Switching between workpiece groups Energy consumption per unit time; This indicates the energy consumption per unit time when the machine is idle. This indicates the maximum number of workpieces in all workpiece groups; This indicates the location index of the workpiece group within the factory. ; This indicates the position index of the workpiece within the group. ; Represents a sufficiently large positive number (the Big M constant); Represents the set of all workpieces; Indicates workpiece group The collection of workpieces in the middle; Indicates the number of operations performed on workpiece j; Indicates the number of operations for all workpieces; Indicates workpiece At the factory machine Start time; Indicates workpiece At the factory machine Completion time; Indicates factory Completion time; This represents the actual processing time of workpiece j on machine m; Indicates the maximum completion time (makespan); Indicates factory China Machinery Completion time; Indicates factory China Machinery Total processing time; Indicates factory China Machinery Total preparation time; Indicates factory China Machinery Total idle time; Indicates processing energy consumption; Indicates the energy consumption to be prepared; Indicates idle energy consumption; This indicates total energy consumption.

[0040] As a further implementation, considering the prior knowledge of DFGSP, five attributes are constructed as the basis for the initialization and optimization strategies of subsequent schemes (i.e., the solutions of the model).

[0041] Specifically, in the proposed DFGSP, the critical factory is defined as the factory with the longest completion time, while all other factories are considered non-critical factories. The critical path is defined as follows: for each factory, the critical path begins with the last operation on the machine with the longest completion time. If there is idle time after the operation, the path moves to the previous operation on the same job; otherwise, it traces back to the previous operation on the same machine. Each factory is an independent production line workshop, and its critical path determines the completion time. This embodiment presents five properties of DFGSP and its critical path as embedded knowledge. The design of these properties provides theoretical support and proof for the effectiveness of subsequent knowledge-based operators. These properties of the critical path include: Attribute 1: Moving a group from an overloaded factory to a less overloaded factory will reduce the global completion time.

[0042] Proof: When a factory becomes a critical factory, it indicates that the number of workpiece groups or the processing load on that factory is too high. By relocating some workpiece groups to factories with lower loads, the processing load among factories can be balanced, thereby shortening the completion time of the critical factory and reducing the overall makespan.

[0043] Attribute 2: Within the same factory, adjusting the processing sequence of workpiece groups can reduce the preparation time between workpiece groups, thereby reducing the maximum completion time.

[0044] Proof: Since switching between different workpiece groups requires additional preparation time, an unreasonable workpiece group order will lead to costly switchover processes. By optimizing the workpiece group order, allowing workpiece groups with shorter preparation times to be processed adjacently, non-processing time can be significantly reduced, thus shortening the factory completion time.

[0045] Attribute 3: If a certain process has idle time afterward, the processing speed of that process can be appropriately reduced without increasing the overall makespan, thereby reducing energy consumption.

[0046] Proof: If a process is completed before its subsequent processes have started, it indicates that the process is not on the critical path and has time redundancy. In this case, reducing the processing speed of this process will not delay the start time of subsequent processes, and therefore will not affect makespan, but it can effectively reduce processing energy consumption and idle energy consumption.

[0047] Attribute 4: Appropriately increasing the processing speed of processes on the critical path can help shorten the overall makespan.

[0048] Proof: There is no time redundancy in the processes on the critical path. Accelerating these processes can directly compress the critical path length, thereby reducing the makespan. However, these processes usually increase processing energy consumption, so a trade-off needs to be made between time and energy consumption targets.

[0049] Attribute 5: For processes in non-critical factories, reducing their processing speed usually does not affect the overall makespan, but can effectively reduce total energy consumption.

[0050] Proof: The completion time of non-critical plants is shorter than the overall makespan, indicating the existence of a time buffer. By reducing processing speed in these plants, energy consumption can be reduced without affecting the overall completion time, achieving energy-saving scheduling.

[0051] In this embodiment, the DFGSP multi-objective mixed integer linear programming model constructed above is solved. First, two independent Q-tables are initialized and trained to estimate action values. Then, the initial population is evaluated using non-dominated sorting and crowding distance sorting, and subsequently divided into two subpopulations. In each generation, new solutions are generated in the two subpopulations using the IG operator, the Q-learning operator, and a mixture of both. The parent and child populations are merged, sorted, and assigned crowding distances, with the best first half of high-quality solutions reserved for the next generation. This process is repeated iteratively until the termination criterion is met. Finally, the non-dominated solutions from the two subpopulations are combined to form the Pareto front, yielding the optimal scheduling scheme. This solution process specifically includes: In step S2, an initial population containing multiple feasible solutions is generated using various initialization strategies. Then, a classification mechanism based on the Pareto front is used to divide the population into multiple subpopulations according to the performance of individual solutions. Each subpopulation is specifically used to optimize the maximum completion time and total energy consumption (TEC).

[0052] Step S2.1, Initialization Strategy. The quality of the initial population not only affects the early search direction but also the ability to explore the global solution space. To improve population quality and maintain diversity, three initialization strategies are adopted to generate feasible solutions, focusing on minimizing the maximum completion time, minimizing total energy consumption, and random initialization of the initial population. That is, the population is divided into three subpopulations: Makespan_pop, TEC_pop, and Rand_pop, thereby balancing heuristic guidance and search space coverage during the initialization process.

[0053] (1) The initial population Makesspan_pop, which focuses on minimizing the maximum completion time, is generated first using a combination of the longest processing time rule and existing mutation heuristic algorithms. This generates a high-quality initial solution, denoted as LPT_v_NEH, which is sorted in descending order of the total processing time of each workpiece group and sequentially allocated to the factory to balance the workload. Then, each remaining group is inserted into all feasible positions within the factory, and the position that generates the minimum completion time is selected to complete the initial allocation, resulting in the initial solution. Next, multiple variant solutions are generated by exchanging or inserting groups into the first solution (i.e., the initial solution), representing the remaining individuals. These perturbations introduced in the above process ensure feasibility while introducing structural diversity, accelerating convergence, and expanding the search space.

[0054] (2) Initial population TEC_pop focused on minimizing total energy consumption: This subpopulation is initialized using three strategies to generate low-energy solutions. The process includes: 1) Load balancing initialization: All workpiece groups are assigned to the lightest-loaded factory at standard speeds to generate corresponding solutions, providing a reference benchmark for energy consumption assessment; 2) Speed ​​perturbation initialization: Controlled random perturbations are applied to the processing speed to generate diverse energy efficiency trade-off solutions (i.e., schemes), providing effective guidance for the Q-learning algorithm; 3) Initialization based on preparation time: First, all workpiece groups are sorted in descending order of their cumulative preparation load. The first F groups are assigned to each factory as initial workpiece groups. The remaining workpiece groups are then inserted using a cyclic greedy insertion strategy, starting from the last workpiece group in each factory and selecting the workpiece group with the shortest preparation time. This process is repeated cyclically according to the factory order until all workpiece groups are assigned, generating corresponding solutions. Finally, several solutions generated by the above three strategies are used as the initial population.

[0055] (3) Randomly initialized initial population Rand_pop: Created through completely random initialization, where group assignment, job sorting, and process speed are all random. This method maximizes the coverage of the search space and enhances the diversity of the population.

[0056] Step S2.2: Merge the above-mentioned initial populations of various types, and use a classification mechanism based on the Pareto front to evaluate the initial populations through non-dominated sorting and crowding distance sorting. Extract the elite solution set of the first Pareto front, and divide the elite solution set into two sub-populations according to the preference of each elite solution on the two optimization objectives. Use the direct objective bias method to allocate the remaining non-elite solutions to balance the size of the two sub-populations.

[0057] Specifically, population classification plays a crucial role in the efficiency and convergence of multi-objective optimization, and an effective classification strategy is essential for generating high-quality subpopulations. Therefore, this embodiment uses a hybrid method combining non-dominated ranking with a direct objective bias strategy to classify the population, such as... Figure 2 As shown, the process is as follows: First, all individuals undergo non-dominated ordination, and the solutions at the first frontier are designated as elite solutions. Subsequently, these elite solutions are divided into two subgroups: one biased towards completion time and the other biased towards TEC. When a solution performs well or poorly on both objectives, it is allocated according to the subpopulation size to maintain subpopulation balance. This ensures that both subgroups contain high-quality solutions. Finally, the remaining individuals are classified using a direct objective bias method to maintain balance and prevent any subpopulation from becoming too small due to randomness.

[0058] In step S3, for each subpopulation, different strategies such as the improved iterative greedy strategy, Q-learning algorithm and its hybrid algorithm are used to generate multiple candidate solutions at the same time to improve the diversity of the search. Then, through continuous iterative optimization, the termination condition is met, and the optimal subpopulation (i.e. the optimal solution set) is obtained.

[0059] Among them, a dual-Q learning mechanism is designed, in which two independent Q tables are constructed during the Q learning process, each Q table is dedicated to optimizing one of the objectives (maximum completion time or total energy consumption). At the same time, multiple features representing the system state are defined and several customized actions are defined to guide the decision-making process of each objective. This approach can effectively capture conflicting optimization directions and improve the stability of convergence. In the improved iterative greedy algorithm, an adaptive destruction strategy is introduced to dynamically adjust the destruction intensity according to the factory size to prevent excessive destruction of the solution.

[0060] (1) Adaptive iterative greedy algorithm is used to generate candidate solutions.

[0061] The standard IG algorithm (iterative greedy algorithm) is used to optimize any random solution (i.e., the current solution) in the population. d workpieces are randomly removed and stored in a temporary set, then reintegrated into the interrupted sequence to create a new solution. It is evident that the magnitude of the damage strength coefficient d affects the algorithm's performance. If d is too small, the perturbation is insufficient, leading to a solution that is too similar to the original solution; if d is too large, the sequence degenerates into random initialization, thus losing the structural advantages of the previous solution. To address this issue, this embodiment proposes an adaptive IG algorithm, which dynamically adjusts the magnitude of the damage strength coefficient d based on (I) the number of groups allocated to each factory and (II) the number of workpieces in each group.

[0062] Specifically, firstly, groups are destroyed and rebuilt. Based on the operating parameters of the distributed assembly line workshop group scheduling scenario, the average number of groups per factory is calculated. The formula for calculating the average number of groups per factory is: (33) The factory size is determined based on the average number of groups, and then the destructive strength coefficient d is dynamically adjusted according to the factory size. That is: if the number of groups in the factory... ≤ If , then it is a small-scale factory, d∈[1,2]; if > If , then it is a large-scale factory, d∈[3,4].

[0063] During the destruction process, d workpiece groups are randomly selected from the factory corresponding to the current solution and stored in a temporary set. During the reconstruction process, the d temporarily stored workpiece groups are inserted one by one into all feasible positions to maximize the improvement of the two optimization objectives, makespan and TEC. The insertion scheme that makes the two optimization objectives optimal is selected, and the group allocation and sorting are reconstructed to generate a better individual, i.e., a scheduling scheme.

[0064] Secondly, the destruction and reconstruction of workpieces are performed. Workpieces within a group cannot be split, so destruction and reconstruction occur within the same group. Specifically, the factory with the longest completion time among all factories is designated as the critical factory. The group with the largest number of workpieces in the critical factory is identified. From this group, d1 workpieces are removed according to the group size, where d1 is half the total number of workpieces in the current group. These removed workpieces are temporarily stored. Then, the order of the remaining workpieces is perturbed to create a new partial sequence. The removed workpieces are re-inserted into all possible positions within their group, and the optimal position is selected based on improvements from Makespan and TEC. The workpiece allocation and sorting within the group are reconstructed, resulting in a refined individual scheduling scheme.

[0065] Finally, a new solution is formed by reconstructing the solution in the above manner. The new solution is then verified to ensure that it satisfies all scheduling constraints, and candidate solutions are output.

[0066] (2) The Q-learning algorithm is used to generate candidate solutions.

[0067] In DFGSP, the complexity of multiple objectives and constraints makes fixed perturbation strategies ineffective at different optimization stages. To improve adaptability, this embodiment integrates the Q-learning mechanism from RL to guide action selection. The Q-learning mechanism dynamically adjusts the perturbation based on continuous feedback, avoiding premature convergence and enhancing population diversity.

[0068] First, in Q-learning, state features are crucial. Well-designed state features help capture the impact of perturbations on solution performance, enabling the agent to learn from experience and gradually learn the optimal policy. In this embodiment, considering that multi-objective scheduling often does not optimize a single objective but needs to find a balance among multiple objectives, two dimensions are used to represent each state to balance multiple objectives: Makespan change ( )for: (34) TEC change ( )for: (35) Based on the two dimensions mentioned above, a total of nine states are defined, as shown in Table 2 below.

[0069] Table 2. Nine states composed of two dimensions

[0070] Secondly, in the Q-learning framework, actions represent operations that an agent can perform in a specific state to perturb the current scheduling solution. These actions are mainly divided into two types: MS operations and TS operations, totaling 20 operations (MS1-MS11, TS1-TS9), and also include three hybrid operators (A1-A3), which simultaneously optimize the scheduling structure and processing speed, improving the overall quality of the solution.

[0071] MS operations are used to adjust the scheduling structure, such as inter-plant group exchange, critical plant intra-plant group sorting adjustment, and intra-group job exchange, specifically including: •MS1: Randomly select two factories, then randomly select a pair from each factory and swap them.

[0072] •MS2: Randomly select a factory, then select two groups from it and swap their positions.

[0073] •MS3: Randomly select two groups within the critical factory and swap them.

[0074] •MS4: Randomly select a group from the critical factories and insert it into all positions of all factories, selecting the position that minimizes the completion time.

[0075] •MS5: Randomly select a group from the critical factories and randomly insert it into another location of that factory.

[0076] •MS6: Based on attribute 3, remove a group from the critical factory and insert it into the factory with the shortest completion time, selecting the best insertion point (the position with the smallest makespan).

[0077] •MS7: Randomly select a group from the non-critical factories and insert it into the critical factory.

[0078] •MS8: Swap all groups in the critical factory and find the swap locations that minimize completion time.

[0079] •MS9: Randomly select a group from the critical plants, then select a workpiece within that group. Swap this workpiece with another workpiece in the same group, choosing the swap location that minimizes completion time.

[0080] •MS10: Select a group from the critical plants and insert it into another location within the same plant, choosing the location that minimizes completion time.

[0081] •MS11: Randomly select a group from the critical factories, select a workpiece from the group, and insert it into all positions within the group, selecting the position that minimizes the completion time.

[0082] TS operations are used to adjust processing speeds, such as disrupting the speed sequence in factories with the highest energy consumption or slowing down non-critical processes. Specifically, this includes: •TS1: Randomly select a machine in the factory and randomly swap the processing speeds of the two processes.

[0083] •TS2: Randomly select two different factories, then select one process from each factory and swap their processing speeds.

[0084] •TS3: Select the factory with the highest energy consumption, then select a machine and reverse the processing speed sequence between two random locations on that machine.

[0085] •TS4: Select the factory with the highest energy consumption, then select a machine and shuffle all the processing speeds on that machine.

[0086] •TS5: To reduce TEC, for factories with the minimum completion time, reduce the process speed on the last machine without affecting the overall completion time.

[0087] •TS6: Based on attribute 2, reduce speed for a specific process to lower TEC.

[0088] •TS7: Select a group from the plants with the highest energy consumption and insert it into the plant with the lowest energy consumption, selecting the position that minimizes TEC.

[0089] •TS8: In the most energy-intensive plants, select the two sets of exchanges that minimize TEC.

[0090] •TS9: Select two workpieces from a set from the factory with the highest energy consumption and swap them at the location with the lowest TEC.

[0091] Hybrid operators A1-A3 are used to simultaneously optimize scheduling structure and processing speed, such as adjusting speed and group allocation based on the critical path, specifically including: • Action A1: Designed based on attribute 2. When idle time exists, A1 reduces TEC without increasing completion time by decreasing the processing speed of the previous operation. Figure 3 Indicates the initial solution, such as Figure 4 As shown, reduce the process in group 3 The speed will reduce the TEC in Factory 2 from 195.910 to 187.672.

[0092] • Action A2: Designed based on attributes 3 and 5. A2 moves a group from the critical plant to the plant with the shortest completion time, then reduces the processing speed of non-critical plants to simultaneously reduce makespan and TEC. Figure 5 As shown, transferring group 5 from factory 2 to factory 1 reduces the completion time from 50.100 to 44.579, and also reduces the TEC. Therefore, the solution generated by A2 dominates the initial solution.

[0093] • Action A3: Designed based on attributes 4 and 5. A3 adjusts the processing speed of operations on the critical path and in non-critical plants to optimize completion time and TEC. For example... Figure 6 As shown, adjusting the three steps reduces both completion time and TEC (Transmission Time Capacity). Therefore, the solution generated by action A3 dominates the initial solution. These actions enhance the diversity and feasibility of solutions, provide a structured action space for strategy selection, strengthen the learning mechanism, and improve the overall performance of the algorithm.

[0094] Next, the reward is the core element driving the agent's learning behavior strategy. For multi-objective optimization problems, the reward function must balance and coordinate multiple objectives to guide the policy towards the global optimum. Therefore, the reward function designed in this embodiment is as follows: (36) Here, ω1 and ω2 are the weight parameters for the two objectives. If the reward is greater than 0, the action improves the overall solution (both objectives are improved, or the improvement in one objective is greater than the deterioration in the other objective); if the reward is equal to 0, there is no change (or the improvement in one objective offsets the deterioration in the other objective); if the reward is less than 0, the action will worsen the solution (both objectives will worsen, or the deterioration in one objective exceeds the improvement in the other objective).

[0095] Finally, Q-Learning training is performed. That is, a baseline instance matching the actual scheduling scenario is selected, and a large number of random initial solutions are generated as the initial population through the above method. This covers different combinations of group allocation, sorting, and processing speed. Then, the initial population is used as the training sample set. Based on each initial solution in the training sample set, a cyclic iterative optimization process is performed until the Q-table converges, thus completing Q-Learning training.

[0096] Specifically, in the DQ-IG framework, each Q-table consists of nine state features, and actions correspond to MS, TS operations, and A1-A3 hybrid operators. During actual training, any random solution in the population is taken as the current solution. A random perturbation action is performed on the current solution, and the changes in completion time and energy consumption before and after the perturbation are calculated. These changes are then matched with the nine system states represented by two dimensions to determine the current state. The current state-action pair is recorded, and the reward value of the action is calculated according to the reward formula to quantify the degree of target optimization. The reward value is used to correct and update the Q-value of the current state-action pair in the Q-table, and the estimation error of the Q-value is calculated. In this embodiment, the classic Q-learning rule is used to correct and update the Q-table, expressed as: (37) In the above formula, The learning rate is used to control the update step size. This is a discount factor used to balance immediate rewards with future rewards; This is the new state after the action is performed. This is the optimal action in the new state.

[0097] Iterative training is performed in the manner described above. In each iteration, the estimation error of the Q-table (i.e., the deviation between the current Q-value and the set ideal Q-value) is calculated, and an error trend curve is plotted. As training continues, the Q-value estimation error gradually decreases. When the error curve tends to stabilize and approach 0 (e.g., the error within a set number of iterations is less than a certain set minimum value), it indicates that the Q-table has learned the optimal action strategy under different states. At this point, Q-learning training is completed, and the final Q-table is saved to obtain the pre-trained Q-table.

[0098] Preferably, to describe the training process, this embodiment analyzes a distributed scheduling instance consisting of 2 factories, 10 groups, 20 workpieces, and 5 machines. Figure 7 and Figure 8 The training errors of the two Q-tables are shown. The Q-value estimation error decreases with increasing training iterations, indicating convergence. Early training exhibits high randomness, with noticeable oscillations in the error curves; these oscillations stabilize as learning progresses.

[0099] Furthermore, based on the Q-table obtained from the pre-training, during the population iterative optimization process, for the current solution of each subpopulation, the pre-trained Q-table is directly called to make action decisions and generate candidate solutions. That is, any random solution in the population is taken as the current solution and randomly perturbed to generate a better solution on the target. All feasible small perturbations are traversed, such as fine-tuning the processing speed of a group or exchanging the order of two workpieces. The change in completion time and energy consumption of the solution before and after each perturbation is calculated and matched with the nine system states represented by two dimensions to determine the current state. The action with the largest Q value in the pre-trained Q-table under this state is queried and used as the optimal action to actually perturb the current solution, generating a new solution. It is checked whether the generated new solution satisfies all scheduling constraints (such as factory allocation constraints, speed selection constraints, etc.). If the new solution satisfies all constraints, it is output as a valid candidate solution. Otherwise, the second-best action in the Q-table is queried and the perturbation is re-executed until a valid candidate solution is generated.

[0100] (3) A hybrid algorithm combining adaptive iterative greedy algorithm and Q-learning algorithm is used to generate candidate solutions.

[0101] Specifically, by combining the two algorithms mentioned above, candidate solutions are generated, and the process is as follows: First, take any random solution in the population as the current solution, and execute an adaptive iterative greedy algorithm on the current solution. Through the determination of the factory scale, the determination of the damage strength coefficient, the group-level destruction and reconstruction, and the workpiece-level optimization steps, an intermediate solution is generated. Next, the Q-learning algorithm is executed on the intermediate solution. Based on the changes in completion time and energy consumption between the intermediate solution and the current solution, the state of the intermediate solution is determined. Then, the optimal action is queried based on the Q table of the corresponding objective, the intermediate solution is fine-tuned, and a new solution is generated. Finally, verify whether the new solution satisfies the scheduling constraints, and output a candidate solution that combines global search and local optimization.

[0102] In step S3, during each iteration, each individual in each subpopulation generates three candidate solutions using an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two. The subpopulation is then merged with the parent individuals to form a candidate solution set. The optimal solution is then selected using non-dominated sorting and crowding distance sorting to update the subpopulation.

[0103] Among them, non-dominated sorting and crowding distance sorting are: for any two solutions and If the solution The performance is no worse than the solution on all optimization objectives. And it is superior to the solution in at least one objective. Then it is called a solution. Dominant Solution Otherwise, x and y are considered non-dominated relative to each other. In this embodiment, in each iteration, individuals are ranked according to their non-dominated level. For individuals of the same level, the density is quantified by calculating the crowding distance. Selecting individuals with larger crowding distances can maintain population diversity and reduce the possibility of premature convergence.

[0104] In step S4, the solutions of the two subpopulations corresponding to the two optimization objectives obtained above are merged to form the Pareto front solution set, and the optimal scheduling scheme is obtained.

[0105] Preferably, three sets of test cases are used to evaluate the performance of the method proposed in this embodiment. The first set consists of 135 instances with the following parameters: factories {2, 3, 4}, groups {5, 10, 20}, workpieces {20, 40, 80, 100, 150}, and machines {2, 5, 8}. The second set contains 24 smaller instances: factories {2, 3}, groups {2, 3}, workpieces {6, 8, 10}, and machines {2, 3}. The third set includes baseline instances from the first set, all limited to two factories, with the number of groups from {5, 10, 20}, workpieces from {20, 40, 80, 100, 150}, and machines from {2, 5, 8}. For clarity, the symbol "2F5M5G20J" represents an instance with 2 factories, 5 machines, 5 groups, and 20 workpieces. The first set is used for performance comparison between algorithms, the second set is used for verification against the CPLEX solver, and the third set is used for parameter calibration. To ensure fairness, the runtime of each instance is the same across different algorithms, and the CPU runtime of each instance is... Second.

[0106] In addition, the evaluation criteria are designed: the performance of the algorithm is compared and analyzed using two metrics, hypervolume (HV) and inverse generation distance (IGD). In the HV calculation, these two objectives are normalized as follows: (38) in, This represents the i-th normalization target value. This represents the i-th target value. This represents the maximum value among the i-th target values. The baseline point in HV is (3.1, 3.1), calculated as follows: (39) Where λ(·) denotes the Lebesgue measure in the target space, and NS is a non-dominated set. Indicates the first The region dominated by each solution in the target space.

[0107] The formula for calculating IGD is: (40) in, It is a true Pareto frontier. Representing the solution ∈ reconciliation The Euclidean distance ∈NS in the target space. It should be noted that larger HV and smaller IGD representations have better convergence and diversity.

[0108] In addition, the relative percentage increment (RPI) is used to compare the results of different algorithms, calculated using the following formula: (41) in, This represents the average value obtained through a given algorithm, while This represents the best average value among all algorithms.

[0109] In the DQ-IG algorithm, the weighting parameters w1 and w2 in the reward function have a significant impact on the learning behavior of the Q-table and the overall performance. By assigning different weights to each objective, the DQ-IG framework guides Q1 towards "TEC-oriented" optimization and Q2 towards "Makespan-oriented" optimization. To identify appropriate values ​​for w1 and w2, parameter calibration is performed using a third set of benchmark instances. Each instance is executed 15 times, and the average HV and IGD are recorded. Figure 9 Tables (a) and (b) show the effect of varying w1 from 0.0 to 1.0 on Q1. The highest HV and lowest IGD are observed when w1 = 0.2. Similarly, Figure 10 Figures (a) and (b) illustrate the changes in HV and IGD in Q2 as w1 increases from 0.0 to 1.0. The best performance is achieved when w1 = 0.7, indicating that Q2 tends to improve completion time. Therefore, for subsequent experiments, the optimal configuration for Q1 is w1 = 0.2, w2 = 0.8, and the optimal configuration for Q2 is w1 = 0.7, w2 = 0.3.

[0110] Next, the effectiveness of random initialization is verified. To evaluate the effectiveness of the proposed knowledge-based initialization strategy, a DQ-IG algorithm using purely random initialization is used as a comparison. The analysis of variance (ANOVA) results of the RPI values ​​of HV and IGD are as follows: Figure 11 As shown in (a) and (b), statistical analysis demonstrates that the knowledge-based initialization strategy outperforms random initialization.

[0111] Validating the effectiveness of the Q-Learning module. To investigate the contribution of the Q-learning component, an algorithm variant was created by removing the Q-learning module and retaining only the IG process. In this setting, the perturbation operator is based on a fixed pattern selection and cannot adapt to constantly changing scheduling states. Figure 12In the figure, (a) and (b) represent the variance analysis results of the RPI values ​​of HV and IGD, respectively. The results show that the DQ-IG algorithm outperforms the improved algorithm without the Q learning module.

[0112] Validating the effectiveness of the IG module. To evaluate the importance of the IG module, a variant of the DQ-IG algorithm was constructed in which the IG component was removed, making the search rely solely on the Q-learning mechanism. Figure 13 In the figure, (a) and (b) represent the variance analysis results of the RPI values ​​of HV and IGD, respectively. The experimental results show that the DQ-IG algorithm outperforms the improved algorithm without the IG module.

[0113] To verify the effectiveness of random action selection, a comparative experiment was conducted using R-action, a variant of DQ-IG (where actions in DQ-IG are replaced with random actions). Figure 14 In the figure, (a) and (b) represent the RPI value analysis results of HV and IGD, respectively. The results show that the scheduling performance of DQ-IG is better than that of R-action, indicating that DQ-IG can select appropriate actions under various scheduling conditions.

[0114] Furthermore, to further evaluate the performance of the proposed DQ-IG algorithm, four existing competing algorithms—tIGA, GCCEA, PBIGA, and RLMA—were compared under the same experimental settings and termination conditions. Although these algorithms target different scheduling problems, their structural design and optimization mechanisms are still closely related to those of DQ-IG. tIGA and RLMA are specifically tailored for DFGSP; tIGA uses a two-stage IG framework to optimize maximum completion time, while RLMA combines Sarsa's RL algorithm to autonomously select a scheduling strategy under velocity constraints, thus simultaneously optimizing maximum completion time and TEC. GCCEA targets the single-factory flow shop group scheduling problem, employing a greedy mechanism to optimize completion time, TEC, and total process time. PBIGA is a hybrid algorithm combining Q-learning and IG to minimize completion time and TEC, used to solve the distributed assembly flexible job shop scheduling problem. Although these algorithms target different scheduling problems, their structure and optimization mechanisms are closely related to the DQ-IG framework. Figure 15 The RPI values ​​of HV and IGD shown demonstrate that DQ-IG outperforms these algorithms in solving DFGSP instances.

[0115] To further validate the MILP model, performance comparisons were performed using the IBM ILOG CPLEX 12.7 solver. The objective function of CPLEX is w·Cmax + (1 w)·TEC, where w is set to {0.1, 0.2, ..., 0.9}, thus generating nine optimized results for each instance. The CPLEX time limit for each instance is 1 hour. For comparison, the DQ-IG algorithm was run independently 15 times on the same instance, and the best HV and IGD values ​​were recorded. Table 3 below shows the results for all 24 test instances. The results in Table 3 show that in almost all cases, DQ-IG outperforms CPLEX in both HV and IGD. However, in some cases, such as 3F2G6J2M and 3F2G8J2M, CPLEX exhibits a slightly higher HV value. This is because the weighted sum method in CPLEX may overemphasize one objective (e.g., completion time), leading to extreme solutions. In contrast, DQ-IG aims to produce a uniformly distributed Pareto set, resulting in slightly lower HV but better IGD in these cases. In summary, DQ-IG provides a competitive solution to complex distributed scheduling problems and achieves better balance on the Pareto front, demonstrating its excellent generalization ability.

[0116] Table 3 Comparison of results with the accurate CPLEX solver

[0117] This embodiment addresses the distributed flow shop scheduling problem with the objectives of minimizing completion time and total energy consumption. It proposes a multi-objective energy-saving distributed flow shop scheduling method, employing a population classification mechanism-based DQ-IG algorithm. This mechanism divides the population into two subpopulations with different optimization preferences based on individual performance on specific objectives. Based on this division, two independent Q-tables focus on different objectives, effectively mitigating learning instability caused by reward conflict in multi-objective environments. Furthermore, five knowledge-based attributes are designed to enhance the optimization capabilities of DQ-IG. Experimental results on large-scale benchmark instances demonstrate the significant advantages of DQ-IG. Comparative experiments show that both DQ and IG components improve the performance of the DQ-IG algorithm, achieving a superior solution compared to other competing algorithms. This proposed method possesses scalability and industrial applicability, and can be further applied to complex scheduling scenarios such as dynamic task arrival, hierarchical energy management, and multi-factory collaborative scheduling.

[0118] Example 2 This embodiment provides a multi-objective energy-saving distributed assembly line workshop scheduling system, including: The data acquisition and model building module is used to acquire the operating parameters of the distributed assembly line workshop group scheduling scenario and build a DFGSP multi-objective mixed integer linear programming model with the optimization objective of minimizing the maximum completion time and total energy consumption. The population initialization module is used to generate an initial population containing multiple feasible solutions using various initialization strategies. It adopts a classification mechanism based on the Pareto front to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. The iterative optimization module is used to construct a dual Q-learning mechanism. It establishes two independent Q-tables to optimize two objectives respectively. In each subpopulation, it uses an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two to generate candidate solutions. Then, it uses non-dominated sorting and crowding distance sorting to select the optimal solution to update the subpopulation. It iterates and updates until the termination condition is met. The scheduling scheme generation module is used to merge the solutions of the two subpopulations to form a Pareto front solution set, thereby obtaining the optimal scheduling scheme.

[0119] Example 3 This embodiment provides an electronic device, including: a memory for storing executable instructions; and a processor for executing the executable instructions stored in the memory to implement the method provided in this embodiment.

[0120] Example 4 This embodiment also provides a computer-readable storage medium storing executable instructions, which, when executed by a processor, will cause the processor to execute the method described above in this embodiment.

[0121] Example 5 This embodiment provides a computer program product including executable instructions, which are computer instructions; the executable instructions are stored in a computer-readable storage medium. When the processor of an electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the electronic device performs the method described in this embodiment.

[0122] The steps and methods involved in Embodiments 2 to 5 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0123] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0124] The above description is only a preferred embodiment of the present invention. Although the specific implementation of the present invention has been described in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that, based on the technical solution of the present invention, various modifications or variations that can be made by those skilled in the art without creative effort are still within the scope of protection of the present invention.

Claims

1. A multi-objective energy-saving distributed flow workshop group scheduling method, characterized in that, include: Obtain the operating parameters of the distributed assembly line workshop group scheduling scenario, and construct a DFGSP multi-objective mixed integer linear programming model with the optimization objectives of minimizing the maximum completion time and total energy consumption; Multiple initialization strategies are used to generate an initial population containing multiple feasible solutions. A classification mechanism based on the Pareto front is used to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. A dual Q-learning mechanism is constructed, two independent Q-tables are established to optimize two objectives respectively, and candidate solutions are generated in each subpopulation using an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two. Then, non-dominated sorting and crowding distance sorting are used to select the optimal solution to update the subpopulation. The process is iterated and updated until the termination condition is met. The solutions from the two subpopulations are merged to form the Pareto front solution set, which yields the optimal scheduling scheme.

2. The multi-objective energy-saving distributed flow workshop group scheduling method as described in claim 1, characterized in that, The operating parameters include the number of factories, the number of machines configured in each factory, the number of workpiece groups, and the number of workpieces in each workpiece group. In the multi-objective mixed-integer linear programming model, the optimization objectives include the maximum completion time and total energy consumption, and the total energy consumption includes processing energy consumption, preparation energy consumption, and idle energy consumption. The decision variables include the group sorting sequence assigned to each factory, the workpiece sorting sequence within each group, and the processing speed of each workpiece on the machine; the constraints include factory allocation constraints, group sorting constraints, machine processing sequence constraints, speed selection constraints, time coordination constraints, and energy consumption calculation constraints.

3. The multi-objective energy-saving distributed flow workshop group scheduling method as described in claim 1, characterized in that, The process of population initialization and subpopulation division is as follows: Multiple initialization strategies are adopted to generate feasible solutions, focusing on minimizing the maximum completion time, minimizing the total energy consumption, and randomly initializing the initial population respectively. Then, the initial populations of the various types are merged. A Pareto front-based classification mechanism is adopted. The initial population is evaluated by non-dominated sorting and crowding distance sorting. The elite solution set of the first Pareto front is extracted. According to the preference of each elite solution on the two optimization objectives, the elite solution set is divided into two subpopulations. The remaining non-elite solutions are allocated by the direct objective bias method to balance the size of the two subpopulations.

4. The multi-objective energy-saving distributed flow workshop group scheduling method as described in claim 1, characterized in that, Candidate solutions are generated using an adaptive iterative greedy algorithm, including: Based on the operating parameters in the distributed assembly line workshop group scheduling scenario, the average number of groups in each factory is calculated to determine the scale of the factory, and then the magnitude of the damage intensity coefficient d is dynamically adjusted according to the factory scale. In the factory corresponding to the current solution, randomly remove d workpiece groups and temporarily store them in a temporary set. Insert the temporarily stored d workpiece groups one by one into all feasible positions, select the insertion scheme that makes the two optimization objectives optimal, and reconstruct the group allocation and sorting. The factory with the longest completion time among all factories is identified as the critical factory. The work group with the most workpieces in the critical factory is identified. Half of the total number of workpieces in the current group are randomly removed and temporarily stored. After perturbing the order of the remaining workpieces in the group, the temporarily stored workpieces are inserted into the optimal position in the group, and the workpiece allocation and sorting in the group are reconstructed. Reconstruct to form a new solution, verify whether the new solution satisfies all scheduling constraints, and output candidate solutions.

5. The multi-objective energy-saving distributed flow workshop group scheduling method as described in claim 4, characterized in that, Candidate solutions are generated using the Q-learning algorithm, including: Take any random solution in the population as the current solution and perform random perturbation. Calculate the change in completion time and energy consumption of the solution before and after the perturbation, and match them with the nine system states represented by two dimensions to determine the current state. Based on the pre-trained Q-table corresponding to the optimization objective, query the optimal action corresponding to the current state, and use the optimal action to perturb the current solution to generate a new solution; Verify whether the generated new solution satisfies the scheduling constraints, and output the solutions that satisfy the constraints as candidate solutions; The pre-training process of the Q-table is as follows: Take any random solution in the population as the current solution and perform random perturbation. Calculate the change in completion time and energy consumption of the solution before and after the perturbation, and match them with the nine system states represented by two dimensions to determine the current state. Record the current state-action pair, calculate the reward value of the action according to the reward formula to quantify the degree of target optimization, use the reward value to correct and update the Q value of the current state-action pair in the Q table, and calculate the estimation error of the Q value; Continuously iterate the training until the Q-value estimation error tends to stabilize and approach 0, complete the Q-Learning training, save the final Q table, and obtain the pre-trained Q table.

6. The multi-objective energy-saving distributed flow workshop group scheduling method as described in claim 5, characterized in that, A hybrid algorithm combining adaptive iterative greedy algorithm and Q-learning algorithm is used to generate candidate solutions, including: Taking any random solution in the population as the current solution, an adaptive iterative greedy algorithm is used to optimize the current solution. Through the steps of determining the factory scale, determining the damage strength coefficient, group-level destruction and reconstruction, and workpiece-level optimization, an intermediate solution is generated. The Q-learning algorithm is used to optimize the intermediate solution. The changes in completion time and energy consumption between the intermediate solution and the current solution are calculated to determine the state of the intermediate solution. Then, the optimal action is queried based on the Q table of the corresponding objective to fine-tune the intermediate solution and generate a new solution. Verify whether the new solution satisfies the scheduling constraints, and output a candidate solution that combines global search and local optimization.

7. A multi-objective energy-saving distributed assembly line workshop scheduling system, characterized in that, include: The data acquisition and model building module is used to acquire the operating parameters of the distributed assembly line workshop group scheduling scenario and build a DFGSP multi-objective mixed integer linear programming model with the optimization objective of minimizing the maximum completion time and total energy consumption. The population initialization module is used to generate an initial population containing multiple feasible solutions using various initialization strategies. It adopts a classification mechanism based on the Pareto front to divide the initial population into two subpopulations that optimize the maximum completion time and total energy consumption, respectively. The iterative optimization module is used to construct a dual Q-learning mechanism. It establishes two independent Q-tables to optimize two objectives respectively. In each subpopulation, it uses an adaptive iterative greedy algorithm, a Q-learning algorithm, and a hybrid algorithm of the two to generate candidate solutions. Then, it uses non-dominated sorting and crowding distance sorting to select the optimal solution to update the subpopulation. It iterates and updates until the termination condition is met. The scheduling scheme generation module is used to merge the solutions of the two subpopulations to form a Pareto front solution set, thereby obtaining the optimal scheduling scheme.

8. An electronic device, characterized in that, include: Memory, used to store executable instructions; The processor, when executing executable instructions stored in the memory, implements the multi-objective energy-saving distributed flow workshop group scheduling method according to any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, The device stores executable instructions that, when executed by a processor, implement the multi-objective energy-saving distributed flow workshop group scheduling method according to any one of claims 1-6.

10. A computer program product, characterized in that, The computer program product includes executable instructions stored in a computer-readable storage medium; When the processor of the electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, it implements the multi-objective energy-saving distributed flow workshop group scheduling method according to any one of claims 1-6.

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