A distributed hybrid interleaved batch scheduling method and system for parallel production lines

Abstract

This invention, entitled "A Distributed Hybrid Interleaved Batch Scheduling Method and System for Parallel Production Lines," belongs to the field of production line scheduling. It addresses the problem of minimizing inventory and stockout costs. The method includes: complete delivery based on vehicle model bill of materials; each production line comprises multiple sequentially connected stages, with the same set of dies only allowed on one production line at any given time; during the stamping stage, only one production line can utilize the stamping die to process the corresponding sub-batch; a consistent sub-batch partitioning strategy is adopted for the same batch, ensuring that the quantity and corresponding dimensions of each sub-batch remain unchanged in subsequent processing stages after partitioning; the dimensions between different sub-batches can be determined by the selected sub-batch partitioning rules; alternating processing of sub-batches from different batches is allowed on the same production line; each sub-batch can only be processed by one production line in each stage; the scheduling objective is to minimize the total sum of inventory and stockout costs while achieving complete delivery.

Description

Technical Field

[0001] This invention relates to the field of production line scheduling technology, and in particular to a distributed hybrid staggered batch scheduling method and system for parallel production lines. Background Technology

[0002] A smart manufacturing workshop is an upstream and downstream chain, meaning that each stage must match the progress and resource usage of other stages. Intermediate inventory serves as a critical buffer for the flow of materials and information, helping to maintain rhythm, smooth demand changes, and reduce disruptions.

[0003] In the automotive manufacturing industry, the stamping workshop represents a typical upstream stage, providing various structural components for downstream processes such as painting and final assembly. A stamping workshop typically consists of multiple parallel production lines. The blanking line is responsible for coil unwinding, straightening, and cutting; while the stamping line usually consists of 4-6 stamping presses connected in series, completing processes such as drawing, trimming, and punching. Due to strict cycle time requirements and the uninterrupted nature of this process, the entire stamping line is usually modeled as a single operation in production scheduling. Detailed processing workflows are as follows... Figure 1 As shown.

[0004] As is well known, production scheduling is a core component of manufacturing system operation. It is a key means to coordinate resources, balance logistics pace, reduce work-in-process inventory, shorten delivery cycles, and ensure production resilience. A reasonable and effective scheduling strategy can not only significantly alleviate bottlenecks and reduce inventory and work-in-process levels, but also improve equipment utilization and plan fulfillment rates, thereby enhancing the shop floor's responsiveness and overall competitiveness.

[0005] Traditional process shop models are characterized by indivisible jobs and strict sequential operations. In contrast, the LFSP (Batch Flow Scheduling Problem) allows batches to be divided into multiple sub-batches. This mechanism significantly compresses production cycles and reduces work-in-process levels, making it a key research area for improving scheduling efficiency. Approaches to solving this problem typically involve determining batch partitioning, sub-batch sequencing, and associated resource constraints, while simultaneously optimizing multiple objectives, such as completion time and total delay.

[0006] Therefore, the core idea of ​​LFSP provides an important modeling foundation for batch production scheduling in complex manufacturing systems. Especially in production environments with multiple stages, multiple resources, and strong process constraints, sub-batch partitioning can enhance the parallelism and flexibility of the production process, enabling closer connections between upstream processing, process transfer, and downstream assembly. However, batch flow scheduling is not entirely the same across different industry scenarios; their resource constraints, material matching relationships, and production organization methods often exhibit distinct industry characteristics. For automotive stamping workshops, stamped parts typically need to be supplied in complete sets according to vehicle model requirements and are influenced by factors such as production line capacity, mold occupancy, mold changeover processes, and downstream welding rhythm. Therefore, it is necessary to further analyze the modeling approaches and shortcomings of existing stamping scheduling research in light of the actual process characteristics of stamping workshops.

[0007] Stamping workshop scheduling is a typical production scheduling problem involving the scheduling of multiple tasks under various process constraints. Existing research has explored stamping workshop scheduling from different perspectives: some studies have developed multi-constraint scheduling models for stamping workshops under time-of-use electricity pricing, considering shift availability, line-side inventory status, and safety stock, describing daily scheduling scenarios for a single workshop with multiple production lines to minimize completion time and total electricity cost; some studies have constructed scheduling models based on enhanced genetic algorithms to solve operation sequencing, die changing, and batch decision-making problems, highlighting issues related to die changing and cycle time control in large-scale batch production; other studies focus on the die scheduling problem in a dual-crane stamping workshop, optimizing the handling process under constraints such as crane travel distance, die positioning, and priority, with crane lifting and die transportation as objectives; still other studies explore multi-workshop collaborative scheduling from the perspective of vehicle production, coordinating the production rhythm of different workshops by buffering and connecting stamping, welding, and final assembly, and establishing an energy consumption assessment framework for stamping workshops based on discrete event simulation methods, simulating and comparing the trade-offs between energy consumption and completion time under various production scenarios and operating conditions.

[0008] In terms of application scenarios, existing research in the field of batch flow scheduling mainly focuses on collaborative decision-making regarding factory allocation, sub-batch division, and processing sequence. However, such research generally treats a single job as an independent sub-batch division unit and regards different batches as independent autonomous entities.

[0009] Regarding solutions, existing research typically employs batch-level coding to optimize objectives such as completion time, energy consumption, or installation time. Among these, metaheuristic methods such as the Two-Role Co-evolutionary Algorithm (BRCE) and the Multi-Strategy Iterative Greedy Algorithm (MSIG) are widely applied to various distributed flow shop scheduling problems. However, these methods require decoding and feasibility checks of the entire plan at each iteration, leading to high computational costs and insufficient utilization of the scheduling structure.

[0010] Based on a systematic review of relevant research, existing studies have the following main shortcomings when applied to actual stamping scenarios: (1) Existing research mainly focuses on efficiency indicators such as completion time, without clearly describing the costs associated with complete delivery, inventory, and shortages. This problem is exacerbated when shared parts compete for limited resources.

[0011] (2) Existing methods treat sub-batch splitting and production line scheduling as separate tasks, ignoring the impact of sub-batch size and count on mold conflict and workload balance.

[0012] (3) Algorithm configuration still heavily relies on human expertise. In scenarios with specialized process characteristics, there is a lack of data-driven methods to automatically configure components and parameters. Summary of the Invention

[0013] The purpose of this invention is to provide: A distributed hybrid staggered batch scheduling method for parallel production lines, and related technologies, are proposed to solve technical problems such as minimizing inventory costs and shortage costs, or a combination thereof, under the premise of complete delivery.

[0014] Terminology Explanation: Unless otherwise defined, all technical terms in this embodiment have the same meaning as commonly understood by one of ordinary skill in the art to which the subject matter of the claims pertains. Unless otherwise stated, all patents, patent inventions, and disclosures referenced throughout this embodiment are incorporated herein by reference in their entirety. If there are multiple definitions for terms in this embodiment, the definitions in this chapter shall prevail.

[0015] It should be understood that the above brief description and the following detailed description are exemplary and for illustrative purposes only, and do not limit the subject matter of the invention in any way. In this invention, the singular is used in conjunction with the plural unless otherwise specifically stated. It should also be noted that, unless otherwise stated, the use of “or” or “or” means “and / or”. Furthermore, the use of the term “comprising” and other forms such as “including,” “containing,” and “contains” are not limiting.

[0016] Unless otherwise stated, conventional methods within the scope of the art shall be used.

[0017] In a first aspect, the present invention provides: a distributed hybrid staggered batch scheduling method for parallel production lines, which addresses the distributed flow shop interleaving scheduling problem (DHILSP), including: Deliver complete sets according to the bill of materials for each vehicle model, set spare parts inventory levels, and ensure safety stock. Each production line includes multiple processing stages connected in sequence; the mold set used to process a specific batch of parts is unique, and at any given time, the same mold set can only be configured on one production line and cannot be occupied by multiple production lines in parallel; the external mold change for the next batch can be carried out in parallel with the processing of the current sub-batch. In the stamping stage, for sub-batches that require processing with the same mold set, at any given time, only one sub-batch is allowed to be processed on one production line. A consistent sub-batch division strategy is adopted for the same batch. After the sub-batch division is completed, the quantity and corresponding size of each sub-batch remain unchanged in subsequent processing stages; the size between different sub-batches can be determined by the selected sub-batch division rule. Any sub-batch must pass through all production stages sequentially and processing cannot be preempted, but alternating processing of sub-batches from different batches is allowed on the same production line; each machine can process at most one sub-batch at any given time, and any sub-batch can only be processed by one production line at each stage. The scheduling objective is to minimize the sum of inventory costs and stockout costs while achieving complete delivery.

[0018] The above solution adopts a consistent sub-batch division strategy based on the Bill of Materials (BOM) under the constraints of complete delivery. This results in strong coupling between sub-batch structures of different batches under the same vehicle model at the levels of cross-parts, cross-processes, and even cross-production lines. This solves the problem of minimizing inventory costs and shortage costs under the premise of complete delivery.

[0019] Based on further solutions to the technical problems of the present invention, or simultaneous solutions to multiple technical problems, the preferred solution in the technical solution provided in the first aspect of the present invention includes: In the first preferred scheme, in DHILSP, switching between different batches of adjacent processing requires setting a time, while two adjacent sub-batches of the same batch do not have a time set by default; after a sub-batch completes processing in the previous stage and reaches the downstream stage, the sub-batch can only start processing when the target machine is available and the necessary settings are completed.

[0020] More preferably, DHILSP is also subject to the following basic assumptions: At the start of the scheduling process, all production lines are idle and all resources are available. No preemption or interruption is permitted during the entire processing period; Allow machine idle time, assuming the buffer capacity between stages is infinite; Any sub-batch can only proceed to the next stage after the current stage of processing is completed; The number of any vehicle model delivered must not exceed the demand. The size of any sub-batch must be greater than or equal to the smallest sub-batch size.

[0021] More preferably, the DHILSP mathematical model is as follows: The objective function is expressed as follows: (1); The constraints are as follows: (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); n Total number of batches, i.e., total number of parts; m Total number of production stages; h Total number of vehicle models; F Total number of production lines; L : Maximum number of sub-batches per batch; H Daily processing time; i Stage index; ; j， Batch index; , Indicates different j batches; k Vehicle model index; ; f Production line index; ; e, Sub-batch index; , Indicates different e Sub-batch; G : Represents a sufficiently large positive integer; Model k Required batch j The quantity, i.e., the batch j The corresponding number of parts; : k Demand for this type of vehicle; : No. j Batch No. i Processing time for each stage; : No. j Batch No. i The time frame for setting the stage, among which Indicates batch j The setup time for Phase 1; : No. j Unit inventory cost per batch; Model k The unit out-of-stock cost; : No. j Historical inventory of batches; : No. j A batch of historical shortages; : No. j sub-batch e The minimum size is determined by the external mold change time; : No. j The daily target production quantity for the batch; Model k The number of deliveries; : No. j Batch inventory; : No. i Phase 1 j Sub-batch of a batch e The start time, of which Indicates batchj The start time of sub-batch 1 in stage 1; : No. i Phase 1 j Sub-batch of a batch e End time; : No. j Sub-batch of a batch e Size; Auxiliary variable ; : A binary variable, if the production line f The first j Sub-batch of a batch e It is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0. : A binary variable, if the first j Sub-batch of a batch e Assigned to production line f If the result is positive, then it equals 1; otherwise, it equals 0. : A binary variable, if the first j Sub-batch of a batch e If activated, it equals 1; otherwise, it equals 0. Indicates batch j Sub-batch 1 is activated; : A binary variable, if the first j Sub-batch of a batch e The stamping stage is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0.

[0022] Equation (1) minimizes the total cost, which includes inventory cost and stockout cost. Equation (2) represents the vehicle model. k The actual number of delivered units. Due to matching delivery requirements, the delivery quantity is determined by the smallest batch that meets the matching constraints, and must not exceed the demand. Equation (3) determines the target production quantity of batch j, requiring that the sum of the sub-batch sizes of a given batch equals its target batch size. Equation (4) defines the final inventory level of the j-th batch. Equation (5) defines the... linearization Equation (6) ensures that each active sub-batch is accurately assigned to a line. Equation (7) ensures that the start time of the first batch of each active batch in the first stage is greater than or equal to its set time. Equation (8) ensures that the processing of each active sub-batch is not interrupted. Equation (9) ensures that in the final stage (i.e., the stamping stage), due to the uniqueness of the die, any batch is processed on only one line at a time. Equation (10) ensures that the processing of the next stage only begins after the current stage is completed. Equation (11) ensures that if two sub-batches are assigned to the same machine in a given stage, the next sub-batch only begins after the previous sub-batch is completed. Equation (12) ensures that each active sub-batch is greater than or equal to its minimum size. Equation (13) ensures that each sub-batch is processed only within the available time window and sets the processing time of inactive sub-batches to zero. Equation (14) establishes the relationship between two decision variables. Equation (15) guarantees that the preceding sub-batches are given priority in accommodating items. Equation (16) establishes the relationship between two decision variables. Equation (17) establishes the relationship between the two decision variables.

[0023] More preferably, a configurable constructor-improvement algorithm is used to determine the scheduling scheme during the DHILSP scheduling process. This includes employing a constructor-then-improvement strategy, initially generating feasible initial solutions using a configurable constructor method, and then performing deep optimization using an improved architecture; including: Determine the initial scheduling scheme: A configurable constructivist heuristic approach is adopted, including sequential execution of sub-batch partitioning, production line routing, and job assignment planning, to determine the initial scheduling scheme; Scheduling scheme optimization: During the scheduling process, a disruptive batch perturbation rule based on iterative greedy IG and variable neighborhood descent VND is introduced to adaptively adjust the existing scheduling scheme. Specifically, IG is used to merge the selected scheduled sub-batches with the unscheduled set and then perform greedy reconstruction. VND is used to perform deep local search through multiple complementary neighborhoods to determine the optimized scheduling scheme.

[0024] It further enables the efficient construction of feasible initial scheduling schemes from original orders that meet complex constraints.

[0025] More preferably, a MILP sub-model is embedded during the search process to quantify the trade-off between inventory costs and shortage costs.

[0026] More preferably, in the packaged delivery mode, a two-stage evaluation strategy is adopted for the allocation of common parts shortages. First, unavoidable shortages caused by dedicated parts are identified. Then, the allocation decision for common parts is transformed into a mixed-integer linear programming (MILP) sub-model. The MILP sub-model establishes boundary constraints based on the actual order quantity and the calculated part shortage quantity, and uses the CPLEX solver to determine the optimal allocation scheme that minimizes the total cost. This further addresses the uncertainty problem of common parts shortage allocation in the packaged delivery mode.

[0027] More preferably, the MILP sub-model modeling process is as follows: Based on stock shortage ,Sure k Inevitable shortage of certain car models : ; in, J k For vehicle model k A collection of specialized parts; make The objective function of the MILP sub-model, representing the remaining shortage attributable to shared parts, is as follows: ; The constraints of the MILP sub-model are as follows: ;

[0028] More preferably, the sub-batch division adopts the following rules: Equal Quantity Rule (ES): This rule aims to generate sub-batches of uniform size while satisfying the minimum sub-batch size constraint. Bill of Materials (BOM) Kit Alignment Rule (BKA): Designed for kit delivery, this rule aligns sub-lot dimensions with kit requirements. Sub-lot dimensions are defined as follows: ,in A positive integer, representing a sub-batch. The complete set of multiples, and , For batch j The smallest sub-batch size, d k for k Demand for this type of vehicle; Gradually Increasing Sub-Batch Size Rule (IS): This rule uses the smallest possible sub-batch size. Use this as an initial base number to gradually increase the size of sub-batches; Fibonacci Increasing Rule (FI): In the rule of progressively increasing sub-batch size, the Fibonacci sequence is further used as weights to generate a sub-batch structure with increasing sub-batch sizes; the weights are defined as the following sequence: That is, starting from the third sub-batch, the size of each sub-batch is the sum of the sizes of the first two sub-batches; where, The weight of the first sub-batch, The weight of the second sub-batch, Let be the weight of the e-th sub-batch; While ensuring that the minimum sub-batch size is met, the upper limit of the number of sub-batches within the same batch should be increased as much as possible. That is, choose the largest possible Make

[0029] Sub-batch dimensions ; u As an intermediate variable; In all the above rules, if the remainder is greater than or equal to when dividing into sub-batches... If yes, then it forms a separate sub-batch; otherwise, it is merged into the last sub-batch. in, Model k Required batch j The quantity, i.e., the batch j The corresponding number of parts; : k Demand for this type of vehicle; e Sub-batch index.

[0030] More preferably, the production line routing selection adopts at least one of the following rules: Earliest Available Time (EAT): Select the production line with the earliest available time among all production lines and assign the sub-batch to that production line; Minimum Total Processing Load (MTPL): Select the production line with the shortest total processing time among all production lines and assign the sub-batch to that production line; Minimum Workload (MWL): Select the production line that processes the fewest sub-batches and assign the sub-batches to that production line.

[0031] More preferably, the job assignment planning adopts rules based on processing time, bottleneck stage load, and mold activation status, including: Shortest / Longest Processing Time (SPT / LPT): Prioritize the sub-batch with the shortest / longest total processing time; Shortest / Longest Phase 1 Processing Time (SPTF / LPTF): Prioritize the sub-batch with the shortest / longest Phase 1 processing time; Shortest / Longest Processing Time to Stamping Stage (SPTS / LPTS): Prioritize the sub-batch with the shortest / longest processing time at the stamping stage; Mold priority: Prioritize the batch that requires the current mold to be activated, and then use the above rules to determine the specific sub-batch; Cost priority: Prioritize the sub-lot with the highest stockout cost, and calculate the stockout cost by multiplying the average unit stockout cost by the sub-lot quantity; Unit cost priority: Prioritize the sub-batch with the highest unit stockout cost, and calculate the unit stockout cost by dividing the stockout cost by the processing time.

[0032] More preferably, the configurable construct-improvement algorithm employs a feedback-based destruction strategy, including: An adaptive closed-loop feedback control mechanism sensitive to the search state is established, and the IG process is modeled as a dynamic control system with a failure rate. The cost difference between adjacent iterations is a control variable. As a feedback signal, the formula is as follows:

[0033] in, t For iterative index, This is the gain coefficient, manually set to amplify the feedback signal; This represents the objective function value generated by the destruction-reconstruction operation in the previous iteration; The reaction gain coefficient is derived from subsequent DG-IFRace optimization.

[0034] More preferably, the cost difference between adjacent iterations of the feedback signal As shown in the formula below: When the current scheduling scheme is better than the previous scheduling scheme, Take the negative value; otherwise ; This represents the objective function value generated by the destruction-reconstruction operation in the current iteration.

[0035] More preferably, during the destruction phase, based on the current scheduling sub-batch length and destruction rate Calculate the amount removed Subsequently, use one of the following rules to select from the current scheduled sub-batch sequence. Select q Remove each target sub-batch: Late-sequence destructive (LSD): from the currently scheduled sub-batch sequence Starting from the end, select in reverse order. q Remove individual batches; Early sequence destruction (ESD): from the currently scheduled sub-batch sequence Select the head in sequence q Delete individual batches; Shortage-related disruption of SCD: from the current scheduled sub-batch sequence Prioritize removing those that cannot be completed within the specified time. H The set of sub-batches that are scheduled within the time limit If the number of sub-batches in the same batch is less than [a certain number], then [the following applies]. q If all are removed, a new batch is randomly selected from the remaining scheduled batches to make up the total. q Otherwise, start from the current scheduled sub-batch sequence. Randomly selected from q Remove individual batches.

[0036] More preferably, during the reconstruction phase, the greedy strategy sequentially modifies the sub-batch sets to be inserted. Sub-batch insertion into the existing schedule Then sort them according to the following rules: Maximum Stockout Cost Priority (MSCF): This rule sorts sub-lots in descending order of potential stockout cost, calculated by multiplying the average unit stockout cost by the sub-lot quantity. Longest Duration Priority (LDF): This rule sorts sub-batches in descending order of total processing time, calculated by multiplying the unit processing time by the number of sub-batches. Highest Value Density Priority (HVDF): This rule sorts sub-lots in descending order of value rate, which is defined as total stockout cost divided by total processing time.

[0037] More preferably, VND is embedded in IG as a local search module after each "destruction-reconstruction" phase. The VND algorithm establishes a hierarchical improvement mechanism, prioritizing production time compression over total cost. The VND algorithm includes: First, within the currently scheduled sub-batch sequence, the order is rearranged and fine-tuned sequentially through same-batch position exchange, same-batch size adjustment, and cross-batch position exchange. This stage uses production line-level time indicators as evaluation criteria and adopts the sum of completion times of each production line as the objective function. Subsequently, the sub-batch structure neighborhood is activated: sub-batches from the unscheduled set are inserted into the scheduled sequence, or sub-batches are swapped between the scheduled sequence and the unscheduled set, and then sub-batches that cannot be scheduled within the specified time are removed from the scheduled sequence again. HFor sub-batches that are scheduled within a given timeframe, the improvement criterion for both the scheduled sequence and the unscheduled set is the total cost objective. Lower.

[0038] More preferably, the calibration of parameters in all algorithms uses the improved I / F-Race (Iterative Competition Configuration Method) algorithm DG-IFRace (Dispersion-Guided Iterated F-Race). The DG-IFRace algorithm introduces a hard instance batching mechanism into the I / F-Race algorithm. It uses the performance dispersion of candidate configurations on the same batch of instances as the distinguishing index. At the end of each iteration, the instance batch with the most significant distinguishing index is selected and reused as the stress test set at the beginning of the next iteration.

[0039] More preferably, in terms of generating new configurations, the DG-IFRace algorithm normalizes the values ​​of objects and discards candidate objects that are statistically worse than at least one competitor in the Wilcoxon rank-sum test after sorting by average cost; after each competition, a new configuration is drawn from the learning model that favors elite configurations in the group, and a parent configuration is selected based on rank-based probability. The classification parameters are generated through adaptive hybrid updates, which combine the global probability model with the parent-oriented approach as the competition index increases, and the numerical parameters are drawn from a truncated normal distribution centered on the parent value.

[0040] More preferably, the total tuning budget B in the DG-IFRace algorithm is measured by the number of configuration instances evaluated, where one experiment evaluates the configuration on a NOI tuning instance, based on the total tuning budget. B and the budget consumed Allocate the budget for each competition from the remaining budget. The budget will be focused on candidates with greater potential.

[0041] On the other hand, the present invention also provides a distributed hybrid staggered batch scheduling system for parallel production lines, including a processor capable of executing a computer program, which, when executed, enables the implementation of the above-described distributed hybrid staggered batch scheduling method for parallel production lines.

[0042] In this invention, Embodiment 1 at least supports the protection scope of the above-described distributed hybrid staggered batch scheduling method for parallel production lines.

[0043] Compared with the prior art, the present invention has at least the following beneficial effects: (1) The present invention adopts a sub-batch boundary alternating processing strategy and establishes a related MILP model accordingly, thereby minimizing inventory costs and shortage costs under the premise of complete set delivery.

[0044] (2) The CCI (Configurable Constructive-Improvement) proposed in this invention is a configurable constructive-improvement framework that integrates construct initialization, feedback-controlled destruction and reconstruction search and two-stage cost calculation. It achieves strong coupling between material availability and production decision-making, maintains competitiveness under tight time budgets, and has a more concentrated performance distribution, indicating strong robustness, low sensitivity to scale changes, and support for wide applicability to different workshop settings.

[0045] (3) The DG-IFRace proposed in this invention can reduce configuration costs and improve screening quality. Compared with FFD (Full Factorial Design), DG-IFRace avoids exhaustive enumeration and identifies strong configurations with fewer evaluations. Compared with F-IFRace (Iterative Race Configuration Based on Friedman Test) and W-IFRace (Iterative Race Configuration Based on Wilcoxon Test), DG-IFRace improves early identification capabilities by using distributed guided hard instance batches, thereby eliminating poor configuration combinations more quickly and selecting elites more stably. Overall, DG-IFRace provides an effective configuration combination, and the resulting CCI provides reliable and scalable performance for the stamping scheduling problem studied. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of the automotive stamping workshop process according to an embodiment of the present invention.

[0047] Figure 2 This is a schematic diagram illustrating a solution according to an embodiment of the present invention.

[0048] Figure 3 This is a flowchart of a configurable construction improvement algorithm according to an embodiment of the present invention.

[0049] Figure 4 This is a schematic diagram of closed-loop feedback control of IG-VND adaptive failure rate according to an embodiment of the present invention.

[0050] Figure 5 This is a schematic diagram of the N1-N5 neighborhood structure according to an embodiment of the present invention.

[0051] Figure 6This is a comparative schematic diagram of DG-IFRace, W-IFRace and F-IFRace according to an embodiment of the present invention; wherein, (a) is a 95% confidence interval plot and (b) is a box plot.

[0052] Figure 7 This is a schematic diagram comparing CCI variants according to an embodiment of the present invention; wherein, (a) is a 95% confidence interval plot and (b) is a box plot.

[0053] Figure 8 This is a comparative diagram of CCI with BRCE (Dual Role Co-evolution), DHICCA (Dynamic Heterogeneous Identity Co-evolution Algorithm), Jaya and MSIG (Multi-Strategy Iterative Greedy Algorithm) according to an embodiment of the present invention; wherein, (a) is a 95% confidence interval plot and (b) is a box plot.

[0054] Figure 9 This is a schematic diagram illustrating the performance trends of CCI, BRCE, DHICCA, Jaya, and MSIG across different problem scale factors according to an embodiment of the present invention; wherein, (a) is the algorithm performance trend of different number models, (b) is the algorithm performance trend across different number of rows, (c) is the algorithm performance trend of different number of stages, and (d) is the algorithm performance trend under different demand levels. Detailed Implementation

[0055] The following non-limiting embodiments are intended to enable those skilled in the art to gain a more comprehensive understanding of the present invention, but do not limit the invention in any way. The following content is merely an exemplary description of the scope of protection claimed by the present invention, and those skilled in the art can make various changes and modifications to the present invention based on the disclosed content, and such changes should also fall within the scope of protection claimed by the present invention.

[0056] The present invention will be further described below by way of specific embodiments. Any aspects not covered in detail are implemented using existing technology and are not the focus of the present invention.

[0057] Example 1 This invention provides a distributed hybrid staggered batch scheduling method for parallel production lines, which uses the distributed flow shop batch interleaving scheduling problem (DHILSP) for scheduling. In DHILSP, there are a total of F Several parallel production lines, each consisting of sequentially connected components. m It consists of several stages; it exists n One reason A batch consisting of identical parts, among which The calculation is based on a combination of historical inventory information, historical stockout information, and demand from downstream workshops. Each batch must go through all stages in sequence and processing cannot be preempted.

[0058] A consistent sub-batch strategy is employed for splitting, maintaining a consistent number and size of sub-batches throughout the process. A minimum sub-batch size is enforced to prevent idle waiting. Sub-batches from different batches are allowed to be interleaved on the same machine. Switching between batches processing adjacent processes requires a time setting; two adjacent sub-batches within the same batch do not have a time setting by default. Sub-batches are processed in stages. i After completion and arrival at the downstream stage, a sub-batch can only begin processing when the target machine is available and the necessary setup is complete. Regarding capacity and allocation, each machine can process at most one sub-batch at any given time, and any sub-batch must be processed by only one machine at each stage. Furthermore, considering the uniqueness constraint of the die, a single-line occupancy strategy is enforced in the final stamping stage: at any given time, only one production line can utilize the die to process its corresponding sub-batch. Each batch has a unit expected inventory cost. Since delivery compliance is assessed on a vehicle-by-vehicle basis, shortages are also calculated at the vehicle level, with each vehicle having a unit expected shortage cost. In summary, the optimization objective is expanded to minimize the sum of inventory holding costs and stockout costs, while ensuring complete product availability.

[0059] Furthermore, DHILSP is subject to the following fundamental assumptions: At the start of the scheduling process, all production lines are idle and all resources are available. At any given time, each sub-batch can be processed by at most one production line; No preemption or interruption is permitted during the entire processing period; For any given batch, it can only be processed on one production line at any given time during the final stage (i.e., the stamping stage); Different batches of sub-batches may have set times during continuous processing; Allow machine idle time, assuming the buffer capacity between stages is infinite; Any sub-batch can only proceed to the next stage after the current stage of processing is completed; The number of any vehicle model delivered must not exceed the demand. Any sub-batch must be greater than or equal to its smallest sub-batch; Each sub-batch must go through all stages of processing.

[0060] The mathematical model for DHILSP is as follows: The objective function is expressed as follows: ; The constraints are as follows: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; in, n Total number of batches, i.e., total number of parts; m Total number of production stages; h Total number of vehicle models; F Total number of production lines; L : Maximum number of sub-batches per batch; H Daily processing time; i Stage index; ; j， Batch index; , Indicates different j batches; k Vehicle model index; ; f Production line index; ; e, Sub-batch index; , Indicates different e Sub-batch; G : Represents a sufficiently large positive integer; Modelk Required batch j The quantity, i.e., the batch j The corresponding number of parts; : k Demand for this type of vehicle; : No. j Batch No. i Processing time for each stage; : No. j Batch No. i The time frame for setting the stage, among which Indicates batch j The setup time for Phase 1; : No. j Unit inventory cost per batch; Model k The unit out-of-stock cost; : No. j Historical inventory of batches; : No. j A batch of historical shortages; : No. j sub-batch e The minimum size is determined by the external mold change time; : No. j The daily target production quantity for the batch; Model k The number of deliveries; : No. j Batch inventory; : No. i Phase 1 j Sub-batch of a batch e The start time, of which Indicates batch j The start time of sub-batch 1 in stage 1; : No. i Phase 1 j Sub-batch of a batch e End time; : No. j Sub-batch of a batch e Size; Auxiliary variable ; : A binary variable, if the production line f The first j Sub-batch of a batch e It is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0. : A binary variable, if the first j Sub-batch of a batch e Assigned to production line f If the result is positive, then it equals 1; otherwise, it equals 0. : A binary variable, if the first j Sub-batch of a batch e If activated, it equals 1; otherwise, it equals 0. Indicates batch j Sub-batch 1 is activated; : A binary variable, if the first j Sub-batch of a batch e The stamping stage is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0.

[0061] This embodiment uses an illustrative example with two production lines. This example considers two vehicle models and five batches. Vehicle demand is specified as follows: And the corresponding unit stockout cost and unit inventory cost BOM Matrix As shown below. Historical inventory is... Initial shortage Assume it to be zero. Specify the minimum subbatch size. The deadline is 10. The value is 500. Key production data is as follows, including unit processing time. and unit set time .

[0062] ; ; .

[0063] Figure 2 A graphical representation of the solution is provided. In stage 5 (stamping stage), taking batch 3 as an example, the sub-batch on production line 2... The mold is occupied first; therefore, the sub-batch on production line 1 We must wait for the sub-batch Processing can only begin after the mold has been released and finishing is complete. The five batches are 25, 26, 48, 22, and 26 units respectively. Deliverable quantities are determined through "setup validation" (based on current production output and initial inventory). (Together determined) Calculated If vehicle model A cannot meet the complete set requirements due to insufficient limiting components, a shortage is recorded, incurring shortage costs; the remaining parts are carried forward as inventory. These data and diagrams together illustrate scheduling, delivery, and inventory. The first three batches are general-purpose parts, while the last two batches are dedicated parts for vehicle models A and B, respectively.

[0064] Based on complete verification and requirement constraints, namely and Model A resulted in a shortage of 8 units, leading to a total stockout cost of 120. Historical inventory... With batch-level distribution quantities of 30, 0, 40, 0 and 30 (from and Combining the obtained costs, the inventory levels for the five batches were 0, 32, 16, 10, and 0, respectively. Based on the unit inventory cost, the inventory cost was 58. In summary, the total cost for this period is the sum of the stockout cost and the inventory cost, i.e. .

[0065] As is well known, designing a single, fixed algorithm capable of effectively solving a wide variety of problem instances is extremely challenging. Therefore, this invention proposes a configurable construct-improve algorithm that employs a hierarchical "construct first, improve later" strategy. Initially, a configurable construction method is used to quickly generate feasible initial solutions, followed by deep optimization using an improved architecture.

[0066] This method integrates three core components: (1) A configurable constructive heuristic method was developed to transform raw orders into feasible initial scheduling schemes by sequentially executing sub-batch size, production line routing selection and job assignment planning.

[0067] (2) A configurable feedback-controlled destruction and reconstruction mechanism was developed to balance global exploration and local development.

[0068] (3) Embed MILP sub-models during the search process to accurately quantify the total cost trade-off between inventory and shortage.

[0069] Figure 3 The overall algorithm framework is outlined. Among them, For a set of batches, These represent the sets of the total sub-batch and the non-insertable sub-batch, respectively. f * , SL*This refers to production lines and sub-batches selected according to specific rules; These represent the scheduling schemes before and after optimization, respectively; They are respectively The total cost. Detailed operational instructions are as follows.

[0070] 1. Configurable constructive heuristics: To effectively construct feasible initial scheduling schemes from raw orders that satisfy complex constraints, such as mold uniqueness, minimum sub-batch size, and complete delivery, this invention proposes a Configurable Constructive Heuristic (CCH). This algorithm decouples highly coupled decision-making into three consecutive stages: configurable sub-batch partitioning, configurable production route routing, and configurable scheduling. By providing multiple optional operators at each stage, CCH can flexibly adapt to the characteristics of different instances, providing a rich policy search space for subsequent CCI based on AAD (Automatic Algorithm Design). The specific stages are described below.

[0071] 1.1 Configurable sub-batch partitioning stages: In the scheduling of multiple production lines with consistent subbatches, subbatching serves as the starting point for constructive heuristics, directly impacting production line load balancing, mold occupancy dynamics, stockouts, and inventory cost-effectiveness. To achieve flexible adjustments based on BOM structure and minimum subbat size constraints, this embodiment proposes a class of subbatching rules. These rules generate subbat structures with distinct characteristics through various combinations of parameters and operators. Specifically, four representative rules are as follows: Equal Quantity (ES) Rule: This rule aims to generate sub-batches of uniform size while satisfying the minimum sub-batch size constraint. It effectively mitigates load imbalances and frequent setups caused by sub-batches that are too large or too small, and promotes balanced distribution across multiple production lines.

[0072] BOM Kit Alignment (BKA) Rule: Designed for kit delivery, this rule aligns sub-lot sizes with kitting requirements to minimize structured inventory and maximize insertability. Sub-lot size is defined as... ,in It is a positive integer, and .

[0073] Increase Size (IS) rule: This rule uses the smallest sub-batch size. As an initial base, the sub-batch size is gradually increased (set to 1.3 in this embodiment) to generate a series of sub-batches.

[0074] Fibonacci Increasing (FI) Rule: Using the Fibonacci sequence as weights to generate an increasing sub-batch structure: The weights are defined as follows: (18) At the same time, while meeting the "minimum sub-batch" condition, the sub-batch size should be increased as much as possible. That is, choose the largest possible Make: (19) That is, the size of the sub-batch .

[0075] In all the rules above, if the remainder is greater than or equal to If the result is positive, it forms a separate sub-batch; otherwise, it is merged into the final sub-batch.

[0076] 1.2 Configurable Production Line Routing Stage: After sub-batching, a suitable production line must be selected for each sub-batching to achieve load balancing and maximize multi-line capacity utilization under mold and process constraints. To this end, this embodiment designs several candidate routing rules that use various time- and load-oriented metrics to evaluate currently available production lines to determine the assigned production line. f Specifically, three representative routing strategies are proposed, as follows: Earliest Available Time (EAT): Select the production line with the earliest available time among all production lines and assign it as... f .

[0077] Minimum Total Processing Load (MTPL): Select the production line with the shortest total processing time among all production lines and allocate it as... f .

[0078] Minimum Working Load (MWL): Select the production line that processes the fewest sub-batches and assign it to [the production line]. f .

[0079] 1.3 Configurable Scheduling Scheme Optimization Phase: After determining the production line to which a sub-batch is assigned, the next step is to select items for immediate processing from the pool of candidate sub-batch waiting on that line. This decision directly impacts the internal operational sequence, work-in-process inventory, and bottleneck stage utilization. Therefore, this embodiment designs multiple candidate scheduling rules that prioritize candidate sub-batch based on different processing times and resource usage characteristics to select the appropriate production line. f The next sub-batch to be processed. Specifically, 14 different scheduling strategies were considered, including rules based on processing time, bottleneck stage load, and mold activation status, as described below.

[0080] Shortest / Longest Processing Time (SPT / LPT): Prioritize the sub-batch with the shortest / longest total processing time.

[0081] Shortest / Longest Stage 1 Processing Time (SPTF / LPTF): Prioritize the sub-batch with the shortest / longest stage 1 processing time.

[0082] Shortest / Longest Processing Time to Stamping Stage (SPTS / LPTS): Prioritize the sub-batch with the shortest / longest processing time at the stamping stage.

[0083] Mold priority: Prioritize the batch that requires the current mold to be activated, and then use the above rules to determine the specific sub-batch.

[0084] Cost priority: Prioritize the sub-lots with the highest stockout costs (calculated by multiplying the average unit stockout cost by the sub-lot quantity).

[0085] Unit cost priority: Prioritize the sub-batch with the highest unit out-of-stock cost (calculated by dividing the out-of-stock cost by the processing time).

[0086] In summary, this algorithm transforms the original order into a feasible schedule that satisfies all process constraints by sequentially executing the sub-batch partitioning, production line routing, and scheduling stages, thus generating a complete initial solution. It is worth noting that... Figure 2 The Gantt chart in the example illustrates a solution generated using a specific combination of rules (BKA+EAT+SPTF).

[0087] 2. Configurable feedback control destruction and reconstruction mechanism CCH generates an initial schedule. However, due to production limitations, some sub-batches may not be able to complete within the allotted time. To improve the quality of local scheduling and minimize unscheduled sub-batches, this invention introduces a configurable feedback-controlled disruption and reconstruction perturbation mechanism, in which several components need to be instantiated. Specifically, this mechanism is triggered if unscheduled sub-batches still exist after the initial decoding. The algorithm runs simultaneously on both scheduled and unscheduled sub-batches, adapting the existing scheduling scheme through an adaptive improvement process by integrating Iterative Greedy (IG) and Variable Neighborhood Descent (VND). IG is used to perform large, structured perturbations, escaping local optima and rapidly reorganizing critical sub-batches under tight time budgets, while VND is used to reinforce the search through multiple complementary neighborhoods, thus achieving efficient local optimization.

[0088] 2.1 IG-VND Framework Algorithm 1 of this invention proposes an IG-VND program with embedded closed-loop feedback control, aiming to repair unscheduled subbatches and optimize the total system cost within a limited budget. The algorithm employs the IG operator to merge selected scheduled subbatches with the unscheduled set, then performs greedy reconstruction to merge these subbatches, and uses a deep local search based on VND to improve the quality of the solution. A core feature is determined by the gain coefficient. A dominant feedback control mechanism that bases performance differences between adjacent iterations. Dynamically adjust the damage rate This allows for an adaptive balance between global exploration and local development during periods of stagnation or improvement, such as... Figure 4 As shown. Algorithm 1. Pseudocode for IG-VND: Input: Initial scheduling scheme

[0089] Completed sub-batch and unavailable sub-batch set A set of defined adjacent structures ,

[0090] A set of rule violations R and a set of sorting rules S Rule violation rate and number of iterations t Reaction gain coefficient

[0091]

[0092] When the current CPU time is less than the termination time

[0093]

[0094] For each sub-batch :

[0095] Terminate loop

[0096] if

[0097]

[0098] Termination judgment

[0099]

[0100] Terminate loop Output: .

[0101] in , and , These represent the optimal scheduling scheme, the previous scheduling scheme, the current scheduling scheme, and a partial scheduling scheme, respectively. This indicates that the sub-batch operation will be destroyed. This indicates the removal of a sub-batch operation. Indicates a sorting operation. This indicates a greedy insertion operation. and They represent and The resulting objective function value.

[0102] 2.2 Feedback-based sabotage strategies Currently, traditional Inductively Coupled Algorithms (IG) typically employ an "open-loop control" strategy, using a fixed destruction intensity regardless of the search phase or solution structure. Therefore, they lack responsiveness to search states, often indiscriminately destroying high-quality structures in later stages, and are unable to adapt to the dynamic evolution of the solution space.

[0103] To address this issue, this invention introduces the concept of closed-loop feedback control and establishes an adaptive mechanism sensitive to the search state. The IG process is modeled as a dynamic control system with a failure rate... Cost difference between successive iterations, as a control variable As a feedback signal, the destruction rate is set to 30% during initialization and then adjusted online to achieve adaptive regulation of the destruction intensity, thereby achieving a robust balance between exploration and development. Therefore, the formula is as follows.

[0104] (20) In equation (20), t For iterative index, This is the gain coefficient, which is manually set to amplify the feedback signal. This represents the objective function value generated by the destruction-reconstruction operation in the previous iteration. To quantify the improvement between successive operations, the feedback signal is defined as Equation (21). It is worth noting that when the current solution is better than the previous solution, Take the negative value; otherwise ; (twenty one) Defined as the reaction gain coefficient, it is derived from subsequent DG-IFRace optimization; This represents the objective function value generated by the destruction-reconstruction operation in the current iteration. It governs the algorithm's response to signals. The response sensitivity. Specifically, This determines the maximum increment in which the perturbation operator is applied when a search stall is detected. Due to the optimal value... In fitness terrain features that are highly dependent on specific problem instances, manual tuning is challenging. Therefore, This is merged into a single, critical real-valued parameter that requires appropriate configuration and incorporated into the I / F-Race configuration space. C The algorithm automatically searches for the statistically optimal gain value for the current workshop scheduling scenario through offline training, thereby achieving the best match between algorithm behavior and problem characteristics.

[0105] Although the feedback mechanism is calculated The required perturbation strength was precisely determined, but the specific target was not specified. To dynamically... This is translated into specific scheduling adjustment operations and gives the algorithm differentiated search directions, introducing many disruptive rules. During the disruptive phase, the algorithm adjusts the scheduling based on the current scheduling length. and Calculate the amount removed Then, it uses one of the following three rules from the current sequence. Select q Remove each target sub-batch.

[0106] Late Sequence Destruction (LSD): This rule aims to alleviate congestion in the later stages of scheduling while maintaining stability in the earlier stages. It starts from the end of the current sequence and selects in reverse order. q Remove individual batches. Early Sequence Destruction (ESD): In contrast to LSD, this rule selects sequentially from the beginning of the current sequence. q Each sub-batch is deleted. Because the timing changes of the first sub-batch will propagate to the start times of all subsequent tasks through the "spreading effect", ESD will cause severe structural disturbances.

[0107] Shortage-related disruption (SCD): This is a targeted strategy focused on repairing shortages. It prioritizes identifying and removing disruptions that cannot be addressed within a specified timeframe. H The set of sub-batches that are scheduled within the time limit This refers to scheduled sub-batches belonging to the same batch. Specifically, if the number of sub-batches in the same batch is less than... q If all are removed, a new batch is randomly selected from the remaining scheduled batches to make up the total. q Otherwise, a random sample will be drawn from that group.q Sub-batches. This approach aims to disrupt the existing structure of a specific batch, significantly enhancing the pluggability of critical unplanned sub-batches.

[0108] 2.3 Priority-based Greedy Reconstruction Strategy During the reconstruction phase, the greedy strategy sequentially modifies the set to be inserted. Sub-batch insertion into the existing schedule In the greedy algorithm, due to path dependency, the insertion order determines the quality of feasible solution generation: higher-priority sub-batches have more time windows and production line selection rights, while subsequent sub-batches face stricter remaining space constraints. To address this issue, several sorting rules are designed to sort the set before reconstruction, guiding the greedy process towards a favorable outcome. The specific rules are as follows: Maximum Stockout Cost Priority (MSCF): This rule sorts sub-lots in descending order of potential stockout cost (calculated by multiplying the average unit stockout cost by the sub-lot quantity). This rule prioritizes sub-lots with high stockout costs to ensure they have priority access to limited capacity resources, thereby minimizing the high penalties for insertion failures.

[0109] Longest Duration First (LDF): This rule sorts sub-batches in descending order of total processing time (calculated by multiplying the unit processing time by the number of sub-batches). This rule draws on the Longest Processing Time (LPT) principle, prioritizing long tasks to mitigate the risk of long task insertion failures due to time fragmentation, thereby improving the overall fill rate.

[0110] High Value Density First (HVDF): This rule ranks sub-batches in descending order of value rate (defined as total stockout cost divided by total processing time). This rule aims to achieve the greatest reduction in the objective function value with the least time consumption, reflecting a cost-effectiveness-first restructuring strategy.

[0111] 2.4 VND-based reinforcement strategy Despite the introduction of a closed-loop feedback mechanism during the destruction phase, the classic IG framework retains structural limitations affecting solution quality and stability: reliance on lightweight local search leads to limited reinforcement capabilities. This results in two typical phenomena: First, reconstructed solutions often reside near "suboptimal but stable" local structures, hindering continued structural reinforcement. Second, while increasing destruction intensity to escape local optima may help break out of local optima, it can also destroy already established high-quality structures. Without sufficient local reconstruction and re-optimization support, the algorithm struggles to achieve effective quality consolidation and local convergence in new regions, thus amplifying the problem. The sensitivity to search behavior and performance reduces overall robustness. To address this shortcoming of local exploitation, VND is explicitly embedded in the framework as a local search module after each "destruction-reconstruction" phase, as shown in Algorithm 2.

[0112] Algorithm 2. Pseudocode for VND: Input: Initial scheduling scheme

[0113] A set of defined neighborhood structures ,

[0114] Local search depth

[0115]

[0116] when hour,

[0117] for :

[0118] if

[0119] if

[0120]

[0121] Termination judgment Other situations if

[0122]

[0123] Termination judgment Termination judgment if

[0124]

[0125] Interruption Termination judgment Terminate loop

[0126] Terminate loop Output: Updated solution .

[0127] in and Let represent the scheduling scheme after local search and the optimal scheduling scheme, respectively. To improve signage, This is a local search operation. and They represent and respectively and The resulting objective function value.

[0128] The VND framework establishes a hierarchical improvement mechanism, prioritizing production time compression over total cost. The algorithm first processes each scheduled sub-batch sequentially... (Position swapped within the same batch) (Scale adjustment in the same batch) and (Cross-batch position swapping) involves reordering and fine-tuning. This stage utilizes production line-level time metrics as the evaluation standard, using the sum of completion times for each production line as the objective function. This is used to compress the overall scheduling span. Then, the structural neighborhood is activated: Insert the sub-batch from the unscheduled set into the scheduled sequence. Subbatches are swapped between the scheduled sequences and the unscheduled sets, structurally removing unavailable subbatches. The improvement criterion for these two neighborhoods is switched to the total cost objective. This effectively reduces costs within the capacity space released from the preceding neighborhood. In summary, ~ Responsible for releasing capacity and stabilizing sequence structure, while and Responsible for optimizing costs using the released capacity. For example... Figure 5 As shown.

[0129] 3. Two-stage cost calculation mechanism To address the uncertainty in allocating common parts shortages under the complete set delivery model—where common parts shortages must be allocated among vehicle models with different penalty weights under constraints of varying order demands and real-time supply, making simple algebraic rules insufficient to determine the theoretical minimum cost—this invention proposes a two-stage precise evaluation strategy based on a mathematical model.

[0130] First, the strategy identifies unavoidable shortages caused by dedicated parts. Then, the complex allocation decisions for shared parts are transformed into a mixed-integer linear programming (MILP) sub-model. This model establishes boundary constraints rigorously based on actual order quantities and calculated part shortages to determine the optimal allocation scheme that minimizes total cost. Thanks to the high computational efficiency of this process, the strategy ensures a rigorous and fair evaluation of the economics of each scheduling scheme without affecting the iterative efficiency of the destruction and reconstruction processes.

[0131] Based on stock shortage Equation (22) determines the car model. k Inevitable shortage ,make This indicates the remaining shortage quantity attributable to shared components. Other symbols are defined above.

[0132] (twenty two) Objective function: (twenty three) constraint: (twenty four) (25) Equation (23) defines total cost as the sum of inventory cost and stockout cost. Equation (24) states that total stockout cannot exceed demand. Equation (25) calculates the cost per unit. j Inventory Specifically, the actual output of workpiece j is defined as demand minus component-level shortage: The actual delivery volume of model k was 。 Therefore, the actual total delivery quantity of component j is Accordingly, inventory Obtained by subtracting actual delivery from actual output: .constraint Ensure that the demand for parts released by the shortage decision at the vehicle level effectively covers the shortage at the parts level, thereby preventing negative inventory.

[0133] 4. DG-IFRace Algorithm In the algorithms described above, several classification and numerical parameters require proper calibration. The classic I / F-Race performs continuous instance evaluation and removes candidates based on statistical evidence; however, when the early instance set has weak discriminative power, the survivor set may shrink slowly, and a large portion of the budget may be spent on significantly poor configurations. To improve screening efficiency, this invention proposes DG-IFRace, an improvement on the I / F-Race algorithm, which introduces a hard instance batching mechanism. Using the dispersion of candidate performance as a discriminative metric, the most significant instance batch is selected at the end of each iteration and reused as a stress test set at the beginning of the next iteration to accelerate early differentiation between candidate configurations. Object values ​​are normalized using Max-Min to mitigate instance-related scale differences, and candidates statistically inferior to at least one competitor (in the Wilcoxon test) are gradually discarded after being sorted by average cost. After each competition, new configurations are drawn from a learning model biased towards a set of elite configurations. A parent configuration is selected based on rank-based probability to support better elitism. Classification parameters are generated through an adaptive hybrid update that blends the global probability model with parent-oriented parameters as the competition index increases. Numerical parameters are drawn from a truncated normal distribution centered on the parent value, with its standard deviation gradually decreasing, transitioning from exploration to development.

[0134] Definition of key program parameters and constructive heuristic automatic design for a reconfigurable distributed flowshop groups scheduling problem (Zhang, B., Meng, L., Lu, C., Han, Y., & Sang, H. (2024). Computers & Operations Research , 161 (106432. https: / / doi.org / 10.1016 / j.cor.2023.106432). This aligns with [reference to a previous point]. The total tuning budget B is measured by the number of configuration instances evaluated, where one "experiment" evaluates the configuration on a single NOI tuning instance. [Based on...] B and the budget consumed Allocate the budget for each competition from the remaining budget. The minimum number of surviving configurations is expressed as... The number of elite configurations used to build the learning model is NOE, which is typically set to be less than MinSurvival (i.e., ...). This is to bias the sampling towards promising candidates.

[0135] Algorithm 3: One race iteration ( k-th Pseudocode for the process: Input: The collection of Candidate configurations:

[0136] Tuning example set: 1 Adjusting the budget:

[0137] A collection of difficult instances:

[0138]

[0139] and

[0140] if

[0141]

[0142] other

[0143] Termination judgment

[0144] for

[0145] for

[0146]

[0147] Update LB and UB of Example and

[0148] Terminate loop for

[0149]

[0150]

[0151] Terminate loop Terminate loop for

[0152]

[0153]

[0154] Terminate loop

[0155] if

[0156]

[0157] Termination judgment for

[0158] for

[0159] if

[0160]

[0161] Termination judgment Terminate loop Terminate loop

[0162]

[0163] Terminate loop for

[0164]

[0165] Terminate loop

[0166] for

[0167] configuration c Substitution

[0168] Terminate loop Output: .

[0169] in Indicates the candidate configuration number. With configuration c Another candidate configuration number to compare. Indicates the budget already used, disp and These represent the dispersion and the optimal dispersion, respectively. Indicates the first k Number of candidate configurations in the next race iteration. Indicates the selection of instance operation. Indicates the first In the next racial iteration, configuration In the example The performance values ​​obtained are shown above. `ins` represents the instance number, and `LB` and `UB` represent the lower and upper bounds of the evaluation results for all candidate configurations on the current instance. This indicates a normalization operation. This represents the average normalized cost of configuration c in the current instance batch. This indicates that configuration c is used for the cumulative ranking or cumulative evaluation value of the final elite selection. This indicates the operation of calculating the standard deviation. ,express Rank-sum test operation, This indicates an operation to delete poor configuration settings. Indicates configurationc The cumulative evaluation values ​​are averaged. This indicates the configuration of the sorting function. This indicates that the smaller value will be selected for operation.

[0170] Algorithm 3 describes in detail the first step in DG-IFRace k The competition process within the next iteration. Given a budget. The algorithm selects some NOI instances to evaluate the current candidate configuration set. If a set of historically difficult instances exists. If a difficult instance set is selected, the initial evaluation is performed using that set; otherwise, a new set of instances is sampled from the instance generator. For each instance, the solver (CCH) is executed across all configurations to obtain the target value, and then the instances are normalized by max-min to eliminate differences in instance size. Subsequently, the normalized results are averaged across the set of instances to derive the average cost per configuration. AvgCost The algorithm calculates the dispersion (disp) of the average performance of each configuration across the set of instances and records the set with the highest dispersion as the "most discriminative" hard instance candidate for the current iteration. During the elimination phase, the algorithm performs pairwise Wilcoxon rank-sum tests based on the performance distribution of each configuration across the set of instances. Clearly inferior configurations are marked and discarded while ensuring the minimum survival count (MinSurvival), thus concentrating the budget on more promising candidates. This process iterates until the budget is exhausted or the survival size reaches a lower bound. Finally, the most difficult instance set recorded in this iteration is used to rank the instances. The system is updated, and surviving configurations are sorted based on cumulative performance, with the top few forming an elite group. .

[0171] This embodiment also includes experimental verification of the algorithm: First, a comprehensive dataset containing extensive tuning and test instances was collected, and the experimental setup was outlined. Next, the tuning phase of constructing the heuristic was detailed. Subsequently, the performance of the heuristic was evaluated in the testing phase. Finally, the effectiveness and efficiency of the designed CCI were verified. Specifically, this includes: 1. Optimization and test data preparation To comprehensively evaluate the algorithm's performance and scalability, we generated two sets of experimental cases of different scales: a small-scale set and a large-scale set. The complexity of the cases was controlled by four key parameters: baseline requirement. r Number of car models c Number of production lines l and number of processing stagess The actual demand for each vehicle model is at the baseline value. r Nearby fluctuations are generated. Given l In actual stamping workshops, c and s Both are constrained by physical facilities and product planning, resulting in relatively limited fluctuations. Therefore, this article mainly relies on baseline order demand. r The magnitude difference is used to define the problem size, supplemented by... l and s A moderate expansion. For the small group (Group 1), r Take a value from {20, 30, 40}. c Take a value from {2,3,4}. l Take a value from {2,3}. s =5, producing 18 different combination 。 For large groups （ Group 2) r Take a value from {80, 90, 100}. c Take values ​​from {5, 6, 7}. l Take a value from {4, 5}. s Taking values ​​from {6, 7} produces 36 different results. Combination. Since the objective is to minimize the total cost within a specified timeframe, the definition of the production time window is crucial. To ensure consistent time constraints across different scales of computational studies, the available time is dynamically calculated using equation (26): (26) The tuning data is used in the tuning phase to determine the optimal configuration using AAD. A series of tuning instances with different problem sizes are randomly generated as tuning data. These instances constitute the set in Algorithm 3. I The test data was generated by the same instance generator but with different random seeds; these were used only for the final evaluation after tuning. The test data included both small-scale and large-scale instances. For each small-scale problem, we generated one instance. For each large-scale problem, we generated 10 instances, each resolved in 5 independent runs. The processing time budget for each run was set to... millisecond. Based on the survey, reasonable ranges for processing time, shortage / inventory costs, and minimum sub-batch size are established as follows: Minimum sub-batch size starts from a uniform distribution. Extracted from [the relevant source]. Unit inventory cost and stockout cost for each model are [determined from...]. Extracted from the middle. The processing and preparation time of the stamping stage are respectively taken from... and The processing and preparation time for other stages are taken from... and The required quantities of parts for the car model are as follows: .

[0172] 2 Experimental Apparatus Design The computational budget is used as an experimental metric during the tuning phase, one experiment being applying a specific configuration to a NOI tuning instance. The NOI value needs to be set appropriately. This is because smaller values ​​generate insufficient statistical evaluation data, while larger values ​​consume excessive computational resources. Regarding the total budget B, a larger value allows for a more detailed evaluation of more candidate configurations. Control parameters The number of candidate configurations for each game is determined, allowing users to adjust the ratio between the total budget and the configuration count. Therefore, both parameters must be set appropriately to ensure sufficient configurations are evaluated effectively. Regarding NOE and MinSurvival, appropriate values ​​help sample the optimal configuration. A larger NOE slows down convergence, while a smaller NOE compromises global optimality. Similarly, a larger MinSurvival fails to adequately evaluate candidate configurations, while a smaller MinSurvival cannot guarantee global optimality. Typically, Through preliminary experiments, suitable parameter values ​​were determined. , , and Under these settings, the convergence and global optimality of the DG-IFRace procedure are satisfactory; that is, the selected elite configuration changes very little as the total budget increases.

[0173] To evaluate the performance of heuristic algorithms during the testing phase, this invention uses the relative deviation index (RDI) as a metric for evaluating algorithm performance: (27) in, This represents the target value obtained through a given algorithm, while and Let represent the best and worst objective values ​​obtained by all comparison algorithms, respectively. This assumes that all algorithms produce the same result (i.e., ...). RDI is defined as 0. Clearly, the lower the RDI, the higher the quality of the solution and the closer it is to the optimal solution. For each algorithm, the average RDI (ARDI) is calculated as follows: first, the RDI values ​​from multiple runs of each instance are averaged; then, these instance-level averages are averaged again across all instances of the same size. All algorithms in this invention are implemented in C++ and executed on an Intel Core i7 3.60 GHz processor.

[0174] 3. Optimization Phase Table 1 summarizes the key processing statistics of the DG-IFRace tuning phase. The entire tuning process consisted of four rounds, evaluating a total of 1,904 candidate configurations to select 10 elite configurations. During this period, the algorithm utilized 190 tuning instances to evaluate configuration performance. Notably, as the rounds progressed, the average number of evaluations allocated to each surviving configuration (i.e., the ratio of computational budget to the number of candidate configurations) showed a significant upward trend. This is attributed to the performance homogenization of the remaining elite configurations in the later stages. After four rounds of competition and rigorous evaluation of 190 instances, the DG-IFRace process identified 10 elite configurations, as shown in Table 2. The data reveals significant differences in component selection and parameter values ​​among these elite configurations, indicating that high-performance heuristic strategies can be constructed through flexible combinations of different components. Specifically: Initial solution generation: Sub-batch splitting rules are highly consistent, with all elite configurations using BKA. Production line routing strategies exhibit diversity, including EAT, MTPL, and MWL. Regarding scheduling rules, cost priority dominates, with only the fourth-ranked configuration using LPT.

[0175] IG core components: Although ESD appears most frequently as a violation rule, the optimal configuration uses SCD; at the same time, the sorting rule converges to MSCF.

[0176] Numerical parameter: gain coefficient (Real number) and VND local search depth (Integers) exhibit a wide range of values ​​under different configurations, reflecting the algorithm's flexibility in balancing the breadth and depth of the search.

[0177] Table 1 DG-IFRace Process Parameters

[0178] In summary, the configuration with the highest ranking in Table 2 was selected as the final solution for resolving DHILSP in the subsequent testing phase.

[0179] Table 2. Optimal Elite Configuration Output for DG-IFRace

[0180] 4. Evaluation of DG-IFRace In the DG-IFRace proposed in this invention, a combination mechanism of difficult instance preservation and Wilcoxon rank-sum test is adopted to perform more discriminative statistical screening of candidate configurations, rather than the typical classic I / F-Race strategy that relies solely on Friedman test.

[0181] To validate the effectiveness of these design choices, two baseline variants were implemented under the same overall budget, in contrast to DG-IFRace: W-IFRace, which used only the Wilcoxon rank-sum test for screening, and F-IFRace, which employed a screening procedure based on the Friedman test. These two variants were then compared to the full DG-IFRace framework. Table 3 summarizes the process parameters and relevant statistics for the three variants across all competitions. The results show that in each competition, DG-IFRace and W-IFRace discarded significantly more configurations than F-IFRace, indicating that the Wilcoxon test has higher sensitivity and stronger discriminative power in identifying statistically poor configurations. Therefore, it can eliminate poor candidates earlier and allocate the evaluation budget more effectively to more promising configurations.

[0182] Table 3 Process parameters for DG-IFRace, W-IFRace, and F-IFRace

[0183] Furthermore, Table 4 presents the screening results for DG-IFRace and W-IFRace after setting up the first instance in each game. It can be observed that, benefiting from the preservation of difficult instances, DG-IFRace exhibits a higher drop rate in this early stage, i.e., a stronger "early compression" effect. Since difficult instances serve as a highly discriminative stress test set, they widen the performance gap between candidate configurations in the initial stage and accelerate the elimination of poor configurations, leading to faster convergence of the surviving configuration set. A direct result is that, with the same total budget, DG-IFRace can allocate more evaluation to a smaller but higher-quality pool of surviving configurations, allowing for more thorough and reliable comparisons across more instances, ultimately improving the overall robustness and generalization of the selected elite configurations.

[0184] To facilitate a fair assessment of how different statistical screening mechanisms affect the final algorithm performance, the configurations of the two comparison variables were first fixed based on their tuning outputs. Specifically, the configuration used for W-IFRace was the rank-1 elite configuration given in Table 5, and the configuration used for F-IFRace was the rank-1 elite configuration given in Table 6. These selected configurations were then evaluated together with the proposed DG-IFRace on the same large-scale test set, and the corresponding ARDI and standard deviation (Sd) results are summarized in Table 7. Furthermore, to provide an intuitive statistical comparison in terms of central tendency and stability, Figure 6 (a) A plot of the average of the 95% confidence intervals for ARDI is given. Figure 6 (b) provides a box plot.

[0185] From Table 7 and Figure 6It is clearly observed that DG-IFRace consistently achieves lower ARDI levels, more concentrated confidence intervals, and a more compact box-shaped distribution than W-IFRace and F-IFRace, indicating better solution quality and less performance variation on large-scale instances. In contrast, W-IFRace and F-IFRace both exhibit higher ARDI values ​​and greater dispersion, suggesting that relying on a single screening scheme or employing a Friedman-based mechanism weakens the discriminative power during configuration selection, thereby reducing generalization performance. Overall, these results confirm that combining the Wilcoxon rank-sum test (and hard instance saving) in DG-IFRace can more effectively distinguish and eliminate inferior configurations, thus improving the overall performance and robustness of the final elite configuration.

[0186] DG-IFRace can be further compared with full factorial baselines to assess effectiveness and efficiency. In this framework, FFD (Full Factorial Design) is constructed by combining parameter values ​​that achieve the best average performance at the factor level, while FFD* corresponds to a single configuration with the best average performance across the entire configuration space (or an enumerated subset thereof). A key finding is that a factor-wise "good" combination is not guaranteed to be the globally optimal configuration, and DG-IFRace-style competition can identify robust combinations when evaluating far fewer configurations than exhaustively enumerated. Notably, the configuration space in the embodiments of this invention contains 166,320 different configurations. Therefore, computationally prohibiting full factorial evaluation by running on every configuration for every tuning instance is practically infeasible due to the enormous time consumption. Furthermore, Zhang et al. proposed that W-IFRace-style automated design achieves better overall performance than the FFD baseline, indicating that competition-based selection is not only more efficient but also more effective in identifying high-quality configurations, and the DG-IFRace proposed in this paper adds a mechanism for retaining difficult instances on top of W-IFRace. In the setup of this embodiment, DG-IFRace evaluated 1,904 candidate configurations in 4 competitions using 190 tuned instances (Table 1). This has demonstrated a strong efficiency advantage of DG-IFRace compared to any exhaustive or near-exhaustive design, while also generating stable elite configurations for subsequent testing.

[0187] Table 4 Results of DG-IFRace and W-IFRace after setting up the first instance in each competition

[0188] Table 5. W-IFRace's Best Elite Configuration Output

[0189] Table 6. F-IFRace's Best Elite Configuration Output

[0190] Table 7 shows the ARDI values ​​obtained from DG-IFRace, W-IFRace, and F-IFRace.

[0191] 5. Evaluation of CCI components (using DG-IFRace-tuning configuration) To quantify the contribution of each key component, we break down the proposed method into three parts: (1) A configurable constructive heuristic that generates a constrained feasible initial scheduling scheme; (2) IG Disruption-Reconstruction provides global perturbation and feasible reconstruction; (3) Embedded VND module, which enhances local convergence through multi-neighborhood enhancement.

[0192] Because the IG in this invention employs a closed-loop feedback mechanism to adaptively adjust the destruction rate, thereby balancing exploration and structural preservation, compared to the traditional open-loop IG with a fixed destruction intensity, it further introduces an ablation variant, CCI-RD, in which only the dynamic destruction rate scheme is replaced by a fixed rate (30%), while all other components remain unchanged. This allows for strict isolation of the performance gain brought about by the adaptive destruction rate. Furthermore, we constructed CCI-RVND, CCI-RIG, and CCI-RIGVND by removing VND, IG, and both IG and VND respectively, to examine the individual and synergistic effects of IG backbone search and local enhancement on the RDI metric.

[0193] Wherein, CCI represents the complete method, while CCI-RD, CCI-RVND, CCI-RIG, and CCI-RIGVND are ablation variants obtained by replacing the adaptive destruction rate scheme with a fixed rate, while simultaneously removing VND, IG, and IG+VND, respectively. To ensure fair comparison, all variants were evaluated on the same benchmark instance set under the same experimental settings, including time budget, stopping criteria, and random seed strategy. As shown in the RDI results in Table 8, the complete CCI consistently achieves the best performance at most problem scales, confirming the effectiveness of the IG backbone, embedded VND enhancement, and adaptive destruction rate mechanism.

[0194] Table 8. Variation results of different CCI components

[0195] like Figure 7As shown, under the ARDI metric, the proposed method exhibits statistically significant superiority and robust stability. Figure 7 The confidence interval plot in (a) shows that the method achieves the lowest ARDI within a narrow uncertainty range, demonstrating a consistent ability to maintain high solution quality across different instances. In contrast, all ablation variants exhibit varying degrees of performance degradation once specific components are removed. In particular, variants without IG-based destruction-reconstruction backbones produce significantly higher ARDIs with wider confidence intervals, implying that removal of global perturbations and feasible reconstruction severely weakens search efficiency. Furthermore, CCI-RD, replacing the adaptive destruction mechanism with a fixed destruction rate, performs slightly worse than CCI, with increased ARDI and a slightly wider uncertainty range, indicating that closed-loop adaptation better balances the exploration and utilization of different search stages and instance features. Figure 7 The box plot in (b) further highlights the robustness of each algorithm. The proposed method exhibits a more compact box with shorter whiskers and a more concentrated median, reflecting lower dispersion and less performance variability, while CCI-RD shows a moderately increased diffusion. Overall, the ablation study confirms the synergistic effect of the main modules: the IG destruction-reconstruction mechanism provides the main global search capability, while the embedded local improvements improve solution quality and stabilize performance; the adaptive destruction strategy further improves search efficiency and robustness compared to using a fixed destruction rate.

[0196] 6. CCI Assessment To comprehensively evaluate the overall performance of the proposed CCI, two sets of comparative experiments were conducted. First, in the first set, CCI was compared with the mathematical solver CPLEX and four representative metaheuristic algorithms: BRCE (Bi-roles co-evolution for energy-efficient distributed heterogeneous permutation flow shop scheduling with flexible machine speed, Huang, K., Li, R., Gong, W., Wang, R., & Wei, H. (2023).) Complex & Intelligent Systems , 9(5), 4805–4816. https: / / doi.org / 10.1007 / s40747-023-00984-x), DHICCA (Dynamic and heterogeneous identity-based cooperative co-evolution for distributed lot-streaming flowshop scheduling problem, Wang, J., Zhang, G., Li, X., & Feng, Y. (2025). Complex System Modeling and Simulation , 5 (1), 86–106. https: / / doi.org / 10.23919 / CSMS.2024.0025, Jaya (An improved meta-heuristic for solving distributed lot-streamingpermutation flow shop scheduling problems, Pan, Y., Gao, K., Li, Z., & Wu, N. (2023a). IEEE Transactions on Automation Science and Engineering , 20 (1), 361–371. https: / / doi.org / 10.1109 / TASE.2022.3151648, MSIG (Advanced metaheuristics forbi-criteria optimization in a distributed blocking flow shop problem with setup times, Sayah, A., Aqil, S., & Lahby, M. (2025). Operations Research Forum , 6(4), 152. https: / / doi.org / 10.1007 / s43069-025-00559-1) was compared to verify its solution quality and accuracy at manageable problem sizes. Subsequently, on the second group, where exact solutions became computationally forbidden, CCI benchmarked four metaheuristics to evaluate their efficiency and robustness in expanded and highly constrained search spaces. Given that no previous studies specifically addressed DHILSP as considered in this work, these algorithms were selected as literature-representative baselines and reimplemented using compatible encoding / decoding procedures to accommodate consistent sub-batch, mold uniqueness feasibility, and kit-based cost evaluation mechanisms. Furthermore, to ensure rigorous and fair comparisons, all neighborhood operators and local improvement components used in this invention were consistently applied to all compared metaheuristic methods, along with the same stopping criteria and random seed strategy, thereby eliminating operator-induced bias and enabling accurate evaluation of each algorithmic framework.

[0197] 6.1 Benchmarking CCI using CPLEX and metaheuristics on the first group The purpose of this experiment is twofold: first, to verify the effectiveness of the proposed MILP formula in solving small-scale instances of the stamping shop scheduling problem (Group 1) under study, where constraints caused by consistent sub-batches and die uniqueness can still be handled by an exact solver; second, to compare the solution quality and computational efficiency of CCI, CPLEX, and their competing metaheuristic algorithms on Group 1. For the CPLEX solver, the objective value (total cost) and corresponding computation time obtained for each instance are recorded. For each metaheuristic algorithm, each instance is executed 5 times independently under the same termination criterion, and the best objective value among these 5 runs is reported; for each instance, the overall best result of all methods is highlighted in bold.

[0198] As shown in Table 9, the optimal objective values ​​(total cost) obtained on the first set of small-scale instances differ significantly among the compared methods. Under a uniform 1800-second time constraint, CPLEX generally provides a high-quality reference solution, demonstrating its advantage in precise optimization on small instances. Meanwhile, as the problem size increases, the coupling between constraints and the complexity of the search space become increasingly apparent, and the advantages of CCI gradually emerge: on several relatively large instances, CCI can obtain optimal total costs comparable to or even better than those returned by CPLEX within the same time budget, demonstrating stronger search efficiency and better scalability under complex constraints. In contrast, BRCE and DHICCA can achieve relatively competitive results in some cases, but generally fluctuate significantly, while Jaya and MSIG typically produce higher objective values, indicating poorer solution quality and stability for this problem setting.

[0199] Table 9 Comparison results of the first group of examples

[0200] 6.2 Benchmarking the metaheuristic CCI on the second group In section 6.1, the effectiveness of the proposed CCI was validated on a first set of small-scale instances by benchmarking CPLEX and representative metaheuristics. This section further evaluates the performance of the CCI on a second set of large-scale instances through a comprehensive comparison with four state-of-the-art metaheuristic algorithms. Table 10 reports the ARDI and standard deviation (Sd) obtained by each algorithm on 36 large-scale instances.

[0201] As shown in Table 10, CCI achieves the best ARDI across all large-scale instances, as indicated by the bold entries, demonstrating its consistently superior solution quality in complex search spaces. For each problem size, 10 instances are generated, each solved independently 5 times for each algorithm; the reported ARDI (Sd) values ​​are based on these repeated runs. Compared to BRCE, DHICCA, Jaya, and MSIG, CCI produces significantly smaller ARDI values ​​across the entire test set. For example, CCI for 7-5-7-90 has an ARDI of 0.0724 and an Sd of 0.0618, while the corresponding ARDI values ​​for BRCE, DHICCA, Jaya, and MSIG are 0.2405, 0.4395, 0.8214, and 0.3114, respectively. This significant difference directly demonstrates CCI's superior optimization capabilities when handling large-scale instances with tightly coupled constraints. Furthermore, CCI typically reports a lower standard deviation, indicating more stable performance. Among the comparative metaheuristic methods, BRCE and MSIG showed relatively competitive results in some cases, while Jaya produced the largest ARDI value in most cases, indicating lower search efficiency in this problem setting.

[0202] Figure 8 The average relative deviation index of the five algorithms was compared from the perspectives of confidence interval plots and box plots. Figure 8 The confidence interval plot in (a) shows that CCI has the lowest average bias, approximately 0.11, and a narrow confidence interval, indicating that its solution quality is superior and the results are stable. The biases of BRCE and DHICCA gradually increase, while MSIG is close to DHICCA but slightly lower. Jaya has the highest average bias, close to 0.8, which is significantly worse than the other algorithms. Figure 8(b) The box plot further illustrates the result distribution. CCI has the smallest box and is located at the lowest position, indicating its smaller fluctuation range; Jaya's box is located at the highest position overall, indicating the worst performance. Overall, CCI performs best in both solution quality and stability.

[0203] To further examine the robustness of each comparison algorithm across different problem dimensions, this section analyzes... Figure 9 The interactive graphs shown (a) represent the algorithm performance trends for different numbers of models, (b) for different numbers of rows, (c) for different numbers of stages, and (d) for different demand levels. It is clear that the CCI curve remains at the bottom across all factor settings (i.e., number of rows, model, stage, and demand level), exhibiting high stability and indicating low sensitivity to changes in problem size and constraint strength. Specifically, the average performance of CCI remains almost unchanged when the number of rows (4→5) and stages (6→7), while even showing a slight improvement with increasing model numbers (5→7) and demand levels (80→100). In contrast, competing metaheuristics exhibit more pronounced factor-dependent changes: BRCE tends to worsen with increasing model size, DHICCA increases with increasing line / stage number, and MSIG shows significant fluctuations across different model settings. Meanwhile, Jaya consistently produces the largest average across all configurations, reflecting limited search efficiency in this problem setting. In summary, these interaction trends provide strong empirical evidence that CCI maintains superior solution quality and robust generalization across diverse large-scale scenarios.

[0204] Table 10 Comparison results of the second group of examples

[0205] This invention examines the automotive stamping shop DHILSP under BOM-driven assembly and die set uniqueness constraints. To address the strong coupling between material availability and production decisions, this invention proposes CCI, a configurable constructive improvement framework that integrates constructive initialization, feedback-controlled destructive reconstruction search, and two-stage cost calculation. In both small and large test sets, CCI consistently outperforms competing heuristics and remains competitive under tight time budgets. Its performance distribution is also more concentrated, demonstrating robustness and low sensitivity to scale variations, supporting broad applicability to different shop floor settings.

[0206] To reduce configuration costs and improve screening quality, this invention develops DG-IFRace. Compared to FFD, DG-IFRace avoids exhaustive enumeration and identifies strong configurations with fewer evaluations. Compared to F-IFRace and W-IFRace, DG-IFRace improves early identification capabilities by using distributed-guided batches of difficult instances, thereby faster elimination of poor configuration combinations and more stable selection of elites. Overall, DG-IFRace provides an efficient configuration combination, and the resulting CCI provides reliable and scalable performance for the studied stamping scheduling problem.

[0207] Finally, it should be noted that the above content is only used to illustrate the technical solution of the present invention, and is not intended to limit the scope of protection of the present invention. Simple modifications or equivalent substitutions made by those skilled in the art to the technical solution of the present invention do not depart from the essence and scope of the technical solution of the present invention.

Claims

1. A distributed hybrid staggered batch scheduling method for parallel production lines, characterized in that, The DHILSP (Distributed Flowshop Interleaving and Mixed Insertion) scheduling problem is used for scheduling, including: Deliver complete sets according to the bill of materials for each vehicle model, set spare parts inventory levels, and ensure safety stock. Each production line includes multiple processing stages connected in sequence; the mold set used to process a specific batch of parts is unique, and at any given time, the same mold set can only be configured on one production line and cannot be occupied by multiple production lines in parallel; the external mold change for the next batch can be carried out in parallel with the processing of the current sub-batch. In the stamping stage, for sub-batches that require processing with the same mold set, at any given time, only one sub-batch is allowed to be processed on one production line. A consistent sub-batch division strategy is adopted for the same batch. After the sub-batch division is completed, the quantity and corresponding size of each sub-batch remain unchanged in subsequent processing stages; the size between different sub-batches can be determined by the selected sub-batch division rule. Any sub-batch must pass through all production stages sequentially and processing cannot be preempted, but alternating processing of sub-batches from different batches is allowed on the same production line; each machine can process at most one sub-batch at any given time, and any sub-batch can only be processed by one production line at each stage. The scheduling objective is to minimize the sum of inventory costs and stockout costs while achieving complete delivery.

2. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 1, characterized in that, In DHILSP, switching between different batches in adjacent processing requires setting a time, while two adjacent sub-batches in the same batch do not have a time set by default. After a sub-batch completes processing in the previous stage and reaches the downstream stage, the sub-batch can only start processing when the target machine is available and the necessary settings are completed.

3. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 2, characterized in that, DHILSP is also subject to the following fundamental assumptions: At the start of the scheduling process, all production lines are idle and all resources are available. No preemption or interruption is permitted during the entire processing period; Allow machine idle time, assuming the buffer capacity between stages is infinite; Any sub-batch can only proceed to the next stage after the current stage of processing is completed; The number of any vehicle model delivered must not exceed the demand. The size of any sub-batch must be greater than or equal to the smallest sub-batch size.

4. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 3, characterized in that, The mathematical model for DHILSP is as follows: The objective function is expressed as follows: ； The constraints are as follows: ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； in, n Total number of batches, i.e., total number of parts; m Total number of production stages; h Total number of vehicle models; F Total number of production lines; L : Maximum number of sub-batches per batch; H Daily processing time; i Stage index; ; j , Batch index; , Indicates different j batches; k Vehicle model index; ; f Production line index; ; e, Sub-batch index; , Indicates different e Sub-batch; G : Represents a sufficiently large positive integer; Model k Required batch j The quantity, i.e., the batch j The corresponding number of parts; : k Demand for this type of vehicle; : No. j Batch No. i Processing time for each stage; : No. j Batch No. i The time frame for setting the stage, among which Indicates batch j The setup time for Phase 1; : No. j Unit inventory cost per batch; Model k The unit cost of stockouts; : No. j Historical inventory of batches; : No. j A batch of historical shortages; : No. j sub-batch e The minimum size is determined by the external mold change time; : No. j The daily target production quantity for the batch; Model k The number of deliveries; : No. j Batch inventory; : No. i Phase 1 j Sub-batch of a batch e The start time, of which Indicates batch j The start time of sub-batch 1 in stage 1; : No. i Phase 1 j Sub-batch of a batch e End time; : No. j Sub-batch of a batch e Size; Auxiliary variable ; : A binary variable, if the production line f The first j Sub-batch of a batch e It is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0. : A binary variable, if the first j Sub-batch of a batch e Assigned to production line f If the result is positive, then it equals 1; otherwise, it equals 0. : A binary variable, if the first j Sub-batch of a batch e If activated, it equals 1; otherwise, it equals 0. Indicates batch j Sub-batch 1 is activated; : A binary variable, if the first j Sub-batch of a batch e The stamping stage is the first Sub-batch of a batch If the predecessor is , then the value is 1; otherwise, it is 0.

5. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 4, characterized in that, In the DHILSP scheduling process, a configurable constructor-improvement algorithm is used to determine the scheduling scheme. This includes a constructor-then-improvement strategy. Initially, a configurable constructor method is used to generate feasible initial solutions, and then an improved architecture is used for deep optimization. This includes: Determine the initial scheduling scheme: A configurable constructivist heuristic approach is adopted, including sequential execution of sub-batch partitioning, production line routing, and job assignment planning, to determine the initial scheduling scheme; Scheduling scheme optimization: During the scheduling process, a disruptive batch perturbation rule based on iterative greedy IG and variable neighborhood descent VND is introduced to adaptively adjust the existing scheduling scheme. Specifically, IG is used to merge the selected scheduled sub-batches with the unscheduled set and then perform greedy reconstruction. VND is used to perform deep local search through multiple complementary neighborhoods to determine the optimized scheduling scheme.

6. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 5, characterized in that, A mixed-integer linear programming (MILP) sub-model is embedded during the search process to quantify the trade-off between inventory costs and shortage costs.

7. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 6, characterized in that, In the complete set delivery model, the allocation of common parts shortages adopts a two-stage evaluation strategy. First, the unavoidable shortages caused by dedicated parts are identified. Then, the allocation decision of common parts is transformed into a mixed integer linear programming (MILP) sub-model. The MILP sub-model establishes boundary constraints based on the actual order quantity and the calculated part shortage quantity, and uses the CPLEX solver to determine the optimal allocation scheme that minimizes the total cost.

8. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 7, characterized in that, The MILP sub-modeling process is as follows: Based on stock shortage ,Sure k Inevitable shortage of certain car models : ； in, J k For vehicle model k A collection of specialized parts; make The objective function of the MILP sub-model, representing the remaining shortage attributable to shared parts, is as follows: ； The constraints of the MILP sub-model are as follows: ； 。 9. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 5, characterized in that, The sub-batch partitioning follows these rules: Equal Quantity Rule (ES): This rule aims to generate sub-batches of uniform size while satisfying the minimum sub-batch size constraint. Bill of Materials (BOM) Kit Alignment Rule (BKA): Designed for kit delivery, this rule aligns sub-lot dimensions with kit requirements. Sub-lot dimensions are defined as follows: ,in A positive integer, representing a sub-batch. The complete set of multiples, and , For batch j The smallest sub-batch size, d k for k Demand for this type of vehicle; Gradually Increasing Sub-Batch Size Rule (IS): This rule uses the smallest possible sub-batch size. Use this as an initial base number to gradually increase the size of sub-batches; Fibonacci Increasing Rule (FI): In the rule of progressively increasing sub-batch size, the Fibonacci sequence is further used as weights to generate a sub-batch structure with increasing sub-batch sizes; the weights are defined as the following sequence: That is, starting from the third sub-batch, the size of each sub-batch is the sum of the sizes of the first two sub-batches; where, The weight of the first sub-batch, The weight of the second sub-batch, Let be the weight of the e-th sub-batch; While meeting the condition of being greater than or equal to the "smallest sub-batch", the upper limit of the number of sub-batches within the same batch should be increased as much as possible. That is, choose the largest possible Make Sub-batch dimensions ; u As an intermediate variable; In all the above rules, if the remainder is greater than or equal to when dividing into sub-batches... If yes, then it forms a separate sub-batch; otherwise, it is merged into the last sub-batch. in, Model k Required batch j The quantity, i.e., the batch j The corresponding number of parts; : k Demand for this type of vehicle; e Sub-batch index.

10. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 5, characterized in that, Production line routing uses at least one of the following rules: Earliest Available Time (EAT): Select the production line with the earliest available time among all production lines and assign the sub-batch to that production line; Minimum Total Processing Load (MTPL): Select the production line with the shortest total processing time among all production lines and assign the sub-batch to that production line; Minimum Workload (MWL): Select the production line that processes the fewest sub-batches and assign the sub-batches to that production line.

11. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 5, characterized in that, The job assignment planning adopts rules based on processing time, bottleneck stage load, and mold activation status, including: Shortest / Longest Processing Time (SPT / LPT): Prioritize the sub-batch with the shortest / longest total processing time; Shortest / Longest Phase 1 Processing Time (SPTF / LPTF): Prioritize the sub-batch with the shortest / longest Phase 1 processing time; Shortest / Longest Processing Time to Stamping Stage (SPTS / LPTS): Prioritize the sub-batch with the shortest / longest processing time at the stamping stage; Mold priority: Prioritize the batch that requires the current mold to be activated, and then use the above rules to determine the specific sub-batch; Cost priority: Prioritize the sub-lot with the highest stockout cost, and calculate the stockout cost by multiplying the average unit stockout cost by the sub-lot quantity; Unit cost priority: Prioritize the sub-batch with the highest unit stockout cost, and calculate the unit stockout cost by dividing the stockout cost by the processing time.

12. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 5, characterized in that, The configurable construct-improvement algorithm employs a feedback-based destruction strategy, including: An adaptive closed-loop feedback control mechanism sensitive to the search state is established, and the IG process is modeled as a dynamic control system with a failure rate. The cost difference between adjacent iterations is used as a control variable. As a feedback signal, the formula is as follows: in, t For iterative index, This is the gain coefficient, manually set to amplify the feedback signal; This represents the objective function value generated by the destruction-reconstruction operation in the previous iteration; The reaction gain coefficient is derived from subsequent DG-IFRace optimization.

13. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 12, characterized in that, Cost difference between adjacent iterations of the feedback signal As shown in the formula below: When the current scheduling scheme is better than the previous scheduling scheme, Take the negative value; otherwise ; This represents the objective function value generated by the destruction-reconstruction operation in the current iteration.

14. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 12, characterized in that, During the destruction phase, based on the current scheduled sub-batch length and destruction rate Calculate the amount removed ; Subsequently, one of the following rules is used to select from the current scheduled sub-batch sequence. Select q Remove each target sub-batch: Late-sequence destructive (LSD): from the currently scheduled sub-batch sequence Starting from the end, select in reverse order. q Remove individual batches; Early sequence destruction (ESD): from the currently scheduled sub-batch sequence Select the head in sequence q Delete individual batches; Shortage-related disruption of SCD: from the current scheduled sub-batch sequence Prioritize removing those that cannot be completed within the specified time. H The set of sub-batches that are scheduled within the time limit If the number of sub-batches in the same batch is less than [a certain number], then [the following applies]. q If all are removed, a new batch is randomly selected from the remaining scheduled batches to make up the total. q Otherwise, start from the current scheduled sub-batch sequence. Randomly selected from q Remove individual batches.

15. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 12, characterized in that, During the reconstruction phase, the greedy strategy sequentially modifies the sub-batch sets to be inserted. Sub-batch insertion into the existing schedule Then sort them according to the following rules: Maximum Stockout Cost Priority (MSCF): This rule sorts sub-lots in descending order of potential stockout cost, calculated by multiplying the average unit stockout cost by the sub-lot quantity. Longest Duration Priority (LDF): This rule sorts sub-batches in descending order of total processing time, calculated by multiplying the unit processing time by the number of sub-batches. Highest Value Density Priority (HVDF): This rule sorts sub-lots in descending order of value rate, which is defined as total stockout cost divided by total processing time.

16. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 12, characterized in that, VND is embedded in IG as a local search module after each "destruction-reconstruction" phase. The VND algorithm establishes a hierarchical improvement mechanism, prioritizing production time compression over total cost. The VND algorithm includes: First, within the currently scheduled sub-batch sequence, the order is rearranged and fine-tuned sequentially through same-batch position exchange, same-batch size adjustment, and cross-batch position exchange. This stage uses production line-level time indicators as evaluation criteria and adopts the sum of completion times of each production line as the objective function. Subsequently, the sub-batch structure neighborhood is activated: sub-batches from the unscheduled set are inserted into the scheduled sequence, or sub-batches are swapped between the scheduled sequence and the unscheduled set, and then sub-batches that cannot be scheduled within the specified time are removed from the scheduled sequence again. H For sub-batches that are scheduled within a given timeframe, the improvement criterion for both the scheduled sequence and the unscheduled set is the total cost objective. Lower.

17. The distributed hybrid staggered batch scheduling method for parallel production lines according to any one of claims 5-16, characterized in that, The calibration of parameters in all algorithms uses the improved I / F-Race algorithm, DG-IFRace. The DG-IFRace algorithm introduces a hard instance batching mechanism into the I / F-Race algorithm. It uses the performance dispersion of candidate configurations on the same batch of instances as the distinguishing index. At the end of each iteration, the instance batch with the most significant distinguishing index is selected and reused as the stress test set at the beginning of the next iteration.

18. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 17, characterized in that, In generating new configurations, the DG-IFRace algorithm normalizes the values ​​of objects and discards candidates that are statistically worse than at least one competitor in the Wilcoxon rank-sum test after sorting by average cost. After each competition, new configurations are drawn from the learning model that favors elite configurations in the group, and a parent configuration is selected based on rank-based probability. Classification parameters are generated through adaptive hybrid updates that combine the global probability model with parent guidance as the competition index increases. Numerical parameters are drawn from a truncated normal distribution centered on the parent value.

19. The distributed hybrid staggered batch scheduling method for parallel production lines according to claim 18, characterized in that, In the DG-IFRace algorithm, the total tuning budget B is measured by the number of configuration instances evaluated. One experiment evaluates the configuration on a NOI tuning instance, based on the total tuning budget. B and the budget consumed Allocate the budget for each competition from the remaining budget. The budget will be focused on candidates with greater potential.

20. A distributed hybrid staggered batch scheduling system for parallel production lines, characterized in that, The system includes a processor capable of executing a computer program that, when executed, implements the distributed hybrid staggered batch scheduling method for parallel production lines as described in any one of claims 1-19.

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