Fuzzy remanufacturing system scheduling method based on improved migratory bird optimization algorithm

Abstract

The application discloses a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm, belongs to the technical field of remanufacturing system scheduling, and solves the problems that the prior art ignores the fuzzy uncertainty of processing time, uses a deterministic model, leads to disconnection with actual production, and does not establish a suitable mathematical model and a solving framework for the special structure of a 3T-RSSP. The method comprises the following steps: establishing a fuzzy mixed integer linear programming model, and solving the fuzzy mixed integer linear programming model based on the improved migratory bird migration optimization algorithm. The application constructs a fuzzy mixed integer linear programming model suitable for the special structure of a 3T-RSSP, accurately quantifies fuzzy completion time based on the conventional operation rules of triangular fuzzy numbers, and accurately depicts the strong coupling logic of "job sequence-workstation allocation" by using the improved migratory bird migration optimization algorithm, so that the global optimization ability and convergence stability of the algorithm in a fuzzy environment are significantly improved, and the efficient scheduling demand of a remanufacturing system in an uncertain environment is met.

Description

Technical Field

[0001] This invention belongs to the field of remanufacturing system scheduling technology, specifically relating to a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. Background Technology

[0002] Remanufacturing, as a core component of green and sustainable manufacturing, holds significant practical importance in alleviating resource scarcity, reducing environmental pollution, and promoting the development of a circular economy. With the continuous increase in the amount of End-of-Life (EOL) products recycled, the remanufacturing industry is experiencing rapid growth. Scheduling, as a core technology in remanufacturing production, directly determines the operational efficiency of the three-stage remanufacturing system (3T-RSSP) through the rational allocation of production resources for the three core operations of dismantling, reprocessing, and reassembly, and has become a hot topic of common concern.

[0003] Existing research on 3T-RSSP mainly focuses on the following directions:

[0004] Deterministic scheduling models: Existing studies often assume that processing times are precisely known values ​​(150 seconds, 8 minutes, etc.) and use precise solvers such as CPLEX and GUROBI, or simple heuristic rules to solve the model. These models ignore the inherent differences in the quality of recycled end-of-life products, as well as the uncertainties brought about by external environmental factors such as machine flexibility fluctuations and differences in personnel operation, resulting in poor adaptability to actual production scenarios.

[0005] Two-layer decision structure processing: 3T-RSSP involves two strongly coupled decision problems: "Job Sequencing (JS)" and "Workstation Assignment (WA)". Existing technologies mostly use single-layer coding schemes or phased independent optimization strategies to handle this coupled problem, which makes it difficult to fully express the associated decision logic of "job sequence - disassembling workstation - reassembling workstation", and easily leads to the solution results deviating from the global optimum.

[0006] Application of fuzzy scheduling technology: In response to the uncertainties in the production process, existing technologies often use triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers to describe the fuzzy characteristics of processing time. However, most related studies focus on traditional scheduling scenarios such as flexible job shop scheduling (FJSP) and flexible flow shop scheduling (FFSP), and rarely conduct fuzzy modeling research on the special structure of three-stage remanufacturing systems.

[0007] Metaheuristic optimization algorithms: For scheduling problems with NP-hard characteristics, Migrating Birds Optimization (MBO), as an emerging metaheuristic algorithm, has demonstrated good search performance in problems such as secondary allocation problems and flexible job shop scheduling by simulating the "lead bird-follower bird" information sharing mechanism of migratory bird V-formation. However, the standard MBO algorithm still has obvious limitations: there is a lack of effective information interaction between individuals in the left and right columns of the V-formation, which easily leads to "information silos"; the algorithm lacks a targeted reset mechanism, and is prone to getting trapped in local optima when facing complex problems such as 3T-RSSP, resulting in insufficient solution stability.

[0008] Chinese patent CN116774657A discloses a robust optimization-based dynamic scheduling method for remanufacturing workshops, comprising a pre-scheduling stage and a dynamic scheduling stage. In the pre-scheduling stage, a discrete scenario set is used to describe the arrival time and processing time of jobs in the remanufacturing workshop. A robust optimization objective function is established based on minimizing the completion time of the pre-scheduling scheme and the difference in completion time under different scenarios. A biogeographical optimization algorithm is used to output the pre-scheduling scheme. In the dynamic scheduling stage, if a disturbance event occurs, the pre-scheduling scheme is used as the input scheme, and a job right-shift rescheduling strategy is executed to obtain the dynamic scheduling scheme. If the dynamic scheduling scheme is equal to or does not dominate the input scheme, the dynamic scheduling scheme is output; otherwise, a dynamic scheduling objective function is established by minimizing the efficiency index and the robustness index, and a biogeographical optimization algorithm is used to output the dynamic scheduling scheme. This invention considers the impact of multiple uncertainties and disturbance events, resulting in a better scheduling scheme. However, existing methods use a robust optimization framework to handle uncertainties and describe disturbances through discrete scene sets, failing to fully utilize the ability of fuzzy mathematics to characterize continuous uncertainties. The biogeographical optimization algorithm (BBO) it employs lacks coding design for the two-layer decision structure of job sorting and workstation allocation, making it difficult to effectively handle the strong coupling characteristics of JS and WA in 3T-RSSP. Furthermore, the migration mechanism of the BBO algorithm is fundamentally different from the V-shaped queue information sharing mechanism of MBO, and it cannot achieve a balance between directional exploration and development of the solution space through the cooperative structure of leader-follower birds.

[0009] Existing technologies, when dealing with the scheduling problem of three-stage remanufacturing systems, either ignore the fuzzy uncertainty of processing time and adopt deterministic models, leading to a disconnect from actual production. Although fuzzy numbers are introduced, no suitable mathematical model or solution framework is established for the special structure of 3T-RSSP. At the same time, existing metaheuristic algorithms have obvious limitations in terms of encoding schemes, information sharing mechanisms, and local optimum escape strategies, making it difficult to obtain high-quality and stable scheduling schemes within an acceptable time. They cannot meet the actual needs of efficient scheduling of remanufacturing systems under uncertain environments. To address the above problems, we propose a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. Summary of the Invention

[0010] The purpose of this invention is to address the shortcomings of existing technologies by providing a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. This method solves the problems of existing technologies that ignore the fuzzy uncertainty of processing time and adopt deterministic models, leading to a disconnect from actual production. Furthermore, although fuzzy numbers are introduced, no suitable mathematical model or solution framework is established for the special structure of 3T-RSSP.

[0011] This invention is implemented as follows: a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm, the method comprising:

[0012] S10, Problem initialization and data input;

[0013] Input the number of scrapped products n and the number of defective parts included in the scrapped products m;

[0014] Enter the configuration for the three-stage remanufacturing system;

[0015] Input the fuzzy processing time;

[0016] S20. Establish a fuzzy mixed-integer linear programming model based on conventional triangular fuzzy number operations. Use TFN addition to calculate the fuzzy completion time, use maximum operation to determine the fuzzy start time of the task, and use sorting operation to compare the quality of solutions in order to minimize the maximum fuzzy completion time. To optimize the objective;

[0017] The conventional operation rules for triangular fuzzy numbers are expressed as follows:

[0018] Define triangular fuzzy number and triangular fuzzy number ;

[0019] Addition operation:

[0020] Subtraction operation:

[0021] Multiplication operation:

[0022] Division operation:

[0023] Sorting operations: calculation and The value of can have the following three different cases:

[0024] if ,So ;

[0025] if So, comparing and .if ,So ;

[0026] if and So, comparing and .if ,So ;

[0027] Maximum operation:

[0028]

[0029] S30, Solve the fuzzy mixed integer linear programming model based on the improved migratory bird migration optimization algorithm;

[0030] S40: Obtain the optimal scheduling scheme obtained from the improved migratory bird migration optimization algorithm, and output the maximum completion time in TFN form, including job sorting, machine allocation, and the maximum completion time. The optimal scheduling scheme.

[0031] Preferably, in step S10, when inputting the configuration of the three-stage remanufacturing system, the input parameters include the number of disassembly workstations (pp) and the number of reprocessing lines. and the number of reprocessing workstations on each processing line l j The number of reassembled workstations is qq;

[0032] When inputting fuzzy processing time, triangular fuzzy numbers are used to represent the processing time of each stage;

[0033] The fuzzy processing time is expressed as t=(t a ,t b ,t c ), t a ,t b ,t c These represent the most optimistic time, the most likely time, and the most conservative time, respectively.

[0034] The processing time during the disassembly stage is expressed as follows:

[0035] The processing time in the reprocessing stage is expressed as follows:

[0036] The processing time in the reassembly stage is expressed as:

[0037] in, and It is the time scaling factor.

[0038] Preferably, the method for solving the fuzzy mixed-integer linear programming model based on the improved migratory bird migration optimization algorithm includes:

[0039] The encoding and decoding adopts a two-layer encoding scheme to represent the solution. The first layer is the job sequence layer (JS), which represents the order in which scrapped products enter the remanufacturing system; the second layer is the workstation allocation layer (WA), which represents the workstation allocation scheme for scrapped products in the disassembly and reassembly stages.

[0040] Population initialization: An initial population is generated using a hybrid strategy;

[0041] Iterative optimization: Population evolution is carried out through joint neighborhood search, cooperative sharing mechanism and reset mechanism. Among them, joint neighborhood search designs the neighborhood structure of the left and right queues differently according to the V-shaped queue structure. The cooperative sharing mechanism uses large-scale crossover operator and small-scale crossover operator triggered by adaptive probability to promote information interaction between individuals. The reset mechanism reinitializes continuously unimproved individuals.

[0042] Termination decision: Output the optimal scheduling scheme after reaching the maximum number of iterations K.

[0043] Preferably, the two-layer coding scheme includes:

[0044] First layer (JS layer): Job sequence layer, represented as {π1,...,π} i ,...,π n}, where π i The number of the scrapped product at the i-th processing position determines the order in which the products enter the remanufacturing system;

[0045] The second layer (WA layer): the workstation allocation layer, represented as {D1,...,D...} i ,...,D n ,A1,...,A i ,...,A n}, where D i ∈{1,2,...,pp} represents the disassembly workstation assigned to workpiece i, A i∈{1,2,...,qq} represents the reassembly workstation assigned to workpiece i.

[0046] Preferably, when using a hybrid strategy to generate the initial population, the hybrid strategy includes a heuristic initialization strategy based on the NEH rule, a random initialization strategy, and a load-balanced initialization strategy, with a population size of PS;

[0047] Of these, 10% of the individuals were generated based on a heuristic initialization strategy using the NEH rule. An improved NEH heuristic for the 3T-RSSP-FPT problem was used to calculate the total fuzzy processing time for each product.

[0048]

[0049] Arranged in descending order of TPT and inserted step by step into the optimal position; workstation allocation layers are randomly generated.

[0050] 45% of the individuals were generated using a random initialization strategy. Both the job sequence layer and the workstation allocation layer were generated randomly. The generation of the workstation allocation layer was constrained by the number of disassembled workstations (pp) and the number of reassembled workstations (qq).

[0051] 45% of the individuals are generated using a load balancing initialization strategy. The workstation allocation layer distributes the products as evenly as possible to each disassembly and reassembly workstation, while the job sequence layer generates them randomly.

[0052] Preferably, the joint neighborhood search, based on the differentiated design of the neighborhood structures of the left and right queues according to the V-shaped queue structure, includes:

[0053] Left queue neighborhood structure: used to optimize job sequence, including Swap_1 operation (randomly select two different positions and swap elements) and Insert operation (randomly select two different positions and move the element at position 2 before the element at position 1).

[0054] Right queue neighborhood structure: used to optimize workstation allocation, including the Adjust operation (randomly selecting four different positions with probability P). e ≥0.5 triggers workstation index adjustment) and Swap_2 operation (randomly selects four different positions, swaps the elements of position 1 and position 2 and the elements of position 3 and position 4, and simultaneously applies to disassembling and reassembling workstation configuration).

[0055] Leader bird processing: In the first iteration, the neighborhood structure of the left or right queue is randomly selected. In subsequent iterations, the corresponding neighborhood structure is selected according to the source queue of the leader bird.

[0056] Preferably, the collaborative sharing mechanism includes:

[0057] Six differentiated crossover operators are designed, and the crossover operators are adaptively selected based on the evolutionary iteration stage:

[0058] Large-scale crossover operator (used for early-stage algorithm exploration, probabilistic) ):

[0059] CO1: Exchange the job sequence portion or workstation allocation portion of two individuals;

[0060] CO2: Exchange the reassembly workshop configuration in the allocation section of the two individual workstations;

[0061] CO3: Exchange the dismantling workshop configuration in the allocation section of the two individual workstations;

[0062] Small-scale crossover operator (used in later algorithm development, probabilistic) ):

[0063] CO4 (Partially Mapping Crossover, PMX): Partially mapped crossover is used in the job sequence layer. It determines the crossover segment based on two random positions and establishes an element mapping table to avoid conflicts.

[0064] CO5 (Linear Order Crossover, LOX): Linear order crossover is used in the job sequence layer. It determines the crossover segment based on two random positions, while maintaining the relative order of the remaining elements.

[0065] CO6 (Multiple Position Crossover, MPX): Multi-position crossover is used for workstation allocation layers, selecting round (0.4×n) positions for exchange in both the disassembly and reassembly sections.

[0066] Preferably, the adaptive triggering strategy for the crossover operator, based on adaptive selection during the evolutionary iteration phase, is as follows:

[0067] The formula for calculating the trigger probability is expressed as follows:

[0068]

[0069] in, This represents the current iteration index, where K is the maximum number of iterations. Early iterations focus on large-scale cross-referencing to quickly locate high-quality regions, while later iterations focus on small-scale cross-referencing for refined searching.

[0070] The collaborative sharing process is as follows: Each generation randomly selects pairs of individuals at the same position in the left and right queues (preferred). ), perform a crossover operation to replace the parent with the better child.

[0071] Preferably, when solving the fuzzy mixed-integer linear programming model using the improved migratory bird migration optimization algorithm, the improved iterative optimization method includes:

[0072] Leader bird evolution: Generate kk neighborhood solutions. If the best neighborhood solution is better than the current leader bird, replace it. Sort the remaining neighborhood solutions by fitness and fill them into the left and right shared sets. and In this context, each shared set contains s solutions;

[0073] Evolution by following the bird: The left and right queues each update their solutions by combining their own kk-ss neighborhood solutions with the ss solutions in the set shared by the previous bird;

[0074] Execution reset mechanism check and collaborative sharing mechanism;

[0075] Leader bird replacement: Move the current leader bird to the back of the formation and move the bird after it to the front of the formation;

[0076] Determine if the current iteration count has reached the maximum iteration count K. If so, terminate the algorithm; otherwise, return to the leader bird evolution step.

[0077] Compared with the prior art, the embodiments of this application have the following main advantages:

[0078] This invention constructs a fuzzy mixed-integer linear programming model adapted to the special structure of 3T-RSSP. Based on the conventional operation rules of triangular fuzzy numbers, it accurately quantifies the fuzzy completion time, solving the problem of the disconnect between fuzzy modeling and the three-stage process. At the same time, in response to the shortcomings of the standard MBO algorithm, such as weak information interaction and susceptibility to local optima, the improved migratory bird optimization algorithm (ECMBO) accurately describes the strongly coupled logic of "job sequence-workstation allocation" through "JS-WA double-layer coding", breaks the information silos of the V-shaped queue by using a "collaborative sharing mechanism", and introduces a reset mechanism to enhance the ability to escape local optima. This significantly improves the algorithm's global optimization ability and convergence stability in fuzzy environments. It can output a high-quality scheduling scheme that includes workpiece sorting, machine allocation and maximum completion time in TFN form within an acceptable time, effectively meeting the high-efficiency scheduling requirements of remanufacturing systems in uncertain environments.

[0079] The improved migratory bird optimization algorithm (ECMBO) proposed in this invention achieves significant benefits through five core technological improvements. First, it adopts a two-layer encoding scheme of job sequence layer and workstation allocation layer to fully express the coupled decision information of job sorting and workstation allocation in the three-stage remanufacturing system, overcoming the limitation of traditional single-layer encoding in effectively expressing complex decision structures. Second, it introduces a hybrid initialization strategy combining NEH rule, random strategy, and load balancing strategy, ensuring that the initial population is evenly distributed in the solution space and contains high-quality solutions, laying a good foundation for subsequent iterative optimization. Third, it designs four differentiated neighborhood structures for JS and WA respectively and introduces a joint neighborhood search strategy, realizing the co-evolution of the left and right queues based on the V-shaped queue structure, fully exploring different dimensions of the solution space. Furthermore, it integrates a collaborative sharing mechanism containing six crossover operators, effectively breaking down information silos between individuals in the left and right of the V-shaped queue by dynamically switching between large-scale and small-scale crossover operators through adaptive probability triggering, promoting the transmission and recombination of excellent features. Finally, it sets a reset mechanism triggered by a continuous unimproved threshold to promptly replace stagnant individuals to maintain population diversity and avoid premature convergence and getting trapped in local optima. The synergistic effect of the above improvements makes ECMBO significantly superior to standard MBO and other comparative algorithms in both solution quality and efficiency. It can obtain a high-quality and stable scheduling scheme within an acceptable time and effectively meet the high-efficiency scheduling requirements of three-stage remanufacturing systems under uncertain environments.

[0080] The joint neighborhood search strategy proposed in this invention aligns with the V-shaped cluster structure unique to the MBO algorithm. By configuring a neighborhood structure for the job sequence layer (JS) for the left queue follower birds (FBs) and a neighborhood structure for the workstation allocation layer (WA) for the right queue FBs, decoupling and specialized exploration of the search space are achieved. Simultaneously, a dynamic selection mechanism designed for the leader bird (LB) randomly selects the search direction in the first iteration to consider global exploration, while subsequent iterations lock the corresponding neighborhood structure based on the source queue to maintain search continuity, thus effectively balancing the algorithm's exploration and development capabilities. This strategy not only fully exploits the neighborhood potential of the JS and WA layers but also promotes synergistic effects between different neighborhood structures, significantly improving the algorithm's optimization efficiency and convergence stability in complex solution spaces. It provides strong local search support for solving NP-hard problems with strong two-layer coupling, such as 3T-RSSP-FPT. Attached Figure Description

[0081] Figure 1 This is a schematic diagram illustrating the implementation process of the fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm provided by the present invention.

[0082] Figure 2 A schematic diagram of the architecture of the three-stage remanufacturing system (3T-RSSP) in an embodiment of the present invention is shown.

[0083] Figure 3 An example diagram of the ECMBO two-layer coding scheme in an embodiment of the present invention is shown.

[0084] Figure 4 This is a schematic diagram of the left queue neighborhood structure provided by the present invention.

[0085] Figure 5 This is a schematic diagram of the right queue neighborhood structure provided by the present invention.

[0086] Figure 6 A schematic diagram of the large-scale crossover operator in an embodiment of the present invention is shown.

[0087] Figure 7 A schematic diagram of a small-scale crossover operator is shown in an embodiment of the present invention.

[0088] Figure 8 The box plot comparison results of ECMBO and ECMBO-HIS in the embodiments of the present invention are shown.

[0089] Figure 9 The diagram shows a comparison of the probability densities of ECMBO and ECMBO-HIS in an embodiment of the present invention. Detailed Implementation

[0090] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs; the terminology used herein in the specification of the application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application; the terms "comprising" and "having," and any variations thereof, in the specification, claims, and foregoing drawings of this application are intended to cover non-exclusive inclusion. The terms "first," "second," etc., in the specification, claims, or foregoing drawings of this application are used to distinguish different objects, not to describe a particular order.

[0091] Existing technologies, when dealing with the scheduling problem of three-stage remanufacturing systems, either neglect the fuzzy uncertainty of processing time and adopt deterministic models, leading to a disconnect from actual production; and although fuzzy numbers are introduced, no suitable mathematical model or solution framework is established for the special structure of 3T-RSSP. Furthermore, existing metaheuristic algorithms have significant limitations in terms of encoding schemes, information sharing mechanisms, and local optimum escape strategies, making it difficult to obtain high-quality, stable scheduling solutions within an acceptable timeframe, and failing to meet the actual needs of efficient scheduling of remanufacturing systems under uncertain environments. To address these issues, we propose a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. In short, the method first initializes the problem and inputs data; then, a fuzzy mixed-integer linear programming model is established based on conventional operations using triangular fuzzy numbers; fuzzy completion time is calculated using TFN addition; the fuzzy start time of the task is determined using maximum operation; and the superiority of solutions is compared using sorting operations to minimize the maximum fuzzy completion time. To optimize the objective, the fuzzy mixed-integer linear programming model is solved using an improved migratory bird migration optimization algorithm. Finally, the optimal scheduling scheme obtained from the improved migratory bird migration optimization algorithm is obtained, and the output includes job sorting, machine allocation, and the maximum completion time in TFN form. The optimal scheduling scheme is found. In this embodiment of the invention, by using triangular fuzzy numbers to describe the uncertainty of processing time and establishing a fuzzy mixed integer linear programming model based on conventional operations, the time fuzziness caused by the quality differences in the recycling of scrapped products and the fluctuations in the production environment can be accurately characterized, generating a feasible scheduling scheme with time buffer, which significantly improves the scheduling robustness and stability in actual production. Secondly, a two-layer coding scheme combining the job sequence layer and the workstation allocation layer is designed to fully express the coupled decision relationship between workpiece sequencing and workstation allocation in the three-stage remanufacturing system, effectively supporting global optimization in an unrelated parallel machine environment. Thirdly, based on the standard MBO, a hybrid initialization strategy, joint neighborhood search, cooperative sharing mechanism, and reset mechanism are introduced, using NEH heuristics, random strategies, and negative... The combination of load balancing strategies improves the quality of the initial solution. Based on the V-shaped queue structure, the neighborhood structure of the left and right queues is designed differently to achieve bidirectional collaborative exploration of the solution space. Six differentiated crossover operators are adopted and triggered adaptively according to the evolution stage to break through information silos. A continuous unimproved threshold triggering reset mechanism is set to avoid premature convergence. Thus, the maximum completion time is significantly reduced while ensuring solution stability. Finally, experimental verification shows that the proposed ECMBO algorithm can obtain the same optimal solution as CPLEX in small-scale instances. In medium and large-scale instances, the mean is significantly better than the standard MBO and other comparative algorithms. The solution efficiency is greatly improved and the standard deviation is significantly reduced. It can provide effective support for production managers to quickly determine the most suitable scheduling scheme based on the current operation.

[0092] In summary, this invention constructs a fuzzy mixed-integer linear programming model adapted to the special structure of 3T-RSSP, accurately quantifies the fuzzy completion time based on the conventional operation rules of triangular fuzzy numbers, and solves the problem of the disconnect between fuzzy modeling and the three-stage process. At the same time, in response to the shortcomings of the standard MBO algorithm, such as weak information interaction and susceptibility to local optima, the improved migratory bird optimization algorithm (ECMBO) accurately characterizes the strongly coupled logic of "job sequence-workstation allocation" through "JS-WA double-layer encoding", breaks the information silos of the V-shaped queue by using a "collaborative sharing mechanism", and introduces a reset mechanism to enhance the ability to escape local optima. This significantly improves the algorithm's global optimization ability and convergence stability in fuzzy environments, and can output a high-quality scheduling scheme that includes workpiece sorting, machine allocation and maximum completion time in TFN form within an acceptable time, effectively meeting the high-efficiency scheduling requirements of remanufacturing systems in uncertain environments.

[0093] Example 1

[0094] This invention provides a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. Figure 1 A schematic diagram illustrating the implementation flow of a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm is shown. The fuzzy remanufacturing system scheduling method based on the improved migratory bird migration optimization algorithm specifically includes:

[0095] S10, Problem initialization and data input;

[0096] In this embodiment of the invention, 3T-RSSP-FPT (Three-Stage Remanufacturing System Scheduling Problem with Fuzzy Processing Times) represents a three-stage remanufacturing system scheduling problem considering fuzzy processing times. 3T-RSSP-FPT studies how to optimize the scheduling of a remanufacturing system comprising three stages—disassembly, reprocessing, and reassembly—under the actual background of uncertain quality of recycled end-of-life products and fluctuations in the production environment, in order to minimize the maximum completion time. The processing times are described using triangular fuzzy numbers. The problem initialization and data input stages of 3T-RSSP-FPT are defined as follows:

[0097] Input the number of scrapped products n, {1,…,i,…n}, and the number of defective parts contained in the scrapped products m. For each scrapped product, it has m components {1,…,j,…,m} and each defective component must be processed in a dedicated RL (reprocessing line).

[0098] It should be noted that in the embodiments of the present invention, RL represents the reprocessing line, DW represents the disassembly workstation, RW represents the reprocessing workstation, AW represents the reassembly workstation, JS represents the job sequence layer, WA represents the workstation allocation layer, TFN (Triangular Fuzzy Number) represents the triangular fuzzy number, ECMBO represents the improved migratory bird migration optimization algorithm, LB represents the lead bird, and FBs represents the follower bird. For ease of description, the English abbreviations appearing in this specification all have fixed Chinese meanings, and the English abbreviations mentioned in the whole text correspond one-to-one with the above Chinese meanings, and will not be repeated one by one later.

[0099] Enter the configuration for the three-stage remanufacturing system;

[0100] Input the fuzzy processing time;

[0101] When configuring the three-stage remanufacturing system, the input parameters include the number of disassembly workstations (DW) (pp) and the number of reprocessing lines (RL). and the number of reprocessing workstations (RW) on each processing line. j The number of reassembled workstations is qq;

[0102] It should be noted that in the dismantling workshop, scrapped products only need to be dismantled by selecting one machine from pp DW{1,…,s,…,pp}. In the post-processing workshop, the separated parts are sent to a dedicated RL, where a series of l j Each RWs{1,…,k,…,l} j} Waiting to repair them. In the reassembly workshop, remanufacturing parts only requires selecting one machine from qq AW{1,…,v,…,qq} to perform the last operation, i.e., the reassembly operation. The processing time of a workpiece in the three-stage production system is represented by a TFNs, i.e. (t a ,t b ,t c ), where t a , t b and t c These represent the shortest (optimistic) processing time, the most likely (reasonable) processing time, and the longest (pessimistic) processing time, respectively.

[0103] in, Figure 2 A schematic diagram of the architecture of a three-stage remanufacturing system (3T-RSSP) according to an embodiment of the present invention is shown. Figure 2In this embodiment of the invention, the three-stage remanufacturing system (3T-RSSP) includes one disassembly shop, one reprocessing shop, and one reassembly shop. The disassembly shop contains three unrelated parallel disassembly workstations, DW1, DW2, and DW3. After disassembly, the scrapped product P1 is used to obtain component C. 11 C 12 C 13 (3 defective parts), after disassembling scrapped product P2, component C was obtained. 21 C 22 C 23 (3 defective parts), the reprocessing shop contains 3 parallel assembly line-type reprocessing lines: RL1, RL2, and RL3. RL1: corresponds to component type C. 11 C 21 etc., including serial workstation RW 11 RW 12 RW 13 RL2: Corresponds to the processing component type C 12 C 22 etc., including serial workstation RW 21 RW 22 RW 23 RL3: Corresponds to the processing component type C 13 C 23 etc., including serial workstation RW 31 RW 32 Each RL (Reassembly Line) has a specific characteristic: a particular type of defective component can only enter the corresponding reassembly line. The reassembly shop contains AW1 and AW2: two unrelated parallel reassembly workstations that reassemble the remanufactured components completed on each reassembly line into remanufactured products. Queue is the queue of workpieces waiting to be processed in the workstation buffer, representing the queue of workpieces waiting to be processed in front of the workstation. Machine represents the workpiece being processed by the workstation, representing the processing equipment on each workstation. When the three-stage remanufacturing system (3T-RSSP) is working, scrapped products P1, P2, etc. in the EOL (Extended Online Product) set first enter the disassembly shop, where one of the unrelated parallel disassembly workstations (DW1~DW3) performs the disassembly operation to obtain various types of defective components (C). 11 ~C 13 C 21 ~C 23(etc.); then, each defective component is sent to the corresponding dedicated remanufacturing line (RL1~RL3) in the reprocessing workshop according to its type, and is repaired sequentially by the serial reprocessing workstation (RW) on the line; finally, all remanufactured components are reassembled into remanufactured products by one of the unrelated parallel reassembly workstations (AW1-AW2) in the reassembly workshop, completing the entire three-stage remanufacturing process.

[0104] When inputting fuzzy processing time, the Triangular Fuzzy Number (TFN) is used to represent the processing time of each stage;

[0105] The fuzzy processing time is expressed as t=(t a ,t b ,t c ), t a ,t b ,t c These represent the most optimistic time, the most likely time, and the most conservative time, respectively.

[0106] The processing time during the disassembly stage is expressed as follows:

[0107] The processing time in the reprocessing stage is expressed as follows:

[0108] The processing time in the reassembly stage is expressed as:

[0109] in, and This is the time scaling factor, where i represents the index of scrapped products and s represents the index of dismantling workstations. To determine the most likely processing time for disassembly, The most optimistic dismantling time, The most conservative disassembly time is given, where D represents the disassembly stage; j is the index of the reprocessing line, and k is the index of the reprocessing workstation. R represents the most likely processing time for the reprocessing stage; v represents the reassembly workstation index; and A represents the reassembly stage. This represents the most likely processing time during the reassembly stage.

[0110] S20. Establish a fuzzy mixed-integer linear programming model based on conventional operations of triangular fuzzy numbers. Use TFN (triangular fuzzy number) addition to calculate the fuzzy completion time, use maximum operation to determine the fuzzy start time of the task, and use sorting operation to compare the quality of solutions to minimize the maximum fuzzy completion time. To optimize the objective;

[0111] Furthermore, the goal of 3T-RSSP-FPT is to minimize the maximum completion time (i.e., by rationally handling the sub-problems (i) job sequences and (ii) workstation allocation in these workshops) In addition, some additional assumptions are required, as follows:

[0112] DW, AW, and RW are ready, and the workpieces were prepared from the beginning.

[0113] Operations including disassembly, reprocessing, and reassembly must not be interrupted.

[0114] The choice of workpiece and machine / workstation is unique.

[0115] The transportation time of the workpiece is not taken into account.

[0116] The buffers for DW, AW, and RW are large enough.

[0117] Random workstation failures are not considered.

[0118] Obtain all necessary data in advance.

[0119] The conventional operation rules for triangular fuzzy numbers are expressed as follows:

[0120] Define triangular fuzzy number and triangular fuzzy number ;

[0121] Addition operation:

[0122] Subtraction operation:

[0123] Multiplication operation:

[0124] Division operation:

[0125] Sorting operations: calculation and The value of can have the following three different cases:

[0126] if ,So ;

[0127] if So, comparing and .if ,So ;

[0128] if and So, comparing and .if ,So ;

[0129] Maximum operation:

[0130]

[0131] In this embodiment of the invention, the "addition" operation calculates the fuzzy end time of the operation based on the fuzzy start time and fuzzy processing time. Furthermore, the "maximum" operation is used to determine the fuzzy start time of a specific task on a particular workstation, taking into account not only the fuzzy completion time of the previous task but also the fuzzy completion time of the most recently assigned task to that workstation. Additionally, the "sorting" operation is used to sort the completion times of the tasks and determine the final completion time.

[0132] In this embodiment of the invention, Table 1 shows the meanings of the parameters and variables in the fuzzy mixed integer linear programming model established based on the conventional operation of triangular fuzzy numbers.

[0133] Table 1

[0134]

[0135] The fuzzy mixed-integer linear programming model uses the nine decision variables (including Boolean and continuous decision variables) listed in Table 1. The model has an optimization objective and constraints, and can be viewed as a common computation-based mathematical model.

[0136] The optimization objective of the model is expressed as:

[0137] The constraints of the fuzzy mixed-integer linear programming model are as follows:

[0138] (1)

[0139] (2)

[0140] (3)

[0141] (4)

[0142] (5)

[0143] (6)

[0144] (7)

[0145] (8)

[0146] (9)

[0147] (10)

[0148] (11)

[0149] (12)

[0150] (13)

[0151] Constraints (1)-(4) discuss the situation in the disassembly workshop. Specifically, constraint (1) ensures that any disassembly workstation (DW) in the disassembly workshop can only process one scrap product at a time. Constraints (2) and (3) ensure that the disassembly operation of the disassembly workstation is not interrupted. Constraint (4) ensures that the initial reprocessing repair operation of the defective component cannot be performed before its corresponding disassembly operation is completed. In addition, the situation in the reprocessing workshop is described in constraints (5)-(8). Specifically, constraint (5) aims to regulate that the reprocessing repair operation must be performed sequentially. Constraints (6) and (7) ensure that the reprocessing operation cannot be interrupted. Constraint (8) aims to establish the relationship between the reassembly operation and the final reprocessing operation. In addition, constraints (9)-(11) describe the situation in the reassembly workshop. Specifically, constraint (9) stipulates that remanufactured parts cannot be assembled by multiple reassembly workstations (AW) at the same time. Constraints (10) and (11) ensure that the reassembly operation of the AW cannot be interrupted. Finally, constraint (12) determines the maximum completion time, while constraint (13) ensures that the dismantling start time of all waste products is greater than or equal to zero.

[0152] In the model analysis, we set pp=0, qq=1, and l j =1 (for all j), the optimization objective problem [P] degenerates into a typical NP-hard two-stage assembly line scheduling problem. Therefore, [P] is also NP-hard, and its solution requires careful design. In fact, [P] is also a typical sequential decision problem, which can be regarded as a Markov decision process, in which the agent should determine the scheduling objective and workshop resource allocation at each decision point.

[0153] S30, Solve the fuzzy mixed integer linear programming model based on the improved migratory bird migration optimization algorithm;

[0154] S40: Obtain the optimal scheduling scheme obtained from the improved migratory bird migration optimization algorithm, and output the maximum completion time in TFN form, including job sorting, machine allocation, and the maximum completion time. The optimal scheduling scheme.

[0155] Among them, workpiece sorting (JS) is the optimal sequence for scrapped products to enter the system;

[0156] Machine allocation (WA): Workstation allocation scheme for each scrapped product during the disassembly and reassembly stages;

[0157] Fuzzy Completion Time: Maximum completion time C expressed in TFN form max .

[0158] This invention constructs a fuzzy mixed-integer linear programming model adapted to the special structure of 3T-RSSP. Based on the conventional operation rules of triangular fuzzy numbers, it accurately quantifies the fuzzy completion time, solving the problem of the disconnect between fuzzy modeling and the three-stage process. At the same time, in response to the shortcomings of the standard MBO algorithm, such as weak information interaction and susceptibility to local optima, the improved migratory bird optimization algorithm (ECMBO) accurately describes the strongly coupled logic of "job sequence-workstation allocation" through "JS-WA double-layer coding", breaks the information silos of the V-shaped queue by using a "collaborative sharing mechanism", and introduces a reset mechanism to enhance the ability to escape local optima. This significantly improves the algorithm's global optimization ability and convergence stability in fuzzy environments. It can output a high-quality scheduling scheme that includes workpiece sorting, machine allocation and maximum completion time in TFN form within an acceptable time, effectively meeting the high-efficiency scheduling requirements of remanufacturing systems in uncertain environments.

[0159] To better handle NP-hard problems, various measures were taken to improve the original Management Buyout (MBO), resulting in the design of an effective cooperative MBO called ECMBO. Compared to the original MBO, ECMBO has the following five contributions:

[0160] Each bird or individual is represented by two layers, both of which are equipped with full information about the job sequence (JS) and workstation assignment (WA).

[0161] A hybrid initialization strategy (i.e., a hybrid strategy) that combines three different rules (including the NEH rule and two heuristic rules) is introduced to improve the quality of the initial solution.

[0162] Four neighborhood structures were designed for JS and WA, and a joint neighborhood search strategy was introduced to promote collaboration between the structures.

[0163] The collaborative sharing mechanism employs six cross-operators to trigger perturbations of different scales.

[0164] A reset mechanism is set up for individuals that are not updated in time to prevent the algorithm from converging too early and getting stuck in local optima.

[0165] The improved migratory bird optimization algorithm (ECMBO) proposed in this invention achieves significant benefits through five core technological improvements. First, it adopts a two-layer encoding scheme of job sequence layer and workstation allocation layer to fully express the coupled decision information of job sorting and workstation allocation in the three-stage remanufacturing system, overcoming the limitation of traditional single-layer encoding in effectively expressing complex decision structures. Second, it introduces a hybrid initialization strategy combining NEH rule, random strategy, and load balancing strategy, ensuring that the initial population is evenly distributed in the solution space and contains high-quality solutions, laying a good foundation for subsequent iterative optimization. Third, it designs four differentiated neighborhood structures for JS and WA respectively and introduces a joint neighborhood search strategy, realizing the co-evolution of the left and right queues based on the V-shaped queue structure, fully exploring different dimensions of the solution space. Furthermore, it integrates a collaborative sharing mechanism containing six crossover operators, effectively breaking down information silos between individuals on the left and right sides of the V-shaped queue by dynamically switching between large-scale and small-scale crossover operators through adaptive probability triggering, promoting the transmission and recombination of excellent features. Finally, it sets a reset mechanism triggered by a continuous unimproved threshold to promptly replace stagnant individuals to maintain population diversity and avoid premature convergence and getting trapped in local optima. The synergistic effect of the above improvements makes ECMBO significantly superior to standard MBO and other comparative algorithms in both solution quality and efficiency. It can obtain a high-quality and stable scheduling scheme within an acceptable time and effectively meet the high-efficiency scheduling requirements of three-stage remanufacturing systems under uncertain environments.

[0166] It should be noted that traditional MBO is a smart metaheuristic approach inspired by the "V" formation of migratory birds during long-distance flights. The flock resembles the letter "V" and consists of a lead bird (LB) and several follower birds (FB) evenly distributed on the left and right sides. This flight formation achieves better energy efficiency through a positive correlation between wingtip spacing (WTS) and depth, or a constant V-angle.

[0167] MBO involves five key algorithmic parameters, including population size [PS], the maximum number of neighborhood solutions a bird can generate [kk], the maximum number of neighborhood solutions a bird can share with the next bird [ss], the cycle count [G], and the maximum number of iterations [K]. The first four parameters correspond to the number of birds in the flock, the flock's flight speed, WTS, and the frequency of queue changes, respectively. Generally, MBO consists of the following four main steps from step 1 to step 4:

[0168] Step 1: Flock Initialization. Generate a flock of birds, including one LB and several FBs.

[0169] Step 2: LB Evolution. An attempt is made to improve the LB by generating and evaluating kk neighborhood solutions.

[0170] Step 3: FB Evolution. Attempt to improve FB by evaluating its own (kk-ss) neighborhood solutions and its ss best neighborhood solutions that were not utilized by its predecessor birds.

[0171] Step 4: LB Replacement. Move LB to the back of the queue and move the bird behind it (which may be in the left or right queue) to the front.

[0172] In this embodiment of the invention, ECMBO (Enhanced / Cooperative Migrating Birds Optimization) refers to a metaheuristic algorithm that improves upon the standard Migrating Birds Optimization (MBO) algorithm by introducing a two-layer coding scheme, a hybrid initialization strategy, a joint neighborhood search, a cooperative sharing mechanism, and an individual reset mechanism. It is used to solve the three-stage remanufacturing system scheduling problem (3T-RSSP-FPT) considering fuzzy processing time. Throughout this text, ECMBO refers to this improved migrating bird optimization algorithm; its English abbreviation can be understood as the corresponding Chinese meaning of "Cooperative / Cooperative Migrating Birds Optimization," which will not be elaborated further here.

[0173] This invention provides a method for solving the fuzzy mixed-integer linear programming model based on an improved migratory bird migration optimization algorithm. The method specifically includes:

[0174] S101, encoding and decoding, uses a two-layer encoding scheme to represent the solution. The first layer is the job sequence layer (JS), which represents the order in which scrapped products enter the remanufacturing system; the second layer is the workstation allocation layer (WA), which represents the workstation allocation scheme for scrapped products in the disassembly and reassembly stages.

[0175] In this embodiment of the invention, ECMBO employs a two-layer encoding scheme to represent latent individuals. The first layer is the JS part, and the second layer is the WA part. This encoding scheme contains sufficient information about the problem under study, which is beneficial to the evolution of ECMBO. Figure 3 This diagram illustrates an example of the ECMBO two-layer coding scheme in an embodiment of the present invention. An individual or a bird uses {π1,...,π} i ,...,π n ;D1,...,D i ,...,D n ,A1,...,A i ,...,A n} represents, where π i P represents the i-th scrapped product. i At the π iOne position was processed, D i Indicates assignment to P i DW index, A i Indicates assignment to P i The AW index. For example, Figure 3 The encoded string [3 4 1 2; 3 1 3 2 1 2 1 1] indicates that the scrapped products will enter the remanufacturing system (first layer) in the order of P3, P4, P1, P2. P1, P2, P3, P4 are processed in DW3, DW1, DW3, DW2 of the disassembly workshop and AW1, AW2, AW1, AW1 (second layer) of the reassembly workshop, respectively.

[0176] Decoding is the inverse process of encoding; it transforms the encoded solution into a feasible scheduling scheme. Clearly, the JS and WA information from the disassembly workshop, and the WA information from the reassembly workshop, can be directly obtained from... Figure 3 The encoding scheme shown is used to obtain the JS in the reprocessing and reassembly workshops. In the embodiment of the present invention, the First Come First Service (FCFS) heuristic method is used for decoding when processing the JS in the reprocessing and reassembly workshops. It should be noted that: (1) Due to the special characteristics of RL, WA does not need to make additional decisions. (2) Unlike the second workshop (i.e., the reprocessing workshop), when executing the FCFS heuristic algorithm, the workstations in the reassembly workshop must wait for all remanufacturing components of the same product to be processed before they can perform the assembly operation.

[0177] The two-layer coding scheme includes:

[0178] First layer (JS layer): Job sequence layer, represented as {π1,...,π} i ,...,π n}, where π i The number of the scrapped product at the i-th processing position determines the order in which the products enter the remanufacturing system;

[0179] The second layer (WA layer): the workstation allocation layer, represented as {D1,...,D...} i ,...,D n ,A1,...,A i ,...,A n}, where D i ∈{1,2,...,pp} represents the disassembly workstation assigned to workpiece i, A i ∈{1,2,...,qq} represents the reassembly workstation assigned to workpiece i.

[0180] S102, Population initialization: An initial population is generated using a hybrid strategy;

[0181] The quality of the initial population has a significant impact on the overall performance of ECMBO. Compared to a single initialization strategy (which in most cases refers to a random strategy), a mixed or combined initialization strategy can help improve the convergence speed and solution quality of the algorithm. This invention provides a mixture of three different initialization rules to generate the initial population for ECMBO.

[0182] When using a hybrid strategy to generate the initial population, the hybrid strategy includes a heuristic initialization strategy based on the NEH rule, a random initialization strategy, and a load-balanced initialization strategy, with a population size of PS.

[0183] Of these, 10% of the individuals were generated based on a heuristic initialization strategy using the NEH rule. An improved NEH heuristic for the 3T-RSSP-FPT problem was used to calculate the total fuzzy processing time for each product.

[0184]

[0185] Arranged in descending order of TPT and inserted step by step into the optimal position; workstation allocation layers are randomly generated.

[0186] 45% of the individuals were generated using a random initialization strategy. Both the job sequence layer and the workstation allocation layer were generated randomly. The generation of the workstation allocation layer was constrained by the number of disassembled workstations (pp) and the number of reassembled workstations (qq).

[0187] 45% of the individuals are generated using a load balancing initialization strategy. The workstation allocation layer distributes the products as evenly as possible to each disassembly and reassembly workstation, while the job sequence layer generates them randomly.

[0188] When 45% of the individuals are generated using a load balancing initialization strategy, the number of scrapped products allocated to each dismantling workstation (DW) and reassembly workstation (AW) is made as even as possible during the initial population generation. This avoids some workstations being overloaded while others are idle, thereby improving the quality of the initial solution and the convergence efficiency of subsequent optimizations. Specifically, for the dismantling workstation allocation segment in the WA layer, scrapped products are sequentially allocated to the DW with the fewest currently allocated products; that is, the DW with the lowest load is selected for allocation each time. If multiple DWs have the same load, one is randomly selected. Similarly, for the reassembly workstation allocation segment in the WA layer, the same minimum load priority principle is used to evenly distribute scrapped products to each AW. The JS layer uses a random generation strategy to ensure the diversity of the job sequence. This strategy, through an explicit load balancing mechanism, enables the workstation allocation scheme in the initial population to have a more reasonable resource distribution in the solution space, avoiding the problem of extreme imbalance in workstation load that may be caused by pure random initialization. It provides a better initial search starting point for subsequent joint neighborhood search and collaborative sharing mechanisms, thereby effectively improving the overall convergence speed and solution stability of the algorithm.

[0189] S103, Iterative optimization: Population evolution is carried out through joint neighborhood search, cooperative sharing mechanism and reset mechanism. Among them, joint neighborhood search designs the neighborhood structure of the left and right queues differently according to the V-shaped queue structure. The cooperative sharing mechanism uses large-scale crossover operator and small-scale crossover operator triggered by adaptive probability to promote information interaction between individuals. The reset mechanism reinitializes continuously unimproved individuals.

[0190] S104, Termination Decision: Output the optimal scheduling scheme after reaching the maximum number of iterations K.

[0191] In this embodiment of the invention, the joint neighborhood search, based on the differentiated design of the neighborhood structures of the left and right queues according to the V-shaped queue structure, includes:

[0192] Left queue neighborhood structure: used to optimize job sequence, including Swap_1 operation (randomly select two different positions and swap elements) and Insert operation (randomly select two different positions and move the element at position 2 before the element at position 1).

[0193] In this embodiment of the invention, Figure 4 The diagram illustrates the left queue neighborhood structure. For the JS part, two neighborhood structures are designed: Swap_1 and Insert, where:

[0194] Swap_1: Randomly selects two different positions (marked as position 1 and position 2) and swaps the elements at these two positions. For example... Figure 4 As shown, the elements 3 and 1 in the first and third positions are swapped.

[0195] Insert: Randomly select two different positions (marked as position 1 and position 2) and move the element at position 2 before the element at position 1. Figure 4 Element 1 is shown to be moved forward and eventually placed before element 3.

[0196] Right queue neighborhood structure: used to optimize workstation allocation, including the Adjust operation (randomly selecting four different positions with probability P). e ≥0.5 triggers workstation index adjustment) and Swap_2 operation (randomly selects four different positions, swaps the elements of position 1 and position 2 and the elements of position 3 and position 4, and simultaneously applies to disassembling and reassembling workstation configuration).

[0197] In this embodiment of the invention, Figure 5 The diagram illustrates the right queue neighborhood structure. Two neighborhood structures are designed for the WA (Web Application) part: Adjust and Swap_2. Unlike the neighborhood structure of the JS (Web Script) part, the WA part needs to select more positions. This is because the WA contains more elements than the JS. The two structures for the WA part are as follows:

[0198] Adjust: Randomly select four different positions, labeled as position 1, position 2, position 3, and position 4. Elements at these positions will be assigned probabilities P. e Trigger adjustment operation. If the generated random number (range 0 to 1) is greater than P. e The element will then be replaced with another workstation index (the index limit is the total number of workstations, which may be pp or qq first introduced in Table 1). For example... Figure 5 As shown, element 3 in position 1 and element 1 in position 4 are replaced with 1 and 2 respectively, while the elements in positions 2 and 3 remain unchanged. In ECMBO, P e Set it to 0.5.

[0199] Swap_2: Randomly selects four different positions, labeled as position 1, position 2, position 3, and position 4. Swaps the elements at positions 1 and 2, and the elements at positions 3 and 4. Unlike the swap structure in the JS section (i.e., Swap_1), this operation applies to both the DW and AW configurations. Figure 5 The corresponding operation result of the WA string [21331112] is displayed.

[0200] Leader bird processing: In the first iteration, the neighborhood structure of the left or right queue is randomly selected. In subsequent iterations, the corresponding neighborhood structure is selected according to the source queue of the leader bird.

[0201] In this embodiment of the invention, considering the unique V-shaped cluster structure of the original MBO algorithm, a joint neighborhood search strategy is also proposed. In this joint neighborhood search strategy, the FBs in the left queue neighborhood structure adopt the JS partial neighborhood structure, while the FBs in the right queue neighborhood structure use the WA partial neighborhood structure. There are two cases for the LB: (i) in the first iteration, the LB can choose either the JS partial or WA partial neighborhood structure; (ii) in subsequent iterations, the LB adopts the corresponding neighborhood structure based on its source queue (left queue or right queue).

[0202] This joint neighborhood search strategy has the following significant advantages:

[0203] (i) The neighborhood structure of the JS part and the WA part can be fully explored.

[0204] (ii) It can effectively promote collaborative cooperation among different neighborhood structures.

[0205] The joint neighborhood search strategy proposed in this invention aligns with the V-shaped cluster structure unique to the MBO algorithm. By configuring a neighborhood structure for the job sequence layer (JS) for the left queue follower birds (FBs) and a neighborhood structure for the workstation allocation layer (WA) for the right queue FBs, decoupling and specialized exploration of the search space are achieved. Simultaneously, a dynamic selection mechanism designed for the leader bird (LB) randomly selects the search direction in the first iteration to consider global exploration, while subsequent iterations lock the corresponding neighborhood structure based on the source queue to maintain search continuity, thus effectively balancing the algorithm's exploration and development capabilities. This strategy not only fully exploits the neighborhood potential of the JS and WA layers but also promotes synergistic effects between different neighborhood structures, significantly improving the algorithm's optimization efficiency and convergence stability in complex solution spaces. It provides strong local search support for solving NP-hard problems with strong two-layer coupling, such as 3T-RSSP-FPT.

[0206] To facilitate interaction between individuals in the left and right queue neighborhood structures, this embodiment of the invention introduces a cooperative sharing mechanism, the operation of which is as follows: Randomly select pairs of individuals (migratory birds) at the same position in the left and right queues, perform a crossover operation on them, and replace their parent individuals with the superior offspring individuals. This mechanism not only promotes cooperation between the two different types of neighborhood structures but also allows offspring individuals to inherit the superior characteristics of their parent individuals, thereby guiding the algorithm to explore the solution space that has not yet been fully explored. In this embodiment of the invention, the cooperative sharing mechanism includes:

[0207] Six differentiated crossover operators were designed to achieve a better balance between solution quality and solution efficiency. These operators were further divided into large-scale adjustment and small-scale adjustment categories, and the crossover operators were adaptively selected based on the evolutionary iteration stage.

[0208] Large-scale crossover operator (used for early-stage algorithm exploration, probabilistic) ):

[0209] CO1: Exchange the job sequence portion or workstation allocation portion of two individuals. For example, the child LB_Child inherits the JS portion [3 4 1 2] of the parent LB, and at the same time obtains the WA portion [2 3 1 1 2 1 2 2] from RB; the child RB_Child is generated in the same way.

[0210] CO2: Exchange the reassembly workshop configuration in the allocation section of the two individual workstations;

[0211] CO3: Exchange the dismantling workshop configuration in the allocation section of the two individual workstations;

[0212] in, Figure 6 A schematic diagram of a large-scale crossover operator in an embodiment of the present invention is shown. Figure 6 In this context, LB_Child refers to the child individuals or population of the left queue, while RB_Child refers to the child individuals or population of the right queue. Large-scale adjustment indicates a large-scale adjustment. The crossover operators CO1, CO2, and CO3 are specifically designed for large-scale crossover operations. Their functional areas are determined according to... Figure 6 The individual coding forms shown are used to distinguish them. Obviously, CO1, CO2, and CO3 cause significant disturbances to the population (bird flock), which helps the algorithm escape local optima.

[0213] Small-scale crossover operator (used in later algorithm development, probabilistic) ):

[0214] CO4 (Partially Mapping Crossover, PMX): Partially mapped crossover is used in the job sequence layer. It determines the crossover segment based on two random positions and establishes an element mapping table to avoid conflicts.

[0215] CO5 (Linear Order Crossover, LOX): Linear order crossover is used in the job sequence layer. It determines the crossover segment based on two random positions and maintains the relative order of the remaining elements.

[0216] CO6 (Multiple Position Crossover, MPX): Multi-position crossover is used for workstation allocation layers, selecting round (0.4×n) positions for exchange in both the disassembly and reassembly sections.

[0217] Figure 7The diagram illustrates small-scale crossover operators in an embodiment of the present invention. Operators CO4, CO5, and CO6 are specifically designed for small-scale crossover operations. On one hand, since the JS part uses workpiece sorting encoding, which is prone to element collisions, Partially Mapping Crossover (PMX) and Linear Order Crossover (LOX) are selected as CO4 and CO5, respectively. Both PMX and LOX require determining a continuous crossover segment based on two randomly selected positions (i.e., Figure 7 The difference between the two (the yellow block in the middle) lies in the rearrangement of the remaining elements: PMX requires the creation of an element mapping table to avoid conflicts. On the other hand, the WA portion is represented by a set of irregular values ​​constrained by the number of workstations (pp and qq), therefore Multiple Position Crossover (MPX) is chosen as CO6. For example... Figure 7 As shown, this operator selects three locations in both the disassembly and reassembly workshops for exchange, with "Small-scale adjustment" representing a small-scale adjustment. This invention defines the number of multiple locations in ECMBO as round(0.4×n) (where n is the number of scrapped products).

[0218] In this embodiment of the invention, the adaptive triggering strategy of the crossover operator based on the adaptive selection during the evolutionary iteration stage is as follows:

[0219] Clearly, compared to CO1, CO2, and CO3, CO4, CO5, and CO6 cause relatively less disturbance to the population. To fully utilize the aforementioned crossover operators, the trigger probabilities of the large-scale adjustment operator and the small-scale adjustment operator are respectively determined by […]. ]and[ The formula for calculating the trigger probability of control is as follows:

[0220]

[0221] in, This represents the current iteration index, where K is the maximum number of iterations. Early iterations focus on large-scale cross-referencing to quickly locate high-quality regions, while later iterations focus on small-scale cross-referencing for refined searching.

[0222] The collaborative sharing process is as follows: Each generation randomly selects pairs of individuals at the same position in the left and right queues (preferred). The crossover operation is performed, replacing the parent with a superior offspring. Therefore, in the early stages of ECMBO evolution, CO1, CO2, and CO3 have a higher execution probability, which helps the algorithm quickly locate potential optimal solution regions; in the later stages of evolution, the execution probability of CO4, CO5, and CO6 increases, which helps the algorithm refine its search and approach the optimal solution. Based on preliminary experimental results, this invention will use the probability [ ]and[ The value should be controlled within the range of [0.20, 0.80].

[0223] In this embodiment of the invention, when solving the fuzzy mixed integer linear programming model using the improved migratory bird migration optimization algorithm, the improved iterative optimization method of the migratory bird migration optimization algorithm includes:

[0224] S201, Leader Bird Evolution: Generate kk neighborhood solutions; if the best neighborhood solution is better than the current leader bird, replace it; sort the remaining neighborhood solutions by fitness and fill them into the left and right shared sets. and In this context, each shared set contains s solutions;

[0225] S202, following the bird's evolution: the left queue and the right queue follow the bird and update their own kk-ss neighborhood solutions with the ss solutions in the set shared by the previous bird;

[0226] S203, Implement reset mechanism check and collaborative sharing mechanism;

[0227] The reset mechanism sets a threshold of 3 for consecutive unimproved solutions. If a migratory bird fails to improve after 3 consecutive iterations, the current solution is abandoned, and a new solution is generated and replaced using a hybrid initialization strategy (NEH / random / load balancing). The reset mechanism effectively avoids premature convergence and getting stuck in local optima. Combined with the co-evolution and collaborative sharing of the V-shaped structure, it significantly improves population diversity and algorithm stability.

[0228] In the collaborative sharing mechanism, the information transmission mechanism is controlled by parameters kk (number of neighborhood solutions) and ss (number of shared solutions); the following bird receives the ss optimal neighborhood solutions that the preceding migratory bird has not used, thus realizing information sharing.

[0229] S204, Leader Bird Replacement: Move the current leader bird to the back of the formation and move the bird after it to the front of the formation;

[0230] S205, determine whether the current iteration count has reached the maximum iteration count K. If so, terminate the algorithm; otherwise, return to the bird-leading evolution step S201.

[0231] Example 2

[0232] In this embodiment of the invention, a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm is provided. The method specifically includes:

[0233] S10: Initialization

[0234] S101: Problem Parameter Initialization. Set all problem parameters required for a specific test instance, such as the number of scrapped products n, the number of reprocessing lines m, the number of disassembly workstations pp, and the number of reassembly workstations qq.

[0235] S102: Algorithm Parameter Initialization. Set all parameters required for the algorithm to run, including PS, kk, ss, G, K, and Flag. For easy reference, these parameters are marked with parentheses when they first appear.

[0236] S103: Population Initialization. An initial population initialization strategy is generated based on a hybrid strategy, producing an initial population of size PS. The best individual is selected as the leader bird (LB), and the remaining (PS-1) individuals are arranged in a V-shaped queue in an alternating left-right order.

[0237] S20: Evolution of the Circuit

[0238] Repeat the cycle process including steps S201 to S204 for a total of G times.

[0239] S201: Leader Bird Evolution. The leader bird generates kk neighborhood solutions based on its own queue (based on a joint neighborhood search strategy) and the neighborhood structures of the left and right queues. If the best solution among these neighborhood solutions is better than the current leader bird, the leader bird is replaced by the best neighborhood solution; otherwise, the leader bird remains unchanged. The remaining kk (or kk-1) neighborhood solutions are sorted in descending order of fitness and distributed sequentially to the two shared sets Ω. L and Ω R This continues until each set contains ss individuals.

[0240] S202: Evolved following the bird.

[0241] S2021: The left queue evolves following the bird. Each bird in the left queue generates kk-ss neighborhood solutions, forming a set. If the set Ψ∪Ω L If the optimal solution in the set is better than the current follower bird, then the follower bird is updated by the optimal solution, and the set Ω is cleared. L The s optimal solutions in the merged set will form a new Ω. L .

[0242] S2022: The right queue follows the bird's evolution. This process is similar to step S2021, and the specific details will not be repeated.

[0243] S203: Follower bird status check. If a follower bird fails to improve after several consecutive Flag iterations, a reset mechanism is executed.

[0244] S204: Implement the described collaborative sharing mechanism.

[0245] S30: Leader Bird Replacement

[0246] Move the lead bird to the back of the line, and move the bird behind it (which may be in the left or right line) to the front.

[0247] S40: Check if the termination condition K is met.

[0248] If yes, proceed to step S50; otherwise, return to step S20.

[0249] Step S50: The ECMBO algorithm terminates.

[0250] Output the optimal solution / scheduling scheme for the 3T-RSSP-FPT problem.

[0251] Example 3

[0252] This invention provides performance testing for generating initial populations using a hybrid strategy. The P5-C2-D2 / A2 instance (the smallest instance) is selected as the test object. The ECMBO without the hybrid initialization strategy is denoted as ECMBO-HIS, which stands for "ECMBO without Hybrid Initialization Strategy," meaning the benchmark comparison algorithm without the hybrid strategy has been removed. ECMBO, on the other hand, represents the improved migratory bird migration optimization algorithm provided in this invention, i.e., the complete improved algorithm proposed in this patent. Due to the maximum completion time C... max These are triplet parameters containing three different elements. Experimental results are presented in three dimensions, characterized by -L, -H, and -M. Specifically, -L, -H, and -M represent the optimistic, most conservative, and most likely scenarios, respectively.

[0253] in, Figure 8 The figure shows a comparison of box plots between ECMBO and ECMBO-HIS in an embodiment of the present invention. Figure 9 The graph shows a comparison of the probability densities of ECMBO and ECMBO-HIS in an embodiment of the present invention. Figure 8 and Figure 9Boxplot comparisons and probability density comparisons of ECMBO and ECMBO-HIS are presented respectively. Specifically: ECMBO-HIS-L: Test results of the ECMBO algorithm without a hybrid initialization strategy under optimistic conditions. ECMBO-L: Test results of the ECMBO algorithm with a hybrid initialization strategy under optimistic conditions. ECMBO-HIS-M: Test results of the ECMBO algorithm without a hybrid initialization strategy under the most probable condition. ECMBO-M: Test results of the ECMBO algorithm with a hybrid initialization strategy under the most probable condition. ECMBO-HIS-H: Test results of the ECMBO algorithm without a hybrid initialization strategy under the most conservative condition. ECMBO-H: Test results of the ECMBO algorithm with a hybrid initialization strategy under the most conservative condition. ECMBO-HIS-E: Results of the ECMBO algorithm without a hybrid initialization strategy on the expected optimal value dimension. ECMBO-E: Results of the ECMBO algorithm with a hybrid initialization strategy on the expected optimal value dimension.

[0254] On the one hand, Figure 8 In the results, ECMBO-HIS outperformed ECMBO across all three dimensions (-L, -M, -H), and even produced several outliers in the -M and -H dimensions, hindering algorithm convergence. The figure clearly shows that ECMBO generates superior individuals compared to ECMBO-HIS, validating the effectiveness of the introduced hybrid strategy. On the other hand, in... Figure 9 In the above analysis, the probability density distribution of ECMBO is superior to that of ECMBO-HIS. However, in the -H dimension, ECMBO-HIS exhibits a larger peak value than ECMBO, which is detrimental to the rapid optimization of the algorithm.

[0255] Example 4

[0256] A comparison of the ECMBO algorithm and the CPLEX solver shows that when the problem / instance size is small, the CPLEX solver can obtain the optimal solution for mixed integer linear programming (MILP) models, which can be used as a benchmark for algorithm comparison or model validation. Considering computational burden, P5-C2-D2 / A2 (5 scrapped products, 2 defective components, 2 disassembly workstations, 2 reassembly workstations), P10-C2-D2 / A2 (10 scrapped products, 2 defective components, 2 disassembly workstations, 2 reassembly workstations), and P15-C2-D2 / A2 (15 scrapped products, 2 defective components, 2 disassembly workstations, 2 reassembly workstations) were selected for CPLEX validation. It is well known that when the MILP model is too complex, CPLEX requires a significant amount of solution time. Therefore, a solution time cap of 10 minutes was set. In other words, when the solution time reaches this threshold, CPLEX outputs the current solution. However, the fuzzy mixed-integer linear programming model in this embodiment of the invention (corresponding to the model in Table 2) does not conform to the conventional MILP framework and cannot be directly solved using CPLEX. To use CPLEX on the fuzzy mixed-integer linear programming model, this paper inputs the three elements of the triangular fuzzy number (i.e., the most optimistic value Min (the most optimistic time of the triangular fuzzy number), the most likely value Mid (the most likely time of the triangular fuzzy number), and the most conservative value Max (the most conservative time of the triangular fuzzy number)) into CPLEX to obtain the corresponding results. Table 2 shows the comparison results of the ECMBO algorithm and the CPLEX solver based on the fuzzy mixed-integer linear programming model in this embodiment of the invention. The table includes indicators such as test instance, algorithm (including the CPLEX exact solver and the ECMBO algorithm of this invention), solution quality, solution efficiency, and CPU time (the central processing unit running time required for the algorithm to solve).

[0257] Table 2

[0258]

[0259] Table 2 shows that CPLEX obtains C max The values ​​are consistent with those obtained by ECMBO. This indicates that the proposed fuzzy mixed-integer linear programming model is feasible, and that the proposed metaheuristic algorithm ECMBO can find the true optimal maximum completion time. Furthermore, for smaller instances, CPLEX can find the optimal solution or optimal scheduling scheme within an acceptable time. However, as the instance size increases, CPLEX can find the optimal solution, but its solution efficiency is lower than that of ECMBO.

[0260] In summary, compared with the CPLEX solver, in small-scale instances such as P5-C2-D2 / A2 and P10-C2-D2 / A2, the ECMBO algorithm described in this invention can find optimal solutions consistent with the exact algorithm. However, in medium-to-large-scale instances such as P15-C2-D2 / A2, the solution time of CPLEX increases exponentially: from approximately 0.3 seconds for P5-C2-D2 / A2 to approximately 2 seconds for P10-C2-D2 / A2, and then to approximately 11-25 seconds for P15-C2-D2 / A2. In contrast, ECMBO offers a significant advantage in solution efficiency while maintaining equivalent solution quality (e.g., only approximately 0.03 seconds for the P15-C2-D2 / A2 example). Therefore, for large-scale 3T-RSSP-FPT problems, the metaheuristic algorithm ECMBO is far superior to the exact solver CPLEX in terms of solution efficiency. The above results demonstrate that the synergistic effect of the dual-layer encoding scheme, hybrid initialization strategy, joint neighborhood search strategy, cooperative sharing mechanism, and reset mechanism gives ECMBO a significant advantage in solution efficiency.

[0261] Compared to deterministic scheduling methods, this invention describes the uncertainty of processing time using triangular fuzzy numbers and establishes a fuzzy mixed integer linear programming model based on conventional operations. This model can handle the inherent uncertainty of the quality of recycled scrap products and external fluctuations in the production environment, generating feasible scheduling schemes with time buffers. The triplet representation of the most optimistic time, most likely time, and most conservative time allows production managers to determine the most suitable scheduling scheme based on the current operating conditions, improving the robustness and stability of scheduling in actual production.

[0262] In summary, this invention provides a fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm. This invention constructs a fuzzy mixed-integer linear programming model adapted to the special structure of 3T-RSSP, and accurately quantifies the fuzzy completion time based on the conventional operation rules of triangular fuzzy numbers, solving the problem of the disconnect between fuzzy modeling and the three-stage process. Simultaneously, addressing the shortcomings of the standard MBO algorithm, such as weak information interaction and susceptibility to local optima, the improved migratory bird migration optimization algorithm (ECMBO) accurately characterizes the strongly coupled logic of "job sequence-workstation allocation" through "JS-WA double-layer encoding," breaks down the information silos of the V-shaped queue using a "collaborative sharing mechanism," and introduces a reset mechanism to enhance the ability to escape local optima. This significantly improves the algorithm's global optimization capability and convergence stability in fuzzy environments, enabling it to output a high-quality scheduling scheme that includes workpiece sorting, machine allocation, and the maximum completion time in TFN form within an acceptable time, effectively meeting the efficient scheduling requirements of remanufacturing systems under uncertain environments.

[0263] It should be noted that, for the sake of simplicity, the foregoing embodiments are all described as a series of actions. However, those skilled in the art should understand that the present invention is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to the present invention. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to the present invention.

[0264] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit the scope of protection of the invention. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on these embodiments, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art can still combine, add, delete, or otherwise adjust the features of the various embodiments of the present invention according to the circumstances without conflict or creative effort, thereby obtaining different technical solutions that do not fundamentally depart from the concept of the present invention. These technical solutions also fall within the scope of protection of the present invention.

Claims

1. A fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm, characterized in that, The method includes: S10, Problem initialization and data input; Input the number of scrapped products n and the number of defective parts included in the scrapped products m; Enter the configuration for the three-stage remanufacturing system; Input the fuzzy processing time; S20. Establish a fuzzy mixed-integer linear programming model based on conventional triangular fuzzy number operations. Use TFN addition to calculate the fuzzy completion time, use maximum operation to determine the fuzzy start time of the task, and use sorting operation to compare the quality of solutions in order to minimize the maximum fuzzy completion time. To optimize the objective; S30, Solve the fuzzy mixed integer linear programming model based on the improved migratory bird migration optimization algorithm; S40: Obtain the optimal scheduling scheme obtained from the improved migratory bird migration optimization algorithm, and output the work ordering, machine allocation, and maximum completion time in TFN form. The optimal scheduling scheme.

2. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 1, characterized in that: In step S10, when inputting the three-stage remanufacturing system configuration, the input parameters include the number of disassembly workstations (pp) and the number of reprocessing lines. and the number of reprocessing workstations on each processing line l j The number of reassembled workstations is qq; When inputting fuzzy processing time, triangular fuzzy numbers are used to represent the processing time of each stage; The fuzzy processing time is expressed as t=(t a ,t b ,t c ), t a ,t b ,t c These represent the most optimistic time, the most likely time, and the most conservative time, respectively. The processing time during the disassembly stage is expressed as follows: The processing time in the reprocessing stage is expressed as follows: The processing time in the reassembly stage is expressed as: in, and It is the time scaling factor.

3. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 2, characterized in that: A method for solving the fuzzy mixed-integer linear programming model based on an improved migratory bird migration optimization algorithm includes: The encoding and decoding adopts a two-layer encoding scheme to represent the solution. The first layer is the work sequence layer, which represents the order in which scrapped products enter the remanufacturing system; the second layer is the workstation allocation layer, which represents the workstation allocation scheme for scrapped products in the disassembly and reassembly stages. Population initialization: An initial population is generated using a hybrid strategy; Iterative optimization: Population evolution is carried out through joint neighborhood search, cooperative sharing mechanism and reset mechanism. Among them, joint neighborhood search designs the neighborhood structure of the left and right queues differently according to the V-shaped queue structure. The cooperative sharing mechanism uses large-scale crossover operator and small-scale crossover operator triggered by adaptive probability to promote information interaction between individuals. The reset mechanism reinitializes continuously unimproved individuals. Termination decision: Output the optimal scheduling scheme after reaching the maximum number of iterations K.

4. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 3, characterized in that: The two-layer coding scheme includes: First layer: Job sequence layer, represented as {π1,...,π} i ,...,π n }, where π i This represents the scrap product number at the i-th processing location; Second layer: Workstation allocation layer, represented as {D1,...,D i ,...,D n ,A1,...,A i ,...,A n }, where D i ∈{1,2,...,pp} represents the disassembly workstation assigned to workpiece i, A i ∈{1,2,...,qq} represents the reassembly workstation assigned to workpiece i.

5. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 4, characterized in that: When using a hybrid strategy to generate the initial population, the hybrid strategy includes a heuristic initialization strategy based on the NEH rule, a random initialization strategy, and a load-balanced initialization strategy, with a population size of PS; Of these, 10% of the individuals were generated based on the heuristic initialization strategy of the NEH rule; 45% of the individuals were generated using a random initialization strategy; 45% of the individuals were generated using a load balancing initialization strategy.

6. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 4, characterized in that: The joint neighborhood search, based on the V-shaped queue structure, differentiates the neighborhood structures of the left and right queues, including: Left queue neighborhood structure: used to optimize job sequences; Right queue neighborhood structure: used to optimize workstation allocation; Leader bird processing: In the first iteration, the neighborhood structure of the left or right queue is randomly selected. In subsequent iterations, the corresponding neighborhood structure is selected according to the source queue of the leader bird.

7. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 6, characterized in that: The collaborative sharing mechanism includes: Six differentiated crossover operators are designed, and the crossover operators are adaptively selected based on the evolutionary iteration stage: Large-scale crossover operator: CO1: Exchange the job sequence portion or workstation allocation portion of two individuals; CO2: Exchange the reassembly workshop configuration in the allocation section of the two individual workstations; CO3: Exchange the dismantling workshop configuration in the allocation section of the two individual workstations; Small-scale crossover operator: CO4: Used in the job sequence layer, it determines the intersection segment based on two random positions and establishes an element mapping table to avoid conflicts; CO5: Used for the job sequence layer; CO6: Used for workstation allocation layer.

8. The fuzzy remanufacturing system scheduling method based on the improved migratory bird migration optimization algorithm as described in claim 7, characterized in that: The adaptive triggering strategy of the crossover operator, based on adaptive selection during the evolutionary iteration stage, is as follows: The formula for calculating the trigger probability is expressed as follows: in, This represents the current iteration index, and K is the maximum number of iterations.

9. The fuzzy remanufacturing system scheduling method based on an improved migratory bird migration optimization algorithm as described in claim 8, characterized in that: When solving the fuzzy mixed-integer linear programming model using the improved migratory bird migration optimization algorithm, the improved iterative optimization method includes: Leader bird evolution: Generate kk neighborhood solutions. If the best neighborhood solution is better than the current leader bird, replace it. Sort the remaining neighborhood solutions by fitness and fill them into the left and right shared sets. and In this context, each shared set contains s solutions; Evolution by following the bird: The left and right queues each update their solutions by combining their own kk-ss neighborhood solutions with the ss solutions in the set shared by the previous bird; Execution reset mechanism check and collaborative sharing mechanism; Leader bird replacement: Move the current leader bird to the back of the formation and move the bird after it to the front of the formation; Determine if the current iteration count has reached the maximum iteration count K. If so, terminate the algorithm; otherwise, return to the leader bird evolution step.

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