Three-stage remanufacturing blocking scheduling method based on machine degradation and maintenance strategy

Abstract

The invention belongs to the technical field of remanufacturing scheduling, and discloses a three-stage remanufacturing blocking scheduling method based on machine degradation and a maintenance strategy, which comprises the following steps: taking a rate regulation activity as a maintenance strategy of a reprocessing machine in a reprocessing stage, and constructing a degradation model considering a time-dependent degradation problem and the maintenance strategy at the same time; on the basis of a degradation model of a reprocessing machine, minimizing completion time is taken as a target, and constraints of a disassembling stage, constraints of a reprocessing stage and constraints of a reassembling stage are set respectively, so that a three-stage remanufacturing blockage scheduling model is constructed; and solving the three-stage remanufacturing blocking scheduling model by adopting a king butterfly optimization algorithm, outputting an optimal three-stage remanufacturing blocking scheduling scheme, and updating sub-populations by adopting a greedy acceptance strategy in the king butterfly optimization algorithm. According to the invention, a high-quality three-stage remanufacturing blocking scheduling scheme can be obtained.

Description

Technical Field

[0001] The present invention belongs to the technical field of remanufacturing scheduling, and particularly relates to a three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies. Background Art

[0002] A classic remanufacturing system consists of three core stages: disassembly, reprocessing, and reassembly. When a batch of end-of-life products enters the remanufacturing system, they are first disassembled into individual components in the disassembly workshop, and then these components are restored to a nearly new state through different operations in the reprocessing workshop. Finally, all reprocessed components are assembled into remanufactured products in the reassembly stage. Compared with single-stage or two-stage remanufacturing systems, the three-stage remanufacturing system is more complex and widely used because it needs to comprehensively optimize the performance of the entire system.

[0003] Machine blocking refers to the phenomenon that the workpiece remains on the current machine after the machine has completed the workpiece processing. Due to reasons such as buffer capacity limitations, technical limitations, and production characteristics, blocking phenomena frequently occur in actual scheduling. Blocking will significantly reduce the overall scheduling efficiency, so it is of great practical significance to study how to reduce or eliminate blocking phenomena in actual scheduling.

[0004] In traditional scheduling problems, the processing time is usually regarded as a fixed parameter. However, due to factors such as worker fatigue, tool misalignment, and machine deterioration, the processing time often changes in the actual production environment. Especially under the influence of time-dependent deterioration, the later the workpiece is processed, the longer its processing time will be. If not controlled, the time-dependent deterioration effect will gradually accumulate and have a negative impact on the processing efficiency. Therefore, decision-makers usually implement maintenance activities to alleviate the influence of time-dependent deterioration. Rate adjustment activities (RMAs) are a common maintenance activity. By implementing RMAs, the machine can be restored to an "almost new state" or a "less deteriorated state".

[0005] At present, the remanufacturing scheduling problem, especially the three-stage remanufacturing scheduling problem, has received extensive attention. Existing technologies have studied the integrated process planning and scheduling problem in the three-stage remanufacturing process and proposed an improved spider monkey optimization algorithm to solve this problem. In addition, considering that the reprocessed parts can be used for the reassembly of other products, a three-stage remanufacturing model considering part generality is proposed. Some scholars have also proposed a three-stage remanufacturing scheduling model based on batch flow production mode, taking the completion time and total energy consumption as optimization objectives. Or a scheduling problem of a three-stage remanufacturing system based on energy awareness is constructed, and the total energy consumption is minimized by adopting a power-on / off strategy. However, the above research on remanufacturing scheduling often ignores complex actual factors, especially the comprehensive influence of actual factors such as machine blocking and deterioration. Summary of the Invention

[0006] The object of the present invention is to provide a three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies, which can obtain a high-quality three-stage remanufacturing blocking scheduling scheme.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies, the three stages include a disassembly stage, a reprocessing stage and a reassembly stage, and the three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies includes:

[0009] Taking the rate adjustment activity as the maintenance strategy of the reprocessing machine in the reprocessing stage, and constructing a deterioration model that simultaneously considers the time-dependent deterioration problem and the maintenance strategy;

[0010] Based on the deterioration model of the reprocessing machine, with the goal of minimizing the makespan, and respectively setting the constraints of the disassembly stage, the constraints of the reprocessing stage and the constraints of the reassembly stage, so as to construct a three-stage remanufacturing blocking scheduling model;

[0011] Using the monarch butterfly optimization algorithm to solve the three-stage remanufacturing blocking scheduling model, and outputting the optimal three-stage remanufacturing blocking scheduling scheme, and the monarch butterfly optimization algorithm updates the sub-population using a greedy acceptance strategy.

[0012] A three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies provided by the present invention, in order to more effectively guide the remanufacturing scheduling in the real scenario, proposes a class of blocking scheduling problems that simultaneously consider time-dependent deterioration and take rate adjustment activities (RMAs) as the machine maintenance strategy. To solve the above problems, a deterioration model integrating RMAs is first proposed to determine the actual reprocessing time and the RMA execution strategy. On this basis, a new blocking scheduling model is constructed to minimize the makespan. And the monarch butterfly optimization algorithm is used to obtain a high-quality three-stage remanufacturing blocking scheduling scheme within a reasonable time. Brief Description of the Drawings

[0013] Figure 1 It is the overall layout of the three-stage remanufacturing system studied by the present invention;

[0014] Figure 2 It is the flowchart of a three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies of the present invention;

[0015] Figure 3 It is the Gantt chart of the solution of the remanufacturing process implementing RMAs and the Gantt chart of the solution of the remanufacturing process without implementing RMAs in the example of the present invention;

[0016] Figure 4 This is the flowchart of using the monarch butterfly optimization algorithm to solve the three-stage remanufacturing blocking scheduling model in the present invention;

[0017] Figure 5 This is the schematic diagram of the individual represented by the three-layer structure proposed in the present invention;

[0018] Figure 6 This is the schematic diagram of the product change assigned to the reassembly machine before and after the present invention executes the MLB strategy;

[0019] Figure 7 This is an example schematic diagram of the present invention performing type-I exchange mutation on the individuals of sub-population 1;

[0020] Figure 8 This is an example schematic diagram of the present invention performing type-II exchange mutation or insertion mutation on the first layer of the individuals in sub-population 2;

[0021] Figure 9 This is the C of each parameter at different levels in the experiment of the present invention max Average value;

[0022] Figure 10 This is the box plot of five algorithms in the present invention on medium and large test instances. Detailed implementation manners

[0023] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0024] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments, and are not intended to limit the present invention.

[0025] Figure 1The overall layout of the three - stage remanufacturing system studied in the present invention is shown. It consists of a disassembly workshop, a reprocessing workshop, and a reassembly workshop. In the disassembly workshop, parallel disassembly machines (DWs, also known as disassembly workstations) are set up. In the reprocessing workshop, an assembly - line - type reprocessing line (RLs) is set up, and reprocessing machines (RWs) are set on each reprocessing line. In the reassembly workshop, parallel reassembly machines (AWs, also known as reassembly workstations) are set up. Different colors are used to distinguish end - of - life (EOL) products with different defect states, and different geometric shapes are used to represent different parts of the products. First, a batch of end - of - life products with similar structures and remanufacturing value enter the system. In the first stage, these products are disassembled into corresponding parts. It should be noted that the disassembly tasks can be assigned to any one of the parallel disassembly machines. In the second stage, a series of reprocessing operations are carried out on the parts on the parallel assembly - line - type reprocessing lines. It is worth noting that different reprocessing lines are equipped with a series of specific reprocessing machines to process specific types of parts. In the third stage, the reprocessed parts are reassembled into remanufactured products. Similar to the disassembly workstations, the reassembly tasks can be assigned to any one of the parallel reassembly machines.

[0026] The problem description is as follows: Currently, there are I end - of - life products, each product consists of J parts, and they need to be assigned to the remanufacturing system for processing. The system consists of G parallel DWs, J parallel dedicated RLs, and R parallel AWs. The j - th RL contains K j reprocessing machines. Assume that the buffer capacity between reprocessing machines is zero, that is, when the adjacent next machine is unavailable, even if the part has completed the reprocessing operation, it cannot be released from the current machine (i.e., it will be blocked on the current machine). In addition, to solve the problem of time - dependent deterioration, each reprocessing machine can perform a rate - adjustment activity (the rate - adjustment activity is used as a maintenance strategy, RMA) after processing a part, and the operation time of each RMA is fixed. During the part reprocessing or when the part is blocked, the RMA cannot be executed. Executing the RMA may affect the operation of the previous machine because the part may be blocked on the previous machine, making that machine unavailable before the RMA is completed.

[0027] The method of the present invention first determines the execution strategy of the rate - adjustment activity. On this basis, it assigns disassembly machines to end - of - life products and determines the disassembly order of end - of - life products on each disassembly machine, then determines the processing order of parts in the reprocessing workshop, and finally assigns reassembly machines to remanufactured products and determines the reassembly order of remanufactured products on each reassembly machine to minimize the makespan.

[0028] In addition, the following assumptions need to be met: all machines are in a normal and available state initially. The same machine cannot process two or more parts or products simultaneously, and each part or product can only be processed on a specific machine once. Once the disassembly, reprocessing, and reassembly operations start, they cannot be interrupted in the middle. The setup time is independent of the operation sequence. The transportation time is ignored.

[0029] As Figure 2 shown, the three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy in this embodiment includes the following steps:

[0030] Step 1: Take the rate adjustment activity as the maintenance strategy of the reprocessing machines in the reprocessing stage, and construct a deterioration model that simultaneously considers the time-dependent deterioration problem and the maintenance strategy.

[0031] To determine the actual reprocessing time and the execution strategy of RMAs, this embodiment proposes a deterioration model integrating RMAs. It should be noted that each reprocessing machine needs to reprocess I products. Considering the extreme case, that is, RMA is executed after each component is reprocessed. Since executing RMA after the reprocessing of the last component does not affect the completion time, at most I - 1 times of RMA can be executed on each reprocessing machine. Here, I - 1 represents the maximum number of RMAs that each RM may execute, because in most cases, it is not necessary to execute I - 1 times of RMA.

[0032] The actual processing time of the workpiece is usually expressed as a linear function of the interval between the end time of the most recent RMA (or the start time of the first job in this RM) and the start processing time of this workpiece. In this embodiment, the actual reprocessing time of the parts is also defined in a similar way, as shown in Equation (1).

[0033]

[0034] Obviously, for those parts that are first reprocessed on the RM jk or are reprocessed immediately after RMAs, the actual reprocessing time is equal to the expected reprocessing time because is zero. Otherwise, due to the deterioration factor, the actual reprocessing time of the parts will exceed the expected reprocessing time. Calculated by formula (2).

[0035]

[0036] Where is the completion time of the RMA jk0 , that is, the assumed 0th RMA on each RM jk . By assuming RMA jk0For any part to be reprocessed on the RM, there will always be a nearest RMA for mathematical modeling of the actual reprocessing time of the part. Equal to the reprocessing start time of the part first reprocessed on the RM jk as shown in formula (3):

[0037]

[0038] Formula (4) ensures that the completion times of potential RMAs are meaningful only when they are actually executed:

[0039]

[0040] λ jkl and λ ij ′ kl are related as shown in formula (5):

[0041]

[0042] Formulas (6) and (7) specify the number of RMAs to be executed on each RM. Formula (8) ensures that there is one and only one nearest RMA to any part to be reprocessed on the RM before the part is reprocessed on the RM.

[0043]

[0044] In the extreme case, RMAs can be executed after each part is reprocessed, but this may lead to significant waste of resources. Therefore, as shown in formula (9), RMAs are executed on the RM jk only when the reprocessing time of the part to be reprocessed on the RM due to deterioration is greater than t RMA . Then, the part will be blocked on the previous RM jk until the RMA is completed. Formula (10) further ensures that for parts reprocessed on the RM, the reprocessing time due to deterioration is less than t j(k-1) RMA RMA .

[0045]

[0046]

[0047] Step 2: Based on the deterioration model of the reprocessing machine, with the goal of minimizing the makespan, and setting the constraints of the disassembly stage, the reprocessing stage, and the reassembly stage respectively, a three-stage remanufacturing blocking scheduling model is constructed.

[0048] (1) Objective function:

[0049]

[0050] (2) Constraints in the disassembly stage:

[0051]

[0052]

[0053] Equation (12) ensures that each scrapped product can only be disassembled on one DW. Equations (13) and (14) specify the disassembly order of the scrapped products. Equation (15) stipulates that when two scrapped products are assigned to the same DW, they should be disassembled in sequence.

[0054] (3) Constraints in the reprocessing stage:

[0055]

[0056]

[0057]

[0058] Equation (16) stipulates that the reprocessing of a product must start after the disassembly of the product is completed. Equations (17) and (18) specify the relationship between the departure time and the completion time of the parts by considering the blocking effect. Equation (19) explains that the start time of the reprocessing of a part on the RM is equal to its departure time on the previous RM. Equation (20) ensures the time relationship between the parts that need to be reprocessed on the same RM by considering the blocking effect. Equation (21) stipulates the time relationship between the RMA and the part in front of it. Equation (22) stipulates the reprocessing order between the parts.

[0059] (4) Constraints in the reassembly stage:

[0060]

[0061]

[0062]

[0063] Equation (23) stipulates that each remanufactured product can only be reassembled on one AW. Equation (24) stipulates the reprocessing and reassembly order of the product. Equation (25) stipulates that two remanufactured products cannot be reassembled on one AW at the same time. Equations (26) and (27) stipulate the reassembly order of the product.

[0064] In addition, the constraints of the binary variables are shown in Equations (28)-(31):

[0065]

[0066]

[0067]

[0068] In this embodiment, the definitions of the parameters and variables are as follows: i and i' are product indices and i, i' ∈ {1,..., I}; j is a part / RL index, j ∈ {1,..., J}, and since there is a corresponding relationship between parts and the reprocessing line, the same index is used; g is a DW index, g ∈ {1,..., G}; k is an RM index, k ∈ {1,..., K j}; k is an RM index, k ∈ {1,..., K j}; r is an AW index, r ∈ {1,..., R}; l is an RMA index, l ∈ {0, 1,..., I - 1}; P i 、P i′ is the i-th or i'-th scrapped product or remanufactured product. In this embodiment, the product is called a scrapped product in the disassembly stage and the reprocessing stage, and a remanufactured product in the reassembly stage; C ij is the j-th part of P i ; DW g is the g-th DW; RL j is the j-th RL; RM jk is the k-th RM of RL j ; AW r is the r-th AW; RMA jkl is the l-th RMA of RM jk ; t RMA is the execution time (duration) of RMA; M is an infinitely large positive number; α is the deterioration rate of the machine; is the disassembly time (duration) of P i ; is the expected reprocessing time (duration) of C ij on RM jk ; is the reassembly time of P i ; C max is the completion time (moment) of the remanufacturing schedule; is the actual reprocessing time (duration) of C ij on RM jk ; FT i D is the completion time (moment) of the disassembly of P i ; is the start time (moment) of the reprocessing of C ij on RM jk ; is the RMA jklCompletion time (at a moment); is C ij In RM jk Completion time of reprocessing on it (at a moment); FT i A is P i Completion time of reassembly; η ig Is a decision variable, expressed as if P i Is disassembled on DW g Then η ig = 1; Otherwise η ig = 0; η i ′ i'g Is a decision variable, expressed as on DW g If P i Is disassembled before P i′ Then η' ii'g = 1; Otherwise η' ii' = 0; λ jkl Is a decision variable, expressed as if RMA jkl Is executed on RM jk Then λ jkl = 1; Otherwise λ jkl = 0; λ ij ′ kl Is a decision variable, expressed as before reprocessing C on RM jk If RMA ij Is the most recently executed RMA, then λ' jkl = 1; Otherwise λ' ijkl = 0; γ ijkl Is a decision variable, expressed as on RM ii′jk If C jk Is reprocessed before C ij Then γ i′j = 1; Otherwise γ ii′jk = 0; μ ii′jk Is a decision variable, expressed as if P ir Is reassembled on AW i Then μ r = 1; Otherwise μ ir = 0; μ ir ′ i ′ i'r Is a decision variable, expressed as on AW r If P i Is reassembled before P i′ Then μ' ii'r = 1; Otherwise μ' ii'r = 0.

[0069] This embodiment illustrates the advantages of RMA for remanufacturing scheduling with an example:

[0070] Suppose there are three scrapped shock mounts, denoted as P1, P2, and P3 respectively. Among them, two related parts - the plate and the bushing - are evaluated to have the value of remanufacturing after assessment. The remanufacturing operation process for the plate is cleaning → laser processing → general milling, and the remanufacturing process for the bushing is cleaning → laser processing. For this purpose, two DWs and two AWs are provided. In the reprocessing workshop, RL1 has a cleaning machine (RM 11 ), a laser machine (RM 12 ) and a CNC milling machine (RM 13 ), and RL2 has a cleaning machine (RM 21 ) and a laser machine (RM 22 ). The deterioration rate of the machine is set to 0.1, and t RMA is set to 5, and other parameters are shown in Table 1.

[0071] A feasible solution is as follows: In the disassembly stage, disassemble P2 and P1 on DW1, and the disassembly order is P2 → P1, and disassemble P3 on DW2; the order in the reprocessing stage is P2 → P3 → P1; in the reassembly stage, reassemble P2 and P1 on AW1, and the reassembly order is P2 → P1, and reassemble P3 on AW2. If the reprocessing time caused by deterioration is greater than t RMA , then execute RMAs.

[0072] Parameter values of the example in Table 1

[0073]

[0074]

[0075] To visually show the impact of RMA on the scheduling results, in this embodiment, two Gantt charts are respectively drawn for the solutions considering the execution of RMA, as shown in Figure 3 . Among them, the C max value of the solution of the remanufacturing process with RMAs executed is 227.8, which is better than the C max value of 230.98 of the solution of the remanufacturing process without RMAs executed, indicating that the RMA execution strategy adopted in this embodiment is effective. It should be noted that the number of scrapped products in this example is small. As the number of scrapped products further increases, if an appropriate RMA strategy is not adopted, the negative impact of deterioration on the scheduling efficiency will continue to increase. Thus, it can be seen that RMA is of great significance for improving the remanufacturing scheduling efficiency.

[0076] This example proposes a three-stage remanufacturing system blocking scheduling problem, considering time-dependent degradation and RMA execution strategy. To solve this problem, a degradation model integrating RMAs is proposed to describe the dynamic changes of actual processing time and determine the location and number of RMAs on each machine during the remanufacturing process. In addition, a new blocking scheduling model is established to minimize the completion time by considering the time waste caused by blocking.

[0077] Step 3: Use the monarch butterfly optimization algorithm to solve the three-stage remanufacturing congestion scheduling model and output the optimal three-stage remanufacturing congestion scheduling solution. Figure 4 As shown, the specific steps are as follows:

[0078] Step 3.1, set the population size to N. To achieve a balance between the quality and diversity of the solution, use the constructive heuristic method and machine load balancing strategy to initialize N / 2 individuals, and randomly initialize the remaining N / 2 individuals. Each individual represents a three-stage remanufacturing blocking scheduling scheme, that is, each monarch butterfly individual represents a feasible scheduling solution.

[0079] In order to encode the solution including product / part sequence and workstation allocation, the individuals in this embodiment are represented by a three-layer structure. The individuals can be represented as The first, second and third layers can be expressed as π={π1,π2,...,π I}、ψ={ψ1,ψ2,...,ψ I} and ξ={ξ1,ξ2,...,ξ I}. Element π h , h , h (h∈{1,...,I}) represent product index, DW index and AW index respectively. h is a non-repeating integer selected from {1,...,I}, where ψ in ψ h and ξ in h are integers selected from {1,...,G} and {1,...R} respectively. The elements in the first and second layers with corresponding positions indicate that the product is disassembled on the disassembly machine. If the elements with the same value in the second layer correspond to multiple elements in the first layer, it means that multiple products are disassembled on the disassembly machine, and the products with the front position are disassembled first; the elements in the first and third layers with corresponding positions indicate that the product is reassembled on the reassembly machine. If the elements with the same value in the third layer correspond to multiple elements in the first layer, it means that multiple products are reassembled on the reassembly machine, and the products with the front position are reassembled first.

[0080] Figure 5 Give an example of understanding, namely The decoding process is as follows: In the disassembly stage, P5 and P4 are disassembled on DW2, P3 and P2 are disassembled on DW1, and P1 is disassembled on DW3. The disassembly sequences at the disassembly stations are P5→P4 and P3→P2 respectively. In the reassembly stage, P3, P2, and P1 are reassembled on AW2, and P5 and P4 are reassembled on AW1 and AW3 respectively. In addition, considering the connection between the three stages, the first-come-first-served (FCFS) heuristic method is adopted to determine the order of components in the reprocessing stage and the order of remanufactured products in the reassembly stage. Due to the specificity of RL, the allocation of RL does not need to be considered. Only when the reprocessing time caused by deterioration exceeds t RMA will RMA be executed.

[0081] The quality of the initial solution has a huge impact on the performance of the metaheuristic algorithm. Compared with the completely random initialization method, the initialization method based on specific problems can often achieve better results within an acceptable time. Therefore, this embodiment proposes a new constructive heuristic method and MLB strategy to generate high-quality initial solutions.

[0082] A. First, use the constructive heuristic method to generate the first and second layers of individuals.

[0083] Sort the I EOL products according to specific sorting rules and represent them as a sequence θ = {P θ(1) ,..., P θ(G) ,..., P θ(I)}. P θ(h) represents the h-th product in the sequence θ, where h ∈ {1,..., I}. Secondly, assign the first G products in the sequence θ to G parallel DWs in turn. Then, for the remaining products in the sequence θ, from P θ(G+1) to P θ(I) , check all possible insertion positions of this product on the DWs in turn, and calculate the maximum departure time in the reprocessing stage caused by the current insertion position Finally, insert P θ(h) into the position that makes the maximum departure time the smallest. The four sorting rules used in this embodiment are as follows. Select one to execute:

[0084] (1) LTT (Longest Total Time) rule: Sort the products in non-increasing order according to the total processing time TT i of the products. The calculation formula is shown in formula (32):

[0085]

[0086] (2) STT (Shortest Total Time) rule: Sort the products in non-decreasing order according to the total processing time TT i of the products.

[0087] (3) LSDT (Long - Short Disassembly Time) rule: First, sort the products according to their disassembly times in non - increasing order, and then iteratively select the first product and the last product from the sorted sequence until all products are selected. This rule attempts to reduce idle time by assigning products with significantly different disassembly times to different DWs. For example, if the products are sorted in non - increasing order of disassembly time as 123456, the final sorting obtained is 162534.

[0088] (4) LSTT (Long - Short Total Time) rule: The only difference between this rule and the LSDT rule is that it sorts the products in non - increasing order according to their total times rather than disassembly times. That is, first sort the products according to their total processing times in non - increasing order, and then iteratively select the first product and the last product from the sorted sequence until all products are selected to generate the final sorting.

[0089] B. Secondly, adopt the machine load balancing strategy to generate the third layer of the individual.

[0090] Through the above - mentioned constructive heuristic method, the allocation scheme of DWs and the disassembly order of products in each initial solution can be determined. However, in the third layer of the solution, if the elements representing AW allocation are randomly generated, it may reduce the quality of the solution and the convergence speed of the algorithm. In some extreme cases, most of the remanufactured products may be assigned to the same AW, resulting in other AWs being completely idle. Therefore, this embodiment proposes a machine load balancing (MLB) strategy to balance the load of AWs, and its main steps are as follows:

[0091] (1) Randomly generate the elements ξ of the third layer from the range of reassembly machine indices {1,..,R} in sequence h , h ∈ {1,...,I}, where R is the total number of reassembly machines and I is the total number of products.

[0092] (2) Calculate the total reassembly time of each reassembly machine, and denote the reassembly machine with the maximum total reassembly time as AW max , and denote the reassembly machine with the minimum total reassembly time as AW min ; if the total reassembly times of multiple AWs are the same, randomly select one of them.

[0093] (3) If AW max has assigned the most products and AW min has assigned the fewest products, or if AW max has assigned the fewest products and AW min has assigned the fewest products, then execute step (4); otherwise, end the process, output the third layer, and complete the individual initialization;

[0094] (4) Randomly select a product from AW max and assign it to AW min , and return to step (2).

[0095] The following is an example of implementing the MLB strategy. Initially, P1, P3, P5, P6 are assigned to AW1, and P2, P4 are assigned to AW3. The reassembly times of P1, P2, P3, P4, P5, P6 are 35, 25, 38, 37, 30, 32 respectively. As Figure 6 shown, after implementing the MLB strategy, the number of products assigned to AW1 is reduced from 4 to 2. Obviously, the utilization rate of AW in the reassembly workshop has been improved.

[0096] Step 3.2: Calculate the fitness value of each individual in the population. Since a smaller C max value represents a better solution, in this embodiment, the fitness value is taken as the reciprocal of the minimum completion time C max .

[0097] Step 3.3: Sort the entire population in non-increasing order based on the fitness value, and divide the population into the first sub-population (sub-population 1) and the second sub-population (sub-population 2) according to the sorting result. The fitness value of individuals in the first sub-population is greater than that of individuals in the second sub-population. Let sub-population 1 be the first N1 better individuals, and sub-population 2 be the remaining N - N1 individuals. N1 = Ceil(p·N), where p represents the proportion of butterflies from sub-population 1. Ceil(p·N) represents the smallest integer not less than p·N.

[0098] Step 3.4: Execute the migration operator on the first sub-population and update the first sub-population using the greedy acceptance strategy; execute the adjustment operator on the second sub-population and update the second sub-population using the greedy acceptance strategy.

[0099] In each iteration of the basic monarch butterfly optimization algorithm (MBO), each element of each butterfly in sub-population 1 is updated through the migration operator. However, since the quality of most individuals in sub-population 1 is relatively high, modifying elements at three levels simultaneously may lead to a decrease in the quality of most individuals. To reduce excessive perturbation, the following improvements are made to the migration operator in this embodiment.

[0100] First, perform type-I exchange mutation on each individual in sub-population 1. Specifically, based on the elements of the first and second layers, randomly select a disassembly machine, and it is required that there are two or more products being disassembled on this disassembly machine. Subsequently, randomly select two products on this disassembly machine and exchange their positions in the first layer. As Figure 7As shown, select the disassembly machine with index 2. The products being disassembled on this disassembly machine are products 4 and 5. Therefore, swap the positions of products 4 and 5.

[0101] Secondly, update the elements of the third layer using formula (33).

[0102]

[0103] Among them, represents the h-th element of the third layer of the s-th individual in sub-population 1 in the (t + 1)-th generation. and respectively represent the h-th element of the third layer of the s1-th individual and the s2-th individual in sub-population 1 in the t-th generation, where s1 and s2 represent randomly selected individuals in sub-population 1 and sub-population 2 respectively. p is a parameter representing the proportion of butterflies from sub-population 1. r = rand * peri, where rand is a random number generated from [0, 1], and peri is the migration period.

[0104] Finally, apply the greedy acceptance strategy to each individual in sub-population 1. Calculate the fitness value of each individual after the migration operation. If the fitness value after migration is higher than the fitness value before migration, select the updated individual to enter the next generation; otherwise, retain the original individual to enter the next generation.

[0105] Different from sub-population 1, the individual quality of sub-population 2 is relatively low. Therefore, the improved butterfly adjustment operator adopts a large-range element perturbation strategy to efficiently search the solution space.

[0106] First, randomly perform type-II exchange mutation or insertion mutation on the first layer of each individual in sub-population 2 to change the product sequence, as Figure 8 (a) and Figure 8 (b) shown. For type-II exchange mutation, randomly select two different elements 3 and 1 and swap their positions. For insertion mutation, randomly select element 4 and insert it into a random position between elements 5 and 3.

[0107] Secondly, update the elements of the second layer using formula (34) and round them:

[0108]

[0109] Among them, represents the h-th element of the second layer of the s'-th individual in sub-population 2 in the (t + 1)-th generation. represents the h-th element of the second layer of the best individual in the entire population in the t-th generation. It represents the h-th element of the second layer of the s2-th individual randomly selected from the sub-population 2. BAR is the adjustment rate. λ is set to 1.5 in the Lévy flight. ε represents the weight coefficient, and its definition is shown in formula (35):

[0110] ε = S max / t 2 (35)

[0111] where S max is the maximum step size, which is set to 1 in this embodiment.

[0112] Then, the same update operation as that of the second layer is performed on the elements of the third layer, that is, the elements of the second layer are updated using formula (34) and rounded off.

[0113] Finally, the greedy acceptance strategy is applied to each individual in the sub-population 2.

[0114] Step 3.5: Merge the updated first sub-population and the second sub-population as the new population. If the iteration end condition (the end condition can be reaching the maximum number of iterations) is not met, return to Step 3.3 to continue the iteration; otherwise, output the individual with the largest fitness value in the new population as the optimal three-stage remanufacturing blocking scheduling scheme.

[0115] In this embodiment, based on the monarch butterfly optimization algorithm, a constructive heuristic method based on the problem characteristics and a new machine load balancing strategy are introduced to generate high-quality initial solutions. In addition, two improved operators adapted to the solution representation mechanism are designed to achieve a comprehensive search of the solution space.

[0116] To demonstrate the advantages of the improved monarch butterfly optimization (MMBO) algorithm proposed in the present invention, the following related experiments are conducted. First, experiments are carried out to determine the best algorithm parameter combination and sorting rule to optimize the performance of the algorithm as much as possible. Then, the branch and bound solver GUROBI is called to solve the proposed model, and the results are compared with those of the MMBO algorithm. Next, the MMBO algorithm is compared with other benchmark algorithms to verify its superiority. The MMBO algorithm is written in the Python language, and the experiments are run on a PC with a 3.20GHz AMD Ryzen 7 CPU, 16GB RAM, and a Windows 11 64-bit system.

[0117] I. Experimental design.

[0118] In this embodiment, a method of randomly generating instances is used to test the comparison algorithm. In the generation of instances, different levels of values are given to four parameters, namely the number of products, the number of parts / RLs, the number of DWs, and the number of AWs, which are respectively \(I\in\{5, 10, 15, 20, 30\}\), \(J\in\{3, 4, 5\}\), \(G\in\{3, 4, 5\}\), \(R\in\{3, 4, 5\}\). Then, they are combined with each other to form different test instances, and the test instance can be expressed as \(t_i(I / J / G / R)\). For each test instance, the \(\alpha\) of all RMs is set to 0.1, and \(t\) RMA is set to 8. \(K\) j 、 The values of are respectively taken as integers randomly generated from \([2, 5]\), \([25, 50]\), \([10, 30]\), \([25, 50]\).

[0119] In this experiment, the relative percentage deviation (RPD) is used to evaluate the experimental results of each test instance, and its calculation formula is as follows:

[0120]

[0121] In the formula, \(C\) max represents the makespan obtained by a certain algorithm on a specific test instance, represents the best \(C\) obtained by all comparison algorithms on the same test instance max .

[0122] II. Algorithm parameter combination.

[0123] The parameters used in the MMBO algorithm include the migration period peri, the migration ratio p, the butterfly adjustment rate BAR, the population size N, and the maximum number of iterations \(N\) it . To determine the best combination of peri, p, and BAR, the Taguchi method is used to grade different values of the parameters, as shown in Table 2. In this experiment, \(t_i(20 / 3 / 4 / 4)\) is used as the test instance, and the algorithm is independently run 5 times on this instance. To ensure the convergence of the algorithm, N is set to 40, and \(N\) it is set to 800. In addition, only the LTT sorting rule is selected for testing.

[0124] Table 3 shows the average value of \(C\) for different parameter level combinations. max Table 4 shows the average value of \(C\) for each parameter at different levels. max The larger the range value, the greater the impact of the parameter on the algorithm performance. From Figure 9 it can be seen that Figure 9 in the broken line from left to right correspond to the parameters peri, p, and BAR respectively. When the levels of peri, p, and BAR are 1, 2, and 1 respectively, the MMBO algorithm performs the best, and the corresponding parameter values are 1, 1 / 2, and 5 / 12 respectively. Therefore, this parameter combination will be used in subsequent experiments.

[0125] Table 2 Classification of Different Parameter Values

[0126] Parameter Parameter Level 1 Parameter Level 2 Parameter Level 3 peri 1 65 75 p 512 1 / 2 712 BAR 512 1 / 2 712

[0127] Table 3 C of Different Grade Combinations max Average Value

[0128] Level Combination peri p BAR <![CDATA[Average C max > C1 1 1 1 366.96 C2 1 2 2 367.63 C3 1 3 3 373.44 C4 2 1 3 370.39 C5 2 2 1 367.98 C6 2 3 2 372.06 C7 3 1 2 370.31 C8 3 2 3 368.13 C9 3 3 1 370.45

[0129] Table 4 C of Parameters at Different Grades max Average Value

[0130] Level peri p BAR 1 369.34 369.22 368.46 2 370.14 367.91 370.00 3 369.63 371.98 370.65 Range Value 0.80 4.07 2.19

[0131] III. Evaluation of the Constructive Heuristic Method under Different Sorting Rules

[0132] To determine the best sorting rule for the improved NEH-based heuristic method, this experiment compared four sorting rules by running test instances. The population size was set to 40, and the maximum number of iterations was set to 800. Each test instance was run independently 5 times. Table 5 records the best relative percentage deviation (RPD) values and the average RPD values. The MMBO algorithms using the four sorting rules are denoted as MMBO_LTT, MMBO_STT, MMBO_LSDT, and MMBO_LSTT, respectively.

[0133] As shown in Table 5, for small test instances with 5 products and medium test instances with 10 products, the MMBO algorithms with the four sorting rules all obtained the optimal or near-optimal best RPD. As the size of the test instances increased, in terms of the best RPD, MMBO_LTT and MMBO_LSTT performed relatively poorly, while MMBO_STT and MMBO_LSDT had mixed performances. In terms of the average RPD, MMBO_STT and MMBO_LSDT still had mixed performances on different test instances. However, MMBO_LSDT performed optimally in terms of the sum of the best RPD and the average RPD. Therefore, MMBO_LSDT was selected as the benchmark algorithm for subsequent experiments.

[0134] Table 5 Comparison of MMBO Using Different Sorting Rules

[0135]

[0136] Note: There is a scientific notation symbol e-02 after the values in the table.

[0137] IV. Model Evaluation

[0138] The GUROBI solver is called to solve the problem to verify the effectiveness of the model. Meanwhile, the results obtained by the GUROBI solver are compared with those obtained by the MMBO_LSDT algorithm. The degradation model and the three-stage remanufacturing blocking scheduling model proposed in the present invention can be simplified as:

[0139] min f = C max

[0140] s.t. x ∈ X

[0141] where x is the decision variable vector and X is the feasible region defined by formulas (1)-(31). This model is directly solved using the GUROBI solver. The experiments are conducted on three small test instances (ti(5 / 3 / 3 / 3), ti(5 / 4 / 4 / 4), ti(5 / 5 / 3 / 4)) and three medium test instances (ti(10 / 3 / 4 / 4), ti(10 / 4 / 3 / 4), ti(10 / 5 / 5 / 4)). The maximum CPU time of the GUROBI solver and the MMBO_LSDT algorithm is uniformly set to 1200 seconds.

[0142] Table 6 records the C max values obtained by the GUROBI solver and the MMBO_LSDT algorithm. For the small test instances, the GUROBI solver can obtain the optimal C max value within a relatively short CPU time. However, for the medium test instances, the GUROBI solver fails to obtain the optimal C max value within 1200 seconds for ti(10 / 3 / 4 / 4) and ti(10 / 5 / 5 / 4). In contrast, for both small-scale and medium-scale test instances, the MMBO_LSDT algorithm can obtain the optimal or near-optimal C max value within a reasonable CPU time. Therefore, the GUROBI solver is suitable for solving small-scale problems but not for larger-scale problems. The MMBO algorithm proposed in the present invention is suitable for solving both small-scale and large-scale problems.

[0143] Table 6 C max values obtained by the GUROBI solver and the MMBO_LSDT algorithm within 1200 seconds

[0144]

[0145] Note: The symbol ' / ' in the table indicates no output within the limited CPU time.

[0146] V. Algorithm evaluation.

[0147] To evaluate the effectiveness and superiority of the MMBO algorithm, this experiment compared it with other benchmark algorithms. The selected benchmark algorithms include the Improved Flower Pollination Algorithm (IFPA), the Improved Genetic Algorithm (IMGA), the Hybrid Genetic Algorithm Based on Variable Neighborhood Search (GAVNS), and the Improved Artificial Bee Colony Algorithm (IABC). The reasons for selecting the above algorithms are as follows: First, the problem solved in this experiment is a discrete single-objective problem, so the benchmark algorithms should be designed specifically for solving such problems to avoid excessive modification; Second, the problems processed by the benchmark algorithms should be related to remanufacturing scheduling to ensure a fair comparison within a similar application field; Third, all benchmark algorithms are advanced algorithms whose effectiveness has been verified.

[0148] According to the preliminary experiment, the population size and the maximum number of iterations of all algorithms were set to 40 and 800 respectively. Other parameters were set according to the suggestions in their respective literatures. Each algorithm was independently run 5 times on each test instance.

[0149] The experimental results are recorded in Table 7 and the continued Table 7, where Best, Avg, and Var represent the best RPD value, the average RPD value, and the variance of the RPD value respectively. MMBO_LSDT obtained the best RPD value on all test instances, and in medium and large test instances, no benchmark algorithm could achieve the same best RPD value as MMBO_LSDT. This shows that the MMBO_LSDT algorithm has significant advantages in searching for the optimal solution. In terms of the average RPD value, MMBO_LSDT performed optimally on 14 out of 15 test instances, further verifying its excellent performance in obtaining the best RPD value. In addition, the variance of the RPD value of MMBO_LSDT is small, indicating that it has higher stability in solving this problem.

[0150] To more intuitively show the performance differences between the algorithms, Figure 10 box plots of the five algorithms on medium and large test instances were drawn. From Figure 10 (a) to Figure 10 (l), it can be seen that the lower edge of the box of MMBO_LSDT is always at the lowest position, indicating that it can always obtain the best RPD value. At the same time, the entire box of MMBO_LSDT is always at the lowest position, indicating that it can obtain better solutions in each run. In summary, the MMBO_LSDT algorithm shows higher efficiency and stability in solving this problem, and is significantly superior to other benchmark algorithms.

[0151] Table 7 RPD values of the MMBO algorithm and other benchmark algorithms

[0152]

[0153]

[0154] Continued Table 7 RPD Values of the MMBO Algorithm and Other Benchmark Algorithms

[0155]

[0156] Note: There is a scientific notation symbol e-02 after the best and average RPD values in Table 7 and Continued Table 7.

[0157] The present invention proposes a new remanufacturing scheduling problem for an integrated three-stage remanufacturing system that includes disassembly, reprocessing, and reassembly. This problem simultaneously considers machine blockage, machine deterioration, and rate modification activities (RMAs). The present invention only considers restoring the machine to an "almost brand-new state" to simplify the complexity of the scheduling problem. To solve this NP-hard problem, a new deterioration model is proposed to determine the actual processing time and the location and quantity of RMAs on each machine. Subsequently, a blocking scheduling model is constructed to minimize the makespan. To obtain an efficient and robust solution, an improved monarch butterfly optimization algorithm is proposed in this paper. This algorithm includes a constructive heuristic method and a machine load balancing strategy for solution initialization according to the problem characteristics, and two improved operators are designed to enhance the search ability of the algorithm. Finally, through comparative experiments with four benchmark algorithms, it is verified that the MMBO algorithm has significant effectiveness and superiority in solving the problem proposed by the present invention

[0158] The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in this specification.

[0159] The above-described embodiments only represent several implementation manners of the present invention, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the appended claims.

Claims

1. A three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies, characterized in that, The three stages include a disassembly stage, a reprocessing stage, and a reassembly stage. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategies includes: Taking rate adjustment activities as the maintenance strategy for the reprocessing machines in the reprocessing stage, a deterioration model that simultaneously considers time-dependent deterioration problems and maintenance strategies is constructed. Based on the deterioration model of the reprocessing machines, with the goal of minimizing the makespan, and setting the constraints for the disassembly stage, the reprocessing stage, and the reassembly stage respectively, a three-stage remanufacturing blocking scheduling model is constructed. The monarch butterfly optimization algorithm is used to solve the three-stage remanufacturing blocking scheduling model, and the optimal three-stage remanufacturing blocking scheduling plan is output. The monarch butterfly optimization algorithm updates the sub-population using a greedy acceptance strategy.

2. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 1, characterized in that The construction of the deterioration model that simultaneously considers time-dependent deterioration problems and maintenance strategies includes: The calculation formula for the actual reprocessing time of parts is as follows: In the formula, represents the actual reprocessing time of the j-th part of the i-th scrapped product on the k-th reprocessing machine of the j-th reprocessing line, represents the expected reprocessing time of the j-th part of the i-th scrapped product on the k-th reprocessing machine of the j-th reprocessing line, and α is the deterioration rate of the reprocessing machine, represents the reprocessing time interval of the j-th part of the i-th scrapped product on the k-th reprocessing machine of the j-th reprocessing line. i is the scrapped product index, i ∈ {1,..., I}, I is the total number of scrapped products, j is the part index or reprocessing line index, j ∈ {1,..., J}, J is the number of parts of each scrapped product or the number of parallel reprocessing lines, and k is the reprocessing machine index, k ∈ {1,..., K j}, K j represents the total number of reprocessing machines on the j-th reprocessing line; Among them, the reprocessing time interval The calculation formula is as follows: where \(i\in\{1,\ldots,I\}\), \(j\in\{1,\ldots,J\}\), \(k\in\{1,\ldots,K j \}\), represents the start time of reprocessing the \(j\)-th part of the \(i\)-th scrapped product on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, represents the completion time of the \(l\)-th maintenance strategy of the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, where \(l\) is the maintenance strategy index, \(\lambda ij ' kl is a decision variable. Specifically, before reprocessing the \(j\)-th part of the \(i\)-th scrapped product on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, if the \(l\)-th maintenance strategy of the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line is the most recently executed maintenance strategy, then \(\lambda ij ' kl = 1; otherwise \(\lambda ij ' kl = 0; When the maintenance policy index \(l = 0\), it represents the assumed 0th maintenance policy, which is equal to the start time of reprocessing of the part to be reprocessed first on the \(k\)th reprocessing machine of the \(j\)th reprocessing line, as shown in formula (3), where \(j\in\{1,\ldots,J\}\) and \(k\in\{1,\ldots,K j \}: Equation (4) ensures that the completion time of the maintenance strategy is meaningful only when the maintenance strategy is actually executed, where j ∈ {1,..., J}, k ∈ {1,..., K j}, l ∈ {0, 1,..., I - 1}, M represents a positive number approaching infinity, and λ jkl is a decision variable. Specifically, if the l-th maintenance strategy of the k-th reprocessing machine on the j-th reprocessing line is executed on the k-th reprocessing machine of the j-th reprocessing line, then λ jkl = 1; otherwise λ jkl = 0; and λ jkl and λ ij ′ kl are related as shown in formula (5): λ jkl ≥λ ij ′ kl (5) Equations (6) and (7) define the number of maintenance strategies to be executed on each reprocessing machine, where j ∈ {1,..., J}, k ∈ {1,..., K j}, l ∈ {0, 1,..., I - 2}, λ jk(l+1) is a decision variable. Specifically, if the (l + 1)-th maintenance strategy of the k-th reprocessing machine on the j-th reprocessing line is executed on the k-th reprocessing machine of the j-th reprocessing line, then λ jk(l+1) = 1; otherwise λ jk(l+1) = 0; Equation (8) ensures that there is exactly one maintenance strategy closest to any part before reprocessing the part on the reprocessing machine, where i ∈ {1,..., I}, j ∈ {1,..., J}, k ∈ {1,..., K j}: λ jk(l+1) ≤λ jkl (7) The constraint of Equation (9) is that the reprocessing time of the part to be reprocessed on the k-th reprocessing machine of the j-th reprocessing line due to deterioration is greater than t RMA only when the maintenance strategy is executed on the k-th reprocessing machine of the j-th reprocessing line, where j ∈ {1,..., J}, k ∈ {1,..., K j}, l ∈ {1,..., I - 1}, and t RMA is the execution time of the maintenance strategy, indicating the completion time of the (l - 1)-th maintenance strategy for the k-th reprocessing machine of the j-th reprocessing line: Equation (10) further restricts that for parts reprocessed on the reprocessing machine, the reprocessing time due to deterioration is less than t RMA , where i ∈ {1,..., I}, j ∈ {1,..., J}, k ∈ {1,..., K j}.

3. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 1, characterized in that The construction of the three-stage remanufacturing blocking scheduling model is as follows: (1) The objective function is constructed as follows: where Minf is the objective function, i.e., minimizing the makespan, C max is the makespan of the three-stage remanufacturing blocking scheduling, FT i A is the completion time of the reassembly of the i-th remanufactured product, i is the remanufactured product index, i ∈ {1,..., I}, and I is the total number of remanufactured products; (2) The constraints for the disassembly stage are set as follows: η i ′ i′g + η i ″ ig ≤ 1(13) η ig +η i′g -η i ′ i′g -η i ″ ig ≤1(14) Equation (12) restricts that each end-of-life product can only be disassembled on one disassembly machine, where \(i\in\{1,\ldots,I\}\), \(\eta\) ig is a decision variable. Specifically, if the \(i\)-th end-of-life product is disassembled on the \(g\)-th disassembly machine, then \(\eta\) ig = 1; otherwise \(\eta\) ig = 0; \(g\) is the disassembly machine index, \(g\in\{1,\ldots,G\}\), and \(G\) is the total number of disassembly machines; Equations (13) and (14) specify the disassembly sequence of end-of-life products, where \(i, i'\in\{1,\ldots,I\}\), \(g\in\{1,\ldots,G\}\), \(i'\) is the index of the end-of-life product, and \(\eta\) i ′ i′g is a decision variable. Specifically, on the \(g\)-th disassembly machine, if the \(i\)-th end-of-life product is disassembled before the \(i'\)-th end-of-life product, then \(\eta\) i ′ i′g = 1; otherwise \(\eta\) i ′ i′g = 0; \(\eta\) i ″ ig is a decision variable. Specifically, on the \(g\)-th disassembly machine, if the \(i'\)-th end-of-life product is disassembled before the \(i\)-th end-of-life product, then \(\eta\) i ″ ig = 1; otherwise \(\eta\) i ″ ig = 0; \(\eta\) i′g is a decision variable. Specifically, if the \(i'\)-th end-of-life product is disassembled on the \(g\)-th disassembly machine, then \(\eta\) i′g = 1; otherwise \(\eta\) i′g = 0; Equation (15) restricts that when two end-of-life products are assigned to the same disassembly machine, the disassembly is carried out in sequence, where \(i, i'\in\{1,\ldots,I\}\), \(g\in\{1,\ldots,G\}\). represents the completion time of disassembling the \(i'\)-th end-of-life product. represents the disassembly time of the \(i'\)-th end-of-life product, \(FT\). i D represents the completion time of disassembling the \(i\)-th end-of-life product, and \(M\) is a positive number approaching infinity. (3) The constraints for the reprocessing stage are set as follows: γ ii′jk +γ i′ijk ≤1(22) Equation (16) restricts that the reprocessing of the product must start after the disassembly of the product is completed, where \(i\in\{1,\ldots,I\}\), \(j\in\{1,\ldots,J\}\). represents the start time of reprocessing the \(j\)-th part of the \(i\)-th scrapped product on the first reprocessing machine of the \(j\)-th reprocessing line, \(j\) is the part index or reprocessing line index, and \(J\) is the number of parts of each scrapped product or the number of parallel reprocessing lines; Equations (17) and (18) constrain the relationship between the departure time and the completion time of the parts by considering the blocking effect, where \(i\in\{1,\ldots,I\}\), \(j\in\{1,\ldots,J\}\), \(k\in\{1,2,\ldots,K j -1\}\), represents the completion time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, represents the start time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, represents the actual reprocessing time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, and \(k\) is the reprocessing machine index, \(K j represents the total number of reprocessing machines on the \(j\)-th reprocessing line, represents the completion time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(K j -th reprocessing machine of the \(j\)-th reprocessing line, represents the start time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(K j -th reprocessing machine of the \(j\)-th reprocessing line, represents the actual reprocessing time of the \(j\)-th part of the \(i\)-th scrapped product when being reprocessed on the \(K j -th reprocessing machine of the \(j\)-th reprocessing line; Equation (19) restricts that the start time of reprocessing a part on the reprocessing machine is equal to its departure time on the previous reprocessing machine, where \(i\in\{1,\ldots,I\}\), \(j\in\{1,\ldots,J\}\), \(k\in\{2,3,\ldots,K j \}\), denotes the completion time of reprocessing the \(j\)-th part of the \(i\)-th scrap product on the \((k - 1)\)-th reprocessing machine of the \(j\)-th reprocessing line; Formula (20) constrains the temporal relationship between parts reprocessed on the same reprocessing machine by considering the blocking effect, where \(i, i' \in \{1, \ldots, I\}\), \(j \in \{1, \ldots, J\}\), \(k \in \{1, \ldots, K j \}\), represents the start time of reprocessing the \(j\)-th part of the \(i'\)-th scrapped product on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, and \(\gamma ii′jk is a decision variable. Specifically, on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, if the \(j\)-th part of the \(i\)-th scrapped product is reprocessed before the \(j\)-th part of the \(i'\)-th scrapped product, then \(\gamma ii′jk = 1; otherwise \(\gamma ii′jk = 0; Equation (21) constrains the time relationship between the maintenance strategy and the part before it, where \(i, i' \in \{1, \ldots, I\}\), \(j \in \{1, \ldots, J\}\), \(k \in \{1, \ldots, K j \}\), \(l \in \{1, \ldots, I - 1\}\), represents the completion time of the \(l\)-th maintenance strategy for the \(k\)-th reprocessing machine on the \(j\)-th reprocessing line, \(t RMA is the execution time of the maintenance strategy, \(\lambda ij '\ k(l-1) is a decision variable. Specifically, before the \(k\)-th reprocessing machine on the \(j\)-th reprocessing line reprocesses the \(j\)-th part of the \(i\)-th scrapped product, if the \((l - 1)\)-th maintenance strategy of the \(k\)-th reprocessing machine on the \(j\)-th reprocessing line is the most recently executed maintenance strategy, then \(\lambda ij '\ k(l-1) = 1; otherwise \(\lambda ij '\ k(l-1) = 0; \(\lambda i '' jkl is a decision variable. Specifically, before the \(k\)-th reprocessing machine on the \(j\)-th reprocessing line reprocesses the \(j\)-th part of the \(i'\)-th scrapped product, if the \(l\)-th maintenance strategy of the \(k\)-th reprocessing machine on the \(j\)-th reprocessing line is the most recently executed maintenance strategy, then \(\lambda i '' jkl = 1; otherwise \(\lambda i '' jkl = 0; \(l\) is the maintenance strategy index; Equation (22) constrains the reprocessing order between parts, where \(i, i' \in \{1, \ldots, I\}\), \(j \in \{1, \ldots, J\}\), \(k \in \{1, \ldots, K\ j \}, \gamma i′ijk is a decision variable. Specifically, on the \(k\)-th reprocessing machine of the \(j\)-th reprocessing line, if the \(j\)-th part of the \(i'\)-th scrapped product is reprocessed before the \(j\)-th part of the \(i\)-th scrapped product, then \(\gamma i′ijk = 1\); otherwise \(\gamma i′ijk = 0\); (4) The constraints for the reassembly stage are set as follows: μ i ′ i′r + μ i ″ ir ≤ 1(26) μ ir +μ i′r -μ i ′ i′r -μ i ″ ir ≤1(27) Equation (23) restricts that each remanufactured product can only be reassembled on one reassembly machine, where \(i\in\{1,\ldots,I\}\), \(\mu\) ir is a decision variable. Specifically, if the \(i\)-th remanufactured product is reassembled on the \(r\)-th reassembly machine, then \(\mu\) ir = 1; \(\mu\) ir = 0; \(r\) is the reassembly machine index, \(r\in\{1,\ldots,R\}\), and \(R\) is the total number of reassembly machines; Equation (24) restricts the reprocessing and reassembly sequence of the product, where i ∈ {1,..., I}, is the reassembly time of the i-th remanufactured product; Equation (25) restricts that two remanufactured products cannot be remanufactured on the same remanufacturing machine at the same time, where \(i, i'\in\{1, \ldots, I\}\) and \(r\in\{1, \ldots, R\}\). is the completion time of the remanufacturing of the \(i'\)-th remanufactured product. is the remanufacturing time of the \(i'\)-th remanufactured product, and \(\mu\) i′r is a decision variable. Specifically, if the \(i'\)-th remanufactured product is remanufactured on the \(r\)-th remanufacturing machine, then \(\mu\) i′r = 1; \(\mu\) i′r = 0; \(\mu\) i ' i′r is a decision variable. Specifically, on the \(r\)-th remanufacturing machine, if the \(i\)-th remanufactured product is remanufactured before the \(i'\)-th remanufactured product, then \(\mu\) i ' i′r = 1; otherwise \(\mu\) i ' i′r = 0; Equations (26) and (27) constrain the reassembly sequence of the products, where \(i, i' \in \{1, \ldots, I\}\), \(r \in \{1, \ldots, R\}\), and \(\mu\) i ″ ir is a decision variable, specifically, on the \(r\)-th reassembly machine, if the \(i'\)-th remanufactured product is reassembled before the \(i\)-th remanufactured product, then \(\mu\) i ″ ir = 1; otherwise \(\mu\) i ″ ir = 0.

4. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 1, wherein The use of the monarch butterfly optimization algorithm to solve the three-stage remanufacturing blocking scheduling model and output the optimal three-stage remanufacturing blocking scheduling plan includes: (1) Set the population size to N. Use a constructive heuristic method and a machine load balancing strategy to initialize N / 2 individuals, and randomly initialize the remaining N / 2 individuals at the same time. Each individual represents a three-stage remanufacturing blocking scheduling plan. (2) Calculate the fitness value of each individual in the population. The fitness value is the reciprocal of the minimized makespan. (3) Sort the entire population in non-increasing order based on the fitness value, and divide the population into a first sub-population and a second sub-population according to the sorting result. The fitness value of the individuals in the first sub-population is greater than that of the individuals in the second sub-population. (4) Execute the migration operator on the first sub-population and update the first sub-population using the greedy acceptance strategy; execute the adjustment operator on the second sub-population and update the second sub-population using the greedy acceptance strategy. (5) Combine the updated first sub-population and the second sub-population as the new population. If the iteration end condition is not reached, return to step (3) to continue the iteration; otherwise, output the individual with the largest fitness value in the new population as the optimal three-stage remanufacturing blocking scheduling plan.

5. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 4, wherein The individual is represented by a three-layer structure. The first layer represents the product index, the second layer represents the disassembly machine index, and the third layer represents the reassembly machine index. The number of elements in the three layers corresponds. The elements corresponding in position in the first layer and the second layer indicate that the product is disassembled on the disassembly machine. If multiple elements in the first layer correspond to the same element in the second layer, it means that multiple products are disassembled on this disassembly machine, and the product with the earlier position is disassembled first. The elements corresponding in position in the first layer and the third layer indicate that the product is reassembled on the reassembly machine. If multiple elements in the first layer correspond to the same element in the third layer, it means that multiple products are reassembled on this reassembly machine, and the product with the earlier position is reassembled first.

6. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 5, characterized in that Initializing N / 2 individuals by using a constructive heuristic method and a machine load balancing strategy includes: Generating the first and second layers of an individual by using a constructive heuristic method; Generating the third layer of an individual by using a machine load balancing strategy, including: (1) Randomly generate the elements ξ of the third layer from the reassembly machine index range {1,..,R} in sequence h , h ∈ {1,...,I}, where R is the total number of reassembly machines and I is the total number of products; (2) Calculate the total reassembly time of each reassembled machine, and denote the reassembled machine with the maximum total reassembly time as AW max , and denote the reassembled machine with the minimum total reassembly time as AW min ; (3) If AW max allocates the most products and AW min allocates the fewest products, or if AW max allocates the fewest products and AW min allocates the fewest products, then execute step (4); otherwise, end the process, output the third layer, and complete the individual initialization; (4) Select a product randomly from AW max and assign it to AW min , then return to step (2).

7. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 6, characterized in that, The sorting rules of the constructive heuristic method are as follows: Sorting the products in non-increasing order according to the total processing time of the products; Or, sorting the products in non-decreasing order according to the total processing time of the products; Or, first sorting in non-increasing order according to the disassembly time of the products, and then iteratively selecting the first product and the last product from the sorted sequence in turn until all products are selected to generate a final sorting; Or, first sorting in non-increasing order according to the total processing time of the products, and then iteratively selecting the first product and the last product from the sorted sequence in turn until all products are selected to generate a final sorting.

8. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 5, characterized in that Performing a migration operator on the first sub-population includes: Based on the elements of the first and second layers, randomly selecting a disassembly machine on which two or more products are being disassembled. Subsequently, randomly selecting two products on the selected disassembly machine and swapping the positions of the two selected products in the first layer; Updating the elements of the third layer by using formula (33); Among them, represents the h-th element of the third layer of the s-th individual in the first sub-population in the (t + 1)-th generation, and respectively represent the h-th element of the third layer of the s1-th individual in the first sub-population 1 and the s2-th individual in the second sub-population in the t-th generation, where s1 and s2 respectively represent randomly selected individuals in the first sub-population and the second sub-population, p is a parameter representing the proportion of butterflies from the first sub-population, r = rand * peri is the period parameter, where rand is a random number generated from [0, 1], and peri is the migration period.

9. The three-stage remanufacturing blocking scheduling method based on machine deterioration and maintenance strategy according to claim 5, characterized in that Performing an adjustment operator on the second sub-population includes: Randomly selecting to perform type-II exchange mutation or insertion mutation on the first layer of each individual in the second sub-population. For type-II exchange mutation, randomly selecting two different elements and swapping their positions; for insertion mutation, randomly selecting one element and inserting the randomly selected element into a random position; Updating the elements of the second and third layers of each individual in the second sub-population by using formula (34) and rounding the values; In the formula, represents the h-th element of the second or third layer of the s'-th individual in the second sub-population in the (t + 1)-th generation, represents the h-th element of the second or third layer of the best individual in the entire population in the t-th generation, represents the h-th element of the second or third layer of the s2-th individual randomly selected from the second sub-population, ε represents the weight coefficient, λ is set to 1.5 in Lévy flight, rand is a random number generated from [0, 1], p is a parameter representing the proportion of butterflies from the first sub-population, and BAR is the adjustment rate.

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