A remanufacturing scheduling optimization method for rework risk

Abstract

The application discloses a remanufacturing scheduling optimization method for rework risk, adopts interval grey number to describe uncertain processing time and rework risk caused by quality difference of EOL products in a remanufacturing production process, constructs a mathematical model of maximum completion time through expected processing time containing rework risk, considers rework risk in remanufacturing scheduling, and finally finds an optimal scheduling scheme through an iterated algorithm of innovative design. According to the technical scheme, the uncertainty in the actual production process is accurately described, the performance of finding the optimal scheduling scheme is improved, and the remanufacturing scheduling scheme found has better scheduling effect compared with other methods.

Description

Technical Field

[0001] This application belongs to the field of remanufacturing scheduling technology, and in particular relates to a remanufacturing scheduling optimization method for rework risk. Background Technology

[0002] While creating immense wealth for human society, the manufacturing industry also consumes vast amounts of resources. Remanufacturing, as a resource-saving and environmentally friendly new manufacturing paradigm, has gained widespread attention from academia and industry. Remanufacturing refers to the process of restoring end-of-life (EOL) products to a new state through a series of operations. Compared to traditional manufacturing, remanufactured products not only provide the same quality but also save resources and reduce emissions of toxic substances, thus achieving greater economic and social benefits.

[0003] Reasonable remanufacturing scheduling is crucial for effective resource utilization and improved enterprise efficiency. However, unlike traditional manufacturing, remanufacturing involves end-of-life (EOL) products. EOL products typically experience different usage environments during their lifespan, resulting in varying degrees and forms of damage. This variability in damage makes accurately assessing the quality of EOL products difficult, leading to uncertainties in processing time, routes, and failure rates during remanufacturing. These complex uncertainties not only increase the risk of rework but also make remanufacturing scheduling more difficult and complex. Therefore, to obtain a more practical and feasible remanufacturing scheduling solution, it is necessary to consider the uncertainties caused by the quality variations of EOL products and the risk of rework.

[0004] Although remanufacturing scheduling has been extensively studied, the uncertainties inherent in the remanufacturing process are often overlooked. In recent years, some studies have employed stochastic optimization methods or fuzzy theory to address uncertainties in remanufacturing scheduling. However, describing uncertainty using limited historical remanufacturing data typically yields only an approximate range rather than a precise distribution. This not only increases the difficulty of describing uncertainty as a random variable but also makes defining membership functions in fuzzy theory challenging. Furthermore, none of the aforementioned studies on remanufacturing scheduling have considered the rework risk of end-of-life (EOL) products. Summary of the Invention

[0005] The purpose of this application is to provide a remanufacturing scheduling optimization method oriented towards rework risk, taking into account the uncertainty caused by EOL product quality differences and rework risk, and to achieve more optimized remanufacturing scheduling.

[0006] To achieve the above objectives, the technical solution of this application is as follows:

[0007] A remanufacturing scheduling optimization method for addressing rework risk includes:

[0008] For batch remanufacturing scheduling with I types of remanufactured products and J non-equal parallel processing routes, interval gray numbers are used to describe the variables in the remanufacturing process, and a mathematical model of the maximum completion time is constructed by the expected processing time including the risk of rework.

[0009] A particle representation scheme for solving the mathematical model is constructed. The particle representation scheme includes two parts: the first part encodes the product information assigned to each processing route, and the second part encodes the operation sequencing information.

[0010] The initial population is initialized based on the particle representation scheme, and the mathematical model is iteratively solved to obtain the optimal remanufacturing scheduling scheme.

[0011] Furthermore, the mathematical model is as follows:

[0012]

[0013] in, This indicates the total time required to process all remanufactured products. Indicates the Nth process in the j-th processing route j The end time of processing the i-th product by the machine, the N-th... j The machine is the last machine in the j-th processing route;

[0014]

[0015] in, This represents the start time of the nth machine in the j-th processing route processing the i-th product. Q represents the expected processing time, including the risk of rework, for the nth machine in the j-th processing route to process the i-th product. ijn This represents the quantity of the i-th type of remanufactured product processed on the n-th machine in the j-th processing route;

[0016]

[0017] in, Let k represent the processing time of the nth machine in the jth processing route when it processes the ith product in the kth time, where k = 1, ..., ∞, where k = 1 indicates the first processing, otherwise it indicates a rework process; This represents the probability that the i-th product needs to be processed k times on the n-th machine in the j-th processing route.

[0018] Furthermore, the step of initializing the initial population according to the particle representation scheme, iteratively solving the mathematical model, and obtaining the optimal remanufacturing scheduling scheme includes:

[0019] Step F1: Initialize the population and parameters;

[0020] Step F2: Calculate the current adaptive probability, and select the mutation operator to execute for the first part of the particles based on the adaptive probability. The mutation operator includes DE / rand / 1, DE / rand / 2, and DE / L. best / 1 and DE / U best / 1 Four mutation operators;

[0021] Step F3: Execute the crossover operator;

[0022] Step F4: Update the quantity allocation information of each type of product on each route in the particle;

[0023] Step F5: Perform a position update mechanism on the second part of the particle;

[0024] Step F6: Execute the local search strategy;

[0025] Step F7: Update the path selection and operation sorting information of the products in the particles;

[0026] Step F8: Determine if the restart conditions are met. If they are met, proceed to the next step; otherwise, proceed to step F10.

[0027] Step F9: Execute the restart mechanism to reinitialize part of the population;

[0028] Step F10: Determine if the stopping condition is met. If it is met, stop the iteration and output the optimal scheduling scheme. Otherwise, return to step F2 and start iterating again.

[0029] Furthermore, the DE / L best / 1 and DE / U best The / 1 operator is represented as follows:

[0030]

[0031]

[0032] in and It is an individual randomly selected from the t-th generation population, and b≠c; and These represent individuals in the population with the smallest upper and lower bounds, respectively. It is the i-th individual generated through the mutation operator; It is the scaling factor for the i-th individual, which follows a uniform distribution.

[0033] Furthermore, the second part of the particle is subjected to a position update mechanism, wherein the position update mechanism updates the position of the particle through a mutation operator and a crossover operator. Each time the crossover operator is executed, a crossover operation is performed with the pbest particle according to a first probability and with the gbest particle according to a second probability.

[0034] Furthermore, the execution of the local search strategy includes a scheduling scheme for a predetermined number of items ranked by quality in each iteration, and... and The local search strategy is implemented based on two neighborhood structures: the exchange neighborhood structure and the insertion neighborhood structure.

[0035] Furthermore, the restart mechanism includes:

[0036] If the globally optimal solution does not improve after a preset number of iterations, a preset proportion of particles are randomly selected from the population for re-initialization.

[0037] This application proposes a remanufacturing scheduling optimization method oriented towards rework risk. It uses interval grey numbers to describe the uncertain processing time and rework risk caused by differences in EOL (Extractable Online Products) quality during the remanufacturing process, accurately describing the uncertainties in actual production. A mathematical model of the maximum completion time is constructed by considering the expected processing time including rework risk. This rework risk is then taken into account in the remanufacturing scheduling. Finally, an innovative iterative algorithm is used to find the optimal scheduling scheme. The technical solution proposed in this application not only improves the performance in finding the optimal scheduling scheme but also achieves better scheduling results compared to other methods. Attached Figure Description

[0038] Figure 1 This is a flowchart of the remanufacturing scheduling optimization method for addressing rework risk in this application;

[0039] Figure 2 This is an example of a particle representation scheme in the embodiments of this application;

[0040] Figure 3 This is a schematic diagram of the method for solving the mathematical model according to an embodiment of this application;

[0041] Figure 4 This is an example of the crossover operator in the position update mechanism of this application.

[0042] Figure 5 This is an example of a mutation operator in an embodiment of this application;

[0043] Figure 6 This is a schematic diagram illustrating the relationship between the objective function and the number of iterations in an embodiment of this application;

[0044] Figure 7 This is a schematic diagram illustrating the relationship between the objective function and the population size in an embodiment of this application. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0046] During remanufacturing, the differences in damage to EOL (Extended Online Products) lead to uncertainties in processing routes and rework risks. Specifically, products of different qualities require different processing routes, while processing routes for products of the same quality are selectable. For example, slightly damaged products can be remanufactured to the same state as new products with only a few operations and low rework risk; while severely damaged products require more operations, have higher rework risk, and require more rework. Therefore, this application employs multiple non-equivalent parallel processing routes to address the uncertainty of processing routes. Each processing route includes multiple machines, each capable of performing specific operations. Different processing routes can be selected to remanufacture the same EOL products; however, the time spent and rework risk vary depending on the selected route. Furthermore, given that batch remanufacturing can effectively process different EOL products and improve processing flexibility, this application introduces a batch remanufacturing mode. This application transforms the remanufacturing scheduling problem into a mathematical model, which is then solved to obtain an optimized remanufacturing scheduling scheme. That is, for a batch of EOL products, the processing route selected for each product should be determined, and the processing sequence of each product on each route should be reasonably allocated to obtain the optimal scheduling scheme.

[0047] In the process of converting to a mathematical model, this application uses the following symbols:

[0048] P i Let i represent the i-th type of remanufactured product, where i = 1, 2, 3, ..., I;

[0049] R j This represents the j-th processing route, where j = 1, 2, 3, ..., J;

[0050] M jn This represents the nth machine in the j-th processing route, where n = 1, ..., N. j ; Nth j The machine is the last machine in the j-th processing route;

[0051] This indicates the start time of the nth machine in the jth processing route processing the i-th product;

[0052] This represents the end time of the nth machine in the j-th processing route processing the i-th product;

[0053] This indicates the total time required to process all remanufactured products;

[0054] Let k represent the processing time of the nth machine in the j-th processing route when processing the i-th product in the k-th iteration, where k = 1, ..., ∞, where k = 1 indicates the first processing iteration, otherwise it indicates a rework process. In simulation experiments, the maximum value of k can be set to a value K. ijn ;

[0055] This represents the probability of failure for the nth machine in the j-th processing route to process the i-th product;

[0056] This represents the probability that the i-th product needs to be processed k times on the n-th machine in the j-th processing route. This means that the first k-1 processing attempts all failed and the k-th processing attempt was successful.

[0057] This represents the expected processing time, including the risk of rework, for the nth machine in the j-th processing route to process the i-th product;

[0058] λ represents the rate of decline over time;

[0059] α ij Let α represent a Boolean variable; if the j-th processing route is chosen to process the i-th product, then α ij =1, otherwise α ij =0;

[0060] β ij Let β represent a Boolean variable; if the i-th product is the first to be processed on the j-th processing route, then β... ij =1, otherwise β ij =0.

[0061] In one embodiment, such as Figure 1 As shown, a remanufacturing scheduling optimization method for addressing rework risk is provided, including:

[0062] Step S1: For batch remanufacturing scheduling with I types of remanufactured products and J non-equal parallel processing routes, the variables in the remanufacturing process are described using interval gray numbers, and a mathematical model of the maximum completion time is constructed by the expected processing time including rework risk.

[0063] To effectively describe and handle uncertainties in the remanufacturing process, this application uses interval grey numbers to describe various variables in the remanufacturing process. For example, the start time is represented as... in and These are the lower and upper bounds, respectively; other variables are represented in the same way.

[0064] Total time to process all remanufactured products The required variables can be calculated using formulas (1) and (2):

[0065]

[0066]

[0067] Formulas (1) and (2) respectively represent the values ​​in M jn Up-processing P i The start and end times. Indicates in M j(n-1) Upper processing P i Completion time, Indicates in P i Previously at M jn Processing other P i' Maximum completion time, Q ijn Indicates in M jn Upper processing P i The quantity.

[0068] The time required for the first processing of an EOL product on the current machine can be estimated from limited remanufacturing history data. The rework time will change as multiple rework operations are performed on the same machine. Therefore, the rework time in this application will decrease at a constant rate λ. It can be calculated using formula (3):

[0069]

[0070] Because in M jn Processing P i The probability of failure is Therefore, the success rate is So P i Need to be in M jn The probability of success is obtained only after k processing steps. (That is, the probability that the process fails in the first k-1 times and succeeds only in the kth time) can be calculated using formula (4):

[0071]

[0072] Then, the expected processing time, including the risk of rework. It can be calculated using formula (5):

[0073]

[0074] The maximum completion time based on the gray number of the interval can be calculated by formula (6):

[0075]

[0076] The above formula is subject to the following conditions:

[0077]

[0078]

[0079]

[0080] Constraint (7) guarantees that at least one processing route can be selected for remanufacturing P. i Constraint (8) guarantees M jn Only in its preceding operation M j(n-1) It will only begin after completion, constraint (9) ensures that it starts first in R. j There can be at most one type of product being processed.

[0081] It should be noted that the mathematical model of this application uses the maximum completion time based on the interval gray number calculated by formula (6) as the objective function, and finally optimizes the solution by minimizing the maximum completion time to obtain the variables of the mathematical model, thereby guiding the scheduling. In the mathematical model of this application, the expected processing time with rework risk is considered, so that scheduling optimization can be performed to address the uncertainty of rework risk.

[0082] This application uses interval grey numbers to describe the uncertain processing time and rework risk caused by differences in EOL product quality during remanufacturing, which can accurately describe the uncertainties in the actual production process. Regarding the basic operational rules of interval grey numbers, for example, assuming... and These are two interval gray numbers. It is a relatively mature theory in the field of grey systems, so I will not go into details here.

[0083] Step S2: Construct a particle representation scheme for solving the mathematical model. The particle representation scheme includes two parts: the first part encodes the product information assigned to each processing route, and the second part encodes the operation sorting information.

[0084] An appropriate representation scheme facilitates the construction of the problem model in this application and also helps in solving the route selection subproblem using the DE algorithm and the operation ordering subproblem using the PSO algorithm. However, the original representation schemes of these two basic algorithms cannot be directly used for encoding the problem in this application. Therefore, this application proposes an efficient particle-based representation scheme. A particle consists of two parts: the first part encodes the product information assigned to each route, and the second part encodes the operation ordering information. To better and more intuitively understand the above representation scheme, Figure 2 An illustrative example of the representation scheme is given.

[0085] (1) The first part indicates the routes available for processing each product and the quantity of product allocated to each route. The terminator “*” is used to separate one product from another. For example, before the first terminator... This means that P1 can be processed on R2 or R4, with 2 P1s allocated to R2 and 8 P1s allocated to R4.

[0086] (2) The second part indicates the operation sequence information of the products on each processing route. For example, This indicates that the order of operations on R2 is P5, P4, and P1.

[0087] Step S3: Initialize the initial population according to the particle representation scheme, iteratively solve the mathematical model, and obtain the optimal remanufacturing scheduling scheme.

[0088] The mathematical model problem in this application can be viewed as a hybrid discrete problem, which can be divided into two subproblems: route selection and operation sorting. While the basic DE and PSO algorithms perform well in solving simple continuous problems, they are not suitable for directly solving hybrid discrete problems. The DE algorithm was initially proposed to explore continuous search spaces and has since been widely applied in various combinatorial optimization fields. The solutions of the PSO algorithm are called "particles," each with two characteristics: velocity and position. The quality of a particle is evaluated based on its fitness value, which is calculated from the objective function value. Typically, particles are randomly generated, and their positions change as they propagate through the solution space. During propagation, each particle adjusts its velocity and updates its position based on its best position found so far (i.e., pbest) and the best position found by all particles in the swarm so far (i.e., gbest). All particles propagate in the directions of pbest and gbest in each iteration to obtain better particle quality. In this embodiment, the gbest particle is the one with the shortest maximum completion time among all particles, and the pbest particle is the one with the shortest maximum completion time obtained by each particle during the iteration process. pbest and gbest are mature technologies in this field, and will not be described in detail here.

[0089] Therefore, this application proposes a novel solution method, also referred to as the HDEPSO solution method, which employs adaptive parameters to improve the convergence speed of the DE algorithm, uses a position update mechanism in the PSO algorithm to maintain population diversity, adopts a local search strategy to improve the performance of the hybrid algorithm, and includes a restart mechanism to prevent premature convergence of the hybrid algorithm. The route selection subproblem is solved by the DE algorithm to obtain the optimal route selection solution (RS). opt The operation sorting subproblem is solved by the PSO algorithm to obtain the optimal operation sorting solution (OS). opt ).

[0090] The HDEPSO solution method proposed in this application, such as Figure 3 As shown, the specific steps include the following:

[0091] Step F1: Initialize the population and parameters;

[0092] Step F2: Calculate the current adaptive probability, and select the mutation operator to execute for the first part of the particles based on the adaptive probability. The mutation operator includes DE / rand / 1, DE / rand / 2, and DE / L. best / 1 and DE / U best / 1 Four mutation operators;

[0093] Step F3: Execute the crossover operator;

[0094] Step F4: Update the quantity allocation information of each type of product on each route in the particle;

[0095] Step F5: Perform a position update mechanism on the second part of the particle;

[0096] Step F6: Execute the local search strategy;

[0097] Step F7: Update the path selection and operation sorting information of the products in the particles;

[0098] Step F8: Determine if the restart conditions are met. If they are met, proceed to the next step; otherwise, proceed to step F10.

[0099] Step F9: Execute the restart mechanism to reinitialize part of the population;

[0100] Step F10: Determine if the stopping condition is met. If it is met, stop the iteration and output the optimal scheduling scheme. Otherwise, return to step F2 and start iterating again.

[0101] In step F2, this application uses two basic mutation operators, DE / rand / 1 and DE / rand / 2, as shown in formulas (10) and (11). Furthermore, considering the characteristics of interval gray numbers, this application also proposes two new mutation operators, DE / L.best / 1 and DE / U best / 1 is used to accelerate the convergence speed, as shown in formulas (12) and (13). The algorithm continuously switches the above mutation operators during the iteration process, which can generate a better quality difference vector, thereby balancing the local and global search capabilities of the algorithm. This application uses the binomial distribution criterion to cross over the mutated individuals.

[0102]

[0103]

[0104]

[0105]

[0106] in and It is an individual randomly selected from the t-th generation population, and and V represents the individual in the population with the smallest upper and lower bounds, respectively; i t F is the i-th individual generated through the mutation operator; i t It is the scaling factor for the i-th individual.

[0107] F i t The magnitude of the crossover rate (CR) significantly impacts the performance of the DE algorithm. Compared to a fixed value, F... i t The adaptive parameter method used in CR can better balance the local and global search capabilities of the algorithm during iteration. Therefore, DE / rand / 1 and DE / L best / 1 and DE / U best / 1 of F i t The CR is adaptively based on a random normal distribution during the iteration process, as shown in equations (14) and (15). Furthermore, F in DE / rand / 2... i t They are set to follow a uniform distribution to disturb individuals so that they may escape local optima.

[0108]

[0109]

[0110] Where the parameter μF of the normal distribution in generation t is... t and μCR tAn update will be performed at the end of the iteration, as shown in equations (16) and (17):

[0111]

[0112]

[0113] Where G is the maximum number of iterations, and IF and ICR are μF. t and μCR t The initial value.

[0114] It should be noted that this application continuously switches the aforementioned mutation operators during the iteration process. Based on probability switching, there is initially a 50% chance of randomly selecting the basic mutation operator, either DE / rand / 1 or DE / rand / 2. Then, if the basic mutation operator is selected, there is an 80% chance of selecting DE / rand / 1. If the basic mutation operator is not selected, the mutation operator is switched from DE / rand / 1 to DE / rand / 2. best / 1 and DE / U best Randomly select one from the / 1 operators.

[0115] The operation ordering subproblem in this application is a discrete optimization problem that cannot be directly solved using the basic PSO algorithm. Therefore, this application proposes a novel position update mechanism that updates the particle position through mutation and crossover operators. This mechanism not only allows the PSO algorithm to be directly used to solve the operation ordering subproblem, but also facilitates the particle's propagation in a more favorable direction to find the optimal solution. First, a mutation operator is used to search the space around the particle to prevent getting trapped in local optima. Then, a crossover operator is used to update the particle position based on its own position and the overall optimal position, thus accelerating convergence.

[0116] The position update mechanism in this application employs two types of crossover operators: single-point crossover and two-point crossover. Figure 4 Illustrative examples are provided. For example... Figure 4 As shown in (a), single-point crossover refers to randomly selecting a point from particle A, copying all elements before that point to the corresponding position in the new particle, and deleting these elements in particle B. Subsequently, the remaining elements in particle B are moved sequentially to the empty positions in the new particle. Figure 4 (b) describes the two-point intersection. The two-point intersection operator randomly selects two points from particle A, copies the elements between the two points to the corresponding positions in the new particle, and simultaneously deletes these elements from particle B. Then, the remaining elements in particle B are moved sequentially to the empty positions in the new particle.

[0117] The position update mechanism in this application uses two mutation operators: reverse mutation and exchange mutation. Figure 5Illustrative examples of two mutation operators are given, where reversal mutation reverses the order of elements between two randomly selected points, and exchange mutation swaps the elements between two randomly selected points.

[0118] In this application's position update mechanism, the mutation operator is also based on probability. The selection probability of the reverse mutation operator is 30%, and each particle undergoes a mutation operation. The crossover operator is also performed on each individual, and at each crossover, it crosses with the pbest particle with a 30% probability, resulting in a 50 / 50 crossover operator selection probability. The remaining 70% probability is that it will cross with the gbest particle, also with a 50 / 50 crossover operator selection probability. Traditional PSO algorithms solve continuous optimization problems, while this application's position update mechanism transforms it into a discrete problem. Furthermore, this application's crossover operator operates based on pbest and gbest, which, unlike traditional methods, encourages particles to propagate in a more favorable direction to find the optimal solution.

[0119] This application's local search strategy takes into account the weak local search capability of the basic PSO algorithm. To improve the algorithm's search capability and achieve better convergence, a scheduling scheme with a predetermined number of top-ranked algorithms is used in each iteration. and The system employs two local search strategies based on neighborhood structures: swapping and insertion. Swapping neighborhood structures involves randomly selecting two points from a solution and swapping their corresponding elements. Insertion neighborhood structures involve randomly selecting two points and inserting the former element into the position preceding the latter.

[0120] The HEDPSO algorithm in this application employs a restart mechanism to avoid premature convergence. If the globally optimal solution does not improve after a preset number of iterations, a preset proportion of particles are randomly selected from the population for re-initialization. In a specific embodiment, 10% of the maximum number of iterations can be chosen as the preset number of iterations, or other proportions can be selected, depending on the actual experimental conditions. The globally optimal solution not improving means that the maximum completion time does not decrease. How to determine whether the solution has improved after iterations is a relatively mature technology in the field and will not be elaborated here.

[0121] This application improves the HDEPSO algorithm in five aspects to obtain better quality solutions faster, including: 1) designing an efficient representation scheme suitable for the proposed model; 2) using an adaptive strategy to improve the mutation and crossover operators of the DE algorithm to balance local exploitation and global exploration capabilities; 3) proposing a new position update mechanism for the PSO algorithm; 4) adopting a local search strategy to enhance the convergence and exploration capabilities of the algorithm; and 5) using a restart mechanism to avoid premature convergence of the algorithm.

[0122] Furthermore, this application verifies the effectiveness of the HDEPSO algorithm in solving the URS problem through a series of simulation experiments. A set of instances of different sizes were randomly generated to more realistically simulate the remanufacturing environment. Because the HDEPSO algorithm proposed in this application combines DE and PSO algorithms to solve two different subproblems respectively, for fairness, the simulation experiments employ a strategy of mixing multiple heuristic optimization algorithms to evaluate the performance of the HDEPSO algorithm. This application uses four algorithms: Biogeography-Based Optimization (BBO), DE algorithm, Flower Pollination (FPA) algorithm, and PSO algorithm, and mixes them in pairs to alternately solve the route selection subproblem and the operation ordering subproblem. For ease of understanding, the combination of the above four baseline mixed algorithms is represented as DE-BBO, DE-PSO, FPA-BBO, and FPA-PSO.

[0123] For the experimental parameters, the parameters for each hybrid algorithm were determined after exploratory experiments. For HDEPSO, IF and ICR were set to 0.75 and 1, respectively. For the FPA algorithm, the switching probability was set to 0.8, and the global pollination γ was set to 0.01. To enhance the robustness of the experiments, each algorithm was executed 10 times in the same environment, and the average fitness value was used as the final result. The fitness value of the solution was represented by the objective function value.

[0124] The first experiment used the instance Ins(7 / 50 / 6) to test the evolutionary trajectory of the above algorithm to obtain a reasonable number of iterations, with the initial population size set to 50. This application uses the average of the upper and lower bounds of the fitness values ​​to more intuitively and clearly display the algorithm's evolutionary trajectory. From Figure 6It is clear from the results that the HDEPSO, DE-BBO, and DE-PSO algorithms converge after approximately 900 iterations. Although the FPA-BBO and FPA-PSO algorithms converge faster, their mean values ​​are significantly lower than those of the HDEPSO algorithm. Furthermore, the HDEPSO algorithm achieves a better mean value with fewer iterations. For example, after 100 iterations, the mean value obtained by the HDEPSO algorithm is already better than the mean values ​​obtained by other algorithms after 1000 iterations. Therefore, the algorithm proposed in this application outperforms other hybrid algorithms in solving the problem. Considering that most algorithms converge after 900 iterations, for a fairer comparison, the number of iterations for all algorithms was set to 1000 in subsequent experiments.

[0125] The following experiments continue using the instance Ins(7 / 50 / 6) to test the performance of the above algorithm with a population size between 20 and 70, in order to obtain a reasonable population size. Figure 7 As shown, the HDEPSO algorithm consistently outperforms other algorithms at any population size, indicating that HDEPSO is superior for this instance. Furthermore, the HDEPSO algorithm demonstrates stable performance across various population sizes. In contrast, other algorithms only stabilize at a population size of 50. Therefore, for a fair and effective comparison in subsequent experiments, the initial population size for all algorithms was set to 50.

[0126] To evaluate the practicality and effectiveness of the HDEPSO algorithm, this application compared its performance with other algorithms on 18 instances of different sizes. Tables 1, 2, and 3 show the experimental comparison results.

[0127] Tables 1 and 2 contain two statistical indicators, namely the optimal and average fitness values ​​based on the interval gray number, which are labeled "optimal" and "average" respectively in this application. To more directly and quantitatively compare the stability of the algorithms, this application uses the scalar evaluation index of relative rating (RR), as shown in formula (18). In addition, the total average relative rating (ARR) is the average of the RR in 10 repeated experiments, and the smaller the ARR value, the more stable the algorithm performance.

[0128]

[0129] in, and These are the maximum completion time obtained in each independent run of the algorithm and the minimum maximum completion time obtained in 10 repeated trials, respectively. express Greater than The probability of.

[0130]

[0131] Table 1

[0132]

[0133] Table 2

[0134] As can be seen from Tables 1 and 2, the HDEPSO algorithm achieves better optimal and average fitness values, as well as better lower and upper bounds on the fitness values, in most cases than other algorithms. This indicates that the HDEPSO algorithm outperforms other algorithms in solving the problem proposed in this application.

[0135]

[0136] Table 3

[0137] Table 3 shows that in all cases, the ARR obtained by the HDEPSO algorithm is no worse than that obtained by other algorithms, indicating that the HDEPSO algorithm is more stable than other algorithms. However, due to the adoption of adaptive and local search strategies, the HDEPSO algorithm requires more CPU computation time than other algorithms, but its computation time is within an acceptable range. In the future, the rapid development of cloud computing technology and computer hardware resources will significantly reduce the CPU computation time of the HDEPSO algorithm.

[0138] This application's technical solution addresses the remanufacturing scheduling optimization problem facing rework risk by using interval grey numbers to describe its uncertainty. To effectively resolve the complex uncertainties in remanufacturing scheduling considering rework risk, a hybrid optimization algorithm combining DE and PSO algorithms—the HDEPSO algorithm—is proposed to efficiently solve the mathematical model. Experimental results show that the technical solution of this application can obtain a better remanufacturing scheduling scheme.

[0139] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A remanufacturing scheduling optimization method for addressing rework risk, characterized in that, The remanufacturing scheduling optimization method for addressing rework risk includes: For batch remanufacturing scheduling with I types of remanufactured products and J non-equal parallel processing routes, interval gray numbers are used to describe the variables in the remanufacturing process, and a mathematical model of the maximum completion time is constructed by the expected processing time including the risk of rework. A particle representation scheme for solving the mathematical model is constructed. The particle representation scheme includes two parts: the first part encodes the product information assigned to each processing route, and the second part encodes the operation sequencing information. The initial population is initialized according to the particle representation scheme, and the mathematical model is iteratively solved to obtain the optimal remanufacturing scheduling scheme. The mathematical model is as follows: ； in, This indicates the total time required to process all remanufactured products. In the j-th processing route, the first... The end time of processing the i-th product by the machine, the i-th The machine is the last machine in the j-th processing route; ； in, This represents the start time of the nth machine in the j-th processing route processing the i-th product. Let represent the expected processing time, including the risk of rework, for the nth machine in the jth processing route to process the i-th product. This represents the quantity of the i-th type of remanufactured product processed on the n-th machine in the j-th processing route; ； in, Let represent the processing time of the nth machine in the j-th processing route when processing the i-th product in the k-th iteration, where k = 1, …, Where k = 1 indicates the first processing, otherwise it indicates a rework process; This represents the probability that the i-th product needs to be processed k times on the n-th machine in the j-th processing route; And satisfy the following constraints: ； ； ； in, Let represent a Boolean variable; if the j-th processing route is chosen to process the i-th product, then = 1, otherwise = 0; Let represent a Boolean variable; if the i-th product is the first to be processed on the j-th processing route, then = 1, otherwise = 0.

2. The remanufacturing scheduling optimization method for rework risk according to claim 1, characterized in that, The process of initializing the initial population according to the particle representation scheme, iteratively solving the mathematical model, and obtaining the optimal remanufacturing scheduling scheme includes: Step F1: Initialize the population and parameters; Step F2: Calculate the current adaptive probability, and select the mutation operator to execute for the first part of the particles based on the adaptive probability. The mutation operator includes DE / rand / 1, DE / rand / 2, and DE / L. best / 1 and DE / U best / 1 Four mutation operators; Step F3: Execute the crossover operator; Step F4: Update the quantity distribution information of each type of product on each route in the particle; Step F5: Perform a position update mechanism on the second part of the particle; Step F6: Execute the local search strategy; Step F7: Update the path selection and operation sorting information of the products in the particles; Step F8: Determine if the restart conditions are met. If they are met, proceed to the next step; otherwise, proceed to step F10. Step F9: Execute the restart mechanism to reinitialize part of the population; Step F10: Determine if the stopping condition is met. If it is met, stop the iteration and output the optimal scheduling scheme. Otherwise, return to step F2 and start iterating again.

3. The remanufacturing scheduling optimization method for rework risk according to claim 2, characterized in that, The DE / L best / 1 and DE / U best The / 1 operator is represented as follows: ； ； in and It is an individual randomly selected from the t-th generation population, and ; and These represent individuals in the population with the smallest upper and lower bounds, respectively. It is the i-th individual generated through the mutation operator; It is the scaling factor for the i-th individual, which follows a uniform distribution.

4. The remanufacturing scheduling optimization method for rework risk according to claim 2, characterized in that, The second part of the particle is subjected to a position update mechanism, wherein the position update mechanism updates the position of the particle through a mutation operator and a crossover operator. Each time the crossover operator is executed, a crossover operation is performed with the pbest particle according to a first probability and with the gbest particle according to a second probability.

5. The remanufacturing scheduling optimization method for rework risk according to claim 3, characterized in that, The execution of the local search strategy includes a scheduling scheme for a pre-defined number of top-quality items in each iteration, and and The local search strategy is implemented based on two neighborhood structures: the exchange neighborhood structure and the insertion neighborhood structure.

6. The remanufacturing scheduling optimization method for rework risk according to claim 2, characterized in that, The restart mechanism includes: If the globally optimal solution does not improve after a preset number of iterations, a preset proportion of particles are randomly selected from the population for re-initialization.

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