A distributed assembly blocking flow shop scheduling optimization system

Abstract

This invention proposes a distributed assembly line workshop scheduling optimization system. A distributed estimation algorithm driven by Kalman filtering and a history learning mechanism is designed to optimize the system. Prediction, observation, first repair, and second repair are applied to the workpiece processing stage. Furthermore, enhanced repair information is fed back to the probabilistic model through the history learning mechanism to improve model accuracy. Additionally, an adaptive adjustment strategy is employed to balance the exploration and development capabilities of the optimization system. Experiments comparing the system with several classic optimization systems on a test set demonstrate the competitiveness of the proposed system. Tests on instances with varying numbers of workpieces, machines, factories, and products show that the proposed system can minimize the assembly completion time in the distributed assembly line workshop, thereby improving production efficiency.

Description

Technical Field

[0001] This invention belongs to the field of manufacturing production scheduling and route planning technology, and specifically relates to a distributed assembly line workshop scheduling optimization system, particularly to a distributed estimation algorithm driven by enhanced Kalman filtering and historical learning mechanism to optimize and minimize the assembly completion time of the distributed assembly line workshop scheduling problem. Background Technology

[0002] Complex continuous optimization problems are common in industrial production, economic forecasting, and complex process manufacturing systems, involving the optimization of various indicators with the goal of minimizing the objective and the optimal solution vector. Their computational complexity increases exponentially with the problem's dimensionality. Furthermore, due to inherent characteristics such as variable indivisibility, rotation invariance, and nonlinearity, branch and bound, linear programming, and traditional mathematical methods are ineffective in solving these problems, making it difficult to accurately establish mathematical models for many practical optimization problems.

[0003] The Distributed Blocking Flow Shop Scheduling Problem (DABFSP) is a crucial problem in modern supply chains and complex process industrial systems, and a key factor in the efficient operation of such systems and supply chains. DABFSP is a combination of the Distributed Blocking Flow Shop Scheduling Problem (DBFSP) and the assembly problem. DABFSP is divided into a production phase and an assembly phase. The production phase consists of several identical blocking flow shops. In the assembly phase, there is an assembly machine where work processed in the production plants is transferred to the assembly plant to complete the assembly process. DABFSP is an NP-hard problem.

[0004] Swarm optimization algorithms are based on Darwin's law of natural selection. In each generation, the individual with the highest fitness survives to the next generation and generates new individuals. These algorithms have achieved good results in solving continuous real-valued optimization problems and practical problems. Metaheuristic algorithms are a typical type of swarm-based intelligent optimization algorithm, but exploration and exploitation capabilities affect their performance. Exploration aims to find the potential region where the optimal solution lies, while exploitation aims to focus on the precise location. In the early stages of evolution, global exploration capability is very important, while in the later stages, further refinement and exploitation are needed until the convergence condition is met and the optimal solution is reached. Therefore, for optimization problems, to achieve the best optimization effect, a good balance must be achieved between exploration and exploitation; however, current traditional algorithms cannot efficiently solve this problem.

[0005] Estimation by Distribution (EDA) is a typical metaheuristic method based on statistical probability models. It differs from other metaheuristic methods that generate new solutions through crossover and mutation operations, such as differential evolution, invasive weed optimization, and brainstorming optimization. EDA combines genetic algorithms and statistical learning methods to select dominant individuals in the population for modeling and estimation, generating the next generation of individuals. EDA models are typically based on Gaussian models, i.e., GEDA, which can extract and utilize effective information from hidden relationships between variables during the evolutionary process. The main drawbacks of traditional GEDA are: first, population diversity is easily lost during evolution, leading to local optima; second, the direction of population evolution deviates from the fitness improvement direction, affecting search efficiency; and third, the quality of the previous generation's solution affects the accuracy of the modeling.

[0006] To address the problems of GEDA and improve its efficiency in optimizing distributed assembly line congestion, algorithm improvement is essential. Liang Yongsheng, in "Enhancing Gaussian Estimation of Distribution Algorithm by Exploiting Evolution Direction with Archive," retained a certain number of high-quality solutions generated in previous generations and placed them in an archive. The solutions in the archive were used to estimate the covariance matrix of the Gaussian model, improving population diversity; however, both modeling accuracy and search efficiency were low. Ren Zhigang, in "Anisotropic adaptive variance scaling for Gaussian estimation of distribution algorithm," proposed the Anisotropic Adaptive Variance Scaling (AAVS) technique. This technique addresses the problems of rapid variance decay and poor directionality by adjusting the variance of different feature directions; however, it results in poor population diversity and cannot effectively balance exploration and development capabilities. In "Multimodal estimation of distributional algorithms", Yang Qiang proposed a multimodal EDA method. It improves the modeling accuracy of multimodal problems by using dynamic clustering adjustment strategy, generating offspring by hybrid distribution and adaptive local search strategy. However, the model framework is complex and difficult to implement, and the practicality of the technology is poor. Summary of the Invention

[0007] The purpose of this invention is to address the problems existing in the prior art. With the support of the "Innovation Star" project for postgraduate students funded by the Gansu Provincial Department of Education (Project No. 2023CXZX-476), this invention conducts relevant research and proposes technical solutions. It improves the problem from both the algorithm and problem-solving perspectives, designing a distribution estimation algorithm based on enhanced Kalman filtering and a history learning mechanism to optimize the scheduling problem of a distributed assembly line with congestion. The specific process is as follows: Figure 1 As shown, the improved algorithm effectively solves the problems of population diversity loss, perpendicularity between evolutionary direction and fitness direction, and the impact of differential solutions on modeling accuracy in traditional EDA. The effectiveness of the optimization system has been verified by theory and experiments. Furthermore, it has been applied to a distributed assembly line scheduling system to minimize the assembly completion time of the factory, achieving good optimization results.

[0008] To achieve the above objectives, the technical solution adopted by this invention is as follows: a distributed assembly line scheduling optimization system, comprising an initialization phase, a workpiece processing phase based on a prediction process, a workpiece processing phase based on a repair process, and a product assembly phase, as follows: Figure 2 As shown, multiple different workpieces are rationally assigned to machines in different factories.

[0009] In the production process, n workpieces to be processed are assigned to F factories for processing. Each processing factory is a congested assembly line workshop consisting of m machines, and each workpiece must go through m processes to complete its processing. In the assembly stage, the n processed workpieces are assembled on one assembly machine. Note that the product is only assembled after all workpieces belonging to a particular product have been processed. Figure 3 This is a Gantt chart for the machining and assembly process. Different numbers represent different workpieces, and the length of different workpieces represents the corresponding production and assembly time. This optimizer aims to minimize the assembly completion time.

[0010] The initialization phase of the production process is as described in steps 1-3:

[0011] Step 1: Input the selected selection rate τ, number of individuals NP, and stopping criterion;

[0012] The method of truncation selection is adopted, that is, to find a number of workpieces with the smallest fitness value from the solution space with a certain selection rate, and to find the dominant solution individuals from all workpieces to guide the direction of subsequent iterations.

[0013] Step 2: Randomly initialize individuals;

[0014] Step 3: Perform the first iteration. Kalman filtering requires prior information, so the first iteration is performed separately, specifically including sub-steps A1-A8:

[0015] Step A1: Select the optimal τ*NP individuals as the modeling individuals POP mod (1);

[0016] Step A2: Find the optimal individual x pbest (1);

[0017] Step A3: Build a historical archive to store the selected dominant individuals;

[0018] Step A4: Calculate the fitness value of individuals in the archive;

[0019] Step A5: Calculate the mean μ(k) and covariance matrix C(k) of the individuals in the archive;

[0020]

[0021]

[0022] Where, x i (k) represents the individual in the archive, and k is the iteration number. Here, k = 1.

[0023] The modeling data is repaired by estimating the mean and covariance matrix of the advantageous workpiece set from the historical archive, thereby improving the model accuracy. The advantageous workpieces obtained by different operation methods are put into the archive, which contains a wealth of information on excellent individuals.

[0024] Step A5: Construct a probability model based on the improved mean μ(k) and covariance matrix C(k);

[0025] Step A6: Sampling to obtain the POP of the observed individuals obs (1);

[0026] Step A7: Merge individual POPs in the model mod (1) and observed individual POP obs (1) A new population NEW is formed POP(1) ;

[0027] Step A8: In order to further improve the model accuracy, calculate the repair boosting matrix π(1) according to formula (3), and add the prediction accuracy obtained in each generation to the repair process of the filtering stage;

[0028]

[0029] in, This represents the dominant individuals selected from the first generation of observed individuals.

[0030] Step 4: Start iterating from the second generation until the maximum number of iterations is met;

[0031] Step 5: Perform the workpiece machining stage based on the Kalman filter prediction process:

[0032] The probability distribution used in this invention is based on Gaussian, and the update process of its conditional probability density is minimum variance estimation.

[0033] Making quantitative inferences about variables based on observational data is an estimation problem, especially the state estimation of dynamic behavior, which enables the estimation and prediction of the current state.

[0034] Kalman filtering makes informed predictions about the system's next move, always pointing to the true state.

[0035] The calculation of the original predicted values ​​and observed values ​​is shown in Formula 4-5:

[0036]

[0037] Y(k)=B*X(k-1)+ε(k) Formula 5

[0038] Where X(k-1) is the predicted value at the previous time step, X(k) is the predicted value at the current time step, Y(k) is the observed value at the current time step, and ε(k) is the noise component at the time of observation. Here, A is the prediction error matrix, B is the state transition matrix from the previous state to the current state, and H is the observation coefficient matrix.

[0039] Introduce the best solutions from the archive into the prediction function, where step 5 includes sub-steps B1-B4:

[0040] Step B1: Based on the Kalman filter concept, this invention predicts the relevant individual information from the previous generation to obtain the predicted individual POP. pre (k), predicting an individual's attainment according to Formula 6:

[0041]

[0042] in, It predicts individuals, C is the identity matrix, and x best (k-1) represents the best individual from the previous generation, and D is a diagonal matrix with its diagonal elements randomly generated. π i (k-1) is the amount of repair and improvement for the i-th individual in generation k-1, defined as in Formula 7:

[0043]

[0044] Step B2: Find the optimal individual x from the historical archive. pbest ;

[0045] Step B3: Combine the previous generation individual POP(k-1) with the predicted individual POP pre The advantageous workpieces in (k) are used for modeling;

[0046] Step B4: Obtain the observation workpiece for observation;

[0047] The workpiece processing stage based on the repair process is as follows: Steps 6-8:

[0048] Step 6: If the conditions for the first repair are met, perform the first repair operation;

[0049] During the evolutionary process, a repair operation is required when the following situations occur:

[0050] Step C1: If the covariance matrix is ​​not positive definite, perform a repair operation;

[0051] Step C2: The difference between the maximum and minimum values ​​of each individual variable is less than 10. -4 Perform a repair operation;

[0052] Step C3: Continuously adjust the percentage of each type of individual;

[0053] The specific repair method is as follows: Under normal circumstances, as we know from the previous evolutionary process, it is guided by elitism, selecting the dominant artifact from two types of individuals according to the selection rate τ for modeling, thus influencing the next generation of individuals. If at least one of the above two situations occurs, the selection rate of the predicted individual is increased by 10%, while that of the other type of individual is decreased by 10%. Similarly, if no better solution can be obtained for two consecutive generations, the selection rate of the predicted individual is increased by another 10%, while that of the other type continues to decrease by 10%. In both of these situations, the percentage of the predicted individual increases by 10% each time, while that of the other type of individual decreases by 10%. If the percentage exceeds 100%, 100% is subtracted, and the remaining percentage is taken; or if it is less than 0%, 100% is added.

[0054] For example: if the predicted individual percentage reaches 95% and the other category accounts for 5%, and if the conditions for a repair are met, then the selection rate of the predicted individual after repair is ((95%+10%)-100%)=5%; the selection rate of the other individuals is ((5%-10%)+100%)=95%. And so on.

[0055] Step 7: Perform the second repair operation, specifically including steps D1-D4:

[0056] Step D1: Model the dominant artifacts from the archived predicted individuals, repaired individuals, and previous generation individuals;

[0057] Step D2: Obtain the individual POP for modeling mod ;

[0058] Step D3: Calculate the population size for each generation according to the adaptive adjustment strategy;

[0059] In the early stages of evolution, a larger search space is needed to avoid getting trapped in local optima, requiring better exploration capabilities. In the later stages, the focus shifts to exploitation capabilities, and the population size can be appropriately reduced to accelerate convergence and obtain the optimal solution. Unlike the conventional approach of fixing the population size, adaptive strategies can better balance exploration and exploitation capabilities.

[0060] By adaptively and dynamically adjusting the population size according to the characteristics of different stages, the waste of computational resources is avoided. The adjustment rule is as shown in Formula 8:

[0061]

[0062] Formula 8

[0063] Among them, NP max NP min These represent the maximum and minimum number of values, respectively; nfes and max_nfes represent the number of fitness evaluations for the current generation and the maximum number of evaluations, respectively; round is a function.

[0064] Initially, the population size is maximized, and during the iteration process, it is linearly reduced until it reaches the minimum value.

[0065] Based on the characteristics of GEDA, its covariance matrix contains 0.5*(n 2 +n) estimated parameters; therefore, the minimum value is taken as 0.5*(n) 2 +n), the maximum value is obtained by calibration test according to the parameters in the experimental section.

[0066] Step D4: Sampling to obtain the observed individual POPs obs ;

[0067] Step D5: Merge the dominant artifacts among the modeled individuals, observed individuals, and repaired individuals;

[0068] Step D6: Model the mean and variance based on the improved Kalman filter;

[0069] Using dominant solutions to guide the iteration speed of EDA and reduce the consumption caused by randomness, elites guiding the direction of population evolution is a very effective measure. The improved optimization method considers the specific knowledge information carried by dominant individuals and optimizes the distribution of individuals. To emphasize this idea, a weighted mean is adopted based on the fitness values ​​of dominant individuals in the archive, which enhances the influence of the fitness values ​​of dominant individuals.

[0070] From the definition of the mean in formula (1), the corrected mean is expressed as:

[0071]

[0072] The improved mean can yield a more promising solution space for EDA.

[0073] Based on the improved mean, the corresponding improved formula for the covariance matrix is:

[0074]

[0075] The above revisions further fully utilize the potential search area and accelerate search efficiency and accuracy.

[0076] The purpose of improving the mean is to find potential search centers. This linear operation can lead to better solutions because it follows the direction of fitness value improvement.

[0077] The correction amount for the mean is:

[0078]

[0079] Clearly, the mean will tend to move in a more promising direction. After the above operations, the change in the mean is brought closer to the search center, quickly finding a better solution along the evolutionary direction. Furthermore, the repair phase uses the optimal solution as a guide, accelerating the EDA search process and reducing the time consumed by random searches.

[0080] The improved EDA based on Kalman filtering expands the search space and improves the search direction, thus achieving a better balance between the exploration and development capabilities of EDA. Figure 4 This is the distribution of the solution after Kalman filtering. For example... Figure 4 As shown, FID represents the fitness improvement direction, and ED represents the principal axis direction. The two directions are parallel to achieve the optimal search.

[0081] Step D7: Generate a new population NEW_POP;

[0082] Step D8: Calculate the repair gain coefficient g;

[0083] The search process of EDA is generally not linear and requires transformation. This transformation is achieved by repairing and improving the matrix and the system gain coefficient. The system gain coefficient g is shown in Equation 12.

[0084]

[0085] Where, f(x) pbest (k-1)), These are the fitness values ​​of the previous generation's best individual and the i-th individual, respectively.

[0086] The gain coefficient utilizes information from the repair and improvement matrix and the fitness difference between the previous generation individuals and the best individual as a weighting coefficient for individuals in the repair phase.

[0087] Step D9: Obtain the repaired individual POP rev ;

[0088]

[0089] More specifically, Figure 5 It refers to the composition and evolutionary direction of the solution after repair. For example... Figure 5 As shown, the influence of prediction and repair operations on the distribution of solutions can be seen. The pentagram represents the optimal individual, the square represents the repaired individual, the triangle represents the predicted individual, and the circle represents the original individual.

[0090] Step D10: Calculate the repair lifting matrix π;

[0091] Step D11: Find the global optimal solution x gbest ;

[0092] Step D12: Iterate through the loop until the stopping criterion is met;

[0093] Step 8: Assembly begins immediately after the last workpiece belonging to the product is completed. The assembly sequence is determined based on the completion time of each workpiece.

[0094] To address the following issues that hinder the performance of traditional optimization methods in the distributed assembly line scheduling problem, which result in poor performance and inability to quickly find the optimal workpiece sequence: First, loss of population diversity leads to the system getting trapped in local optima; second, the population evolution direction deviates from the fitness improvement direction, reducing search efficiency; and third, the quality of previous solutions affects the accuracy of modeling. While existing improved EDA variants can solve one or two of these problems, their practicality is limited. This invention introduces Kalman filtering, a digital processing technique used in communication systems, into the EDA framework. Based on the specific characteristics of the distributed assembly line scheduling problem, the Kalman filtering process is further improved, and the advantages of historical learning mechanisms and population adaptive strategies are fully combined to optimize the production processing and assembly processes. Comparison with traditional optimization systems shows that this system can simultaneously and quickly solve the above problems, significantly improving system optimization performance. The specific advantages of the improvements implemented in this system are as follows:

[0095] 1. The introduction of an enhanced Kalman filter mechanism not only embeds observation, prediction, and repair operations into the EDA optimization framework, but also improves the probability model, increases the search range, and adjusts the search direction, making evolution faster and more efficient;

[0096] 2. Historical data archiving was added to the Kalman filter to increase population diversity;

[0097] 3. An enhanced Kalman filter mechanism is introduced into the probabilistic model. An improved enhanced Kalman filter mechanism is designed to address the specific problem characteristics. The prediction, observation, first repair, and second repair processes in the improved filtering process are applied to the workpiece processing stage, thereby improving the efficiency of solving for the optimal workpiece sequence with the minimum assembly completion time.

[0098] 4. An elite strategy was introduced into the repair boosting matrix to guide the evolution of the repair population and improve modeling accuracy. Furthermore, the enhancement information obtained from Kalman filtering was fed back to the EDA model through a history learning mechanism to increase population diversity and further improve the quality of the solution.

[0099] 5. The adaptive adjustment strategy reduces invalid searches, effectively balances the exploration and development capabilities of the optimization system, and is conducive to finding the workpiece sequence with the optimal objective;

[0100] 6. The effectiveness of the proposed optimization system was demonstrated on 30 different optimization problems and instances with varying numbers of factories, machines, workpieces, and products. Theoretically, it was also proven that the enhanced Kalman filter operation is beneficial to improving the search range and direction of EDA. Attached Figure Description

[0101] Figure 1 This is the proposed KFHLEDA flowchart.

[0102] Figure 2 This is a schematic diagram of the distributed assembly line scheduling problem.

[0103] Figure 3 This is an example diagram of a factory, machine, and workpiece in a distributed assembly line scheduling system.

[0104] Figure 4 It is the distribution of the solution after Kalman filtering.

[0105] Figure 5 It is the composition and evolutionary direction of the solution after repair.

[0106] Figure 6 These are the main effects plot and interaction plot of the parameters.

[0107] Figure 7 It is a box diagram (10D) of ten optimized systems.

[0108] Figure 8 It is a box diagram (30D) of ten optimized systems.

[0109] Figure 9 It is a box diagram (50D) of ten optimized systems.

[0110] Figure 10It is a box diagram (100D) of ten optimized systems.

[0111] Figure 11 The ranking is obtained through the Friendman test (10D).

[0112] Figure 12 The ranking (30D) is obtained through the Friendman test.

[0113] Figure 13 This ranking (50D) is obtained through the Friendman test.

[0114] Figure 14 The ranking is obtained through the Friendman test (100D). Detailed Implementation

[0115] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings, embodiments, and illustrative examples. It should be understood that the specific examples and illustrative examples described herein are for illustrative purposes only and are not intended to limit the invention.

[0116] like Figure 1 As shown, Figure 1 This is a flowchart of the optimization system proposed in this invention. In conjunction with the accompanying drawings and technical solutions, the pseudocode of the optimization system KFHLEDA of this invention is further provided below.

[0117]

[0118] A distributed assembly line scheduling optimization system for congested production lines, characterized by comprising: an initialization step, a workpiece processing step based on a prediction process, a workpiece processing step based on a repair process, and a product assembly step, specifically including the following steps:

[0119] Step 1: Initialization steps in the production process, specifically including steps A1-A5:

[0120] Step A1: Using the truncation selection method, find several workpieces with the smallest fitness values ​​from the solution space with a certain selection rate, and find the dominant individuals from all workpieces to guide the direction of subsequent iterations;

[0121] Step A2: Find the optimal solution;

[0122] Step A3: Modeling and sampling to obtain the first generation of observed individuals;

[0123] Step A4: Merge the modeled individuals and the observed individuals to obtain a new individual;

[0124] Step A5: Calculate the first-generation correction and improvement matrix according to Formula 1, and incorporate the prediction accuracy of each generation into the correction process of the filtering stage to improve the model accuracy.

[0125]

[0126] Where π(1) is the first-generation correction and improvement matrix, π i (1) is the correction and improvement amount of the i-th individual in the first generation, x pbest (1) Represents the optimal workpiece of the first generation. Represents the dominant individuals among the first generation of observed individuals;

[0127] Step 2: Randomly initialize individuals;

[0128] Step 3: Perform the first iteration; Kalman filtering requires prior information, and the first iteration is performed separately; specifically, it includes sub-steps B1-B9:

[0129] Step B1: Select the optimal τ*NP individuals as the modeling individuals POP mod (1);

[0130] Step B2: Find the optimal x pbest (1);

[0131] Step B3: Build a historical archive to store the selected dominant individuals;

[0132] Step B4: Calculate the fitness value of individuals in the archive;

[0133] Step B5: Calculate the mean μ(k) and covariance matrix C(k) of the individuals in the archive:

[0134]

[0135] Where |X| is the number of individuals in the archive, x i (k) represents an individual in the archive;

[0136]

[0137] Where, x i (k) represents an individual in the archive, μ(k) is the mean of all individuals in the archive, and k is the number of iterations. Here, k = 1.

[0138] The mean and covariance matrix of the advantageous solution set from the historical archive are estimated to repair the modeling data, and the advantageous workpiece sequences obtained by different operation methods are put into the archive.

[0139] Step B6: Construct a probability model based on the improved mean μ(k) and covariance matrix C(k);

[0140] Step B7: Sampling to obtain the POP of the observed individuals obs (1);

[0141] Step B8: Merge modeled individual POPs mod (1) and observed individual POP obs (1) Forming a new individual NEW POP(1) ;

[0142] Step B9: Calculate the repair and enhancement matrix π(1) according to Formula 4, and add the prediction accuracy obtained in each generation to the repair process in the filtering stage to further improve the model accuracy;

[0143]

[0144] in, This represents the selected dominant workpiece sequence from the first generation of observed individuals;

[0145] Step 4: Iterate from the second generation according to the process from Step 5 to Step 8 until the stopping criterion is met;

[0146] Step 5: Execute the workpiece machining steps based on the Kalman filter prediction process;

[0147] The probability distribution is based on a Gaussian model, and the update process of the conditional probability density is a minimum variance estimation.

[0148] The calculations for the original predicted values ​​and observed values ​​are shown in Formulas 5 and 6:

[0149]

[0150] Y(k)=B*X(k-1)+ε(k) Formula 6

[0151] Where X(k-1) is the predicted value at the previous time step, X(k) is the predicted value at the current time step, Y(k) is the observed value at the current time step, and ε(k) is the noise component at the time of observation. Here, A is the prediction error matrix, B is the state transition matrix from the previous state to the current state, and H is the observation coefficient matrix.

[0152] Incorporate the best workpiece sequences from the archive into the prediction function, including sub-steps C1-C4:

[0153] Step C1: Based on the Kalman filter concept, predict the relevant workpiece information from the previous generation to obtain the predicted individual POP. pre (k), the predicted individual is obtained according to Formula 7:

[0154]

[0155] in, It predicts individuals, C is the identity matrix, and x best (k-1) represents the best individual from the previous generation, and D is a diagonal matrix randomly generated from its diagonal elements; π i (k-1) is the amount of repair and improvement for the i-th individual in generation k-1, defined as in Formula 8:

[0156]

[0157] Step C2: Find the optimal x from the historical archive. pbest ;

[0158] Step C3: Combine the previous generation individual POP(k-1) and the predicted individual OPO pre The dominant workpiece sequence in (k) is used for modeling;

[0159] Step C4: Obtain the observation solution for observation;

[0160] Step 6: Perform workpiece machining steps based on the repair process:

[0161] Step D1: If the conditions for the first repair are met, perform the first repair operation; during the evolution process, when situations B1.1-B1.3 occur, a repair operation is required:

[0162] Step D1.1: If the covariance matrix is ​​not positive definite, perform a repair operation;

[0163] Step D1.2: The difference between the maximum and minimum values ​​of each variable is less than 10. -4 Perform a repair operation;

[0164] Step D1.3: Continuously adjust the percentage of each type of individual;

[0165] Step D1.4: Select dominant individuals from the two populations according to the selection rate τ for modeling, influencing the next generation;

[0166] Step D1.5: If at least one of the above two situations occurs, the selection rate of the predicted individual will increase by 10%, and the selection rate of the other individual will decrease by 10%.

[0167] Step D1.6: If no better solution can be obtained for two consecutive generations, the selection rate of the predicted individual is increased by 10%, while the selection rate of the other group is decreased by 10%.

[0168] Step D1.7: When both of the above situations occur simultaneously, the percentage of the predicted individual increases by 10% each time, while the percentage of the other group of individuals decreases by 10%; if the percentage exceeds 100%, subtract 100% and take the remaining percentage; or if it is less than 0%, add 100%.

[0169] Step D1.8: For step D1.7, if the predicted individual proportion reaches 95%, the other category accounts for 5%; if the conditions for one repair are met, the selection rate of the predicted after repair is ((95%+10%)-100%)=5%; the selection rate of other individuals is ((5%-10%)+100%)=95%, and so on.

[0170] Step D2: Perform the second repair operation, specifically including steps D2.1-D2.14:

[0171] Step D2.1: Model the dominant individuals from the archived predicted individuals, repaired individuals, and previous generation individuals;

[0172] Step D2.2: Obtain the individual POP for modeling mod ;

[0173] Step D2.3: Initially, maximize the size of each individual, and then linearly decrease it during the iteration until it reaches the minimum value;

[0174] Step D2.4: During the iteration process, the adjustment rule is as follows:

[0175]

[0176] Among them, NP max NP min These represent the maximum and minimum number of individuals, respectively; nfes and max_nfes represent the number of fitness evaluations in the current generation and the maximum number of evaluations, respectively; round is a function;

[0177] Step D2.5: Considering the characteristics of GEDA, its covariance matrix contains 0.5*(n 2 +n) estimated parameters; therefore, the minimum value is taken as 0.5*(n) 2 +n);

[0178] Step D2.6: Sampling to obtain the observed individual POPs obs ;

[0179] Step D2.7: Merge the dominant artifacts among the modeled individuals, observed individuals, and repaired individuals;

[0180] Step D2.8: Model the mean and variance based on the improved Kalman filter;

[0181] The corrected mean is expressed as follows:

[0182]

[0183] Where |X| is the number of individuals, x i (k) is the i-th workpiece, x pbest(k) is the global optimum, f(x) i (k) is x i The fitness value of (k), π i (k) is the i-th value in the correction and improvement matrix;

[0184] Based on the improved mean, the corresponding improved formula for the covariance matrix is:

[0185]

[0186] in, It is the corrected mean;

[0187] The correction amount for the mean is:

[0188]

[0189] Where μ(k) is the mean value before correction;

[0190] Step D2.9: Generate a new workpiece sequence NEW_POP;

[0191] Step D2.10: Calculate the repair gain coefficient g;

[0192] The system gain coefficient g is shown in Formula 13:

[0193]

[0194] Where, f(x) pbest (k-1)), These are the fitness values ​​of the previous generation's best individual and the i-th individual, respectively, where m is the number of individuals selected, and π is the fitness value. i (k-1) is the i-th value in the previous generation's correction and improvement matrix, g i (k) is the gain coefficient of the i-th individual in the k-th generation;

[0195] Step D2.11: Obtain the repaired individual POP res ;

[0196]

[0197] i = 1, ..., m Formula 14;

[0198] in, It is the i-th repaired individual in the k-th generation. It is the i-th predicted individual in the k-th generation. It is the fitness of the best individual among all predicted individuals. It is the fitness of the i-th predicted individual, π i (k) is the i-th value in the correction and improvement matrix;

[0199] Step D2.12: Calculate the repair lifting matrix π;

[0200] Step D2.13: Find the globally optimal x gbest ;

[0201] Step D2.14: Iterate repeatedly until the stopping criterion is met;

[0202] Step 7: Assembly begins immediately after the last workpiece belonging to the product is completed; that is, the assembly sequence of the product is determined based on the completion time of each workpiece; the factory with the longest assembly time is designated as the critical factory, and the completion time of the last workpiece of the product is the start time of assembly; specifically, this includes three rules as shown in steps E1-E3:

[0203] Step E1: In each processing plant, all parts of the same product should be processed together, and the start time of assembly of the product should be advanced as early as possible;

[0204] Step E2: Once all parts belonging to the same product have been processed in the processing plant, the assembly process begins immediately;

[0205] Step E3: Workpieces belonging to the same product should be assigned to different factories for parallel processing, which helps to advance the start time of product assembly.

[0206] First, the optimization method proposed in this invention is verified using a demonstrative example.

[0207] The covariance matrix determines the search range and direction. As mentioned earlier, individuals generated by traditional Gaussian distribution EDA tend to move in a worse direction, perpendicular to the direction of fitness improvement. Through the modified enhanced Kalman filter operation described above, potentially repaired solutions can be obtained. It is easy to see from the modified mean and covariance formulas that the principal axis of the probability density ellipsoid of the repaired covariance matrix will move towards... The direction is tilted in the direction of fitness improvement.

[0208] Step 1: The covariance matrix after the enhanced Kalman filter operation is the rank-1 correction of the original covariance matrix;

[0209]

[0210] It is the rank-1 correction of C(k).

[0211] As can be seen from the proof in the following two steps, the estimation method of enhanced Kalman filtering adaptively expands the search range of EDA and adjusts the search direction.

[0212] Step 2: This operation is based on a set of excellent samples. For a given set, if... The coverage of the probability density ellipsoid after Kalman filtering correction is greater than or equal to the coverage of the probability density ellipsoid of the covariance matrix without correction.

[0213] Step F1: In a Gaussian distribution, the principal axes of the probability density ellipsoid are the eigendirections of the covariance matrix, and the length of the semi-axis is equal to the square root of the eigenvalue. The longer the major axis, the larger the coverage of the probability density ellipsoid. The eigenvalues ​​of the improved estimated covariance matrix and the unimproved original covariance matrix are defined as follows: and λ, λ=(λ1, λ2,…,λ i Let i = 1, 2, ..., m. From the definition of the covariance matrix, it is positive semi-definite. Therefore, the covariance matrix can be decomposed into C(k) = ψ(k) * Λ(k) * ψ(k). T Where Λ(k) is the set of eigenvalues. It is an orthogonal matrix;

[0214] Step F2: Let Then we have:

[0215]

[0216] Based on formula Lemma 1.4 from "The Inverse Problem of Rank-1 Modification of Real Symmetric Matrices" (2003), Characteristic polynomial and The characteristic polynomials are equal.

[0217] Step F3:

[0218]

[0219] E is the identity matrix.

[0220] Step F4:

[0221] Let λ = 0, then we have:

[0222]

[0223] This proves that the coverage of the probability density ellipsoid after repair is indeed larger than before.

[0224] Step 3: For the set of selected excellent sample combinations, if Then the angle between Δ(k) and the direction of the probability density ellipsoid after Kalman filtering repair is less than or equal to the angle between the direction of the principal axis of the probability density ellipsoid before repair.

[0225] Step G1: It is necessary to prove that after the repair, the direction of fitness improvement moves towards the optimal solution;

[0226] definition Maximum eigenvalue The corresponding feature vector is The largest eigenvalue λ of Λ(k) m The corresponding eigenvector is φ(k), which is obvious. Λ(k) and Λ(k) are the orthogonal transformations of the repaired covariance matrix and the unrepaired covariance matrix, respectively.

[0227] Therefore, it is only necessary to prove

[0228] Step G2: The angle between vectors can be acute or obtuse. Here, the angle is represented by an acute angle, i.e.

[0229] By the definition of eigenvalues, we have:

[0230]

[0231] Step G3: Transformation;

[0232]

[0233] Step G4: Use λ in Formula 17 replace;

[0234]

[0235] Step G5: Discussion of Case 1:

[0236] when At that time, because Obviously, The statement in step 3 is true;

[0237] Step G6: Discussion of Case 2: When At that time, that is From formula 15 in step 1, we can see that... If Λ(k) is a rank-1 correction and the product of two vectors is not zero, then neither vector can be equal to 0. Therefore:

[0238] Step G7: Transform formula 21, then we have:

[0239]

[0240] Step G8: Transformation:

[0241]

[0242] Step G9: Calculate the angle between the vectors:

[0243]

[0244]

[0245] Step G10: Draw a conclusion;

[0246]

[0247] Clearly, the evolutionary direction of the repaired solution has been improved.

[0248] Secondly, the system proposed in this invention was verified in Example 1.

[0249] The proposed optimized system KFHLEDA was tested on the IEEE CEC (2017) test set, demonstrating its effectiveness.

[0250] Step 1: Parameter analysis;

[0251] As mentioned earlier, this optimization system involves three parameters: the initial population size NP. max The selection rate τ and the size of the historical archive H.

[0252] If the initial population size is too large, the convergence speed will be too slow, wasting computational resources. Conversely, population diversity will decrease, further impairing exploration capabilities. The selection rate τ affects the selection of dominant solutions and directly determines the quality of solutions in each generation. If it is too small, the influence of dominant individuals on subsequent iterations weakens, making it easy to get trapped in local optima and affecting modeling accuracy. If it is too large, the quality and efficiency of the selected solutions will decrease.

[0253] The size of the historical archive H determines how much historical information will be retained; that is, the degree to which advantageous solutions are learned so that solution acquisition depends not only on the current state but also on useful historical information. If the value of H is too large, the search space is enormous, containing too much disadvantageous information, which is detrimental to evolution and slows down the convergence speed. Similarly, if it is too small, the main information in the archive cannot be fully learned.

[0254] Step I1: Different combinations of these three parameters produce different performance results. The total number of parameter combinations is 4*4*4 = 64. Each parameter combination is run on 30 optimization problems until the termination condition is met.

[0255] Step I2: Analyze the differences in the results of the Design of Experiments (DOE) and multivariate analysis of variance (ANOVA) to obtain the optimal combination of parameters;

[0256] Step I3: The value ranges of the three parameters are respectively: NP max ={1000, 2000, 3000, 4000}, τ = {0.15, 0.25, 0.35, 0.45}, H = {100, 200, 300, 400};

[0257] Step I4: After parameter calibration analysis, the main effects plot and interaction plot are obtained as follows: Figure 6 And as shown in Table 1;

[0258] Table 1

[0259]

[0260]

[0261] Step I5: Obtain the optimal parameter combination;

[0262] NP max The p-value is less than 0.05, and the corresponding F-value is also the largest, indicating that within the 95% confidence interval, NP... max It has a significant impact. τ and h greater than 0.05 indicate an interaction. For example... Figure 6 As shown, when NP max The optimal result is obtained when NP = 3000. max When the value is 3000, the worst value is obtained. NP max The smaller the value, the more detrimental it is to the system's detection capability, and the worse the performance. A larger NP... max This will waste computational resources. Optimal results can be obtained when τ = 0.35.

[0263] Therefore, NP max =3000, τ=0.35, H=200 is the optimal combination.

[0264] Step 2: Compare with other optimization methods to demonstrate the effectiveness of the proposed optimization method in continuous optimization problems;

[0265] Step J1: Select the optimization system for comparison;

[0266] Two classic optimization systems were selected: APBIL from "A Short Survey on Population-Based Incremental Learning Algorithm APBIL" and CMA-ES from "A global surrogate assisted CMA-ES".

[0267] Three classic differential evolution-based optimization systems were selected: LSHADE from "Improving the search performance of SHADE using linear population size reduction", jSO from "Single objective real-parameter optimization: Algorithm jSO", and PID-DE from "Aproportional, integral and derivative differential evolution algorithm for global optimization".

[0268] Three recently proposed EDA optimization systems were selected: EDA2 from "Enhancing Gaussian Estimation of Distribution Algorithm by Exploiting Evolution Direction with Archive", IWOEDA from "A hybrid evolutionary approach based on the invasive weed optimization and estimation distribution algorithms", ACSEDA from "An Adaptive Covariance Scaling Estimation of Distribution Algorithm", and AEDDDE from "Adaptive Estimation Distribution: Distributed Differential Evolution for Multimodal Optimization Problems".

[0269] Step J2: Experiments were conducted in 10 (10D), 30 (30D), 50 (50D) and 100 (100D) dimensions, running independently 51 times;

[0270] Step 3: Use the error value (the difference between the optimal solution and the known global optimum in each run) to evaluate the performance of the optimization system. Set the conditions and experimental environment of all optimization systems to be the same;

[0271] Step 4: Obtain the mean and variance results. The comparison results between the proposed optimized system and the comparative optimized system on 10D, 30D, 50D, and 100D are shown in Figures 2-5 respectively;

[0272] Table 2

[0273] Comparison results (10D) between the proposed optimized system (KFHLEDA) and the comparative optimized system.

[0274]

[0275] Table 3 shows the comparison results between the proposed optimization system (KFHLEDA) and the comparative optimization methods (30D).

[0276]

[0277]

[0278] Table 4 shows the comparison results between the proposed optimization system (KFHLEDA) and the comparative optimization methods (50D).

[0279]

[0280] Table 5 shows the comparison results between the proposed optimization system (KFHLEDA) and the comparative optimization methods (100D).

[0281]

[0282]

[0283] Step 5: Analysis of experimental results;

[0284] Step K1: Evaluate the overall system performance using the values ​​in Table 2-5, which lists the mean and variance of all 10 optimization systems for the CEC 2017 test set at 10D, 30D, 50D, and 100D. The bolded numbers in each row indicate that the value for that optimization problem is minimized, corresponding to the best overall performance.

[0285] Step K2: Figure 7-10 Box plots show the stability of the optimized system, such as Figure 7-10 As shown in the figure, the horizontal axis represents all optimization methods, and the vertical axis represents the error between the candidate solution and the optimal solution. The smaller the value of the vertical axis, the better the performance. It can be seen that, compared with other optimization methods, this optimization system (KFHLEDA) has the best overall stability.

[0286] The superior performance of this optimization system (KFHLEDA) is attributed to the following factors: the improved Kalman filter operation enhances modeling accuracy; the elitist strategy employed during the repair phase guides the population towards the optimal direction; and the introduction of a history learning mechanism increases population diversity, thereby improving the convergence accuracy of the optimization method.

[0287] Step K3: Perform the Friedman test.

[0288] The performance of all optimization methods was evaluated using the Friedman test, and statistical conclusions were obtained. The average ranks of the ten optimization methods on 10D, 30D, 50D, and 100D are shown in Table 6-9. Figure 11-14 In the diagram, the horizontal axis represents all optimized systems, and the vertical axis represents the average rank. Figure 11-14 The test results demonstrate the advantages of this optimization system, with rankings across different dimensions showing that this optimization system (KFHLEDA) ranks best. Except for LSHADE and jSO on 10D, EDA2 on 30D, and AEDDDE on 50D, within the 90% and 95% confidence intervals, this optimization system (KFHLEDA) shows no significant difference from most optimization systems on 10D, 30D, and 50D. Furthermore, this optimization system (KFHLEDA) shows no significant difference from other optimization methods on 100D.

[0289] Table 6 Friedman Test (10D)

[0290]

[0291] Table 7 Friedman Test (30D)

[0292]

[0293] Table 8 Friedman Test (50D)

[0294]

[0295] Table 9 Friedman Test (100D)

[0296]

[0297] Step K4: Perform the Wilcoxon test.

[0298] The rankings obtained from the Wilcoxon test are shown in Table 10. The significance level α was set to 0.05 and 0.1, respectively. The symbols “+”, “-”, and “=" indicate that the performance of this optimization system (KFHLEDA) is better than, worse than, or approximately the same as the corresponding optimization system, respectively. R+ and R- indicate that the rank sum of this optimization system (KFHLEDA) is better than or worse than the comparison optimization method. If the p-value is less than α, it indicates that the difference is significant, and is represented by “yes”. As shown in Table 10, except that this optimization system (KFHLEDA) has no significant difference from LSHADE and jSO at 10D (α = 0.05, 0.1) and EDA2 at 30D (α = 0.05), the others have significant differences at 10D, 30D, 50D, and 100D (α = 0.05, 0.1).

[0299] Table 10 Rankings obtained from the Wilcoxon test

[0300]

[0301]

[0302] The results of the two nonparametric tests show that, compared with other comparative optimization systems, our optimization system (KFHLEDA) performs best at 10D, 30D, 50D, and 100D.

[0303] Step K5: Draw a conclusion.

[0304] The experimental results above demonstrate that the optimization system (KFHLEDA) designed based on problem characteristics is effective. The enhanced Kalman filtering mechanism, history learning mechanism, and adaptive adjustment strategy employed in this invention effectively improve the system's performance. The Kalman filter feeds enhanced information back to the EDA model through the history learning mechanism. The improved optimization system enhances the quality of the results and effectively balances exploration and development capabilities. Therefore, it possesses significant competitive advantages.

[0305] Step K6: Analyze the effectiveness of each strategy in this optimization system using effective components.

[0306] This optimization system (KFHLEDA) incorporates three key strategies: Kalman filtering, a history learning mechanism, and an adaptive adjustment strategy.

[0307] The effectiveness of these three strategies is evaluated below. Three KFHLEDA variant systems are proposed, denoted as KFHLEDA1 optimized system, KFHLEDA2 optimized system, and KFHLEDA3 optimized system, representing KFHLEDA with enhanced Kalman filter removed, KFHLEDA with historical learning mechanism removed, and KFHLEDA with adaptive adjustment strategy removed, respectively.

[0308] (1) KFHLEDA1: This variant eliminates the enhanced Kalman filtering strategy while retaining the other operations of the optimized system (KFHLEDA). The effect of the enhanced Kalman filtering is shown by comparison with this optimized system. The mean and variance of the EDA model are updated according to formulas (1)-(2) instead of according to formulas (9)-(10) of the filtering operation, i.e. the mean and variance are guided by the dominant information in the historical archive.

[0309] (2) KFHLEDA2: This variant omits the history learning mechanism but retains other operations of this optimization system (KFHLEDA), namely, the enhanced Kalman filter operation no longer modifies individuals based on historical archives, but based on information obtained from the previous operation.

[0310] (3) KFHLEDA3: This variant removes the adaptive adjustment strategy but retains other operations of the original optimization system (KFHLEDA). Therefore, the population size update is no longer an iterative adaptive adjustment, but a fixed value set to 2000.

[0311] The results of comparing this optimization system (KFHLEDA) with three variants on 30 optimization problems are shown in Table 11.

[0312] Table 11

[0313] Performance comparison of four optimization systems

[0314]

[0315] Compared to the three improved systems, our optimized system (KFHLEDA) exhibits the best overall performance. KFHLEDA1 performs worse than KFHLEDA2 and KFHLEDA3, while both KFHLEDA2 and KFHLEDA3 are worse than our optimized system (KFHLEDA). KFHLEDA1 performs the worst due to the lack of a filtering mechanism, while KFHLEDA3 achieves the best results through Kalman filtering and history learning mechanisms. These results demonstrate the effectiveness and importance of the selected strategy.

[0316] The adaptive adjustment strategy enables the proposed optimization system to benefit both early exploration (avoiding premature convergence) and later development (avoiding ineffective searches and increasing evolutionary speed). The historical learning mechanism helps the population explore hidden information in the archives during evolution, guiding it towards potentially promising regions. In the prediction, observation, and secondary correction operations of the enhanced Kalman filter, an elitist strategy is incorporated to guide the population out of local optima, enriching population diversity and preventing stagnation. All of these strategies balance exploration and development capabilities.

[0317] Finally, the optimization system proposed in this invention was verified in Example 2 with different numbers of factories, machines, workpieces, and products, demonstrating that the optimization system designed in this invention can minimize assembly completion time and improve production efficiency.

[0318] To demonstrate the effectiveness of the distributed assembly line scheduling optimization system designed in this invention, it was verified on test cases with 8, 12, 16, 20, and 24 workpieces; 2, 3, and 4 machines; 2, 3, and 4 factories; and 30, 40, and 50 products. This optimization system was then compared with two existing optimization systems, ILS and IG.

[0319] Step 1: Use the Mean Relative Percentage Deviation (ARPD) for evaluation. ARPD is defined as:

[0320]

[0321] Among them, C i This is the result of the i-th optimized system in the current instance. R is the number of independent runs. C best This is the best result for the current instance.

[0322] Table 12

[0323] Performance comparison of three optimization systems

[0324]

[0325] It can be seen that the ARPD value of the system of the present invention is significantly smaller compared to the other two optimized systems.

[0326] Step 2: The Wilcoxon test was used to analyze that the optimized system designed in this invention is significantly different from the two optimized systems ILS and IG.

[0327] Table 13

[0328] Wilcoxon test of three optimization systems

[0329]

[0330] Step 3: The Friedman test was used to analyze the significant differences between the optimized system designed in this invention and the two optimized systems ILS and IG.

[0331] Table 14

[0332] Friedman test for three optimization systems

[0333]

[0334]

[0335] The test results all demonstrate that the optimized system of this invention is significantly different from the other two optimized systems and can significantly improve production efficiency.

[0336] The basic principles, main features, and advantages of the present invention have been described above in conjunction with the accompanying drawings. For those skilled in the art, modifications or variations can be made without departing from the principles of the present invention, and these improvements are also considered to be within the scope of protection of the present invention.

Claims

1. A distributed assembly blocking flow-shop scheduling optimization system, characterized in that, The workshop scheduling optimization system is used to execute the following steps: initialization step, workpiece processing step based on prediction process, workpiece processing step based on repair process, and product assembly step, specifically including the following steps: Step 1: Initialization steps in the production process, specifically including steps A1-A5: Step A1: Using the truncation selection method, find several workpieces with the smallest fitness values ​​from the solution space with a certain selection rate, and find the dominant individuals from all workpieces to guide the direction of subsequent iterations; Step A2: Find the optimal individual workpiece for the current generation; Step A3: Modeling and sampling to obtain the first generation of observed individuals; Step A4: Merge the modeled individuals and the observed individuals to obtain a new individual; Step A5: Calculate the first-generation correction and improvement matrix according to Formula 1, and incorporate the prediction accuracy of each generation into the correction process of the filtering stage to improve the model accuracy. Equation 1 in, It is the first generation of modified and improved matrix. It is the amount of correction and improvement for the i-th individual in the first generation. The best workpiece representing the first generation. Represents the dominant individuals among the first generation of observed individuals; Step 2: Randomly initialize individuals; Step 3: Perform the first iteration; Kalman filtering requires prior information, so the first iteration is performed separately. Step 4: Iterate from the second generation according to the process from Step 5 to Step 7 until the stopping criterion is met; Step 5: Execute the workpiece machining steps based on the Kalman filter prediction process; The probability distribution is based on a Gaussian model, and the update process of the conditional probability density is a minimum variance estimation. The calculations for the original predicted values ​​and observed values ​​are shown in Formulas 2 and 3: Formula 2 Formula 3 Official 3 in, This is the predicted value from the previous moment. It is the predicted value at the current moment. These are the observations at the current moment. It is the noise component during observation. It is the prediction error matrix. It is the state transition matrix from the previous state to the current state. It is the coefficient matrix of the input variables. It is the observation coefficient matrix; Step 6: Perform workpiece machining steps based on the repair process: Step B1: If the conditions for the first repair are met, perform the first repair operation; during the evolution process, when situations B1.1-B1.3 occur, a repair operation is required: Step B1.1: If the covariance matrix is ​​not positive definite, perform a repair operation; Step B1.2: The difference between the maximum and minimum values ​​of each variable is less than 10. 4 Perform a repair operation; Step B1.3: Continuously adjust the percentage of each type of individual; Step B2: Perform the second repair operation, specifically including steps B2.1-B2.12: Step B2.1: Model the dominant individuals from the archived predicted individuals, repaired individuals, and previous generation individuals; Step B2.2: Acquiring the modeled individual ; Step B2.3: Calculate the number of individuals in each generation according to the adaptive adjustment strategy; Step B2.4: Sampling, obtaining observed individuals ; Step B2.5: Merge the dominant artifacts among the modeled individuals, observed individuals, and repaired individuals; Step B2.6: Model the mean and variance based on the improved Kalman filter; The corrected mean is expressed as follows: Official 4; in, It is the number of individuals. It is the first i One workpiece, It is globally optimal. yes fitness value, It is the first in the modified and improved matrix i One possible value; Based on the improved mean, the corresponding improved formula for the covariance matrix is: Official 5; wherein is the corrected mean value; The correction amount for the mean is: Official 6; in, This is the mean before correction; Step B2.7: Generate a new workpiece sequence ; Step B2.8: Calculate the repair gain coefficient ; System gain coefficient As shown in Equation 7: Equation 7; in, , They are the best individual of the previous generation and the first generation. i The fitness value of each individual It is the number of individuals selected. It is the first in the previous generation's correction and improvement matrix. i Each possible value It is the first k The generation i Gain coefficient for each individual; Step B2.9: Obtaining a repaired individual ; Equation 8; in, It is the i-th repaired individual in the k-th generation. It is the i-th predicted individual in the k-th generation. It is the fitness of the best individual among all predicted individuals. It is the fitness of the i-th predicted individual. It is the i-th value in the improved matrix; Step B2.10: Compute the repair lifting matrix ; Step B2.11 : Find global optimum ; Step B2.12: Iterate repeatedly until the stopping criterion is met; Step 7: Assemble the product as soon as the last workpiece is finished; that is, determine the assembly sequence of the product based on the completion time of each workpiece.

2. The distributed assembly blocking flow shop scheduling optimization system of claim 1, wherein, Step 3 specifically includes sub-steps C1-C9: Step C1 : Selecting the best individual as modeling individual ; Step C2: Find the best ; Step C3: Build a historical archive to store the selected dominant individuals; Step C4: Calculate the fitness value of individuals in the archive; Step C5: Calculate mean of individuals within archive and covariance matrix : Official 9 in, The number of individuals in the archive. For individuals in the archive; Official 10 in, For individuals in the archive, It is the average of all individuals in the archive. k Here, is the number of iterations. k=1 ; The mean and covariance matrix of the advantageous solution set from the historical archive are estimated to repair the modeling data, and the advantageous workpiece sequences obtained by different operation methods are put into the archive. Step C6: Compute the mean values and covariance matrices , construct the probability model; Step C7: Sampling, obtaining observed individuals ; Step C8: Combine modeled individuals and observed individuals to form new individuals ; Step C9: Calculate the repair boosting matrix π(1) according to Formula 11, and add the prediction accuracy obtained from each generation to the repair process in the filtering stage to further improve the model accuracy: Official 11 in, , representing the selected dominant workpiece sequence among the first-generation observed individuals.

3. The distributed assembly blocking flow shop scheduling optimization system of claim 1, wherein, Incorporate the archived sequences of high-quality workpieces into the prediction function. Step 5 includes sub-steps D1-D4: Step D1: Based on the Kalman filter idea, the relevant workpiece information of the last generation is predicted to obtain the predicted individual The predicted individual is obtained according to formula 12: (k-1) Equation 12 in, It is to predict individuals. C It is the identity matrix. He was the best individual of the previous generation. D It is a diagonal matrix, randomly generated from its diagonal elements. (k-1) is the kth i The amount of repair and improvement for each individual in generation k-1 is defined as shown in Formula 13: Official 13 Step D2: Find the best from history archive ; Step D3: Transfer the previous generation individual and predicting individuals The advantageous workpiece sequence is used for modeling; Step D4: Obtain the observation solution for observation.

4. The distributed assembly blocking flow shop scheduling optimization system of claim 2, wherein, The specific repair steps in sub-step B1 of step 6 are as follows: Step E1: Select predicted and observed individuals from the archive according to the selection rate. Selecting dominant individuals for modeling to influence the next generation; Step E2: If the following occurs: the covariance matrix is ​​not positive definite or the difference between the maximum and minimum values ​​of a single variable is less than 10. 4 In either case, the predicted individual's selection rate will increase by 10%, while in the other case it will decrease by 10%. Step E3: If no better solution can be obtained for two consecutive generations, the selection rate of the predicted individual is increased by 10%, while that of the other group is decreased by 10%. Step E4: When both of the above situations occur simultaneously, the percentage of the predicted individual increases by 10% each time, while the percentage of the other group of individuals decreases by 10%; if the percentage exceeds 100%, subtract 100% and take the remaining percentage; or if it is less than 0%, add 100%. Step E5: For step E4, if the predicted individual proportion reaches 95%, and the other category accounts for 5%; if the conditions for one-time repair are met, then the selection rate of the repaired prediction is... The selection rate for other individuals is... And so on.

5. The distributed assembly blocking flow shop scheduling optimization system of claim 1, wherein, In sub-step B2.3 of step 6, an adaptive adjustment strategy is adopted, updating according to steps F1-F3: Step F1: Initially, maximize the size of the individuals, and then linearly decrease it during the iteration until it reaches the minimum value; Step F2: During the iteration process, adjust the rules as shown in Formula 14: Equation 14 in, , These represent the maximum and minimum number of individuals, respectively; , These represent the number of fitness evaluations for the current generation and the maximum number of evaluations, respectively; `round` is a function. Step F3: Combining the characteristics of GEDA, its covariance matrix includes There are several estimated parameters; therefore, the minimum value is taken as... .

6. The distributed assembly line scheduling optimization system according to claim 1, characterized in that, The steps performed by the workshop scheduling optimization system also include: Traditional GEDA-generated individuals evolve in a less desirable direction, perpendicular to the direction of fitness improvement. An enhanced Kalman filter operation can potentially correct this by adjusting the principal axis of the probability density ellipsoid of the corrected covariance matrix, based on the improved mean and covariance matrix. This axis tilts towards the direction shown in Equation 6, which is the direction of fitness improvement. For a given set, when... At that time, the volume of the probability density ellipsoid of the covariance matrix after Kalman filtering correction is greater than or equal to the volume of the probability density ellipsoid without Kalman filtering correction. The specific steps are G1-G6: Step G1: Prove that the covariance matrix improved by the enhanced Kalman filter operation is a rank-1 correction of the original covariance matrix: Equation 15 in, It is the number of workpiece sequences. It is the corrected mean. It is an improvement in quantity. This is the original mean before correction. It is the original covariance matrix; Improved covariance matrix It is the original covariance matrix Rank 1 correction; Step G2: In a Gaussian distribution, the major axis of the probability density ellipsoid is the characteristic direction of the covariance matrix, and the length of the semi-axis is equal to the square root of the eigenvalue. The longer the major axis, the larger the coverage area of ​​the probability density ellipsoid. Step G3: The eigenvalues ​​of the improved covariance matrix estimate and the original estimate are defined as follows: λ′ and λ ,in From the definition of the covariance matrix, it can be proven that the covariance matrix is ​​positive semi-definite. Therefore, the covariance matrix... Decomposed into ,in Composed of eigenvalues, It is an orthogonal matrix; Step G4: When Equation 16 holds; Equation 16 wherein is the modified covariance matrix, is the original covariance matrix, is the increment of the mean before and after modification; Step G5: the characteristic polynomial of is equal to that of Equation 17 wherein represents a unit matrix; Step G6: Let ,get: Equation 18.

7. The distributed assembly blocking flow shop scheduling optimization system of claim 6, wherein, For a selected set of excellent sample combinations, when hour, (k) The principal axis orientation angle of the probability density ellipsoid after Kalman filtering correction is less than or equal to the orientation angle without Kalman filtering. The verification process is as follows: steps H1-H6. Step H1: When hour, (k) When the conclusion that the principal axis orientation angle of the probability density ellipsoid after Kalman filtering correction is less than or equal to the orientation angle before Kalman filtering is true, it is equivalent to proving that the direction of fitness improvement is the direction of movement towards the optimal solution after enhanced Kalman filtering. Step H2: Definition Maximum eigenvalue The corresponding feature vector is , Maximum eigenvalue The corresponding feature vector is According to formula 16, and These are the orthogonal transformations of the corrected covariance matrix and the uncorrected covariance matrix, respectively. Step H3: There must be an acute angle and an obtuse angle between two vectors. Here, we use an acute angle, i.e. , ; Step H4: According to the definition of eigenvalues, Equation 19 holds true: Official 19 Step H5: After transformation, we get: Official 20 Step H6: Let , then Equation 21 Step H6.1 : When Because then the conclusion holds; Step H6.2: When , ; According to the proof of Equation 15, is a rank 1 modification of ; Step H6.3: Equation 21 is transformed into: Equation 22 Step H6.4: Further transformation yields Equation 23: Equation 23 Equation 24 in, Representative vector The cosine of the angle between them E represents the modulus, and E is the identity matrix. Official 25 That is, the conclusion is valid.

8. The distributed assembly line scheduling optimization system according to claim 1, characterized in that: In step 7, the factory with the longest assembly time is designated as the critical factory, and the time when the last workpiece of that product is completed is taken as the start time of assembly; specifically, this includes three rules as shown in steps I1-I3: Step I1: In each processing plant, all parts of the same product should be processed together, and the start time of assembly of the product should be brought forward as much as possible; Step I2: Once all workpieces belonging to the same product have been processed in the processing plant, the assembly process begins immediately; Step I3: Workpieces belonging to the same product should be assigned to different factories for parallel processing, which helps to advance the start time of product assembly.

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