Scheduling method and device applied to multi-stage and multi-level fuzzy flexible job shop

Abstract

The invention discloses a scheduling method and device applied to a multi-stage and multi-level fuzzy flexible job shop, and the method integrates the shape features of a fuzzy number and the multi-dimensional features of a relative position, avoids information loss and sorting deviation through the introduction of a fuzzy preference relation, improves the credibility of a scheduling scheme, and improves the scheduling efficiency. And a memetic algorithm fusing fuzzy fitness gradient analysis and key product and key part identification is designed, and the scheduling problem of the multi-stage and multi-level fuzzy flexible job shop is solved by taking minimization of the maximum fuzzy completion time as an optimization target. According to the algorithm, algorithm operation is dynamically adjusted through a self-adaptive crossover and mutation strategy, and the global and local search capability is improved; by identifying the key products and the key parts, the part which has the greatest influence on the overall completion time is optimized in a targeted manner, the solving quality is improved, and the stability in processing the fuzzy number sorting problem and the effectiveness in solving the multi-stage and multi-level fuzzy flexible job shop scheduling problem are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of resource scheduling, and particularly relates to a scheduling method applied to a multi-stage multi-level fuzzy flexible job shop, and a scheduling device applied to a multi-stage multi-level fuzzy flexible job shop. Background Art

[0002] Multi-level multi-stage complex product flexible systems are widely used in the production of personalized customized products facing multiple varieties, small batches, and complex processes, such as the manufacturing fields of aerospace and precision instruments. The processing and manufacturing environment of such systems has high flexibility, many coupled constraints in the production process, and strong correlations between subsystems. An effective production scheduling scheme is an effective way to improve the production efficiency of complex products and optimize resource allocation. However, due to the uncertainty of processing technologies and times caused by high personalized demands, as well as the unclear regularity of dynamic influencing factors, traditional deterministic scheduling methods are difficult to meet the complex and changeable production requirements. Fuzzy theory plays an important role in solving the uncertainty problems in the production process. Existing research mainly focuses on single workshops or simple multi-stage scheduling problems, and the research on simultaneously handling the scheduling problems of multi-stage multi-level fuzzy flexible job shops is still insufficient. On the one hand, using existing criteria to make comparisons in a hierarchical order may lead to the loss of fuzzy number information, and thus generate sorting biases. Especially when the weighted average value or sorting order cannot represent the shape characteristics of the fuzzy number, such comparisons may draw unreasonable conclusions. On the other hand, in multi-stage multi-level complex scheduling problems, due to the coupling relationships between stages, it is crucial to maintain the coherence and consistency of sequential decisions. Existing criteria are very sensitive to minor numerical changes in fuzzy numbers, which may cause significant changes in sorting results and cannot guarantee the consistency and reliability of sorting results. Furthermore, it affects the coordination of the entire production plan and reduces the credibility of the scheduling scheme. Summary of the Invention

[0003] The main technical problem to be solved by the present invention is to overcome the above-mentioned defects existing in the prior art, and to provide a scheduling method and device applied to a multi-stage multi-level fuzzy flexible job shop, thereby improving the scheduling efficiency and accuracy.

[0004] The technical solution adopted by the present invention to solve its technical problems is as follows: According to one aspect of the present disclosure, there is provided a scheduling method applied to a multi-stage multi-level fuzzy flexible job shop, including the following steps: S1: For the scheduling problem of a multi-stage multi-level fuzzy flexible job shop, triangular fuzzy numbers are used to represent the uncertain processing times of each process, and a scheduling model is constructed. The triangular fuzzy number is expressed as TFN = , ; Wherein: Denote fuzzy numbers as the most optimistic value of Denote fuzzy numbers as the most likely value of with a membership degree of 1, and denote fuzzy numbers as the most pessimistic value; The membership function of the triangular fuzzy number is as shown in Equations 1 - 3, where S2: Based on the design of the addition priority rule of the fuzzy preference relation, solve the scheduling model, including the following sub - steps: S2.1: Consider the relative position weight function of fuzzy numbers defined on the interval and satisfying where the weight function is as shown in Equation 4: where μ is the weight coefficient; S2.2: Define the weighted membership function as shown in Equations 5 - 6: S2.3: Calculate the addition priority calculated as shown in Equation 7: Determine the addition preference degree of fuzzy numbers and as shown in Equation 8: where and are preset addition preference degree values, and are the minimum value function and the maximum value function; The addition priority matrix based on the fuzzy preference relation can be obtained, as shown in Equation 9: S2.4: Calculate to sort the fuzzy numbers according to their magnitudes, and the calculation formula is as shown in Equation 10: S3: The memetic algorithm that integrates fuzzy fitness gradient analysis and the identification of key products and key parts is used to obtain the scheduling strategy for the multi-stage and multi-level fuzzy flexible job shop, including the following steps: S3.1: Randomly initialize the population according to the priority rule. Each individual represents a processing sequence, and the processing sequences of the parts corresponding to the products are generated. S3.2: Adaptive crossover and mutation strategies based on fuzzy fitness: S3.2.1: Calculate the fitness value of each individual in the population ; S3.2.2: Divide the population into three sub-populations with high, medium, and low fitness according to the fitness sorting results; S3.2.3: Different selection, crossover, and mutation operations are adopted for different sub-populations. The crossover probability and the mutation probability are dynamically adjusted according to formulas 11 and 12 respectively: where, , are the upper and lower bounds of the crossover probability respectively, is a constant, ; represents the maximum value of the fitness values in the population, is the average value of the fitness values in the population, represents the fitness value of the parent with the larger fitness value among the parents participating in the crossover, mid(·) represents the most likely probability value, is the improved adaptive mutation probability change; is the improved adaptive crossover probability change; S3.3: Calculate the fuzzy fitness gradient and the average gradient , as shown in formulas 13 - 14: where, g is the generation number of the population; According to judge the search trend, and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, perform local search; otherwise, continue global search; S3.4: Local search based on key products and key parts: S3.4.1: For key products, use operator NS1 and operator NS2 for search; S3.4.2: For key parts, use operator NS3 and operator NS4 for search; Among them, the local search operator is defined as follows: NS1: Perform a reverse operation inside the key product; NS2: Randomly swap the processes at two positions inside the key product; NS3: Randomly select two workpieces of the key product, one of which is a key workpiece, and swap the positions of the two workpieces; NS4: Randomly select two workpieces of the key product, one of which is a key workpiece, and perform a pre-insertion or post-insertion operation.

[0005] In an embodiment of the present disclosure, in step S3.2, the selection operation includes: Calculate the total fitness , where is the population size; Calculate the probability of each individual being selected ; Calculate the cumulative probability: ; Generate a random number by roulette wheel selection , and select an individual according to the relationship between the random number and the cumulative probability .

[0006] In an embodiment of the present disclosure, the objective function of the multi-stage multi-level fuzzy flexible job shop scheduling problem is to minimize the maximum fuzzy completion time: Where: represents the fuzzy assembly completion time of product .

[0007] In an embodiment of the present disclosure, the constraint conditions of the scheduling model include at least one of the following: At any moment, the same processing operation can be carried out on at most one machine; At any moment, the same assembly operation can be carried out on at most one machine; At any moment, the same machine can process at most one operation; The operations of the same workpiece can only be carried out in the next operation after the previous operation is completed; Before the product starts the assembly operation, the required part processing must be completed.

[0008] According to another aspect of the present disclosure, the present disclosure also provides a scheduling device applied to a multi-stage multi-level fuzzy flexible job shop, including: The first processing module is used to construct a scheduling model for the multi-stage and multi-level fuzzy flexible job shop scheduling problem by using triangular fuzzy numbers to represent the uncertain processing time of each process. The triangular fuzzy number is expressed as TFN = , ; Where: represents the most optimistic value of the fuzzy number , represents the most likely value of the fuzzy number with a membership degree of 1, represents the most pessimistic value of the fuzzy number ; The membership degree function of the triangular fuzzy number is as shown in Equation 1 - Equation 3, where represents any fuzzy number: The second processing module is used to solve the scheduling model based on the design of the addition priority rule of the fuzzy preference relation. The second processing module includes a first processing sub-module, a second processing sub-module, a third processing sub-module, and a fourth processing sub-module, where The first processing sub-module is used to consider the relative position weight function defined on the interval and satisfying , where the weight function is as shown in Equation 4: where μ is the weight coefficient; The second processing sub-module is used to define the weighted membership degree function as shown in Equation 5 - Equation 6: The third processing sub-module is used to calculate the addition priority , and the calculation is as shown in Equation 7: Determine the addition preference degree of the fuzzy numbers and , as shown in Equation 8: where and are preset addition preference degree values, and are the minimum value function and the maximum value function; An additive priority matrix based on the fuzzy preference relation can be obtained, as shown in Equation 9: The fourth processing sub-module is used to calculate the magnitude of to sort the fuzzy numbers, and the calculation formula is shown in Equation 10: The third processing module is used to fuse the fuzzy fitness gradient analysis and the memetic algorithm for identifying key products and key parts to obtain the scheduling strategy of the multi-stage and multi-level fuzzy flexible job shop. The third processing module includes a fifth processing sub-module, a sixth processing sub-module, a seventh processing sub-module, and an eighth processing sub-module, where The fifth processing sub-module is used to randomly initialize the population according to the priority rule. Each individual represents a processing sequence, and a processing sequence of the corresponding parts of the product is generated; The sixth processing sub-module is used for the adaptive crossover and mutation strategy based on fuzzy fitness: calculating the fitness value of each individual in the population ; dividing the population into three sub-populations with high, medium, and low fitness according to the fitness sorting result; adopting different selection, crossover, and mutation operations for different sub-populations, and the crossover probability and the mutation probability are dynamically adjusted according to Equations 11 and 12 respectively: where , are the upper and lower bounds of the crossover probability respectively, is a constant, ; represents the maximum value of the fitness values in the population, is the average value of the fitness values in the population, represents the fitness value of the parent with the larger fitness value participating in the crossover, mid(·) represents the most likely probability value, is the improved adaptive mutation probability change; is the improved adaptive crossover probability change; The seventh processing sub-module is used to calculate the fuzzy fitness gradient and the average gradient , as shown in Equations 13 - 14: where g is the generation number of the population; According to Judge the search trend and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, perform local search; otherwise, continue with global search; The eighth processing sub-module is used for local search based on key products and key parts: for key products, use operator NS1 and operator NS2 for search; for key parts, use operator NS3 and operator NS4 for search; Among them, the local search operators are defined as follows: NS1: Perform a reverse operation inside the key product; NS2: Randomly swap the processes at two positions inside the key product; NS3: Randomly select two workpieces of the key product, one of which is a key workpiece, and swap the positions of the two workpieces; NS4: Randomly select two workpieces of the key product, one of which is a key workpiece, and perform a pre-insertion or post-insertion operation.

[0009] The beneficial effects of the present invention are as follows: A fuzzy number sorting method based on fuzzy preference addition priority is provided to solve the problems of fuzzy number sorting information loss and sorting deviation in the multi-stage multi-level fuzzy flexible job shop scheduling problem. The multi-dimensional characteristics of the shape features and relative positions of fuzzy numbers are fused. By introducing fuzzy preference relations, information loss and sorting deviation are avoided, the credibility of the scheduling scheme is improved, and a memetic algorithm that combines fuzzy fitness gradient analysis and key product and key part identification is designed to solve the scheduling problem of the multi-stage multi-level fuzzy flexible job shop with the optimization goal of minimizing the maximum fuzzy completion time. Through adaptive crossover and mutation strategies, the algorithm dynamically adjusts algorithm operations to improve global and local search capabilities; by identifying key products and key parts, the part that has the greatest impact on the overall completion time is optimized specifically to improve the solution quality, enhance the stability in dealing with fuzzy number sorting problems and the effectiveness in solving the multi-stage multi-level fuzzy flexible job shop scheduling problem, improve the credibility of the scheduling scheme, and design a memetic algorithm that combines fuzzy fitness gradient analysis and key product and key part identification to solve the problem, thereby improving the scheduling efficiency. Description of the Drawings

[0010] The present invention will be further described below in conjunction with the drawings and embodiments.

[0011] Figure 1 Show a schematic diagram of the multi-level multi-stage fuzzy flexible job shop scheduling process according to an embodiment of the present invention.

[0012] Figure 2 Show a flowchart of a scheduling method for a multi-stage multi-level fuzzy flexible job shop according to an embodiment of the present invention.

[0013] Figure 3Schematic diagrams of selection, crossover, and mutation operations for different populations in the embodiments of the present invention are shown.

[0014] Figure 4 Schematic diagram of the local search operator in the embodiments of the present invention.

[0015] Figure 5 Gantt chart of the multi-stage multi-level fuzzy flexible job shop scheduling in the embodiments of the present invention.

[0016] Figure 6 Max results of Lei and max results of APFPR for 6 groups of test data before and after random generation of perturbations for fuzzy numbers A and B in the embodiments of the present invention.

[0017] Figure 7 Experimental results corresponding to test data before and after generation of perturbations from small to large in the embodiments of the present invention are shown.

[0018] Figures 8 - 10 Experimental results when different experimental methods are used to solve problems of different scales in the embodiments of the present invention are shown.

[0019] Figure 11 Schematic diagram of the structure of a scheduling device for a multi-process parallel machine workshop for precious metal detection in the embodiments of the present invention. Detailed implementation manners

[0020] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification, but the present invention is not limited in any form. It should be noted that for those of ordinary skill in the art, several changes and improvements can be made without departing from the idea of the present invention. These all belong to the protection scope of the present invention.

[0021] The application scenario of the present invention is multi-level multi-stage fuzzy flexible job shop scheduling. As Figure 1 shown, the basic process of multi-level multi-stage fuzzy flexible operation includes raw material supply, processing, and assembly, etc. Parts are processed and assembled through different processing processes and assembly processes to obtain different products. Different orders are generated for different products, such as Figure 1 Order 1 and Order 2 in. Each order can include multiple products, and the types of products can be the same or different. Before this process, a production scheduling plan for multi-level multi-stage fuzzy flexible operation needs to be completed in the processes to be processed.

[0022] The production scheduling problem is described as follows: According to the orders and the BOM list, raw materials required for processing the corresponding parts of the products are provided by different suppliers. After the raw materials arrive, multi-process machining of the parts is completed in the processing stage, and then product assembly is completed in the mixing and assembling stage. The problem in the processing stage is a multi-process heterogeneous parallel machine scheduling problem with release times and setup times, considering the release times of the parts. The assembly stage is a multi-process heterogeneous parallel machine scheduling problem with part constraints, considering the composition of product parts and assembly process constraints, and the processes can be carried out on any assembly machine.

[0023] Specifically described as: A product is composed of multiple parts or components, and a component may be composed of multiple parts. Each part contains multiple processes. In the fuzzy scheduling problem, the processing time of each process is uncertain. The multi-level multi-stage fuzzy flexible job shop scheduling is considered as a two-stage scheduling problem of processing - assembly, and its problem model is as Figure 1 shown.

[0024] Figure 2 The flowchart of a scheduling method for a multi-process parallel machine workshop for precious metal detection according to an embodiment of the present invention is shown. As Figure 2 shown, in one embodiment, the scheduling method for a multi-process machine workshop for precious metal detection provided by the present disclosure mainly includes the following steps: S1: For the multi-stage multi-level fuzzy flexible job shop scheduling problem, triangular fuzzy numbers are used to represent the uncertain processing time of each process, and a scheduling model is constructed. The triangular fuzzy number is expressed as TFN = , ; Where: represents the most optimistic value of the fuzzy number , represents the most likely value of the fuzzy number with a membership degree of 1, represents the most pessimistic value of the fuzzy number ; The membership degree function of the triangular fuzzy number is as shown in Equation 1 - Equation 3, where, represents any fuzzy number: S2: Based on the design of the addition priority rule of the fuzzy preference relation, the above scheduling model is solved.

[0025] Suppose there are n triangular fuzzy numbers , and each fuzzy number is expressed as , where, . In order to consider the value range of all fuzzy numbers, the support set of the fuzzy number , and take the minimum value of the entire support set of the fuzzy number and the maximum value .

[0026] S2 includes the following sub-steps S2.1 to S2.4: S2.1: Consider the relative position weight function of the fuzzy number , defined on the interval , satisfying , and the weight function reflects the position of the fuzzy number on the real axis. When calculating the addition priority, different fuzzy numbers are weighted by the weight function, so that the largest fuzzy number contributes more to its addition priority, thereby highlighting the relative position characteristics of the fuzzy number.

[0027] Among them, the relative position weight function considering the fuzzy number x is shown in Equation 4: Among them, μ is the weight coefficient; S2.2: Define the weighted membership function: The membership function of the fuzzy number reflects the shape characteristics of the fuzzy number. Combining the membership function of the fuzzy number with the weight function aims to comprehensively consider the shape characteristics and relative position of the fuzzy number and construct a weighted membership function, as shown in Equations 5 - 6: is the value of the weight function corresponding to when the fuzzy number belongs to different intervals.

[0028] S2.3: Calculate the addition priority : The addition priority is the integral of the weighted membership function of the fuzzy number on the support set, considering the overall characteristics of the fuzzy number. Through integral calculation, the shape and relative position of the fuzzy number are fully captured. The calculation of the addition priority is shown in Equation 7: For any two fuzzy numbers and , determine the addition preference degree and of the fuzzy numbers , as shown in Equation 8: Among them, and are preset addition preference degree values, and is the minimum function and the maximum function; The additive priority matrix based on the fuzzy preference relationship can be obtained, as shown in Formula 9: S2.4: Fuzziness ranking: calculation The size of Sort the fuzzy numbers. The calculation formula is shown in Formula 10: S3: Fusion of fuzzy fitness gradient analysis and memetic algorithm for key product and key part identification to obtain a multi-stage and multi-level fuzzy flexible job shop scheduling strategy. S3 includes the following steps S3.1~S3.4: S3.1: Randomly initialize the population according to the priority rule, each individual represents a processing order, and generate the processing sequence of the corresponding parts of the product.

[0029] In S3.1, the population can be generated by random initialization according to the PA rule to ensure the diversity of the population, where each individual represents a solution to a problem, namely the processing sequence. The arrangement of each individual is randomly generated to ensure that the processing sequence of each product is a unique combination, and then the processing sequence of the corresponding parts of the product is generated.

[0030] S3.2: Adaptive crossover and mutation strategy based on fuzzy fitness. Step S3.2 includes S3.21~S3.23, where: S3.2.1: Calculate the fitness value of each individual in the population ; The fuzzy fitness is evaluated using the addition priority rule of formula 7 to obtain the comprehensive fitness value of each individual S3.2.2: According to the fitness ranking results, the population is divided into three sub-populations with high, medium and low fitness; Individuals whose comprehensive fitness values ​​are greater than the first fitness threshold are divided into the high fitness sub-population, individuals whose comprehensive fitness values ​​are less than the second fitness threshold are divided into the low fitness sub-population, and individuals whose comprehensive fitness values ​​are greater than the second fitness threshold and less than the first fitness threshold are divided into the medium fitness sub-population.

[0031] The first fitness threshold is greater than the second fitness threshold, and the specific values ​​of the above thresholds are not specifically limited in this disclosure.

[0032] S3.2.3: Apply different selection, crossover, and mutation operations to different subpopulations.

[0033] For the sub-population in fitness, operations such as selection, adaptive crossover (Formula 11), and adaptive mutation (Formula 12) can be adopted. For individuals with extremely low fitness, after selection, both adaptive crossover and mutation operations are carried out simultaneously.

[0034] Crossover probability and mutation probability are dynamically adjusted according to Formula 11 and Formula 12 respectively.

[0035] Among them, the change of the improved adaptive crossover probability is shown in Formula 11: The change of the improved adaptive mutation probability is shown in Formula 12: Among them, , are the upper and lower bounds of the crossover probability respectively, is a constant, ; represents the maximum value of the fitness values in the population, is the average value of the fitness values in the population, represents the fitness value of the parent with a larger fitness value among the parents participating in the crossover, mid(·) represents the most probable probability value, is the change of the improved adaptive mutation probability; is the change of the improved adaptive crossover probability.

[0036] As Figure 3 shown in (a), the mutation operation is for two individuals PS1 and PS2. The gene fragment of one individual is intercepted, and the genes not existing in the gene fragment are added to the gene fragment by traversing the other individual to form a new individual New_PS.

[0037] As Figure 3 shown in (b), the product sequence exchange operation is to randomly select two positions in the product sequence and exchange them. For example, exchange the positions of gene 2 and gene 3 in PS, or exchange the gene fragments 21 and 96 in OS.

[0038] As Figure 3 shown in (c), the product sequence forward insertion operation is to randomly select two positions in the product sequence, and the product with a larger position number is inserted before the product with a smaller position number. For example, insert gene 3 in PS before gene 2, or insert gene fragment 96 in OS before gene fragment 21.

[0039] As Figure 3As shown in (d), for the backward insertion operation of the product sequence, two positions are randomly selected from the product sequence, and the product with the smaller position number is inserted after the product with the larger position number. For example, insert Gene 2 in PS after Gene 3, or insert Gene Fragment 21 in OS after Gene Fragment 96.

[0040] S3.3: Fuzzy fitness gradient calculation: Assume that in the -th generation population, the fuzzy fitness of individual is , and in the -th generation population, the fuzzy fitness of individual is .

[0041] Calculate the fuzzy fitness gradient , as shown in Formula 13: What is calculated in Formula 13 is the average change amount of the fuzzy fitness between two consecutive generations. Calculate for the three points of the triangular fuzzy number respectively and take the average value to obtain the average gradient . In this way, the overall change trend of the fuzzy number can be comprehensively considered. The average gradient is calculated using Formula 14: where g is the generation number of the population and N is the number of individuals in the population.

[0042] According to judge the search trend, and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, perform local search; otherwise, continue global search; the above preset threshold can be determined according to actual needs, and the present disclosure does not make specific limitations on this.

[0043] S3.4: Local search based on key products and key parts, including S3.4.1~S3.4.2: S3.4.1: For key products, use operator NS1 and operator NS2 for search; First, determine the current solution and search based on key products. Perform NS1 and NS2 operations on the solution respectively to obtain a new solution, and judge whether the new solution is better than the current solution. If so, replace the current solution with the new solution; otherwise, retain the current solution.

[0044] S3.4.2: For key parts, use operator NS3 and operator NS4 for search; Based on S3.4.1, perform local search based on key parts. For the solutions obtained in S3.4.1, perform NS3 and NS4 operations respectively to obtain new solutions, and determine whether the new solutions are better than the current solutions. If so, replace the current solutions with the new solutions; if not, retain the current solutions.

[0045] Among them, the local search operator is defined as follows: NS1: Perform a reverse operation inside the key product; that is, for the key product, perform a reverse operation inside it.

[0046] NS2: Randomly exchange the processes at two positions inside the key product; that is, for the key product, randomly select two positions inside it and exchange them.

[0047] NS3: Randomly select two workpieces of the key product, one of which is a key workpiece, and exchange the positions of the two workpieces.

[0048] NS4: Randomly select two workpieces of the key product, one of which is a key workpiece, and perform a front-insertion or back-insertion operation.

[0049] As Figure 4 shown, NS1: For the key product, perform a reverse operation inside it.

[0050] NS2: For the key product, randomly select two positions inside it and exchange them.

[0051] NS3: Randomly select two workpieces of the key product (one of which is a key workpiece) and exchange their positions.

[0052] NS4: Randomly select two workpieces of the key product (one of which is a key workpiece) and perform an insertion operation on the two.

[0053] As Figure 2 shown, after local search based on key products and parts, determine whether the iteration end condition is met. If so, end; if not, execute the steps of determining the population size, randomly generating a population using the PA rule, and initializing the ranges of the crossover probability and mutation probability until the iteration end condition is met.

[0054] The present disclosure can solve the problems of loss of fuzzy number sorting information and sorting deviation in the multi-stage and multi-level fuzzy flexible job shop scheduling problem, integrates the multi-dimensional features of the shape characteristics and relative positions of fuzzy numbers, avoids information loss and sorting deviation by introducing fuzzy preference relations, improves the credibility of the scheduling scheme, and designs a memetic algorithm that integrates fuzzy fitness gradient analysis and the identification of key products and key parts to solve the scheduling problem of the multi-stage and multi-level fuzzy flexible job shop with the objective of minimizing the maximum fuzzy completion time. The algorithm dynamically adjusts the algorithm operations through adaptive crossover and mutation strategies, improves the global and local search capabilities; by identifying key products and key parts, specifically optimizes the part that has the greatest impact on the overall completion time, improves the solution quality, enhances the stability in dealing with fuzzy number sorting problems and the effectiveness in solving the multi-stage and multi-level fuzzy flexible job shop scheduling problem, improves the credibility of the scheduling scheme, and designs a memetic algorithm that integrates fuzzy fitness gradient analysis and the identification of key products and key parts to solve the problem, and improves the scheduling efficiency.

[0055] In one embodiment, in step S3.2, the selection operation includes: Calculate the total fitness , where is the population size; Calculate the probability of each individual being selected ; Calculate the cumulative probability: ; Generate a random number by roulette wheel selection , according to the random number and the cumulative probability to select an individual.

[0056] Generate N random numbers between 0 and 1 by roulette wheel selection , and select individuals according to the following rules: If , then select the first individual; if , then select the th individual.

[0057] In one embodiment, the objective function of the multi-stage and multi-level fuzzy flexible job shop scheduling problem is to minimize the maximum fuzzy completion time: Where: represents the fuzzy assembly completion time of product , is the maximum fuzzy completion time.

[0058] Based on the product and the part processing sequence, a multi-stage multi-level fuzzy flexible job shop scheduling model is established with the objective of minimizing the makespan under the premise of meeting the constraint conditions, and the constraint conditions of the model are determined; the objective function is shown in Formula 15.

[0059] In one embodiment, the constraint conditions of the scheduling model include at least one of the following: At any moment, at most one machining operation of the same machining process can be carried out on one machine. At any moment, at most one assembly operation of the same assembly process can be carried out on one machine. At any moment, at most one operation can be processed on one machine. The operations of the same workpiece can only be carried out in the next operation after the previous operation is completed. Before the product starts the assembly operation, the required part processing must be completed.

[0060] The scheduling model of the multi-stage multi-level fuzzy flexible job shop is as follows: Equation (16) represents the calculation of the completion time of the workpiece when it is processed for the first time and the processing machine is the one that processes the workpiece for the first time. Equation (17) represents the calculation of the completion time of the workpiece when it is processed for the first time and the processing machine is not the one that processes the workpiece for the first time. Equation (18) represents the calculation of the completion time of the workpiece when it is not processed for the first time and the processing machine processes the workpiece for the first time. Equation (19) represents the calculation of the completion time of the workpiece when it is not processed for the first time and the processing machine is not the one that processes the workpiece for the first time. Equation (20) represents the calculation of the completion time of the assembly process when it is processed for the first time and the assembly machine is the one that processes the process for the first time. Equation (21) represents the calculation of the completion time of the assembly process when it is processed for the first time and the assembly machine is not the one that processes the process for the first time. Equation (22) represents the calculation of the completion time of the assembly process when it is not processed for the first time and the assembly machine processes the process for the first time. Equation (23) represents the calculation of the completion time of the assembly process when it is not processed for the first time and the assembly machine is not the one that processes the process for the first time.

[0061] The definitions of the mathematical symbols in the model are as follows: S2: Solve the scheduling model of the multi-stage multi-level fuzzy flexible job shop, including the following sub-steps: S2.1: Initialize the population: including determining the product processing order; determining the fuzzy processing time of each operation on each machine; setting parameters including the population size and the iteration time.

[0062] In the embodiment, the specific parameter settings are as follows: population size = 20, maximum number of iterations = 50.

[0063] S2.2: Perform an adaptive crossover and mutation strategy based on fuzzy fitness for the initialized population; S2.3: Calculate the fuzzy fitness gradient to determine whether to perform local search. If local search is not performed, continue with the global search; otherwise, enter the local search. S2.4: Determine whether to perform local search based on the magnitude of the fitness gradient. S2.5: Sort the fuzzy numbers according to the results obtained from the search, and select the maximum value of the fuzzy completion time as the fuzzy completion time.

[0064] By randomly generating the part processing information and assembly information of 7 products, a Gantt chart for the multi-stage multi-level fuzzy flexible job shop scheduling is obtained as Figure 5 shown.

[0065] To further verify that the additive priority criterion based on fuzzy preference relations proposed in this paper has a certain stability for fuzzy number ranking, experimental verification is carried out by comparing the additive priority method considering multi-dimensional features proposed in this paper with the literature that only considers numerical features from two aspects: the evaluation between two fuzzy numbers and the evaluation of multiple fuzzy numbers. For fuzzy numbers A and B, 6 groups of test data before and after perturbation are randomly generated, as specifically Figure 6 shown, where represents the fuzzy number before perturbation, and represents the fuzzy number after perturbation. It can be seen from the comparison of the results in the table that after the fuzzy numbers are perturbed and changed, the ranking results of the method in this paper are the same as those before perturbation, indicating that after adding more feature information, although the fuzzy numbers change slightly, it does not affect the evaluation results, making the results more reasonable and reliable. To further verify the stability of the ranking of multiple fuzzy numbers, the Spearman rank correlation coefficient and Kendall's Tau rank correlation coefficient are introduced as the evaluation indicators for the stability of fuzzy number ranking. The Spearman rank correlation coefficient ( ) is used to measure the correlation between the ranks (rankings) of two variables (an explanation of the correlation for the value) is shown in Equation (24). Among them: is the sample size (i.e., the number of objects to be ranked). is the rank difference (ranking difference) of the th object in the two ranking results. When it indicates no correlation and low stability. When it indicates complete consistency and high stability.

[0066] Kendall's Tau rank correlation coefficient is used to evaluate the degree of consistency between the sorting results before and after a slight change in fuzzy numbers. In the two sorting results, two objects at the same sorting position before and after the perturbation are regarded as a pair of numbers. If the relative order of the pair of numbers is consistent, it is counted as a consistent pair of numbers . Otherwise, it is counted as an inconsistent pair of numbers . Thus, the stability value can be obtained . As shown in Equation (25), when represents complete consistency and the highest stability. When represents no correlation and low stability. When represents complete opposition and poor stability

[0067] According to the numerical intervals of triangular fuzzy numbers, four groups of test data before and after the perturbation are generated from small to large. Each group of data includes 5 fuzzy numbers. As Figure 7 shown, where the data with quotes are the experimental results obtained by the present invention, R is the ranking of the fuzzy number array before the perturbation, and S is the ranking of the fuzzy number array after the perturbation

[0068] The experimental results show that when using the addition priority method for fuzzy number sorting, the sorting results before and after the perturbation are completely consistent. The Spearman rank correlation coefficient and Kendall's Tau rank correlation coefficient are both 1, indicating that the method has extremely high stability. In contrast, the Spearman rank correlation coefficient of the Lei criterion is [0.7, 0.8] and the Kendall's Tau rank correlation coefficient is [-0.6, 0.1] under the same conditions, indicating that the sorting results are more sensitive to slight perturbations and the stability is poor. This difference shows that the method used in this paper can provide more stable and reliable sorting results when dealing with fuzzy scheduling problems, which helps to maintain the consistency of multi-stage scheduling decisions

[0069] The beneficial effects of the present invention are as follows: For the multi-stage and multi-level fuzzy flexible job shop scheduling problem, this paper proposes an addition priority rule based on fuzzy preference relations for the ranking and evaluation of fuzzy numbers, and designs a memetic algorithm based on fuzzy fitness gradient and the identification of key products and key parts to solve the problem. First, in order to avoid the loss of fuzzy number information and ranking deviation, an addition priority rule based on fuzzy preference relations is designed to ensure the consistency and stability of the ranking results, and the effectiveness and robustness of this method in dealing with fuzzy number ranking problems are verified through theoretical analysis and experimental results. Secondly, the algorithm designs adaptive crossover and mutation operations on the population using fuzzy fitness information, judges the search convergence trend through the fuzzy fitness gradient, flexibly selects local search or global search strategies, introduces local search based on key products and key parts, and optimizes the parts that have a greater impact on the overall completion time specifically, improving the quality of the solution. The experimental results show that the proposed addition priority rule has higher stability and reliability in fuzzy number ranking. Compared with other algorithms, the present invention shows better performance when solving problems of different scales, especially when the problem scale increases, the advantage is more obvious. For details, see Figures 8 - 10 , which indicates that the method of this paper can effectively handle complex fuzzy scheduling problems and is of great significance for improving the production efficiency of complex products and optimizing resource allocation.

[0070] In the embodiments of the present disclosure, a method for ranking fuzzy numbers based on fuzzy preference addition priority is provided to solve the problems of fuzzy number ranking information loss and ranking deviation in the multi-stage and multi-level fuzzy flexible job shop scheduling problem, improve the credibility of the scheduling scheme, and design a memetic algorithm that integrates fuzzy fitness gradient analysis and the identification of key products and key parts to solve the problem and improve the scheduling efficiency.

[0071] The addition priority rule based on fuzzy preference relations has good stability.

[0072] According to the elaboration of the membership degree of fuzzy numbers and the addition priority membership degree function, first, it is clear that the fuzzy membership degree function For the fuzzy number , is piecewise linear, so it is continuously differentiable for the fuzzy number, and is a strictly increasing and continuous function, then for the fuzzy number , is also continuous. Since the fuzzy membership degree function and the weight function are both continuous functions, their product functions and are also continuous functions. Then for the addition priority For: Since the integrand is continuous and the upper and lower limits of integration are both continuous functions, thus Regarding , is continuous.

[0073] For the three parameters of the fuzzy number: , where is a small perturbation. First, prove , where is very small.

[0074] S3: Perform Taylor expansion on with respect to the parameter as shown in Equation (26): where represents higher-order infinitesimals and can be ignored.

[0075] Next, take as an example and use Leibniz's formula to calculate : Since , we can obtain: Similarly, the partial derivatives of and can be calculated.

[0076] Since and are both continuous functions, and the partial derivatives with respect to the parameters are finite within the integration interval. Therefore, the partial derivatives are finite. Since is very small and the partial derivatives are finite, thus is also very small and can be considered: where is a constant determined by the maximum value of the partial derivatives.

[0077] According to the previous elaboration, it can be known that the addition preference degree with respect to and is continuous, and the and functions included in the definition are linear at non-boundary points. Take the partial derivatives of with respect to and as shown in Equation (29): Therefore, is also tiny. Since is very small, so is also very small. The sorting result is based on the magnitude of the priority vector . When is very small and not enough to change 's relative magnitude relationship with other , the sorting result remains unchanged. Only when accumulates to be large enough such that 's difference from is exceeded, the sorting result may change. Therefore, the addition priority rule is robust to small perturbations of the fuzzy number parameters and can maintain the stability of the sorting result.

[0078] Figure 11 shows a schematic structural diagram of a scheduling device for a precious metal detection multi - process parallel machine workshop provided by an embodiment of the present disclosure. As Figure 11 shown, in one embodiment, a scheduling device of the present disclosure applied to a multi - stage multi - level fuzzy flexible job shop includes: A first processing module 1110, configured to use triangular fuzzy numbers to represent the uncertain processing time of each process for the multi - stage multi - level fuzzy flexible job shop scheduling problem, and construct a scheduling model. The triangular fuzzy number is represented as TFN = , ; Where: represents the most optimistic value of the fuzzy number , represents the most likely value of the fuzzy number with a membership degree of 1, represents the most pessimistic value of the fuzzy number ; The membership function of the triangular fuzzy number is as shown in Equations 1 - 3. Among them, represents any fuzzy number: A second processing module 1120, configured to solve the scheduling model based on the design of the addition priority rule of the fuzzy preference relationship. The second processing module includes a first processing sub - module 1121, a second processing sub - module 1122, a third processing sub - module 1123, and a fourth processing sub - module 1124. Among them, The first processing sub - module 1121 is configured to consider the relative position weight function defined on the interval and satisfy . Among them, the weight function is as shown in Equation 4: Among them, μ is the weight coefficient; The second processing sub-module 1122 is used to define the weighted membership function as shown in Equation 5 - Equation 6: The third processing sub-module 1123 is used to calculate the addition priority , and the calculation is as shown in Equation 7: Determine the fuzzy number and of the addition preference degree , as shown in Formula 8: Among them, and are preset addition preference degree values, and are the minimum value function and the maximum value function; The addition priority matrix based on the fuzzy preference relation can be obtained, as shown in Formula 9: The fourth processing sub-module 1124 is used to calculate to sort the fuzzy numbers according to the size of , and the calculation formula is as shown in Formula 10: The third processing module 1130 is used to fuse the fuzzy fitness gradient analysis and the memetic algorithm for identifying key products and key parts to obtain the scheduling strategy of the multi-stage multi-level fuzzy flexible job shop. The third processing module includes a fifth processing sub-module 1131, a sixth processing sub-module 1132, a seventh processing sub-module 1133, and an eighth processing sub-module 1134. Among them, The fifth processing sub-module 1131 is used to randomly initialize the population according to the priority rules. Each individual represents a processing sequence, and generate the processing sequence of the parts corresponding to the product; The sixth processing sub-module 1132 is used for the adaptive crossover and mutation strategy based on fuzzy fitness: calculate the fitness value of each individual in the population ; divide the population into three sub-populations with high, medium, and low fitness according to the fitness sorting result; adopt different selection, crossover, and mutation operations for different sub-populations, and the crossover probability and the mutation probability are dynamically adjusted according to Formulas 11 and 12 respectively: Among them, , are the upper and lower bounds of the crossover probability respectively, is a constant, ; represents the maximum of the fitness values in the population, is the average of the fitness values in the population, represents the fitness value of the parent with the larger fitness value among the parents participating in crossover, and mid(·) represents the most probable probability value, is the change of the improved adaptive mutation probability; is the change of the improved adaptive crossover probability; The seventh processing sub-module 1133 is used to calculate the fuzzy fitness gradient and the average gradient , as shown in Formula 13 - Formula 14: Among them, g is the generation number of the population; According to to judge the search trend, and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, local search is performed; otherwise, global search continues; The eighth processing sub-module 1134 is used for local search based on key products and key parts: for key products, operators NS1 and NS2 are used for search; for key parts, operators NS3 and NS4 are used for search; Among them, the local search operators are defined as follows: NS1: Perform a reverse order operation inside the key product; NS2: Randomly exchange the processes at two positions inside the key product; NS3: Randomly select two workpieces of the key product, one of which is a key workpiece, and exchange the positions of the two workpieces; NS4: Randomly select two workpieces of the key product, one of which is a key workpiece, and perform a front-insertion or back-insertion operation.

[0079] The above is only a preferred embodiment of the present invention, and does not impose any form of limitation on the present invention. Any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention still fall within the scope of the technical solution of the present invention.

Claims

1. A scheduling method for a multi-stage and multi-level fuzzy flexible job shop, characterized in that: include: S1: For the multi-stage and multi-level fuzzy flexible job shop scheduling problem, triangular fuzzy numbers are used to characterize the uncertain processing time of each process and a scheduling model is constructed. The triangular fuzzy number is expressed as TFN= , ; in: Representing fuzzy numbers The most optimistic value of Representing fuzzy numbers The most likely value of has a membership of 1. Representing fuzzy numbers The most pessimistic value of; the membership function of the triangular fuzzy number is as shown in formula 1-formula 3, where, Represents any fuzzy number: S2: Designing an additive priority rule based on fuzzy preference relations to solve the scheduling model includes the following sub-steps: S2.1: Considering the relative position weight function of fuzzy numbers , defined in the interval On, meet , where the weight function is shown in Formula 4: Among them, μ is the weight coefficient; S2.2: Define the weighted membership function as shown in Formula 5-Formula 6: S2.3: Calculate addition priority , calculated as shown in formula 7: Determine fuzzy number and Additive preference , as shown in Formula 8: in, and is the preset additive preference value, and is the minimum function and the maximum function; The additive priority matrix based on the fuzzy preference relationship can be obtained, as shown in Formula 9: S2.4: Calculation The size of fuzzy numbers is sorted, The calculation formula is shown in Formula 10: S3: Integrating the fuzzy fitness gradient analysis and the memetic algorithm for identifying key products and key parts to obtain the scheduling strategy of the multi-stage and multi-level fuzzy flexible job shop, including the following steps: S3.1: Randomly initialize the population according to the priority rule, each individual represents a processing order, and generate the processing sequence of the corresponding parts of the product; S3.2: Adaptive crossover and mutation strategy based on fuzzy fitness: S3.2.1: Calculate the fitness value of each individual in the population ; S3.2.2: According to the fitness ranking results, the population is divided into three sub-populations with high, medium and low fitness; S3.2.3: Different selection, crossover and mutation operations are applied to different subpopulations. The crossover probability and mutation probability Dynamically adjust according to formula 11 and formula 12 respectively: in, , are the upper and lower bounds of the crossover probability, respectively. is a constant, ; represents the maximum fitness value in the population, is the average fitness value in the population, represents the fitness value of the parent with the larger fitness value among the crossover parents, mid(·) represents the most likely probability value, To improve the adaptive mutation probability change; To improve the adaptive crossover probability change; S3.3: Calculate the fuzzy fitness gradient and the average gradient , as shown in Formula 13-Formula 14: Among them, g is the number of generations of the population; according to Determine the search trend and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, perform local search; otherwise, continue global search; S3.4: Local search based on key products and key parts: S3.4.1: For key products, use operators NS1 and NS2 to search; S3.4.2: For key parts, use operators NS3 and NS4 to search; Among them, the local search operator is defined as follows: NS1: Reverse operation on key products; NS2: Randomly swap two positions of the process within the key product; NS3: Randomly select two artifacts of the key product, one of which is the key artifact, and swap the positions of the two artifacts; NS4: Randomly select two artifacts of the key product, one of which is the key artifact, and perform the pre-insert or post-insert operation.

2. The method according to claim 1, characterized in that: In step S3.2, the selection operation includes: Calculate total fitness ,in, is the population size; Calculate the probability of each individual being selected ; Calculate the cumulative probability: ; Roulette wheel selection to generate random numbers , according to the random number and cumulative probability Select individuals based on their relationship.

3. The method according to claim 1, characterized in that: The objective function of the multi-stage multi-level fuzzy flexible job shop scheduling problem is to minimize the maximum fuzzy completion time: in: Indicates products The fuzzy assembly completion time.

4. The method according to any one of claims 1 to 3, characterized in that: The constraints of the scheduling model include at least one of the following: The same processing operation can only be processed on one machine at any one time; At any one time, the same assembly process can only be assembled on one machine at most; At any one time, the same machine can process at most one process; The next process of the same workpiece can only be carried out after the previous process is completed; Before product assembly begins, the required parts processing must be completed.

5. A scheduling device for a multi-stage and multi-level fuzzy flexible job shop, characterized in that: include: The first processing module is used to construct a scheduling model for the multi-stage multi-level fuzzy flexible job shop scheduling problem by using triangular fuzzy numbers to characterize the uncertain processing time of each process. The triangular fuzzy number is represented by TFN= , ; in: Representing fuzzy numbers The most optimistic value of Representing fuzzy numbers The most likely value of has a membership of 1. Representing fuzzy numbers The most pessimistic value of; the membership function of the triangular fuzzy number is as shown in formula 1-formula 3, where, Represents any fuzzy number: The second processing module is used to design an additive priority rule based on a fuzzy preference relationship to solve the scheduling model, and the second processing module includes a first processing submodule, a second processing submodule, a third processing submodule and a fourth processing submodule, wherein: The first processing submodule is used to consider the relative position weight function of the fuzzy number , defined in the interval On, meet , where the weight function is shown in Formula 4: Among them, μ is the weight coefficient; The second processing submodule is used to define a weighted membership function as shown in Formula 5-Formula 6: The third processing submodule is used to calculate the addition priority , calculated as shown in formula 7: Determine fuzzy number and Additive preference , as shown in Formula 8: in, and is the preset additive preference value, and is the minimum function and the maximum function; The additive priority matrix based on the fuzzy preference relationship can be obtained, as shown in Formula 9: The fourth processing submodule is used to calculate The size of fuzzy numbers is sorted, The calculation formula is shown in Formula 10: The third processing module is used to integrate the fuzzy fitness gradient analysis and the memetic algorithm for identifying key products and key parts to obtain the scheduling strategy of the multi-stage and multi-level fuzzy flexible job shop. The third processing module includes a fifth processing submodule, a sixth processing submodule, a seventh processing submodule and an eighth processing submodule, wherein: The fifth processing submodule is used to randomly initialize the population according to the priority rule, each individual represents a processing order, and generate a processing sequence of the corresponding parts of the product; The sixth processing submodule is used for adaptive crossover and mutation strategy based on fuzzy fitness: calculating the fitness value of each individual in the population ; According to the fitness ranking results, the population is divided into three sub-populations with high, medium and low fitness; different selection, crossover and mutation operations are used for different sub-populations, and the crossover probability and mutation probability Dynamically adjust according to formula 11 and formula 12 respectively: in, , are the upper and lower bounds of the crossover probability, respectively. is a constant, ; represents the maximum fitness value in the population, is the average fitness value in the population, represents the fitness value of the parent with the larger fitness value among the crossover parents, mid(·) represents the most likely probability value, To improve the adaptive mutation probability change; To improve the adaptive crossover probability change; The seventh processing submodule is used to calculate the fuzzy fitness gradient and the average gradient , as shown in Formula 13-Formula 14: Among them, g is the number of generations of the population; according to Determine the search trend and select local search or global search. When the average fuzzy fitness gradient is less than the preset threshold, perform local search; otherwise, continue global search; The eighth processing submodule is used for local search based on key products and key parts: for key products, operators NS1 and NS2 are used for search; for key parts, operators NS3 and NS4 are used for search; Among them, the local search operator is defined as follows: NS1: Reverse operation on key products; NS2: Randomly swap two positions of the process within the key product; NS3: Randomly select two artifacts of the key product, one of which is the key artifact, and swap the positions of the two artifacts; NS4: Randomly select two artifacts of the key product, one of which is the key artifact, and perform the pre-insert or post-insert operation.

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