Flexible job shop scheduling optimization method based on meta-heuristic algorithm

Abstract

The application provides a flexible workshop scheduling optimization method based on a meta-heuristic algorithm, data sets of flexible workshop scheduling are acquired; a differential evolution algorithm based on a trigonometric function mutation is adopted to process the data sets of flexible workshop scheduling and preliminarily form a scheduling scheme by introducing a nonlinear disturbance and a dynamic adjustment mechanism of a companion population; the adaptive hill climbing algorithm is adopted to optimize a current optimal solution and process again to form an optimal scheduling scheme by combining a multiple disturbance mechanism and a disturbance intensity adjustment strategy; and the optimal scheduling scheme achieving the maximum completion time is visually presented; the flexible workshop scheduling optimization method based on the meta-heuristic algorithm can improve the ability to jump out of a local optimum, can significantly reduce the maximum completion time and can improve the solution efficiency of the flexible workshop scheduling problem.

Description

Technical Field

[0001] This invention relates to a flexible shop floor scheduling optimization method based on metaheuristic algorithms, belonging to the field of shop floor scheduling technology. Background Technology

[0002] The main challenges facing Flexible Job-Shop Scheduling (FJSP) lie in its complexity and the dynamic changes in actual needs. These include conflicts between multi-objective optimization goals such as minimizing production time, energy consumption, and cost; the difficulty of selecting multiple process paths; the difficulty in finding the global optimum due to excessive computational complexity; and the uncertainties of the production environment, such as order additions and deletions, machine failures, and processing time fluctuations, all of which necessitate real-time response and scheduling optimization. Furthermore, it is necessary to address the contradiction between equipment utilization and balancing shop floor resource loads to improve overall production efficiency and the adaptability of flexible manufacturing systems.

[0003] In recent years, researchers and enterprises have introduced various advanced technologies to address the scheduling challenges of flexible workshops, including intelligent optimization algorithms, artificial intelligence (AI), digital twins, and big data analytics. While the combination of these technologies has significantly improved the intelligence level and ability to cope with dynamic environments in flexible workshops, several challenges remain. For example, intelligent algorithms face a trade-off between computational efficiency and solution accuracy in large-scale problems; AI relies on large amounts of high-quality data and domain knowledge, but its generalization ability and transparency are insufficient; the deployment of digital twins requires high costs and demands significant integration of multi-source heterogeneous data; and big data analytics and the Industrial Internet of Things (IIoT) face challenges such as data silos, privacy and security, and insufficient real-time performance. Furthermore, the practical implementation of these technologies is limited by the uncertainties of the industrial environment and enterprises' acceptance and operational capabilities of modern intelligent technologies.

[0004] For example, Chinese invention application publication number CN118938840A discloses a flexible shop scheduling method based on the sparrow algorithm, but this application has the following shortcomings: (1) Insufficient maintenance of population diversity. It relies only on simple population classification (discoverers, joiners, and watchers) to maintain population diversity, which may cause the algorithm to converge to a local optimum prematurely during the iteration process. (2) It mainly relies on the basic position update formula, the mutation strategy is simple, and it lacks the ability to fully explore the search space, especially when dealing with complex flexible shop scheduling problems. (3) It mentions that it is not easy to get trapped in local optima, but it relies only on the improved watcher formula, which is insufficient in terms of the refinement of the solution, which may affect the algorithm's ability to find a better scheduling scheme.

[0005] For example, Chinese invention application publication number CN118551966A discloses a method for solving the flexible workshop scheduling problem based on an improved golden jackal optimization algorithm. However, the shortcomings of this application are: (1) It mentions that "under certain conditions, the diversity of golden jackals is limited", which indicates that there is a problem in maintaining the diversity of the population. Simply relying on simple population division and position updates cannot effectively maintain the diversity of the population, which can easily lead to premature convergence and thus fail to obtain the corresponding optimal solution. (2) It mainly focuses on the step of performing crossover operations on the replaced individuals. Although the Cauchy-Gaussian mutation strategy is used, it still has great shortcomings in terms of complexity and effectiveness. (3) It uses a "variable neighborhood search strategy", but it is only a simple neighborhood solution generation and evaluation. The depth and breadth of the local search for optimal solutions are insufficient. Among them, the "variable neighborhood operation strategy based on critical path" lacks an adaptive adjustment mechanism, which to some extent affects the refinement and improvement effect of the solution. Summary of the Invention

[0006] The purpose of this invention is to provide a flexible shop scheduling optimization method based on metaheuristic algorithms to address the problems in existing technologies, such as the lack of ability to escape local optima and the need to improve the solution efficiency of flexible shop scheduling problems.

[0007] The technical solution of this invention is:

[0008] A flexible shop floor scheduling optimization method based on metaheuristic algorithms includes the following steps:

[0009] S1. Obtain the dataset for flexible workshop scheduling;

[0010] S2. Introduce nonlinear perturbation and dynamic adjustment mechanism of the accompanying population, and use differential evolution algorithm based on trigonometric function mutation to process the dataset of flexible workshop scheduling to initially form a scheduling scheme.

[0011] S21. Population initialization: The initial solution set is generated, which includes the main population and the companion population. The main population is generated by the shortest processing time rule, and the companion population is formed by dividing the flexible workshop scheduling dataset into multiple regions as the solution space, and performing random generation, splicing operation and row-level random permutation on each region to form a temporary vector.

[0012] S22. Adaptively adjust parameters and update the main population: Considering the mutation factor F and the crossover probability CR, a trigonometric function nonlinear perturbation mechanism is adopted. The periodicity and nonlinearity of the trigonometric function are used to control the perturbation amplitude of the trigonometric function through the mutation factor F. Mutation and crossover operations are performed on each individual of the main population in the g-th generation, i.e., the target vector, to generate an experimental vector. The experimental vector is compared with the solution of the current generation of the companion population, i.e., the companion vector. The individual with the best fitness among the target vector, experimental vector, and companion vector is selected to update the target vector of the main population in the g+1-th generation.

[0013] S23. Use a diversity selection strategy to select individuals whose fitness is neither the best nor the worst as the companion vector for the g+1th generation.

[0014] S24, Success rate s of the main population g Adaptive adjustment of the mutation factor F and crossover probability CR;

[0015] S25, Iteration steps S22-S24, until the maximum number of iterations T is reached. max When the iteration is terminated, the flexible workshop scheduling scheme is obtained;

[0016] S3. The resulting scheduling scheme is then processed using an adaptive hill-climbing algorithm, combined with multiple perturbation mechanisms and perturbation intensity adjustment strategies, to optimize the current optimal solution and form the optimal scheduling scheme.

[0017] S4. Present the optimal scheduling scheme to achieve the maximum completion time in a visual format.

[0018] Furthermore, in step S1, the dataset for flexible shop floor scheduling includes the number of workpieces and machines, and the workpiece processing data, i.e., the processing procedure information for each workpiece, including the selectable machines for each procedure and the corresponding processing time.

[0019] Further, in step S21, the accompanying population is constructed by dividing the flexible workshop scheduling dataset into multiple regions as the solution space, and performing random generation, splicing, and row-level random permutation on temporary vectors within each region. Specifically:

[0020] S211. Define the accompanying population A, which includes NP individuals: Where i represents the individual index. This represents the 1st, 2nd, ..., i, ..., NPth individual. Each individual, i.e., the chromosome, uses a two-layer encoding method, which includes a machine allocation part and a process sorting part.

[0021] S212. Use the dataset of flexible workshop scheduling as the solution space, i.e., the j-dimensional interval. Divide the region into q equal regions, and define the upper bound of the h-th region. The lower bound of the h-th region :

[0022] ,

[0023] in, These represent the lower and upper bounds of the entire search space, respectively, and h represents the region index;

[0024] S213. Generate NP / q random numbers uniformly within the h-th region to form a random value vector of the j-th dimension within the h-th region. :

[0025] ,

[0026] Where rand represents a random function used to generate random numbers;

[0027] S214. Divide the j-dimensional structure into q sub-segments and concatenate them in order to form the initial concatenation vector of the j-th dimension. :

[0028] ,

[0029] Where j represents the dimension index, Let h represent the random value vector of the j-th dimension in the 1st, 2nd, ..., hth, ..., qth regions, where h represents the region index and q represents the total number of regions after dividing the search space equally.

[0030] S215, the assembled Perform another global permutation to obtain the set of candidate values ​​for the j-th dimension across all regions. :

[0031] ,

[0032] in, This represents a random permutation function used to shuffle the order;

[0033] S216, to By performing random permutations while keeping the set of elements unchanged and only changing their order, we can enhance diversity and obtain a temporary vector of the j-th dimension. :

[0034] ,

[0035] S217, Constructing temporary population matrix B:

[0036] ,

[0037] Where D represents the problem dimension, i.e., the number of temporary vectors;

[0038] S218. Assign the temporary population matrix B to the accompanying population A: A=B.

[0039] Further, step S22 specifically involves,

[0040] S221. Perform mutation operations on the individuals of the main population in the g-th generation. Perturb the solution using sine and cosine functions, and control the perturbation amplitude of the trigonometric functions using the mutation factor F to generate the mutation vector of the i-th individual in the g-th generation. ;

[0041] S222, Perform a cross operation, using the current individual and mutation vector The candidate solutions generated after performing the crossover operation are the trial individuals. :

[0042] ,

[0043] in, To test individuals The j-th dimension component, For the mutation vector The j-th dimension component, For the current individual The j-th dimension component, where rand is a random function and CR is the cross probability, controlling the proportion of cross variables; j rand The dimension index is randomly selected;

[0044] S223. Using a greedy selection strategy, select the individual with the best fitness among the three vectors (target vector, trial vector, and adjoint vector) in the main population as the next generation individual in the main population:

[0045] ,

[0046] in, This represents the i-th individual in the main population of generation g+1. Let represent the target vector of the i-th current individual in the g-th generation of the main population. Let represent the candidate individual, i.e., the trial vector, after the i-th mutation crossover in the g-th generation. Let f represent the companion vector of the i-th individual in the g-th generation companion population, i.e., the g-th generation, min represents the minimum function, and f represents the fitness function.

[0047] Furthermore, in step S221, the mutation vector of the i-th individual in the g-th generation... The j-th dimension component The formula for generating it is:

[0048] ,

[0049] in, Let j be the j-th dimension component of the mutation vector of the i-th individual. These represent three distinct solutions, or individuals, randomly selected from the main population; i represents the individual index, and j represents the dimension index; F is the mutation factor, or scaling factor, used to control the magnitude of mutation. , For nonlinear perturbations based on the difference between two solutions, This indicates rounding down, used to map the result to discrete values ​​within a specified range.

[0050] Further, step S23 specifically includes:

[0051] ,

[0052] in, Let represent the adjoint vector of the i-th individual in the (g+1)-th generation of the adjoint population, and max represent the maximum value function.

[0053] Further, step S24 specifically involves,

[0054] ,

[0055] in, These are the minimum and maximum values ​​of the variation factor, respectively. These are the minimum and maximum values ​​of the crossover probability, respectively. The success rate of the main population is expressed as the percentage of new individuals less than or equal to the number of old individuals out of the total number of iterations. ,in, This represents the i-th individual in the main population of generation g. Let f represent the candidate individual after the i-th mutation crossover in the g-th generation, f represent the fitness function, and NP represent the number of individuals in the companion population.

[0056] Further, step S3 specifically involves,

[0057] S31. Set adaptive parameters including disturbance strength, no improvement counter and maximum number of no improvement, where the disturbance strength is 1 and the no improvement counter is 0;

[0058] S32. Dynamically adjust the perturbation intensity: When the no-improvement counter is greater than the first threshold, increase the perturbation intensity to expand the search range for global exploration; when the no-improvement counter is less than the second threshold, decrease the perturbation intensity for global search; wherein, the first threshold is greater than the second threshold.

[0059] S33. Use the scheduling scheme formed in step S2 as the search space, and randomly generate a solution x from the search space. init As the initial current solution x current =x init Calculate the objective function value f(x) current );

[0060] S34. Perform neighborhood search: Generate the current solution x current The neighborhood N(x) current ): N(x current )={x`|candidate solution x` is obtained from the current solution x through neighborhood operations current generate};

[0061] S35. Calculate the neighborhood N(x) current The objective function values ​​of each candidate solution in the given set are used to select the candidate solution with the smallest objective function value as the new solution x. new ;

[0062] S36, in the new interpretation x new Objective function value f(x) new ) < current solution x current The objective function value f(x) current When x is the solution, then the current solution x is the solution. current =New Solution x new The current solution x current The objective function value f(x) current The new solution x = new Objective function value f(x) new If no improvement is achieved, the no improvement counter is reset to 0; otherwise, the no improvement counter is incremented by 1.

[0063] S37. When the maximum number of iterations is reached or the no-improvement counter exceeds the threshold, the iteration ends and the current solution x is obtained. current If the optimal scheduling scheme is selected, then return to step S32 for the next iteration.

[0064] Further, in step S33, the set of candidate scheduling schemes includes machine allocation space, process sequencing space, and constraints. The machine allocation space is the set of machines that can be allocated to each process; the process sequencing space is the set of process execution sequences that satisfy the process constraints; the constraints include process constraints, machine capacity constraints, and time constraints.

[0065] The beneficial effects of this invention are:

[0066] I. This flexible shop scheduling optimization method based on metaheuristic algorithms addresses the problem that traditional differential evolution algorithms are prone to getting stuck in local optima when dealing with flexible shop scheduling problems and lack the ability to escape local optima. By using nonlinear perturbation (trigonometric function mutation) and the dynamic adjustment mechanism of the accompanying population, it can improve the ability to escape local optima, enabling the algorithm to achieve an effective balance between global search and local exploitation. It can significantly reduce the maximum completion time (makespan) and improve the solution efficiency of flexible shop scheduling problems.

[0067] Second, this invention addresses the problem that traditional scheduling algorithms suffer from insufficient population diversity in the initial stage, which can easily lead to a limited search space. By designing the population initialization and accompanying population, it avoids individual duplication and similarity, ensuring the comprehensiveness of the search space. At the same time, through an adaptive adjustment mechanism, it adapts to scheduling problems of different scales and complexities, demonstrating strong robustness.

[0068] Third, this flexible workshop scheduling optimization method based on metaheuristic algorithms addresses the difficulty in interpreting the results of traditional optimization algorithms and the lack of intuitive display tools. This invention helps users evaluate algorithm performance through visualization tools and detailed performance analysis, and outputs scheduling schemes that can be directly used for workshop management, significantly improving the practicality and interpretability of the algorithm. Attached Figure Description

[0069] Figure 1 This is a flowchart illustrating the flexible workshop scheduling optimization method based on metaheuristic algorithms according to an embodiment of the present invention.

[0070] Figure 2 This is a schematic diagram comparing the maximum completion time of the flexible workshop scheduling optimization method based on metaheuristic algorithms in the embodiment with existing methods including Discrete Particle Swarm Optimization (DPSO), Genetic Algorithm-Random Restart Hill Climb Hybrid Optimization Algorithm GA-RRHC, Teaching Optimization Method (TSO), and Discrete Artificial Bee Colony Algorithm DB-ABC.

[0071] Figure 3 The embodiment is a flexible workshop scheduling optimization method based on metaheuristic algorithms, and the existing methods include Discrete Particle Swarm Optimization (DPSO), Genetic Algorithm-Random Restart Hill Climb Hybrid Optimization Algorithm GA-RRHC, Teaching Optimization Method (TSO), and Discrete Artificial Bee Colony Algorithm (DB-ABC) maximum completion time box plot. Detailed Implementation

[0072] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0073] The embodiment provides a flexible shop floor scheduling optimization method based on metaheuristic algorithms, such as Figure 1 This includes the following steps:

[0074] S1. Obtain the dataset for flexible workshop scheduling.

[0075] In step S1, the dataset for flexible job shop scheduling includes the following information: the number of workpieces (i.e., the number of workpieces to be processed in the shop), the number of machines (i.e., the number of machines available in the shop), and workpiece processing data (i.e., the processing steps for each workpiece, including the available machines and processing time for each step). Relevant data for the Flexible Job Shop Scheduling Problem (FJSP) is read from external files or databases and parsed into structured data that can be directly used by the differential evolution algorithm and adaptive hill-climbing algorithm based on trigonometric function mutation in subsequent steps. The core task of data loading is to ensure the correctness and completeness of the scheduling data, providing reliable input for the execution of subsequent optimization algorithms.

[0076] S2. Introducing nonlinear perturbation and dynamic adjustment mechanism of the accompanying population, and using differential evolution algorithm based on trigonometric function mutation, the dataset of flexible workshop scheduling is processed to initially form a scheduling scheme.

[0077] S21. Population initialization: The initial solution set is generated, which includes the main population and the companion population. The main population is generated by the shortest processing time rule, and the companion population is formed by dividing the flexible workshop scheduling dataset into multiple regions as the solution space, and performing random generation, splicing operation and row-level random permutation on each region to obtain temporary vectors.

[0078] In step S21, the accompanying population is constructed by dividing the flexible workshop scheduling dataset into multiple regions as the solution space, and then performing random generation, concatenation, and row-level random permutation on temporary vectors within each region. Specifically:

[0079] S211. Define the accompanying population A, which includes NP individuals: Where i represents the individual index. This represents the 1st, 2nd, ..., i, ..., NPth individual. Each individual, i.e., the chromosome, uses a two-layer encoding method, which includes a machine allocation part and a process sorting part.

[0080] S212. Use the dataset of flexible workshop scheduling as the solution space, i.e., the j-dimensional interval. Divide the region into q equal regions, and define the upper bound of the h-th region. The lower bound of the h-th region :

[0081] ,

[0082] in, These represent the lower and upper bounds of the entire search space, respectively, and h represents the region index;

[0083] S213. Generate NP / q random numbers uniformly within the h-th region to form a random value vector of the j-th dimension within the h-th region. :

[0084] ,

[0085] Where rand represents a random function used to generate random numbers;

[0086] S214. Divide the j-dimensional structure into q sub-segments and concatenate them in order to form the initial concatenation vector of the j-th dimension. :

[0087] ,

[0088] Where j represents the dimension index, Let h represent the random value vector of the j-th dimension in the 1st, 2nd, ..., hth, ..., qth regions, where h represents the region index and q represents the total number of regions after dividing the search space equally.

[0089] S215, the assembled Perform another global permutation to obtain the set of candidate values ​​for the j-th dimension across all regions. :

[0090] ,

[0091] in, This represents a random permutation function used to shuffle the order;

[0092] S216, to By performing random permutations while keeping the set of elements unchanged and only changing their order, we can enhance diversity and obtain a temporary vector of the j-th dimension. :

[0093] ,

[0094] S217, Constructing temporary population matrix B:

[0095] ,

[0096] Where D represents the problem dimension, i.e., the number of temporary vectors;

[0097] S218. Assign the temporary population matrix B to the accompanying population A: A=B.

[0098] In steps S211-218, through partition initialization, various initialization sets are generated while retaining the vectors obtained from the previous concatenation. Then, in step S215, the concatenated vectors from step S214 are processed... Perform random permutation and step S216 will change the c from step S215. j By performing random permutations, the order of the sequence is shuffled by the two independent row-level random permutations, making the distribution of solutions more uniform and wider in space. Compared with the traditional random generation starting from zero, it is not a simple generation of new solutions, but a modification on the existing basis, which improves the initial diversity.

[0099] In step S21, the size of the accompanying population A is the same as that of the main population. To reduce individual duplication and similarity caused by random initialization, when the accompanying population is initialized, the flexible shop floor scheduling dataset is divided into q regions as the solution space. Each region randomly generates NP / q values, which are then randomly concatenated to form initial accompanying individuals based on dimensions. The accompanying population is initialized using a region partitioning method to ensure population diversity. Population initialization combines random generation with a heuristic rule based on the shortest processing time (SPT) to generate diverse and higher-quality initial solutions, avoiding individual duplication and similarity, and providing a good starting point for subsequent optimization processes. Population initialization supports two solution initialization methods: random solution generation (generating diverse solutions by randomly selecting the available machines for each process) and solution generation based on the shortest processing time rule (prioritizing the selection of the machine with the shortest processing time to generate higher-quality solutions).

[0100] S22. Adaptively adjust parameters and update the main population: Considering the mutation factor F and the crossover probability CR, a trigonometric function nonlinear perturbation mechanism is adopted. The periodicity and nonlinearity of the trigonometric function are used to control the perturbation amplitude of the trigonometric function through the mutation factor F. Mutation and crossover operations are performed on each individual of the main population in the g-th generation, i.e., the target vector, to generate an experimental vector. The experimental vector is compared with the solution of the current generation of the companion population, i.e., the companion vector. The individual with the best fitness among the target vector, experimental vector, and companion vector is selected to update the target vector of the main population in the g+1-th generation.

[0101] S221. Perform mutation operations on the individuals of the main population in the g-th generation. Perturb the solution using sine and cosine functions, and control the perturbation amplitude of the trigonometric functions using the mutation factor F to generate the mutation vector of the i-th individual in the g-th generation. .

[0102] In step S221, the mutation vector of the i-th individual in the g-th generation... The j-th dimension component The formula for generating it is:

[0103] ,

[0104] in, Let j be the j-th dimension component of the mutation vector of the i-th individual. These represent three distinct solutions, or individuals, randomly selected from the main population; i represents the individual index, and j represents the dimension index; F is the mutation factor, or scaling factor, used to control the magnitude of mutation. , For nonlinear perturbations based on the difference between two solutions, This indicates rounding down, used to map the result to a discrete value (such as a machine number) within a specified range. The result is restricted to a valid range (machine number between 1 and the total number of available machines, num_machines).

[0105] In step S221, a trigonometric function mutation method is used to further enhance the mutation capability. The perturbation amplitude of the trigonometric function can be controlled by a scaling factor F, achieving a balance between global search and local exploitation. The mutation process is one of the key steps in the differential evolution algorithm, used to generate mutation vectors. The trigonometric function mutation method, using sine and cosine functions to perturb the solution, dynamically adjusts the mutation factor F to generate diverse mutated solutions, improving global search capability. The mutation formula combines the nonlinear characteristics of sine and cosine functions with the differential perturbation idea of ​​the differential evolution algorithm, enabling the generation of diverse solutions in the solution space, enhancing the algorithm's global search capability and ability to escape local optima. Simultaneously, discretization and range constraints ensure that the generated solutions are applicable to the flexible workshop scheduling problem. Traditional differential mutation may generate solutions outside the valid range, requiring additional repair operations. Trigonometric function mutation, combined with range constraints, directly generates valid solutions, reducing the generation of invalid solutions and improving algorithm efficiency. For flexible workshop scheduling problems with large scale variations, trigonometric function mutation can adapt to problems of different scales by dynamically adjusting the direction and magnitude of the perturbation, demonstrating strong robustness. Furthermore, the introduction of nonlinear perturbation reduces the dependence on parameters, enabling the algorithm to achieve good performance under different parameter settings.

[0106] S222, Perform a cross operation, using the current individual and mutation vector The candidate solutions generated after performing the crossover operation are the trial individuals. :

[0107] ,

[0108] in, To test individuals The j-th dimension component, For the mutation vector The j-th dimension component, For the current individual The j-th dimension component, where rand is a random function and CR is the cross probability, controlling the proportion of cross variables; j randThe dimension index is randomly selected.

[0109] S223. Using a greedy selection strategy, select the individual with the best fitness among the three vectors (target vector, trial vector, and adjoint vector) in the main population as the next generation individual in the main population:

[0110] ,

[0111] in, This represents the i-th individual in the main population of generation g+1. Let represent the target vector of the i-th current individual in the g-th generation of the main population. Let represent the candidate individual, i.e., the trial vector, after the i-th mutation crossover in the g-th generation. Let f represent the companion vector of the i-th individual in the g-th generation companion population, i.e., the g-th generation, min represents the minimum function, and f represents the fitness function.

[0112] S23. A diversity selection strategy is used to select individuals whose fitness is neither optimal nor worst as the companion vector for the (g+1)th generation. Specifically:

[0113] ,

[0114] in, Let represent the adjoint vector of the i-th individual in the (g+1)-th generation of the adjoint population, and max represent the maximum value function.

[0115] In step S23, a diversity selection strategy is used to select the next generation of individuals from the companion population. The aim is to maintain population diversity and avoid premature convergence. A modified selection criterion is used to update the companion population to prevent the premature elimination of valuable individuals. In the original selection operation, excellent individuals were determined by pairwise comparisons of the trial vector and the target vector, rather than ranking all vectors. This resulted in the elimination of some individuals who might still be valuable. Therefore, the implementation method modifies the selection criterion to iteratively update the companion population: for the i-th individual, the fitness values ​​of the target vector, the trial vector, and the companion vector are evaluated. The vector with the best fitness value becomes the new target vector, the vector with the second best fitness value becomes the new companion vector, and the vector with the worst fitness value is eliminated. The implementation method improves the pairwise comparison to a three-entity comparison, and the suboptimal solution is stored and utilized through the companion population.

[0116] S24, Success rate s of the main population g The mutation factor F and crossover probability CR are adaptively adjusted. Specifically,

[0117] ,

[0118] in, These are the minimum and maximum values ​​of the variation factor, respectively. These are the minimum and maximum values ​​of the crossover probability, respectively. The success rate of the main population is expressed as the percentage of new individuals less than or equal to the number of old individuals out of the total number of iterations. ,in, This represents the i-th individual in the main population of generation g. Let F represent the candidate individual after the i-th mutation and crossover in the g-th generation, f represent the fitness function, and NP represent the number of individuals in the companion population. In a specific example, F min =0.3, F max =1.0, CR min =0.1,CR max =0.95.

[0119] In step S24, by adaptively adjusting the mutation factor F and the crossover probability CR, the success rate s of the improved main population can be achieved. g When the value is small, increasing the mutation factor F to expand the step size helps to escape the local optimum; if the success rate s of the improved main population is... g A larger mutation factor F and crossover probability CR reduce the crossover probability to inherit more mutation information, resulting in stronger structural perturbations and helping to escape stagnation. The adaptive adjustment mechanism dynamically adjusts the mutation factor F and crossover probability CR based on changes in population fitness, enabling the algorithm to adapt to flexible workshop scheduling problems of varying scales and complexities, demonstrating strong robustness. The adaptive adjoint population enhances the algorithm's diversity and local exploitation capabilities. By adaptively adjusting the mutation factor F and crossover probability CR, and combining the mutation and crossover operations of differential evolution, the adjoint population is updated, thus preventing the main population from getting trapped in local optima during optimization and improving the algorithm's global search capability.

[0120] S25, Iteration steps S22-S24, until the maximum number of iterations T is reached. max When the iteration ends, the flexible workshop scheduling scheme is obtained.

[0121] In step S2, region initialization ensures that the companion population differs from the main population in the initial stage. The companion population is updated and reinitialized to maintain the characteristics of the suboptimal solution. These mechanisms ensure the value of the companion population. Subsequently, mutation is mainly performed using the companion population with better diversity. During the optimization process, generally selected individuals are considered to have valuable evolutionary information, but under the constraints of many problems, the algorithm may still fall into a local optimum. If individuals eliminated by fitness evaluation can also be selected to participate in evolution, richer evolutionary information can be utilized, which is of great significance for problems that converge prematurely and produce near-stagnation. This invention uses the companion population method to solve this problem, where individuals are called companion vectors. Through adaptive companion population, this invention can effectively improve global search capability and local development capability, significantly improving the solution efficiency and solution quality of the flexible job shop scheduling problem. The region partitioning and dynamic update mechanism of the companion population further enhances the diversity of the population. When the companion population converges with the main population, it is reinitialized around the optimal solution of the main population, avoiding premature convergence of the main population. By introducing nonlinear perturbations (trigonometric function mutation) and a dynamic adjustment mechanism for the accompanying population, the algorithm's global search capability and local exploitation capability are enhanced. Trigonometric function mutation utilizes the nonlinear characteristics of sine and cosine functions to introduce diversity during the mutation process, preventing the population from getting trapped in local optima. The accompanying population, through dynamic updates and reinitialization mechanisms, enhances the algorithm's ability to escape local optima, achieving an effective balance between global search and local exploitation, thereby significantly reducing the maximum completion time (makespan). Through optimization convergence, the fitness difference of all individuals in the population is reduced to less than a certain threshold. It can efficiently perform global optimization on the main population, significantly improving the accuracy and efficiency of solving the flexible job shop scheduling problem.

[0122] S3. The resulting scheduling scheme is then processed using an adaptive hill-climbing algorithm, combined with multiple perturbation mechanisms and perturbation intensity adjustment strategies, to optimize the current best solution and form the optimal scheduling scheme.

[0123] S31. Set adaptive parameters including disturbance strength, no improvement counter and maximum number of no improvement, where the disturbance strength is 1 and the no improvement counter is 0.

[0124] S32. Dynamically adjust the perturbation intensity: When the unimproved counter is greater than the first threshold, such as 5, the perturbation intensity is increased to expand the search range for global exploration. When the unimproved counter is less than the second threshold, such as 2, the perturbation intensity is reduced for global search. The first threshold is greater than the second threshold.

[0125] S33. Use the scheduling scheme formed in step S2 as the search space, and randomly generate a solution x from the search space. init As the initial current solution x current=x init Calculate the objective function value f(x) current ).

[0126] In step S33, the set of candidate scheduling schemes includes machine allocation space, process sequencing space, and constraints. The machine allocation space is the set of machines that can be allocated to each process; the process sequencing space is the set of process execution sequences that satisfy process constraints; the constraints include process constraints, machine capacity constraints, time constraints, etc.

[0127] S34. Perform neighborhood search: Generate the current solution x current The neighborhood N(x) current ): N(x current )={x`|candidate solution x` is obtained from the current solution x through neighborhood operations current generate}.

[0128] S35. Calculate the neighborhood N(x) current The objective function values ​​of each candidate solution in the given set are used to select the candidate solution with the smallest objective function value as the new solution x. new .

[0129] S36, in the new interpretation x new Objective function value f(x) new ) < current solution x current The objective function value f(x) current When x is the solution, then the current solution x is the solution. current =New Solution x new The current solution x current The objective function value f(x) current The new solution x = new Objective function value f(x) new If no improvement is achieved, the no improvement counter is reset to 0; otherwise, the no improvement counter is incremented by 1.

[0130] S37. When the maximum number of iterations is reached or the no-improvement counter exceeds the threshold, the iteration ends and the current solution x is obtained. current If the optimal scheduling scheme is selected, then return to step S32 for the next iteration.

[0131] In step S3, the adaptive hill-climbing algorithm, a local search strategy, is used to further optimize the current optimal solution. Combining perturbation and adaptive adjustment mechanisms, it can perform a fine search within the local region of the solution, thereby improving the solution quality. A perturbation mechanism is introduced: during the operation, a job and one of its operations are randomly selected, and the machine number of that operation is randomly changed, generating a new solution; the number of perturbations is proportional to the perturbation intensity—the greater the perturbation intensity, the more perturbations are performed. The adaptive random hill-climbing algorithm performs a local search on the current optimal solution, combined with a dynamic adjustment mechanism for perturbation intensity, to perform a fine search within the local region of the solution, further optimizing the quality of the current solution. The adaptive hill-climbing algorithm dynamically adjusts the search range based on the perturbation intensity: expanding the search range when there is no improvement, escaping local optima; and narrowing the search range when there is improvement, performing a fine search. Expanding or narrowing the search range based on the number of times there is no improvement enhances the algorithm's local search capability and its ability to escape local optima, allowing the algorithm to effectively improve the solution quality even in later optimization stages. Combining multiple perturbation mechanisms and dynamic adjustment of perturbation intensity, the current optimal solution is further optimized to find a better maximum completion time (makespan). Through multiple perturbations and adaptive adjustments, it can adapt to flexible shop floor scheduling problems of different scales and complexities, and avoids premature convergence and search stagnation in the later stages of the differential evolution algorithm. It exhibits strong robustness across different search stages and problem scales.

[0132] S4. Present the optimal scheduling scheme to achieve the maximum completion time in a visual format.

[0133] In step S4, the data from steps S2 and S3, including the optimal solution and the fitness value for each iteration, are recorded. An analysis report is generated, a fitness value change curve with the number of iterations is plotted, and the optimal scheduling scheme, optimal fitness value, and statistical analysis results are output. The optimization process and results are visually displayed through the fitness value change curve, scheduling Gantt chart, and performance analysis report. The output scheduling scheme can be directly used for flexible shop floor management, improving the algorithm's practicality. The generation of intuitive analysis reports and charts helps users evaluate algorithm performance and outputs the final scheduling scheme and optimization results, comprehensively showcasing the optimization results of the flexible shop floor scheduling problem. It provides users with intuitive and detailed analysis and output, significantly improving the algorithm's practicality and interpretability.

[0134] This flexible shop scheduling optimization method based on metaheuristic algorithms addresses the problem that traditional differential evolution algorithms are prone to getting stuck in local optima and lack the ability to escape local optima when dealing with flexible shop scheduling problems. By using nonlinear perturbation (trigonometric function mutation) and dynamic adjustment mechanism of the accompanying population, it can improve the ability to escape local optima, enabling the algorithm to achieve an effective balance between global search and local exploitation. It can significantly reduce the maximum completion time (makespan) and improve the solution efficiency of flexible shop scheduling problems.

[0135] This invention addresses the problem that traditional scheduling algorithms suffer from insufficient population diversity in the initial stage, which can easily lead to a limited search space. By designing the population initialization and accompanying population, it avoids individual duplication and similarity, ensuring the comprehensiveness of the search space. At the same time, through an adaptive adjustment mechanism, it adapts to scheduling problems of different scales and complexities, demonstrating strong robustness.

[0136] This flexible shop floor scheduling optimization method based on metaheuristic algorithms addresses the challenges of interpreting the results of traditional optimization algorithms and the lack of intuitive visualization tools. By employing visualization tools and detailed performance analysis, this invention helps users evaluate algorithm performance and outputs scheduling schemes that can be directly used for shop floor management, significantly improving the algorithm's practicality and interpretability.

[0137] This flexible shop scheduling optimization method based on metaheuristic algorithms employs an adaptive update mechanism of the accompanying population to improve the algorithm's performance in global search, thereby finding the optimal solution better. An improved sine and cosine mutation formula is used to address the problem that differential evolution algorithms are prone to getting trapped in local optima. Especially for NP-hard problems such as flexible shop scheduling, an improved differential evolution algorithm and an adaptive random hill-climbing algorithm are introduced to balance the efficiency and accuracy of global search and local development, thereby obtaining a better scheduling scheme.

[0138] This flexible shop floor scheduling optimization method based on metaheuristic algorithms can achieve efficient optimization: the improved differential evolution algorithm combined with the adjoint population and random hill-climbing module significantly improves the solution efficiency of the flexible shop floor scheduling problem; it can achieve a balance between global search and local exploitation without compromising safety and versatility: through trigonometric function mutation, adaptive adjoint population, and random hill-climbing module, the global search capability and local exploitation capability are balanced; the results are highly interpretable: through visualization tools and statistical analysis, the optimization process and results are intuitively displayed, enhancing the practicality and interpretability of the method.

[0139] The experimental verification of this flexible shop floor scheduling optimization method based on metaheuristic algorithms in the embodiment is as follows:

[0140] The implementation method and existing methods, including Discrete Particle Swarm Optimization (DPSO), Genetic Algorithm-Random Restart Hill Climb Hybrid Optimization Algorithm (GA-RRHC), Teaching Optimization Method (TSO), and Discrete Artificial Bee Colony Algorithm (DB-ABC), are compared in Table 1 with the maximum completion time of the Brandimarte dataset (test instances Mk01-Mk10) in the Brandimarte dataset of the Flexible Workshop Scheduling Standard Test Dataset.

[0141] Table 1. Comparison of maximum completion time between the embodiment method and existing methods.

[0142]

[0143] From Table 1 and Figure 2 The results show that the implementation method has a significant advantage in terms of maximum completion time across all test cases. For example, in the Mk08 instance, the maximum completion time of the implementation method (526) is 47% lower than that of GA-RRHC (1000.9), and this advantage becomes more pronounced as the complexity of the problem increases. This advantage is maintained even in complex instances such as MK07-MK10, indicating that the implementation method has strong robustness and engineering applicability.

[0144] like Figure 3 The bin size and upper and lower bounds of the implementation method are typically located at lower positions, resulting in significantly shorter completion times compared to DPSO, GA-RRHC, TSO, and DB-ABC algorithms. In terms of bin length, the implementation method's bins are relatively short, implying less data dispersion. Compared to other algorithms, the implementation method exhibits smaller completion time fluctuations across multiple experiments or application scenarios. It consistently maintains a low completion time level with minimal time deviation, demonstrating good stability.

[0145] Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A flexible shop floor scheduling optimization method based on metaheuristic algorithms, characterized in that: Includes the following steps, S1. Obtain the dataset for flexible workshop scheduling; S2. Introduce nonlinear perturbation and dynamic adjustment mechanism of the accompanying population, and use differential evolution algorithm based on trigonometric function mutation to process the dataset of flexible workshop scheduling to initially form a scheduling scheme. S21. Population initialization: The initial solution set is generated, which includes the main population and the companion population. The main population is generated by the shortest processing time rule, and the companion population is formed by dividing the flexible workshop scheduling dataset into multiple regions as the solution space, and performing random generation, splicing operation and row-level random permutation on each region to form a temporary vector. S22. Adaptively adjust parameters and update the main population: Considering the mutation factor F and the crossover probability CR, a trigonometric function nonlinear perturbation mechanism is adopted. The periodicity and nonlinearity of the trigonometric function are used to control the perturbation amplitude of the trigonometric function through the mutation factor F. Mutation and crossover operations are performed on each individual of the main population in the g-th generation, i.e., the target vector, to generate an experimental vector. The experimental vector is compared with the solution of the current generation of the companion population, i.e., the companion vector. The individual with the best fitness among the target vector, experimental vector, and companion vector is selected to update the target vector of the main population in the g+1-th generation. S221. Perform mutation operations on the individuals of the main population in the g-th generation. Perturb the solution using sine and cosine functions, and control the perturbation amplitude of the trigonometric functions using the mutation factor F to generate the mutation vector of the i-th individual in the g-th generation. ; S222, Perform a cross operation, using the current individual and mutation vector The candidate solutions generated after performing the crossover operation are the trial individuals. : ， in, To test individuals The j-th dimension component, For the mutation vector The j-th dimension component, For the current individual The j-th dimension component, where rand is a random function and CR is the cross probability, controlling the proportion of cross variables; j rand The dimension index is randomly selected; S223. Using a greedy selection strategy, select the individual with the best fitness among the three vectors (target vector, trial vector, and adjoint vector) in the main population as the next generation individual in the main population: ， in, This represents the i-th individual in the main population of generation g+1. Let represent the target vector of the i-th current individual in the g-th generation of the main population. Let represent the candidate individual, i.e., the trial vector, after the i-th mutation crossover in the g-th generation. Let f represent the companion vector of the i-th individual in the g-th generation companion population, i.e., the g-th generation; min represents the minimum value function; and f represents the fitness function. S23. Use a diversity selection strategy to select individuals whose fitness is neither the best nor the worst as the companion vector for the g+1th generation. S24, Success rate s of the main population g Adaptive adjustment of the mutation factor F and crossover probability CR; S25, Iteration steps S22-S24, until the maximum number of iterations T is reached. max When the iteration is terminated, the flexible workshop scheduling scheme is obtained; S3. The resulting scheduling scheme is then processed using an adaptive hill-climbing algorithm, combined with multiple perturbation mechanisms and perturbation intensity adjustment strategies, to optimize the current optimal solution and form the optimal scheduling scheme. S4. Present the optimal scheduling scheme to achieve the maximum completion time in a visual format.

2. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 1, characterized in that: In step S1, the dataset for flexible workshop scheduling includes the number of workpieces and machines, and the workpiece processing data, i.e., the processing procedure information for each workpiece, including the selectable machines for each procedure and the corresponding processing time.

3. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 1, characterized in that: In step S21, the accompanying population is constructed by dividing the flexible shop scheduling dataset into multiple regions as the solution space, and then performing random generation, concatenation, and row-level random permutation on temporary vectors within each region. Specifically, S211. Define the accompanying population A, which includes NP individuals: Where i represents the individual index. This represents the 1st, 2nd, ..., i, ..., NPth individual. Each individual, i.e., the chromosome, uses a two-layer encoding method, which includes a machine allocation part and a process sorting part. S212. Use the dataset of flexible workshop scheduling as the solution space, i.e., the j-dimensional interval. Divide the region into q equal regions, and define the upper bound of the h-th region. The lower bound of the h-th region : ， in, These represent the lower and upper bounds of the entire search space, respectively, and h represents the region index; S213. Generate NP / q random numbers uniformly within the h-th region to form a random value vector of the j-th dimension within the h-th region. : ， Where rand represents a random function used to generate random numbers; S214. Divide the j-dimensional structure into q sub-segments and concatenate them in order to form the initial concatenation vector of the j-th dimension. : ， Where j represents the dimension index, Let h represent the random value vector of the j-th dimension in the 1st, 2nd, ..., hth, ..., qth regions, where h represents the region index and q represents the total number of regions after dividing the search space equally. S215, the assembled Perform another global permutation to obtain the set of candidate values ​​for the j-th dimension across all regions. : ， in, This represents a random permutation function used to shuffle the order; S216, to By performing random permutations while keeping the set of elements unchanged and only changing their order, we can enhance diversity and obtain a temporary vector of the j-th dimension. : ， S217, Constructing temporary population matrix B: ， Where D represents the problem dimension, i.e., the number of temporary vectors; S218. Assign the temporary population matrix B to the accompanying population A: A=B.

4. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 1, characterized in that: In step S221, the mutation vector of the i-th individual in the g-th generation... The j-th dimension component The formula for generating it is: ， in, Let j be the j-th dimension component of the mutation vector of the i-th individual. These represent three distinct solutions, or individuals, randomly selected from the main population; i represents the individual index, and j represents the dimension index; F is the mutation factor, or scaling factor, used to control the magnitude of mutation. , For nonlinear perturbations based on the difference between two solutions, This indicates rounding down, used to map the result to discrete values ​​within a specified range.

5. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 1, characterized in that: Step S23, specifically, ， in, Let represent the adjoint vector of the i-th individual in the (g+1)-th generation of the adjoint population, and max represent the maximum value function.

6. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 1, characterized in that: Step S24, specifically, ， in, These are the minimum and maximum values ​​of the variation factor, respectively. These are the minimum and maximum values ​​of the crossover probability, respectively. The success rate of the main population is expressed as the percentage of new individuals less than or equal to the number of old individuals out of the total number of iterations. ,in, This represents the i-th individual in the main population of generation g. Let f represent the candidate individual after the i-th mutation crossover in the g-th generation, f represent the fitness function, and NP represent the number of individuals in the companion population.

7. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in any one of claims 1-6, characterized in that: Step S3, specifically, S31. Set adaptive parameters including disturbance strength, no improvement counter and maximum number of no improvement, where the disturbance strength is 1 and the no improvement counter is 0; S32. Dynamically adjust the perturbation intensity: When the no-improvement counter is greater than the first threshold, increase the perturbation intensity to expand the search range for global exploration; when the no-improvement counter is less than the second threshold, decrease the perturbation intensity for global search; wherein, the first threshold is greater than the second threshold. S33. Use the scheduling scheme formed in step S2 as the search space, and randomly generate a solution x from the search space. init As the initial current solution x current =x init Calculate the objective function value f(x) current ); S34. Perform neighborhood search: Generate the current solution x current The neighborhood N(x) current ): N(x current )={x`|candidate solution x` is obtained from the current solution x through neighborhood operations current generate}; S35. Calculate the neighborhood N(x) current The objective function values ​​of each candidate solution in the given set are used to select the candidate solution with the smallest objective function value as the new solution x. new ; S36, in the new interpretation x new Objective function value f(x) new ) < current solution x current The objective function value f(x) current When x is the solution, then the current solution x is the solution. current =New Solution x new The current solution x current The objective function value f(x) current The new solution x = new Objective function value f(x) new If no improvement is achieved, the no improvement counter is reset to 0; otherwise, the no improvement counter is incremented by 1. S37. When the maximum number of iterations is reached or the no-improvement counter exceeds the threshold, the iteration ends and the current solution x is obtained. current If the optimal scheduling scheme is selected, then return to step S32 for the next iteration.

8. The flexible shop floor scheduling optimization method based on metaheuristic algorithms as described in claim 7, characterized in that: In step S33, the set of candidate scheduling schemes includes machine allocation space, process sequencing space, and constraints. The machine allocation space is the set of machines that can be allocated to each process; the process sequencing space is the set of process execution sequences that satisfy the process constraints; the constraints include process constraints, machine capacity constraints, and time constraints.

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