Robust scheduling method for bad scene set of wait-free flow shop based on memetic algorithm

Abstract

The invention discloses a no-waiting flow shop bad scene set robust scheduling method based on a memetic algorithm, and the method comprises the steps: carrying out the processing time description of a to-be-processed workpiece set and a processing machine set through employing a discrete multi-scene method, defining a threshold value bad scene set according to a preset performance threshold value, and building a robust optimization model of the threshold value bad scene set. According to the method, a threshold value bad scene set robust scheduling model of a wait-free flow shop is established, and an optimization objective function is determined. An initial population is generated by using a scene-based NEH heuristic algorithm. And performing global search on the current population through a genetic operator and a selection mechanism, selecting a target scene by using a strategy related to the model, constructing a neighborhood structure by using knowledge related to problems in the target scene, and performing local search. The whole optimization process is circularly executed under iteration control until the preset maximum iteration number is reached, and finally the optimal scheduling solution in the optimized population is output. The method can effectively cope with the uncertainty of the processing time, improves the robustness and optimization efficiency of no-waiting flow shop scheduling, gives consideration to the global search capability and the local refinement capability, and achieves the quick solving of a high-quality scheduling scheme.

Description

Technical Field

[0001] This invention belongs to the field of workshop scheduling technology, specifically relating to a robust scheduling method for bad scenario sets in a no-wait flow workshop based on the meme algorithm. Background Technology

[0002] Modern production systems face challenges such as high-variety, low-batch, and rapid delivery, placing higher demands on scheduling systems' ability to optimize the time allocation of equipment, personnel, and materials. Flow Shop Scheduling (FSP) is a classic abstraction of this type of problem. However, some processes involve no-wait constraints to ensure that workpieces do not suffer from delays in quality or process continuity between machines. This no-wait flow shop scheduling problem is a strongly NP-hard problem when minimizing the maximum makespan. Traditional exhaustive or rule-based methods struggle to provide high-quality scheduling results on a real-world scale, thus driving the demand for solutions based on intelligent methods.

[0003] On the other hand, real-world production environments present numerous uncertainties, such as equipment fluctuations, operational errors, variations in incoming material quality, and environmental disturbances, making it difficult to accurately predict the processing time of each workpiece on the machine. If scheduling schemes are based solely on deterministic processing times, delays or significant performance degradation are highly likely to occur during actual execution. Therefore, three mainstream methods have emerged in scheduling research: stochastic scheduling, fuzzy scheduling, and robust scheduling. Compared to stochastic and fuzzy methods, robust scheduling does not require knowledge of precise probability distributions or membership functions; its tolerance to uncertainty and stability make it more valuable in industrial applications. Scenario-based methods are commonly used in robust scheduling. They construct scenario sets by discretizing multiple possible processing time values ​​(e.g., empirical sampling or simulated sampling), thereby considering the impact of uncertainty on performance during the scheduling optimization phase.

[0004] In existing publicly available technologies, there are already solutions to scheduling problems using evolutionary algorithms or hybrid metaheuristic algorithms. For example, Chinese patent CN114580743A discloses a flexible job shop scheduling method and system based on a hybrid evolutionary algorithm. This method first constructs a flexible job shop scheduling model that considers setup time and processing time, and uses an integer encoding / decoding strategy to segment and encode the process sequence and machine selection. Then, it generates a new population through operations such as population initialization, neighborhood crossover, insertion mutation, etc., and performs local search (local search) and elite retention based on the two subpopulations. Finally, it determines the final scheduling scheme through the elite population.

[0005] However, this existing technology still has the following limitations, including: 1. Although the hybrid evolutionary algorithm combines mutation and neighborhood search, its local search strategy (local search) is a relatively fixed or predefined neighborhood operation, lacking adaptive adjustments for different risk scenarios (i.e., scenarios where processing time may deviate). 2. This method focuses on classic scheduling objectives (such as makespan), but its hybrid evolutionary algorithm does not explicitly incorporate performance degradation penalties in bad scenarios into the optimization objective. Therefore, this method may lack sufficient robustness to extreme or rare but critical bad scenarios; 3. Traditional robust scheduling methods only focus on a single worst-case scenario, resulting in very conservative robust scheduling solutions.

[0006] Therefore, although existing technologies have achieved certain results in the scheduling of flexible workshops, they are still insufficient to fully address the scheduling risks caused by the uncertainty of processing time, and it is also difficult to guarantee that the generated scheduling schemes are sufficiently robust in high-risk scenarios. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a robust scheduling method for bad scenarios in a no-wait flow shop based on the meme algorithm.

[0008] The objective of this invention can be achieved through the following technical solutions: This invention provides a robust scheduling method for bad scenario sets in a wait-free flow shop based on the meme algorithm, comprising the following steps: The uncertainty of processing time is described using a discrete multi-scenario method for the set of workpieces to be processed and the set of processing machines. Based on the preset performance threshold, a threshold bad scenario set is constructed, and a robust optimization model for the threshold bad scenario set of no-wait flow shop scheduling is constructed to determine the optimization objective function of the no-wait flow shop robust scheduling. The initial population is generated using a scenario-based NEH heuristic algorithm. For the current population, a global search based on genetic operators and selection mechanisms is performed according to the optimization objective function to obtain the updated population; For the updated population, target scenarios are selected based on the optimization objective function using model-related strategies. Then, a neighborhood structure is constructed and a local search is performed within the target scenarios using problem-related knowledge to obtain the optimized population. Update the iteration count, determine whether the preset maximum iteration count has been reached. If not, use the optimized population as the current population, return to global search to continue iterating until the maximum iteration count is reached, and output the optimal scheduling solution in the optimized population.

[0009] Furthermore, the set of workpieces to be processed is represented as follows: ,in, The total number of workpieces to be processed. Indicates the first One workpiece to be processed; The set of processing machines is represented as ,in, This represents the total number of processing machines. Indicates the first Taiwanese processing machine.

[0010] Furthermore, a threshold-bad scenario set is constructed based on a preset performance threshold, specifically including: Collection of workpieces to be processed and processing machines The processing time for all operations is uncertain, using discrete scene sets. This indicates that the discrete scene set Each scene A machining time matrix consisting of the machining time for each workpiece on each machine. , Indicates in the scene Lower workpiece In the machine Processing time on discrete scene sets It includes all possible processing time scenarios, to Representing scene set The number of scenes in; For each scheduling feasible solution According to the processing sequence and processing time of the workpiece Calculate each scene sequentially The maximum completion time is obtained by taking the assigned task completion time. ; Given a performance threshold For each scenario ,like Then the scene Determined as a feasible solution for scheduling In bad scenarios, scheduling feasible solutions All bad scenarios constitute the threshold bad scenario set , represented as: in, Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below.

[0011] Furthermore, the optimization objective function of the no-wait flow shop scheduling model is given by the following formula: in, A feasible solution for scheduling is the arrangement of the processing order of the workpieces on the processing machine; Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below; For scheduling feasible solutions In the scene Maximum completion time; This is the performance threshold; Let represent the feasible solution space of the scheduling problem, and let represent the set of all workpiece processing sequences that satisfy the no-wait flow shop constraint. This indicates the optimization of the objective function value.

[0012] Furthermore, the constraint of the no-wait assembly line is: Workpiece in processing machine assembly The processing must be carried out continuously according to the preset machine sequence, and no waiting time is allowed between any two adjacent machines. Specifically, this includes: The access order of each workpiece on the machine set is fixed, and they must be accessed sequentially on the machines. to Processing must be carried out on the surface, and the processing order must not be changed; Workpiece in machine After processing is completed, it must be immediately placed on the machine. Processing begins on the first step, meaning the waiting time between any two adjacent processes is limited to zero. At any given time, each machine can only process one workpiece, meaning that a machine cannot be occupied by multiple workpieces simultaneously. Once the processing of a workpiece on a machine begins, it must not be interrupted and must be completed continuously.

[0013] Furthermore, the generation of the initial population using the scenario-based NEH heuristic algorithm specifically includes: In the threshold bad scenario set In the middle, for each scenario According to the processing time matrix Calculate the total processing time of the workpiece on all machines; Sort the workpieces according to their total processing time from longest to shortest to obtain the scene. The workpiece sorting sequence below; Based on the workpiece sorting sequence, an insertion-based construction strategy is used to incrementally generate a scheduling sequence, including: The workpiece with the longest total processing time in the workpiece sorting sequence is placed into an empty sequence to form the current sequence; For the remaining workpieces in the workpiece sorting sequence, enumerate all possible insertion positions in the current sequence, and form a new sequence for each candidate insertion position. And calculate the new sequence. In the scene Maximum completion time ; Select the insertion position with the shortest maximum completion time and insert the workpiece into the current sequence until all workpieces have been inserted, thus constructing the scene. Scheduling solution under ; Scene set All scheduling solutions generated for each scenario are added to the initial population. If the initial population The number of solutions in the scheduling process exceeds the set population size. Then, randomly delete any extra individuals. If the number of scheduling solutions in the initial population is insufficient Then in the feasible solution space X A feasible scheduling sequence that satisfies the no-wait flow workshop constraint is randomly generated and added to the initial population.

[0014] Furthermore, the step of performing a global search based on genetic operators and selection mechanisms on the current population according to the optimization objective function to obtain an updated population specifically includes: For the first The generation population uses a binary tournament selection method to generate a globally searchable set of parent generations. : Randomly select two individuals from the current population, based on the threshold bad scenario set respectively. The number of scenarios (BSN), worst-case maximum completion time (WSM), and the value of the objective function are all considered. As a fitness value, individuals with higher fitness are selected for the parent set; a predetermined number of binary tournament selections are performed independently on each of the three fitness types, and the results are then combined to form the parent set. ; Based on parent set The individual in the set performs a two-point crossover operator to generate a new individual: for each individual in the parent set, another individual is randomly selected from the parent set as a pairing individual, two different positions are randomly selected as crossover points, the gene segments of the two parents in the crossover point interval are exchanged, and the missing workpieces outside the crossover interval are filled into the corresponding positions according to the order in which the workpieces appear in the pairing individuals, forming the offspring sequence. Based on the insertion mutation operator on the parent set Individuals in the process undergo mutation operations: according to the mutation probability. A mutated individual is formed by randomly selecting a position in an individual, removing the workpiece corresponding to that position and inserting it before another randomly selected position. All new individuals obtained from the two-point crossover operator and the insertion mutation operator are incorporated into the current population, according to the optimization objective function value. The expanded population is sorted, and individuals with the worst objective function values ​​are removed until the population size is restored to the preset population size. The updated population is obtained.

[0015] Furthermore, the updated population is subjected to a model-related strategy for target scene selection based on the optimization objective function, and a neighborhood structure is constructed and a local search is performed within the target scene using problem-related knowledge to obtain the optimized population. Specifically, this includes: A binary tournament selection mechanism is used to select a local search parent set from the updated population and perform a local search operation. Based on the target scene selection strategy of reinforcement learning, target scenes for local search are determined from the threshold bad scene set, and a single scene neighborhood is constructed for each parent solution in the local search parent set under the target scene, generating a corresponding neighborhood candidate solution set. All neighborhood candidate solutions in the neighborhood candidate solution set are incorporated into the current population. The expanded population is then sorted according to the optimization objective function. Individuals with poor optimization objective function are removed, and the population size is restored to the preset population size. The optimized population was obtained.

[0016] Furthermore, the step of constructing a single-scene neighborhood for each parent solution in the local search parent set under the target scene, and generating a corresponding neighborhood candidate solution set, specifically includes: In the target scenario, a critical path is determined based on the deterministic processing time corresponding to the target scenario. The critical path is the longest chain of operations consisting of consecutive operations that determines the completion time under no waiting constraints. A forward operation is defined as an operation in the critical path that proceeds sequentially according to the processing sequence and extends the completion time; its processing time is a positive value. A reverse operation is defined as an operation in the critical path that generates a reverse dependency due to no waiting constraints and shortens the feasible time of preceding operations; its processing time is a negative value. Operations that are both forward and reverse operations have a processing time of zero. The operation refers to the processing steps of the workpiece on the machine. Based on the critical path, key blocks are identified. A key block is the longest consecutive operation segment on the critical path, located on the same machine and consisting of at least two consecutive operations, without containing reverse operations. The largest key block is selected from all key blocks, and the workpiece corresponding to the leftmost operation within that key block is determined. The rightmost operation corresponds to the workpiece. And the workpiece corresponding to the operation with the longest processing time. As a target workpiece; If there is no critical block in the critical path, then in addition to the workpiece Two workpieces are randomly selected from the remaining workpieces as workpieces. With workpiece , with the workpiece Together they serve as target workpieces; In the sequence of workpieces in the parent solution, for workpieces , workpiece and workpiece Perform pairwise position swaps to generate up to three neighborhood candidate solutions. All the swapped neighborhood candidate solutions constitute the set of neighborhood candidate solutions of the parent solution in the target scene.

[0017] Furthermore, the target scene selection strategy based on reinforcement learning (RL) determines target scenes for local search from a set of threshold bad scenes, specifically including: The first The local search current solution set of the generation is denoted as ; The target scene selection process is modeled as a reinforcement learning (RL) sequential decision problem, where the agent's state... The optimal solution of the previous generation and the current generation Number of bad scenes Completion time in worst-case scenario Variations and combinations The characterization, the formula is: in, , These represent the number of bad scenarios and the worst-case completion time of the previous generation's optimal solution, respectively. , These represent the number of bad scenarios and the worst-case completion time of the current generation's optimal solution, respectively. These represent the changes in the number of bad scenarios and the worst-case completion time, respectively. According to the status use Greedy strategy and action value function for action selection ,in, Indicates the threshold bad scene set Selected target scene ; In the selected target scenario Below, for Each parent solution generates a neighborhood candidate solution set. ,in, This represents the set of neighborhood candidate solutions generated by critical path and critical block operations in a given target scenario; Based on the parent's understanding of the target scenario Calculate the immediate reward for optimizing the objective function value. ,in, This indicates that local search yields the population optimal solution. The optimization objective function is improved. This indicates that there has been no improvement; According to instant rewards Update the action value function : in, The learning rate; Discount factor; For the next generation state, This represents the maximum Q value for all possible target scenarios in the next-generation state; When the group optimal solution If the solution remains unchanged for several consecutive generations, another solution from the population is temporarily selected as a replacement. To calculate the state.

[0018] Compared with the prior art, the present invention has the following advantages: (1) In the prior art, the scheduling problem of the flow shop, especially the scheduling of the flow shop without waiting, is prone to processing delays in actual implementation due to the uncertainty of processing time. This affects the reliability of the production plan. In order to address this technical problem, this invention proposes to describe the uncertainty of processing time through a discrete multi-scenario method and construct a threshold bad scenario set according to a preset performance threshold. This allows the uncertainty of processing time to be explicitly introduced into the scheduling model, identify and constrain potential high-risk scenarios, and enable the scheduling scheme to maintain high robustness under multiple possible processing times. This significantly reduces the risk of exceeding the completion time limit in the actual execution process and improves the reliability and stability of the scheduling scheme.

[0019] (2) In existing technologies, the optimization objective of traditional deterministic no-wait flow shop scheduling models only focuses on minimizing the maximum makespan, failing to effectively consider performance degradation under multiple scenarios with uncertain processing times, resulting in significant performance degradation of the scheduling scheme in some bad scenarios. To address this issue, this invention proposes a PT optimization objective function based on a threshold bad scenario set. By minimizing the sum of squared performance penalties under the threshold bad scenarios, balanced optimization of performance under multiple bad scenarios is achieved, thereby generating a scheduling scheme with lower makespan fluctuations and higher stability under bad scenarios, improving the overall robustness of the scheduling scheme. Replacing the traditional focus on only the worst scenario with a focus on the threshold bad scenario set in the objective function greatly reduces the conservatism of the obtained robust scheduling solution.

[0020] (3) In the prior art, the generation of the initial population usually relies on randomization or simple heuristic methods, which can easily lead to insufficient population diversity or low quality of the initial solution, affecting the global search effect and the convergence speed of the algorithm. This invention proposes a scenario-based NEH heuristic algorithm to generate the initial population. By calculating the total processing time of the workpiece for each scenario in the threshold bad scenario set and using an insertion construction strategy to generate the scheduling sequence, it not only ensures that the initial population contains diverse and high-quality individuals, but also improves the search coverage, thereby enhancing the global search efficiency, reducing the risk of the algorithm getting trapped in local optima, and accelerating the convergence speed.

[0021] (4) In the existing technology, the existing hybrid evolution or metaheuristic algorithms lack a comprehensive evaluation of the robustness index of bad scenarios during the global search process. The selection and crossover operations are not targeted enough, resulting in low search efficiency and difficulty in making full use of the structural features of the scheduled solution. This invention proposes a global search based on three types of fitness values ​​(number of bad scenarios selected by the binary tournament method, worst-case completion time, and PT value), and combines two-point crossover operators and insertion mutation operators to generate new solutions. By screening the parent generation and generating the offspring generation through multi-dimensional indicators, the global search not only focuses on the overall performance but also considers the robustness of bad scenarios, realizing efficient exploration of high-quality and robust solution space and improving the algorithm's solution accuracy and stability.

[0022] (5) In the prior art, traditional local search strategies are mostly fixed or random neighborhoods, which do not make sufficient use of the key path features in different scenarios, resulting in low local search efficiency, especially in large-scale problems where it is difficult to quickly improve scheduling performance. This invention proposes a single-scenario neighborhood operator, which identifies target workpieces and generates neighborhood solutions by identifying key paths and key blocks. It makes full use of the key operation chain structure without waiting constraints and makes local adjustments to the most critical workpieces in bad scenarios, thereby significantly improving local search efficiency and enhancing the optimization capability of key bottleneck links in complex scheduling problems.

[0023] (6) In existing technologies, local search is usually performed uniformly across all scenarios, lacking an intelligent selection mechanism, leading to wasted computational resources and unclear search direction. This invention proposes a target scenario selection strategy based on RL learning. By modeling the selection of target scenarios as a sequential decision problem, and using the changes in BSN and WSM of the swarm optimal solution in bad scenarios as the state, the strategy is developed. - The greedy strategy selects target scenarios and updates the Q-value with immediate rewards, making the local search focus on the key scenarios most sensitive to PT, achieving efficient use of computing resources, and guiding the local search in the most effective direction to improve robustness, significantly improving search performance and solution efficiency. Attached Figure Description

[0024] Figure 1This is a flowchart of the robust scheduling method for a no-wait flow workshop bad scenario set according to an embodiment of the present invention; Figure 2 This is a flowchart of the reinforcement learning-based meme algorithm according to an embodiment of the present invention; Figure 3 This is a schematic diagram of a binary tournament selection mechanism based on multiple fitness values ​​according to an embodiment of the present invention; Figure 4 This is a schematic diagram illustrating the specific process of the intersection of two points according to an embodiment of the present invention; Figure 5 This is a schematic diagram illustrating the specific process of insertion mutation in an embodiment of the present invention; Figure 6 The same scheduling solution in the embodiments of the present invention Diagrams illustrating different critical paths in different scenarios. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0026] Example 1: This embodiment provides a robust scheduling method for bad scenario sets in a wait-free flow shop based on the meme algorithm, such as... Figure 1 As shown, it includes the following steps: Step S1: Describe the uncertainty of processing time using the discrete multi-scenario method for the set of workpieces to be processed and the set of processing machines; In step S1, this invention addresses the no-wait flow shop scheduling problem with uncertain processing times by establishing a robust scheduling model for a no-wait flow shop based on a scenario-based approach and a threshold-bad scenario set. Given a set of workpieces to be processed... With the collection of machines used for processing All jobs are processed sequentially on a set of machines in the same order. The processing of a job on one machine constitutes one operation, and waiting between adjacent machines is not allowed. A deterministic scheduling scheme is a sequence of operations processing all jobs on the machine set to optimize the scheduling objective, makaspan. A feasible scheduling solution... It can be expressed as a pair A permutation of a set of machines accessed for processing of each workpiece; Solution All constraints must be satisfied, and the set of all feasible solutions is the solution space. The processing time for each operation is uncertain; this invention employs a discrete scenario method to describe these processing times. Specifically, let the discrete scenario set... Represents a set of indeterminate processing times, where each scenario Provide a unique set of processing time implementations ,in Indicates in the scene Lower workpiece In the machine The processing time for the above processing operations.

[0027] Step S2: Construct a threshold bad scenario set based on preset performance thresholds, construct a robust optimization model for the threshold bad scenario set of no-wait flow shop scheduling, and determine the optimization objective function for robust scheduling of no-wait flow shop. The threshold-based robust optimization model expands the focus of the traditional min-max model from a single worst-case scenario to a set of worst-case scenarios that includes the worst-case scenario. The set of worst-case scenarios is defined based on a given performance threshold. A reasonable performance threshold varies depending on the computational example to be solved; it can be given by the decision-maker based on experience or pre-calculated and evaluated based on the computational example to be processed. Assume the decision-maker gives a performance threshold of... For scheduling feasible solutions If in the scene makespan Not less than Then the scenario is determined to be a solution. A bad scenario, based on threshold Scheduling solution Threshold for all bad scenarios - bad scenario set : in, Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below, This represents the performance threshold.

[0028] For the solution Due to each bad scenario The performance obtained All are inferior to the threshold. Define a penalty term for each bad scenario. .

[0029] The objective function for the no-wait flow shop scheduling model is given by the following formula: in, A feasible solution for scheduling is the arrangement of the processing order of the workpieces on the processing machine; Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below; For scheduling feasible solutions In the scene Maximum completion time; This is the performance threshold; Let represent the feasible solution space of the scheduling problem, and let represent the set of all workpiece processing sequences that satisfy the no-wait flow shop constraint. This indicates the optimization of the objective function value.

[0030] This invention needs to consider the sorting of the processing order of all workpieces on the machine under uncertain processing times. The goal is to optimize the total penalty for the worst-case scenario set to suppress the decline in scheduling performance under bad scenarios and obtain a robust scheduling solution that performs well in all scenarios. Because this robust scheduling model considers more bad scenarios than the worst-case scenario in its optimization objective, the conservatism of the obtained robust scheduling solution will be weakened. The degree of weakening of conservatism depends on the performance threshold. The size of the threshold bad scenario set. The robust solution obtained by the robust optimization model is called the PT-robust solution.

[0031] Performance threshold The value of should not be better than the best mean performance of the processed examples across all scenarios, to ensure that the obtained robust scheduling solution reflects the decision-maker's risk aversion bias; at the same time, the performance threshold The value of cannot be worse than the best and worst performance of all scenarios in the processed examples. Otherwise, under some elite solutions, the bad scenario set will be empty, the penalty function PT will be zero, and it will be impossible to distinguish these elite solutions. The bad scenario set model will lose its effectiveness.

[0032] This embodiment designs an efficient meme algorithm to solve a robust scheduling model for a threshold-bad scenario set in a no-wait flow workshop.

[0033] like Figure 2 As shown, the Memetic Algorithm (MA) is a metaheuristic algorithm that combines global search and local optimization. Its basic idea originates from Darwin's theory of evolution and the concept of memes. By introducing a problem-related local search mechanism into the genetic algorithm framework, it improves the quality of solutions and search efficiency.

[0034] A typical MA process consists of three phases: initialization, iterative optimization, and iteration termination. The initialization phase generates an initial population and evaluates fitness. The iterative phase generates new individuals through selection, crossover, and mutation, performs local optimization on these individuals, and then merges the optimized individuals back into the population for natural selection. If the iteration termination condition is not met, the next generation cycle begins.

[0035] This algorithm demonstrates strong optimization capabilities through an effective balance between global exploration and local development.

[0036] Step S3: Generate the initial population using the scenario-based NEH heuristic algorithm; Population initialization is crucial in meme algorithms. First, a well-designed initialization strategy can generate diverse and representative initial individuals in the solution space, providing a broad foundation for subsequent global and local searches. Second, diverse initial solutions not only effectively cover potentially high-quality regions but also reduce the risk of premature convergence, increasing the probability of finding the global optimum. Finally, a high-quality initial population accelerates the search process, reducing invalid evaluations of inferior solutions in subsequent iterations, thereby improving the algorithm's convergence speed and solution efficiency. Therefore, in robust optimization problems with no-wait pipeline scheduling, careful design and implementation of population initialization are key steps to ensure the performance and stability of meme algorithms.

[0037] In this invention, population initialization employs a scenario-based NEH heuristic method designed specifically for the problem characteristics. Let the population size be denoted as . Initial population Based on scenario set Generation. Specifically, in each scene. The following uses the NEH heuristic to construct an initial solution: First, sort the workpieces in descending order based on their total processing time across all machines in this scenario. Then, first select the workpiece with the highest total processing time and insert it into an empty sequence as the current sequence. Next, insert each workpiece into the current sequence sequentially, iterating through all possible insertion positions and selecting the position that produces the minimum completion time in this scenario to insert the next workpiece into the sorted workpiece queue. Repeat this process until the sequence of workpieces to be inserted is empty. This process can generate a maximum of [number missing] solutions. There are multiple distinct initial solutions. If the number of initial solutions exceeds... Then, redundant solutions are deleted until the number of initial solutions equals the number of redundant solutions. If the number of solutions in the initial population is insufficient Then, several solutions are randomly generated until the initial solution set contains... There are two different solutions.

[0038] Step S4: For the current population, perform a global search based on genetic operators and selection mechanisms according to the optimization objective function to obtain the updated population; Global search is a crucial stage in meme algorithms, helping to maintain the diversity of solutions in the population and prevent the algorithm from getting trapped in local optima. In the global search phase, individuals are selected from the population through a specific selection mechanism to perform the global search; these selected solutions are called Global Search Parents (GSPs). GSPs contain a series of parent solutions, which are used to generate offspring using genetic operators.

[0039] The specific steps of the global search phase of this invention are as follows: Step S401: Use the binary tournament method to select the global search parent. : Specifically, in a binary tournament, two individuals are randomly selected from the current population and compared according to three fitness values. The individual with the higher fitness value is selected as the next... g The GSP of the era, denoted as . Figure 3 This invention demonstrates three fitness values ​​designed based on robust solution properties, derived from... PT- The robust solution is determined by three abstract metrics: the number of bad scenarios (BSN), the worst-case scenario makespan (WSM), and the physical test (PT) value. For each metric, a binary tournament selection is performed independently, using that metric as the fitness value. Therefore, at most a total of Individuals were selected .

[0040] Step S402: Using the two-point crossover operator and the insertion mutation operator based on The solution in the middle generates a new global search solution: For the two-point intersection operator, for Each individual in , and then from Another individual is randomly selected, and a two-point crossover operation is performed on the two individuals. Specifically, for the two selected parents, two different positions are randomly selected as crossover points, the segments between the two crossover points are swapped, and the missing workpieces outside the crossover points of the first parent are filled in the order of the workpieces in the second parent. Figure 4 This process was demonstrated. Figure 3 In the diagram, intersection 1 is the 3rd position, and intersection 2 is the 5th position. Therefore, the position... Gene fragments are exchanged, and at this time in parent 1, the artifact other than the crossover fragment... and A conflict occurred, and there was a lack of and Therefore, according to parent generation 2 and The generated offspring will be filled in the order specified in the parent generation sequence. The generated offspring will replace the original parent generation 1.

[0041] Insertion-based mutation operations will proceed according to probability. Apply to Individuals within a given context. Insertion-based mutation specifically involves randomly selecting a workpiece at one location and inserting it before moving it to another location. Figure 5 This process is demonstrated. The workpiece at position 5 is selected, removed from the sequence, and inserted in front of position 1.

[0042] Step S403: Incorporate all newly generated solutions In, and will The solutions will be sorted according to their PT values, and finally the worst solutions in the population will be removed until... The population size was restored to .

[0043] Step S5: For the updated population, select target scenarios according to the model-related strategy based on the optimization objective function, and construct a neighborhood structure and perform local search in the target scenarios using problem-related knowledge to obtain the optimized population; In the local search phase, the binary tournament method is first used to select the parent population (LocalSearchParents, LSP) for the local search, denoted as . In the context of Before performing a local search on the parent solutions, the reinforcement learning RL agent selects a scene as the target scene from the scene set. Then, all parent solutions construct a single-scene neighborhood within this target scene and perform a local search. The resulting neighborhood solutions are then incorporated into the target scene. The population is updated in the middle. This neighborhood structure constructed under the target scene determined by reinforcement learning (RL) is called the reinforcement learning RL scene neighborhood structure.

[0044] The single-scenario neighborhood operator is an operator that generates a set of candidate neighborhood solutions for the current solution under a given target scenario. In this operator, for a given target scenario, the processing time of all operations is determined, corresponding to a deterministic no-wait flow shop scheduling problem. The method for generating candidate neighborhood solutions adopts the neighborhood definition corresponding to this deterministic problem. In this method, the single-scenario neighborhood is generated based on the critical path, which is defined as the longest continuous chain of operations with no idle time that determines the project duration. To express the reverse constraint under no-wait conditions, the processing time on the reverse critical path is recorded as a negative value, and the processing time on the forward critical path is recorded as a positive value. If an operation belongs to both the forward and reverse critical paths, its processing time on the critical path is set to 0. Accordingly, the algebraic sum of the processing times on the critical path equals the makespan of the solution.

[0045] For the same solution in different scenarios The critical paths are different, such as Figure 5 As shown, the red line with the arrow is the critical path. Figure 6 , the same solution ( In two different scenarios and There are different critical paths. Therefore, the critical paths obtained will also differ depending on the target scenario chosen. The solutions obtained based on different critical paths... The neighboring candidate solutions will also be different, which can greatly enhance the ability to find solutions around the current solution. Local search.

[0046] The steps for constructing a single-scene neighborhood based on the critical path are: given a scene Next, find the current solution. A critical path is defined, and on this critical path, a "critical block" is defined as the longest consecutive operation segment on the same machine along the critical path, containing at least two operations; operations involving reverse paths are not counted as blocks. The algorithm selects the largest block from all critical blocks, and denotes its leftmost operation as the workpiece. The rightmost operation is for the workpiece. The operation, and the operation with the longest processing time in the critical block is taken as the operation. Let the workpieces of these three operations be the target workpieces; then, the scheduling solution... The three target artifacts are swapped pairwise to generate at most three neighborhood solutions. If no maximum block exists, only the following can be determined. Then in addition to Two operations are randomly selected from the outside to act as and Then perform the exchange. Based on the above exchange in the scenario... The solution is obtained below. All neighborhood solutions constitute Because the critical path differs in different scenarios, the same solution... The neighborhood of a single scene obtained varies depending on the scenario.

[0047] Building upon the single-scene neighborhood structure, this module introduces reinforcement learning to guide target scene selection, thus forming a reinforcement learning (RL) scene neighborhood structure. In this structure, the algorithm treats "selecting a target scene and generating a neighborhood accordingly" as a sequential decision-making process. In each generation, the agent selects a target scene from the scene set and generates a neighborhood within that scene. All parent solutions generate a single-scene neighborhood and perform a local search. This design is centered around Q-learning. The system treats the environment as an evolving process over generations, with the agent maximizing cumulative rewards through trial and error, and continuously refining action value estimates using a standard Q-value incremental update formula. This allows the system to learn the choice of "in which target scene a deep search is more likely to find a better neighborhood solution," rather than relying on fixed or random policies.

[0048] The optimization objective in this method is to more effectively reduce the total penalty PT. Since PT reflects the sum of squared performance degradation across all scenarios, changing the duration of the current solution in any single scenario can potentially decrease PT. QN uses reinforcement learning to select only a "representative" target scenario for each generation to drive local exploration, thereby significantly reducing computational overhead, especially in large-scale examples. To improve discriminative power, the state is not directly defined by changes in PT, but rather based on the structural characteristics of PT-robust solutions, specifically defined by the number of bad scenarios (BSN) and the worst-case makespan (WSM) for each individual. BSN and WSM are considered auxiliary metrics for PT-robust solutions.

[0049] The individual in the population with the optimal PT value is defined as the optimal solution in the population. If there are multiple individuals in the population with the optimal PT value, then the one with the smaller BSN is selected as the optimal solution. .

[0050] To define the state of the agent, the optimal solution for each generation... Different combinations of BSN and WSM change trends are defined as states. The system records the BSN and WSM values ​​of the optimal solution in the previous and current generations respectively, and calculates the changes accordingly: in, , These represent the number of bad scenarios and the worst-case completion time of the previous generation's optimal solution, respectively. , These represent the number of bad scenarios and the worst-case completion time of the current generation's optimal solution, respectively. These represent the changes in the number of bad scenarios and the worst-case completion time, respectively. In the current generation, a total of four possible combinations of variations can be obtained: (a) and Both are negative; (b) only (c) Only (d) Neither is negative. The four combinations of changes constitute four discrete states. Each generation is characterized by one of these four categories. The agent is in a state where it selects an action using an ε-greedy strategy. This action involves choosing a target scene from the scene set, generating a single-scene neighborhood for each parent solution within that target scene, producing candidate neighborhood solutions, and merging them into the target scene. .

[0051] The reward function is based on the improvement of the population optimum. If the contemporary local search improves the population optimum... If improvement is achieved, a positive reward of 2 is given; otherwise, a reward of 0 is given. The system uses this reward to update the Q-table, making the agent more inclined to choose actions that bring improvement in subsequent iterations. Improved Scenario. Because the algorithm's solution acceptance mechanism first compares PT (Placement Time), and if PTs are the same, it prioritizes the solution with the smaller BSN (Body Size Number), even if PT remains unchanged temporarily, changes in BSN will be recorded and affect the state, thus continuing to drive structural improvements in the agent during the PT plateau period. This mechanism ensures that reinforcement learning (RL) still has a guiding role after the PT value tends to stabilize. Based on the obtained rewards, the Q-table can be updated using the formula: in, The learning rate; Discount factor; For the next generation state, This represents the maximum Q value for all possible target scenarios in the next-generation state; To prevent learning from stalling later on, the system employs a stall recovery strategy. When the BSN of the population's optimal solution remains unchanged for several consecutive generations, the system temporarily replaces the "optimal solution" with another solution from the population to calculate the state, thereby altering the state input and breaking the situation where the Q-table remains in the same state for an extended period. This recovery mechanism restores the diversity of action choices and reactivates the exploration of different scenarios.

[0052] Step S6: Update the iteration count and determine whether the preset maximum iteration count has been reached. If not, use the optimized population as the current population, return to global search and continue iterating until the maximum iteration count is reached, and output the optimal scheduling solution in the optimized population.

[0053] Example 2: This embodiment provides a robust scheduling system for bad scenario sets in a wait-free assembly line based on the meme algorithm, including: The data input module is used to receive information about the set of workpieces to be processed and the set of processing machines, and to obtain workpiece processing time uncertainty data to generate a discrete processing time scenario set.

[0054] The threshold bad scenario set construction module is used to determine the maximum completion time of each scheduling feasible solution in each scenario based on a preset performance threshold, and generate the corresponding threshold bad scenario set.

[0055] The initial population generation module adopts the scenario-based NEH heuristic algorithm to generate an initial scheduling solution set under the threshold bad scenario set, ensuring population diversity and initial solution quality.

[0056] The global search module applies genetic operators and selection mechanisms to the current population to generate an updated population, thereby improving global optimization capabilities.

[0057] The local search module constructs a single-scene neighborhood for the parent solution and searches it under the threshold bad scene set, based on the optimization objective function and target scene selection strategy, thereby optimizing the local performance of the solution.

[0058] The iteration control module is used to update the iteration count, determine whether the maximum iteration count has been reached, and output the optimized scheduling solution in the population.

[0059] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0060] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A robust scheduling method for bad scenario sets in a wait-free assembly line based on a meme algorithm, characterized in that, Includes the following steps: The uncertainty of processing time is described using a discrete multi-scenario method for the set of workpieces to be processed and the set of processing machines. Based on the preset performance threshold, a threshold bad scenario set is constructed, and a robust optimization model for the threshold bad scenario set of no-wait flow shop scheduling is constructed to determine the optimization objective function of the no-wait flow shop robust scheduling. The initial population is generated using a scenario-based NEH heuristic algorithm. For the current population, a global search based on genetic operators and selection mechanisms is performed according to the optimization objective function to obtain the updated population; For the updated population, target scenarios are selected based on the optimization objective function using model-related strategies. Then, a neighborhood structure is constructed and a local search is performed within the target scenarios using problem-related knowledge to obtain the optimized population. Update the iteration count, determine whether the preset maximum iteration count has been reached. If not, use the optimized population as the current population, return to global search to continue iterating until the maximum iteration count is reached, and output the optimal scheduling solution in the optimized population.

2. The robust scheduling method for bad scenario sets in a wait-free flow shop based on the meme algorithm according to claim 1, characterized in that, The set of workpieces to be processed is represented as follows: ,in, The total number of workpieces to be processed. Indicates the first One workpiece to be processed; The set of processing machines is represented as ,in, This represents the total number of processing machines. Indicates the first Taiwanese processing machine.

3. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 2, is characterized in that... A threshold-based bad scenario set is constructed based on a preset performance threshold, specifically including: Collection of workpieces to be processed and processing machines The processing time for all operations is uncertain, using discrete scene sets. This indicates that the discrete scene set Each scene A machining time matrix consisting of the machining time for each workpiece on each machine. , Indicates in the scene Lower workpiece In the machine Processing time on discrete scene sets It includes all possible processing time scenarios, to Representing scene set The number of scenes in; For each scheduling feasible solution According to the processing sequence and processing time of the workpiece Calculate each scene sequentially The maximum completion time is obtained by taking the assigned task completion time. ; Given a performance threshold For each scenario ,like Then the scene Determined as a feasible solution for scheduling In bad scenarios, scheduling feasible solutions All bad scenarios constitute the threshold bad scenario set , represented as: in, Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below.

4. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 1, is characterized in that... The objective function of the no-wait flow shop scheduling model is given by the following formula: in, A feasible solution for scheduling is the arrangement of the processing order of the workpieces on the processing machine; Indicates a feasible solution for scheduling. At performance threshold The threshold bad scenario set below; For scheduling feasible solutions In the scene Maximum completion time; This is the performance threshold; Let represent the feasible solution space of the scheduling problem, and let represent the set of all workpiece processing sequences that satisfy the no-wait flow shop constraint. This indicates the optimization of the objective function value.

5. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 4, is characterized in that... The constraint for the no-wait assembly line workshop is: Workpiece in processing machine assembly The processing must be carried out continuously according to the preset machine sequence, and no waiting time is allowed between any two adjacent machines. Specifically, this includes: The access order of each workpiece on the machine set is fixed, and they must be accessed sequentially on the machines. to Processing must be carried out on the surface, and the processing order must not be changed; Workpiece in machine After processing is completed, it must be immediately placed on the machine. Processing begins on the first step, meaning the waiting time between any two adjacent processes is limited to zero. At any given time, each machine can only process one workpiece, meaning that a machine cannot be occupied by multiple workpieces simultaneously. Once the processing of a workpiece on a machine begins, it must not be interrupted and must be completed continuously.

6. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 1, is characterized in that... The initial population is generated using the scenario-based NEH heuristic algorithm, specifically including: In the threshold bad scenario set In the middle, for each scenario According to the processing time matrix Calculate the total processing time of the workpiece on all machines; Sort the workpieces according to their total processing time from longest to shortest to obtain the scene. The workpiece sorting sequence below; Based on the workpiece sorting sequence, an insertion-based construction strategy is used to incrementally generate a scheduling sequence, including: The workpiece with the longest total processing time in the workpiece sorting sequence is placed into an empty sequence to form the current sequence; For the remaining workpieces in the workpiece sorting sequence, enumerate all possible insertion positions in the current sequence, and form a new sequence for each candidate insertion position. And calculate the new sequence. In the scene Maximum completion time ; Select the insertion position with the shortest maximum completion time and insert the workpiece into the current sequence until all workpieces have been inserted, thus constructing the scene. Scheduling solution under ; Scene set All scheduling solutions generated for each scenario are added to the initial population. If the initial population The number of solutions in the scheduling process exceeds the set population size. Then, randomly delete any extra individuals. If the number of scheduling solutions in the initial population is insufficient Then in the feasible solution space X A feasible scheduling sequence that satisfies the no-wait flow workshop constraint is randomly generated and added to the initial population.

7. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 1, is characterized in that... The step of performing a global search based on genetic operators and selection mechanisms on the current population according to the optimization objective function to obtain an updated population specifically includes: For the first The generation population uses a binary tournament selection method to generate a globally searchable set of parent generations. : Randomly select two individuals from the current population, based on the threshold bad scenario set respectively. The number of scenarios (BSN), worst-case maximum completion time (WSM), and the value of the objective function are all considered. As a fitness value, individuals with higher fitness are selected for the parent set; a predetermined number of binary tournament selections are performed independently on each of the three fitness types, and the results are then combined to form the parent set. ; Based on parent set The individual in the set performs a two-point crossover operator to generate a new individual: for each individual in the parent set, another individual is randomly selected from the parent set as a pairing individual, two different positions are randomly selected as crossover points, the gene segments of the two parents in the crossover point interval are exchanged, and the missing workpieces outside the crossover interval are filled into the corresponding positions according to the order in which the workpieces appear in the pairing individuals, forming the offspring sequence. Based on the insertion mutation operator on the parent set Individuals in the process undergo mutation operations: according to the mutation probability. A mutated individual is formed by randomly selecting a position in an individual, removing the workpiece corresponding to that position and inserting it before another randomly selected position. All new individuals obtained from the two-point crossover operator and the insertion mutation operator are incorporated into the current population, according to the optimization objective function value. The expanded population is sorted, and individuals with the worst objective function values ​​are removed until the population size is restored to the preset population size. The updated population is obtained.

8. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 1, is characterized in that... The process of selecting target scenarios based on a model-related strategy according to the optimization objective function for the updated population, and constructing a neighborhood structure and performing a local search within the target scenarios using problem-related knowledge to obtain the optimized population, specifically includes: A binary tournament selection mechanism is used to select a local search parent set from the updated population and perform a local search operation. Based on the target scene selection strategy of reinforcement learning, target scenes for local search are determined from the set of threshold bad scenes, and a single scene neighborhood is constructed for each parent solution in the local search parent set under the target scene, generating a corresponding neighborhood candidate solution set. All neighborhood candidate solutions in the neighborhood candidate solution set are incorporated into the current population. The expanded population is then sorted according to the optimization objective function. Individuals with poor optimization objective function are removed, and the population size is restored to the preset population size. The optimized population was obtained.

9. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 8, is characterized in that... The step of constructing a single-scene neighborhood for each parent solution in the local search parent set under the target scene, and generating a corresponding neighborhood candidate solution set, specifically includes: In the target scenario, a critical path is determined based on the deterministic processing time corresponding to the target scenario. The critical path is the longest chain of operations consisting of consecutive operations that determines the completion time under no waiting constraints. A forward operation is defined as an operation in the critical path that proceeds sequentially according to the processing sequence and extends the completion time; its processing time is a positive value. A reverse operation is defined as an operation in the critical path that generates a reverse dependency due to no waiting constraints and shortens the feasible time of preceding operations; its processing time is a negative value. Operations that are both forward and reverse operations have a processing time of zero. The operation refers to the processing steps of the workpiece on the machine. Based on the critical path, key blocks are identified. A key block is the longest consecutive operation segment on the critical path, located on the same machine and consisting of at least two consecutive operations, without including reverse operations. The largest key block is selected from all key blocks, and the workpiece corresponding to the leftmost operation within that key block is determined. The rightmost operation corresponds to the workpiece. And the workpiece corresponding to the operation with the longest processing time. As a target workpiece; If there is no critical block in the critical path, then in addition to the workpiece Two workpieces are randomly selected from the remaining workpieces as workpieces. With workpiece , with the workpiece Together they serve as target workpieces; In the sequence of workpieces in the parent solution, for workpieces , workpiece and workpiece Perform pairwise position swaps to generate up to three neighborhood candidate solutions. All the swapped neighborhood candidate solutions constitute the set of neighborhood candidate solutions of the parent solution in the target scene.

10. A robust scheduling method for bad scenario sets in a wait-free flow shop based on a meme algorithm, as described in claim 8, is characterized in that... The target scene selection strategy based on reinforcement learning determines target scenes for local search from a set of threshold bad scenes, specifically including: The first The local search current solution set of the generation is denoted as ; The target scene selection process is modeled as a reinforcement learning sequential decision problem, where the agent's state The optimal solution of the previous generation and the current generation Number of bad scenes Completion time in worst-case scenario Variations and combinations The characterization, the formula is: in, , These represent the number of bad scenarios and the worst-case completion time of the previous generation's optimal solution, respectively. , These represent the number of bad scenarios and the worst-case completion time of the current generation's optimal solution, respectively. These represent the changes in the number of bad scenarios and the worst-case completion time, respectively. According to the status use Greedy strategy and action value function for action selection ,in, Indicates the threshold bad scene set Selected target scene ; In the selected target scenario Below, for Each parent solution generates a neighborhood candidate solution set. ,in, This represents the set of neighborhood candidate solutions generated by critical path and critical block operations in a given target scenario; Based on the parent's understanding of the target scenario Calculate the immediate reward for optimizing the objective function value. ,in, This indicates that local search yields the population optimal solution. The optimization objective function is improved. This indicates that there has been no improvement; According to instant rewards Update the action value function : in, The learning rate; Discount factor; For the next generation state, This represents the maximum Q value for all possible target scenarios in the next-generation state; When the group optimal solution If the solution remains unchanged for several consecutive generations, another solution from the population is temporarily selected as a replacement. To calculate the state.

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