Production scheduling and worker allocation joint optimization method for aircraft pulsation assembly line

Abstract

The invention relates to a production scheduling and worker allocation joint optimization method for an aircraft pulsation assembly line, and the method comprises the following steps: obtaining assembly task data and resource data of the aircraft pulsation assembly line, resources being workers; constructing a dictionary sequence dual-objective optimization integer programming model, and taking the minimum pulsation period and the total resource consumption as objective functions; solving the dictionary sequence dual-objective optimization integer programming model to realize joint optimization of worker allocation and multi-mode assembly tasks in an aircraft pulsation final assembly line under a resource exclusive policy: estimating resource-construction period mapping of each workstation, and determining initial resource allocation according to the estimated mapping; searching a scheduling scheme by adopting genetic manipulation; perfecting mapping by adopting a mapping updating strategy; after a predetermined evolution cycle, resource reallocation is performed to adjust and optimize resource allocation. Compared with the prior art, the method has the advantages of high scheduling result production efficiency, calculation overhead saving and the like.

Description

Technical Field

[0001] The invention relates to the technical field of scheduling of an aircraft pulse assembly line, and in particular to a joint optimization method for production scheduling and worker allocation for an aircraft pulse assembly line. Background Art

[0002] In the past few decades, the aircraft manufacturing industry has experienced a rapid development from single production to mass production. As the final stage of aircraft production, the aircraft pulse assembly line is responsible for assembling the aircraft fuselage, including the addition of the nose and tail, wings, engines and landing gear. It is worth noting that this stage accounts for 50% of the total production time and more than 20% of the total production cost. Therefore, improving the efficiency of the aircraft assembly process is crucial to optimizing aircraft production.

[0003] The pulse assembly line of an aircraft consists of a specific number of workstations, each of which has a set of assembly tasks that must obey priority constraints. All assembly tasks must be completed within a predetermined period before the aircraft can move to the next workstation to complete the assembly of the remaining parts. This requires that the completion time of all workstations be balanced. Worker allocation in the pulse assembly line of an aircraft is one of the most important resource optimization problems, because more than 70% of the assembly tasks are performed manually. Flexibility in worker allocation is inherent, because the processing time of a given task is directly affected by the number of workers assigned to it. This allows the assembly of tasks to have multiple modes, each corresponding to a certain worker input and processing time.

[0004] Existing studies on aircraft pulse assembly line scheduling reveal several deficiencies. First, few studies fully address the necessary flexibility in resource allocation. In addition, since the two objectives of minimizing the pulse cycle and minimizing resource costs often conflict, few studies simultaneously optimize these two objectives under limited resource constraints. In addition, the unified pulse cycle characteristic of the aircraft pulse assembly line requires that the production cycle of all workstations remain consistent to reduce resource waste caused by idle periods. These gaps greatly limit the practical application of existing algorithms. Summary of the invention

[0005] The purpose of the present invention is to provide a joint optimization method for production scheduling and worker allocation for an aircraft pulse assembly line, which realizes the joint optimization of worker allocation and production scheduling under a resource exclusive policy by considering the characteristics of the aircraft pulse assembly line, including the constraints of limited resources, the balance of the pulse cycle and the flexibility of resource allocation.

[0006] The purpose of the present invention can be achieved by the following technical solutions:

[0007] A joint optimization method for production scheduling and worker allocation for an aircraft pulse assembly line includes the following steps:

[0008] Step 1) obtaining assembly task data and resource data of an aircraft pulse assembly line, wherein the resource is workers;

[0009] Step 2) constructing a lexicographic order bi-objective optimization integer programming model to minimize the pulsation period and total resource consumption as the objective function;

[0010] Step 3) Solve the lexicographic order bi-objective optimization integer programming model to achieve the joint optimization of worker allocation under the resource exclusive policy and multi-mode assembly tasks in the aircraft pulse assembly line:

[0011] Step 31) estimating the resource-duration mapping of each workstation, and determining the initial resource allocation based on the estimated mapping;

[0012] Step 32) Based on the determined initial resource allocation, a scheduling solution is searched using genetic operations;

[0013] Step 33) Based on the scheduling scheme determined in step 32), a mapping update strategy is used to improve the mapping;

[0014] Step 34) After a predetermined evolution cycle, resource reallocation is performed to adjust and optimize resource allocation, wherein the resource reallocation includes a resource transfer strategy and a resource recovery strategy, wherein the resource transfer strategy increases the resources of the bottleneck workstation by reallocating excess resources of the workstation with excess resources to optimize the pulsation cycle, and the resource recovery strategy recovers the remaining resources from each workstation after completing the pulsation cycle optimization to optimize the total resource consumption.

[0015] The modeling assumptions of the lexicographically ordered bi-objective optimization integer programming model include:

[0016] Each assembly task corresponds to a non-preemptive duration;

[0017] After the assembly task starts, the execution mode cannot be changed;

[0018] During the assembly cycle, resources cannot be transferred between workstations;

[0019] Among them, consider an assembly line workshop with k workstations, denoted by W = {w1,w2,...,w k}, where k is the number of workstations, and each workstation w i Process a set of assembly tasks using J i ={J i,0 ,J i,2 ,...,J i,mi} means, where m i Indicates workstation w i The number of tasks to be processed, the assembly tasks at each workstation is predetermined, J i,jDenotes the assembly task set J i The jth task of the assembly task is j; the scheduling of tasks is subject to the priority constraint, that is, each task can only be started after all its predecessor tasks are completed; each assembly task J i,j Corresponding to multiple execution modes, each mode m∈M i,j Corresponding to a specific worker requirement r ijm and assembly time d ijm , where M i,j represents the assembly task J i,j All assembly mode sets, d ijm represents the assembly task J i,j The assembly time of mode m, r ijm represents the assembly task J i,j The number of assembly workers in mode m.

[0020] The optimization objective of the lexicographic order bi-objective optimization integer programming model is expressed as:

[0021]

[0022] The constraints include:

[0023]

[0024] x ijm ∈{0,1}

[0025]

[0026] Where C represents the pulsation period, n i Indicates that it is assigned to workstation w i The number of resources, n represents the total number of resources, t i,j Represents task J i,j The start time, x ijm Represents the decision task J i,j Whether the assembly mode of is a 0-1 variable of m, A i,j Indicates that at time t i,j The task being assembled, Pre i,j Represents task J i,j All the preceding tasks, Represents a set of integers.

[0027] The estimation of resource-duration mapping for each workstation specifically includes the following steps:

[0028] Determine the resource interval, that is, the workstation w i Minimum resource investment and maximum resource investment in, It represents the minimum resources required for each workstation to complete the assembly task, which is equivalent to the maximum resources required for a single task in the assembly task set, and is executed in the mode with the least resource requirements.

[0029]

[0030] Among them, M i,j represents the assembly task J i,j The set of all assembly modes, r ijm represents the assembly task J i,j The number of assemblers in mode m, m i Indicates workstation w i The number of tasks to be processed;

[0031] For a given assembly task set J i , maximum resource requirement Depends on the assembly task set J i Topology and assembly mode, when resource input exceeds the maximum resource demand The construction period will not be shortened. Represents the maximum resource demand J i All subsets of , where l is the total number of subsets, and each subset S i,p Contains assembly tasks that can be executed in parallel. Assume that S i,p All tasks in S are executed in the mode that requires the most resources, then all subsets S i,p The resource requirement R i,p The maximum value of

[0032]

[0033] To ensure that all workstations can complete their tasks, and assigned to workstation w i The number of workers is not less than Among them, n i Indicates that it is assigned to workstation w i The number of resources, n represents the total number of resources, is allocated to workstation w i The biggest resource It is expressed as:

[0034]

[0035] Then the maximum number of workers to be assigned to a workstation is calculated as follows:

[0036]

[0037] Based on the determined resource interval, a hybrid scheduling rule is established by considering both task selection and mode selection. Based on the hybrid scheduling rule, a task and mode list is generated for the assembly task of each workstation. The scheduling scheme is decoded by the serial scheduling generation method (SSGS) to determine the minimum duration and establish a resource-duration mapping. The scheduling rules for task selection include the shortest assembly time priority, the longest assembly time priority and the priority for the largest number of subsequent tasks. The scheduling rules for mode selection include the shortest assembly time mode priority and the maximum resource utilization mode priority.

[0038] The determining of the initial resource allocation according to the estimated mapping is specifically as follows:

[0039] According to the estimation mapping, the resource allocation problem is formulated as follows:

[0040] minimizeC=max{f1(n1),f2(n2),...,f k (n k )}

[0041]

[0042] Among them, the function f i : To map resource inputs to assembly task set J i The mapping function of the production cycle, f i (n i ) indicates workstation w i The working time, C represents the pulsation cycle;

[0043] Solving the resource allocation problem obtains initial resource allocation.

[0044] The starting point of each constant completion time stage is defined as a critical point, and the resource allocation problem is efficiently solved by sampling the critical point.

[0045] The step 32) is specifically as follows: the assembly task scheduling problem of each workstation is abstracted into a multi-mode resource-constrained project scheduling problem, wherein each workstation operates under a given resource allocation scheme, and the scheduling problems of each workstation are independent of each other after completing the worker resource allocation, and a genetic algorithm is simultaneously executed on each workstation to determine a scheduling scheme, which includes the following steps:

[0046] Step 321) Encoding and decoding:

[0047] Each individual's chromosome consists of two parts: (p, m), where represents a task list, subject to the priority constraint, p j represents the task index at position j, represents a list of task execution patterns subject to resource constraints, where each element specifies the number of workers and processing time required for the corresponding task, m i Indicates workstation w i The number of tasks to be processed;

[0048] Step 322) Population initialization and parent selection:

[0049] Generate an initial population POP0 of size N, where the chromosomes are randomly initialized: for the task list p, a task is selected only when all previous tasks have been scheduled; for the mode list m, determine the feasible modes of the task based on the available resources and randomly select a mode;

[0050] Each individual in the initial population POP0 is a parent, and offspring are generated by randomly dividing into two sets of equal size (P1, P2). A pair of parents is drawn from each set in turn without replacement, generating N / 2 pairs of parents, and each pair of parents produces two offspring;

[0051] Step 323) Crossover and mutation:

[0052] The crossover operator is used with probability p c Applied to the task list p to generate a subtask list, and the corresponding pattern list m is interleaved in the same way;

[0053] The mutation operator is p m The probability of is applied to the task list p and the pattern list m, where the mutation in the task list p is achieved by selecting a task p i And change the gene p i The position of the corresponding model gene m i The mutation in the pattern list m is achieved by selecting a task p i , and use a randomly selected feasible mode m i ′Replace its original pattern m i ;

[0054] Step 324) Fitness calculation and population update:

[0055] According to the individual fitness fit, individuals need to be evaluated before retaining or eliminating offspring:

[0056] fit=C max =C i (p,m,n i )

[0057] Among them, C max represents the completion time solution, C i () indicates that the workstation w iThe individual chromosomes (p,m) are decoded into functions of completion time, n i Indicates that it is assigned to workstation w i The number of resources;

[0058] The offspring are stratified according to fitness, and the best 0.25N individuals out of a total of 2N individuals of the parents and offspring are retained. The remaining 1.75N individuals are divided into two sub-populations at a ratio of 2:1, and then 0.5N and 0.25N individuals are selected from them to form the next population POP. i+1 , repeat step 323) and step 324) to update the population.

[0059] The step 33) is specifically as follows:

[0060] For a given set of assembly tasks J i and resources i , when a better completion time solution C is obtained during the execution of the genetic algorithm max , by mapping assembly progress to resource investment in a non-increasing manner to improve the mapping f i :

[0061]

[0062] The resource transfer strategy is specifically as follows:

[0063] According to the type and resource buffer status of each workstation, a corresponding resource transfer plan is determined for each workstation, wherein the workstation includes bottleneck workstations and non-bottleneck workstations. If its completion time is greater than the pulse cycle, it is defined as a bottleneck workstation, otherwise it is defined as a non-bottleneck workstation; a resource pool is defined to store resources provided by non-bottleneck workstations, and a resource buffer is defined as a temporary storage point for resources owned by each workstation;

[0064] When a workstation is a non-bottleneck workstation and its resource buffer is empty, the workstation stores part of the resources into its resource buffer;

[0065] When a workstation is a non-bottleneck workstation and its resource buffer is not empty, the workstation transfers all resources in its resource buffer to the resource pool and stores some resources transferred out in this round into the resource buffer;

[0066] When a workstation is a bottleneck workstation and its resource buffer is empty, all resources in the resource pool are allocated to the workstation. If there are multiple such workstations, all resources are allocated to the workstation with the smallest index.

[0067] When a workstation is a bottleneck workstation and its resource buffer is not empty, all resources are taken from the resource buffer and evenly distributed to all such workstations.

[0068] The resource recovery strategy is specifically as follows:

[0069] After determining the best resource input of each workstation so far, the resource recovery strategy is implemented by reducing the resource input of each workstation by one unit. At this time, the working time of all workstations will exceed the pulse cycle. After that, the workstation continues to execute the genetic algorithm to find a better scheduling solution to reduce the completion time that does not exceed the pulse cycle.

[0070] Iterate the above steps. If a better scheduling solution cannot be found, keep the resource input and scheduling solution of the previous iteration, and repeat the iteration until the iteration stop condition is met to obtain the optimal solution to the problem.

[0071] Compared with the prior art, the present invention has the following beneficial effects:

[0072] (1) This paper proposes a joint optimization problem involving worker allocation under a resource-exclusive policy and scheduling of multi-mode assembly tasks in an aircraft pulse assembly line. A lexicographically ordered bi-objective optimization integer programming model is constructed to minimize the pulse period and cost for solving a small-scale instance.

[0073] (2) The present invention establishes a resource-duration mapping to facilitate worker allocation, and proposes a mapping-based resource reallocation method, which has two strategies to achieve dynamic allocation of workers. Among them, the resource transfer strategy effectively balances the work span of all workstations by transferring workers before execution, thereby optimizing the pulse cycle. After determining the pulse cycle, a resource recovery strategy is proposed to actively recover excess workers from each workstation, simplify labor allocation, and maintain the optimal pulse cycle.

[0074] (3) The present invention integrates resource reallocation into the evolution process, which does not require a large amount of computing overhead and saves computing resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 is a flow chart of the method of the present invention;

[0076] Figure 2 It is the resource-duration mapping curve;

[0077] Figure 3 A Gantt chart of a feasible scheduling solution in one embodiment;

[0078] Figure 4 It is a schematic diagram of the crossover and mutation operators of the genetic algorithm;

[0079] Figure 5 It is a schematic diagram of resource transfer;

[0080] Figure 6It is the flow chart of RReGA algorithm;

[0081] Figure 7 The comparison results between the present invention and GUROBI are shown in Figure 1, where (a) is the comparison when the total number of workers is sufficient, and (b) is the comparison when the total number of workers is insufficient;

[0082] Figure 8 The comparison results between the present invention and RRGA, where (a) is the comparison when the total number of workers is sufficient, and (b) is the comparison when the total number of workers is insufficient;

[0083] Fig. 9 The present invention is compared with RTGA, wherein (a) is the comparison when the total number of workers is sufficient, and (b) is the comparison when the total number of workers is insufficient;

[0084] Fig.10 is the adjustment process of resource reallocation, where (a) is the adjustment process when there are sufficient total workers, and (b) is the adjustment process when there are insufficient total workers;

[0085] Fig.11 is the updating process of the completion time of each workstation, where (a) is the updating process when the total number of workers is sufficient, and (b) is the updating process when the total number of workers is insufficient;

[0086] Fig.12 Optimize the update process for pulse cycles. DETAILED DESCRIPTION

[0087] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0088] Worker resources are crucial in an aircraft assembly line (aircraft pulse assembly line) consisting of a large number of manual assembly tasks. The aircraft pulse assembly line has the characteristics of resource constraints, station duration balance and flexible resource allocation, which makes it more challenging than traditional scheduling problems. Therefore, this embodiment explores the joint optimization problem of worker allocation and multi-mode task scheduling under the resource exclusive strategy in the aircraft pulse assembly line. The dual objectives of minimizing the pulse cycle and the total investment of workers with lexicographic order are considered, and an integer programming model is established to express this problem. In addition, a resource reallocation embedded genetic algorithm (RReGA) is proposed to effectively solve this optimization problem. First, a hybrid scheduling rule (HDRs) is used to estimate the resource-completion time mapping of each workstation, so as to obtain a high-quality initial resource allocation plan. Using these mappings, a resource reallocation method consisting of a resource transfer strategy and a resource recovery strategy is embedded in the evolutionary process of the genetic algorithm (GA) searching for assembly station scheduling solutions. The resource transfer strategy is responsible for dynamic resource transfer across workstations, and it follows a novel transfer principle to optimize the pulsation period; while the resource recycling strategy aims to meet the duration constraint with the least number of workers to minimize the cost. The effectiveness of the proposed algorithm is verified through comprehensive comparison and ablation experiments as well as experiments on an unbalanced case.

[0089] like Figure 1 As shown, this embodiment provides a joint optimization method for production scheduling and worker allocation for an aircraft pulse assembly line, including the following steps:

[0090] Step 1) Obtain assembly task data and resource data of the aircraft pulse assembly line.

[0091] Step 2) construct a lexicographically ordered bi-objective optimization integer programming model to minimize the pulsation period and total resource consumption as the objective function.

[0092] This embodiment studies the worker allocation and multi-mode assembly task scheduling problems in the aircraft pulse assembly line, and considers the bi-objective optimization of the lexicographic order. Specifically, the primary goal is to minimize the workstation pulse cycle, thereby improving production efficiency. Resource allocation principles are generally divided into two strategies: resource sharing and resource exclusive strategies. Unlike traditional multi-project scheduling problems, in the aircraft pulse assembly line, it is not feasible to quickly and arbitrarily transfer workers between different workstations within a production cycle, and the frequent movement of workers between stations will bring huge time costs, so it is not suitable to adopt the RS policy. Therefore, the worker allocation problem studied in this embodiment assumes the use of a resource exclusive policy. Under the resource exclusive policy, workers are allocated to each workstation in a specific number throughout the production cycle, and are not allowed to transfer to other workstations in a production cycle. The basic multi-mode resource-constrained project scheduling problem (MRCPSP) is considered to be NP-hard. As an extension of MRCPSP, the multi-mode resource-constrained multi-project scheduling problem (MRCMPSP) under the RS strategy can be simplified to a large-scale MRCPSP by integrating all assembly task sets. The problem studied in this embodiment, namely, MRCMPSP under the resource exclusive policy, can be regarded as a combination of multiple coupled MRCPSPs. Therefore, this is also an NP-hard problem.

[0093] The secondary goal is to minimize the total worker investment, aiming to reduce resource costs. Studies have shown that minimizing worker investment helps balance the production cycle of workstations, taking into account assembly tasks and pulsation cycles. It expands the target space and may lead to local optimality. Existing research mainly focuses on resource allocation and scheduling under resource-exclusive policies in aircraft pulsation assembly lines, but has not yet solved the optimization problems of pulsation cycles and resource costs at the same time, which are contradictory goals but often encountered in practice. Based on these challenges and shortcomings, this embodiment proposes the RReGA algorithm to solve this problem. A mapping-based resource reallocation method consisting of a resource transfer strategy and a resource recovery strategy is embedded in the GA evolution process, thereby facilitating dynamic adjustment of worker allocation without adding additional computational burden.

[0094] In this embodiment, "worker" and "resource" are synonyms because this embodiment only considers worker resources. In other parts of this embodiment, these two words can be used interchangeably. The problems to be solved in this embodiment mainly include two aspects: (1) assigning workers to workstations; (2) arranging assembly tasks at each workstation.

[0095] Consider an assembly line workshop with k workstations, denoted by W = {w1,w2,...,w k}, where k is the number of workstations, and each workstation w i Process a set of assembly tasks, using Indicates that, m iIndicates workstation w i The number of tasks to be processed, the assembly tasks at each workstation is predetermined, J i,j Denotes the assembly task set J i The jth task of the assembly task is j; the scheduling of tasks is subject to the priority constraint, that is, each task can only be started after all its predecessor tasks are completed. i,j There may be multiple execution modes, each mode m∈M i,j Corresponding to a specific worker requirement r ij, and assembly time d ijm , where M i,j represents the assembly task J i,j All assembly mode sets, d ijm represents the assembly task J i,j The assembly time of mode m, r ijm represents the assembly task J i,j The number of assembly workers in mode m. In general, an increase in the number of workers leads to a decrease in the processing time of a given task. The total number of available workers is n, and the differences in worker skills are not considered, assuming that the assembly efficiency of workers is constant. The number of workers assigned to each workstation, as well as the start time and execution mode of each task need to be determined to optimize a lexicographically ordered dual objective. The primary objective is to minimize the pulsation period, and the secondary objective is to minimize worker input.

[0096] The modeling assumptions for the problem include:

[0097] Each assembly task corresponds to a non-preemptive duration;

[0098] After the assembly task starts, the execution mode cannot be changed;

[0099] During the assembly cycle, resources cannot be transferred between workstations;

[0100] In order to describe the problem rigorously and use commercial solvers to solve it, a dictionary-order bi-objective optimization integer programming model is given below, and its optimization objective is expressed as:

[0101] Primary goals:

[0102] Minimize C(1) Secondary objective:

[0103]

[0104] Constraints

[0105]

[0106] x ijm ∈{0,1(10)

[0107]

[0108] Where C represents the pulsation period, n i Indicates that it is assigned to workstation w i The number of resources, n represents the total number of resources, t i,j Represents task J i,j The start time, x ijm Represents the decision task J i,j Whether the assembly mode of is a 0-1 variable of m, A i,j Indicates that at time t i,j The task being assembled, pre i,j Represents task J i,j All the preceding tasks, Represents a set of integers.

[0109] Objective function (1) aims to minimize the pulsation period, while objective function (2) aims to minimize the total resource consumption. Since both objectives are integers, in order to reflect the idea of ​​lexicographic order optimization in the solver, this embodiment combines these two objectives into a weighted objective function:

[0110]

[0111] As shown in (13), by adding a weight of 1 / n, the lexicographic dual objective is transformed into a weighted single objective function. Constraint (3) ensures that all tasks are completed within the cycle. Constraint (4) ensures that only one mode is selected for each task. Constraint (5) indicates that the total number of workers assigned to each workstation does not exceed the total number of available workers. Constraint (6) stipulates that the priority constraint must be observed. Constraint (7) means that at any given moment, the demand for workers at each workstation will not exceed the total number of its available workers. Constraint (8) indicates the conditions satisfied by the two tasks executed in parallel. Constraints (9)-(12) define the decision variable t i,j 、x ijm 、n i Attributes.

[0112] Step 3) Solve the lexicographic order bi-objective optimization integer programming model to achieve the joint optimization of worker allocation under the resource exclusive policy and multi-mode assembly tasks in the aircraft pulse assembly line.

[0113] To solve large-scale problems, this embodiment proposes an RReGA algorithm. First, a step is implemented to estimate the resource-completion time mapping for each workstation. Subsequently, the optimal resource allocation and pulsation period are determined based on these mappings, and a method of sampling key mapping points is used to speed up the calculation. Subsequently, genetic operations are used to search for scheduling solutions. Once an improved scheduling solution is determined for a given resource allocation, a mapping update strategy is used to improve the mapping. After a predetermined evolution cycle, resource reallocation is performed to adjust and optimize the resource allocation.

[0114] The specific steps include:

[0115] Step 31) Estimate the resource-duration mapping of each workstation and determine the initial resource allocation based on the estimated mapping.

[0116] Function f i : Map resource inputs to assembly task set J i The production cycle of these mappings {f1,f2,...,f k} helps to obtain the best resource allocation and pulsation cycle. However, due to the huge amount of calculation, it is impractical to achieve the optimal production cycle of all resource investments. Therefore, a scheduling rule-based method is usually used to quickly obtain a feasible solution. However, a single scheduling rule is usually not universally applicable to all problems. To solve this problem, this embodiment uses a hybrid scheduling rule and a serial schedule generation scheme (SSGS) to speed up the estimation of these mappings. It must be admitted that this estimation deviates from the precise mapping and may mislead the optimization of resource allocation and pulsation cycle. Therefore, resource reallocation and mapping update strategies are essential.

[0117] Step 311) Mapping estimation

[0118] Before estimation, the resource interval must be determined, that is, the workstation w i Minimum resource investment and maximum resource investment It represents the minimum resources required for each workstation to complete the assembly task, which is equivalent to the maximum resources required for a single task in the assembly task set, and is executed in the mode with the least resource requirements. For details, see formula (14).

[0119]

[0120] For a given assembly task set J i , maximum resource requirement Depends on the assembly task set J i When resource input exceeds the maximum resource demand The construction period will not be shortened. Represents the maximum resource demand J i All subsets of , where l is the total number of subsets, and each subset S i,p Contains assembly tasks that can be executed in parallel. Assume that S i,p All tasks in S are executed in the mode that requires the most resources, then all subsets S i,p The resource requirement R i,p The maximum value of As shown in formula (15).

[0121]

[0122] To ensure that all workstations can complete their tasks, and assigned to workstation w i The number of workers is not less than Then assigned to workstation w i The biggest resource It is expressed as:

[0123]

[0124] Then the maximum number of workers to be assigned to a workstation is calculated as follows:

[0125]

[0126] Given that The computational difficulty of is NP-hard, so we use To estimate if This approach may bring additional computational burden, but in most cases of aircraft pulsation assembly lines, due to insufficient worker resources, this approximation is still feasible.

[0127] Based on the determined resource interval, a hybrid scheduling rule is established by considering both task selection and mode selection. Based on the hybrid scheduling rule, a task and mode list is generated for the assembly task of each workstation. The scheduling scheme is decoded by the serial schedule generation method (SSGS) to determine the minimum duration and establish a resource-duration mapping. Table 1 shows the details of the hybrid scheduling rule.

[0128] Table 1

[0129]

[0130] Step 312) Determine initial resource allocation based on estimated mapping

[0131] According to the estimation mapping, the resource allocation problem is formulated as follows:

[0132] minimizeC=max{f1(n1),f2(n2),...,f k (n k )} (18)

[0133]

[0134] The optimization objective (18) is to minimize the pulsation period, f i (n i ) indicates workstation w i The optimal solution, i.e. the optimal resource allocation, can be obtained by traversing all allocation schemes, but the computational cost is high. A feature of these mappings can greatly reduce unnecessary iterations, thereby improving computational efficiency. For a given set of assembly tasks, the duration does not immediately decrease as the resource input increases. Instead, there will be periods where the curve remains unchanged. This phenomenon forms a clear step-like pattern on the mapping curve, such as Figure 2 This embodiment defines the starting point of each constant completion time phase as a critical point, marking the minimum resources required to achieve a specific completion time, such as Figure 2 By sampling these critical points, the solution to problem (18) can be obtained efficiently, which only requires evaluating a few resource allocation schemes.

[0135] Step 32) Based on the determined initial resource allocation, a scheduling solution is searched using genetic operations.

[0136] The assembly task scheduling problem of each workstation is abstracted as MRCPSP, where each workstation runs under a given resource allocation scheme. The scheduling problems of each workstation are independent of each other after completing the worker resource allocation. The genetic algorithm is executed simultaneously for each workstation to determine the scheduling scheme, which includes the following steps:

[0137] Step 321) Encoding and decoding:

[0138] Each individual's chromosome consists of two parts: (p, m), where Represents a task list, p j represents the task index at position j, represents a list of task execution patterns, where each element specifies the number of workers and processing time required for the corresponding task. To ensure the feasibility of the solution, the arrangement p must obey the priority constraints, ensuring that each task follows its predecessor. At the same time, the mode selection in m must satisfy the resource constraints, ensuring that the resource requirements of each task do not exceed the available resources.

[0139] After determining the solution represented by the task list p and the pattern list m, the method in step 31) is used to decode through SSGS to determine the start processing time and completion time of each task, and finally obtain the duration. A simplified example is shown in Table 2. The corresponding encoding and decoding results are as follows Figure 3 shown.

[0140] Table 2

[0141] Task Possible assembly modes Immediate tasks 1 (7,4),(1,14) - 2 (5,7),(2,13) - 3 (7,7),(4,12),(3,13) 1 4 (7,5),(6,7),(5,10) 2,3 5 (7,8),(4,11),(2,15) 4 6 (7,3),(6,4),(3,5) 4 7 (7,2),(6,9),(5,10) 5,6

[0142] Step 322) Population initialization and parent selection:

[0143] Generate an initial population POP0 of size N, where the chromosomes are randomly initialized: for the task list p, tasks are selected only when all previous tasks have been scheduled to ensure feasibility; for the mode list m, the feasible modes of the tasks are determined based on the available resources, and a mode is randomly selected; this approach ensures that all solutions are inherently feasible without the need for preprocessing before the population evolves.

[0144] In order to maintain the diversity of offspring, each individual in the initial population POP0 is a parent, and offspring are generated by randomly dividing the population into two sets of equal size (P1, P2). A pair of parents are drawn from the two sets in turn without replacement, generating N / 2 pairs of parents, and each pair of parents produces two offspring.

[0145] Step 323) Crossover and mutation:

[0146] The crossover operator is used with probability p c Applied to task list p to generate subtask list, and the corresponding pattern list m is crossovered in the same way; this embodiment uses two types of two-point crossover operators, which are known to maintain the feasibility of the solution. Although crossover does not directly operate on pattern list m, the coupling of p and m ensures the consistency of genetic genes between offspring.

[0147] The mutation operator is p m The probability of is applied to the task list p and the pattern list m, where the mutation in the task list p is achieved by selecting a task p i And change the gene p within the allowed range i The position of the corresponding model gene m i The mutation in the pattern list m is achieved by selecting a task p i , and use a randomly selected feasible mode m i ′Replace its original pattern m i . Figure 4 The implementation of these operators is presented in detail.

[0148] Step 324) Fitness calculation and population update:

[0149] According to the individual fitness fit, individuals need to be evaluated before retaining or eliminating offspring:

[0150] fit=C max =C i (p,m,n i ) (twenty one)

[0151] Among them, C max represents the completion time solution, C i () indicates that the workstation w i The individual chromosomes (p,m) are decoded into functions of completion time, n i Indicates that it is assigned to workstation w i As shown in formula (21), the fitness of an individual is determined by the completion time obtained after decoding. The lower the value, the better the individual quality.

[0152] From the parent POP i After the offspring are produced, the population size increases to 2N. In order to maintain diversity while improving the overall quality of the population, the offspring will be stratified according to fitness, retaining the best 0.25N individuals among the 2N individuals. The remaining 1.75N individuals are divided into two sub-populations at a ratio of 2:1, and then 0.5N and 0.25N individuals are selected from each of them to form the next population POP. i+1 , repeat step 323) and step 324) to update the population.

[0153] Step 33) Based on the scheduling scheme determined in step 32), a mapping update strategy is used to improve the mapping.

[0154] For a given set of assembly tasks J i and resources i , when a better completion time solution C is obtained during the execution of the genetic algorithm max , by mapping assembly progress to resource investment in a non-increasing manner to improve the mapping f i , as shown in formula (22).

[0155]

[0156] For more than n i The strategy will replace the previous C with the newly determined optimal assembly cycle. max After adopting the mapping update strategy, the mapping of each workstation will be closer to the global optimal value, which may bring closer to the optimal resource allocation and pulsation period for problem (18). This adjustment can guide the subsequent resource reallocation.

[0157] Step 34) After a predetermined evolution cycle, resource reallocation is performed to adjust and optimize resource allocation. The resource reallocation includes a resource transfer strategy and a resource recovery strategy. The resource transfer strategy increases the resources of the bottleneck workstation by reallocating the excess resources of the workstation with excess resources to optimize the pulsation cycle and achieve a balance between the workstations. After completing the pulsation cycle optimization, the resource recovery strategy recovers the remaining resources from each workstation to optimize the total resource consumption, thereby minimizing resource input without affecting the scheduling efficiency.

[0158] The deviations in the mapping estimation caused by the hybrid scheduling rule mean that the effectiveness of the initial resource allocation to solve problem (18) cannot be guaranteed. Since GA optimizes the span of workstations, inappropriate resource allocation schemes will aggravate the imbalance of workstation spans and hinder further optimization of the pulsation period due to insufficient resources of bottleneck workstations.

[0159] In order to strengthen resource allocation and optimize the objective function, this embodiment proposes an efficient resource reallocation method consisting of a resource transfer strategy and a resource recovery strategy. The resource transfer strategy includes increasing the resources of bottleneck workstations by reallocating excess resources of workstations with excess resources. This reallocation temporarily increases the working time of non-bottleneck workstations and achieves a balance in the pulse cycle of the workstations. After the resource transfer, the pulse cycle optimization ends, and the resource recovery strategy recovers the remaining resources from each workstation to manage resource costs. This recovery strategy ensures that each workstation maintains normal working time that does not exceed the pulse cycle with minimal resource investment.

[0160] 1. Resource transfer strategy

[0161] Before resource transfer, the following definitions are first made.

[0162] Definition (1): Bottleneck workstation and non-bottleneck workstation. For a given workstation, if its completion time is greater than the pulse period, it is defined as a bottleneck workstation, otherwise it is defined as a non-bottleneck workstation.

[0163] Definition (2): Resource pool. It stores the resources provided by non-bottleneck workstations.

[0164] Definition (3): Resource buffer: A temporary storage point for resources owned by each workstation.

[0165] Principles to be followed are established for each workstation based on its type and resource buffer status.

[0166] Type 1: When a workstation is a non-bottleneck workstation and its resource buffer is empty, the workstation stores part of the resources in its resource buffer. The workstation has the potential to transfer resources, and the empty resource buffer indicates that the workstation received resources in the previous round of resource transfer, or this is the first round of resource transfer. In this case, if NBWS cannot ensure that it will not deteriorate into a bottleneck workstation after the resource transfer, it will temporarily store part of the resources in the resource buffer to prevent other workstations from obtaining them.

[0167] Type 2: When a workstation is a non-bottleneck workstation and its resource buffer is not empty, the workstation transfers all resources in its resource buffer to the resource pool and stores some of the resources transferred out in this round in the resource buffer. The presence of resources in the buffer indicates that the workstation has contributed resources in the previous round and may transfer more resources in this round. Therefore, all resources allocated in the previous round can be transferred to the resource pool for use by the bottleneck workstation. In addition, the resources transferred out in this round can be stored in the resource buffer.

[0168] Type 3: When a workstation is a bottleneck workstation and its resource buffer is empty, all resources in the resource pool are allocated to the workstation. If there are multiple such workstations, all resources are allocated to the workstation with the smallest index. The idleness of the resource buffer indicates that the workstation has exhausted its reserve resources. However, as a bottleneck workstation, it must obtain more resources. Therefore, it obtains resources from the resource pool that have been fully released by other workstations.

[0169] Type 4: When a workstation is a bottleneck workstation and its resource buffer is not empty, all resources are taken out of the resource buffer and evenly distributed to all such workstations. The fact that the buffer is not empty indicates that the workstation contributed resources in the previous round, and it is this resource transfer that makes it a bottleneck workstation. Therefore, the resources stored in the resource buffer must be retrieved in a timely manner.

[0170] Obviously, only type 3 workstations can use the resources in the resource pool. Suppose there are no such workstations in a certain round of resource transfer. In this case, the resources in the resource pool will continue to accumulate, and the resources circulating between workstations will also decrease accordingly. In order to alleviate this problem, when there are no type 3 workstations, due to the inevitability of the existence of bottleneck workstations, type 4 workstations must exist at this time, and all resources in the resource pool will be evenly distributed to all type 4 workstations. This strategy ensures that the resource pool is emptied after each round of resource transfer, thereby preventing resource backlogs in the resource pool. Figure 5 The resource transfer strategies for different types of workstations are demonstrated.

[0171] During the resource transfer process, all workers participate in the assembly task without considering the minimization of resource costs. Each workstation has an upper limit for resource investment based on the topology of its assembly task, and any resource investment exceeding this upper limit is futile. When a workstation reaches its upper limit but is still a bottleneck workstation, the resource transfer strategy will continue to allocate additional resources to it, especially when total resources are abundant. Although this practice does not affect the optimization of the pulsation cycle, it is not conducive to controlling the resource cost of the workstation and cannot explore scheduling solutions under low resource conditions. Therefore, the subsequent resource recovery strategy is designed to solve these problems.

[0172] 2. Resource recovery strategy

[0173] After completing the resource transfer and reaching the optimal pulsation period, the temporary minimum resource requirement of the workstation is determined. At this stage, the resource allocation solution has been obtained for the optimization of the primary objective, namely the pulsation period. After the pulsation period is optimized, the resource investment can be further reduced to optimize the secondary objective function.

[0174] Throughout the resource transfer process, some workstations may have been acting as bottleneck workstations to acquire new resources. Therefore, their genetic algorithms are always executed when resources are abundant. However, when resources are limited, these workstations lack sufficient opportunities to update the mapping of this interval. In fact, they may be able to achieve an interval no greater than the pulsation period with reduced resource input. Based on this core concept, the resource recovery strategy aims to allow all workstations to tap their own potential and complete all assembly tasks within the pulsation period with fewer resources than currently allocated.

[0175] Specifically, after determining the best resource input of each workstation so far, the resource recovery strategy is implemented by reducing the resource input of each workstation by one unit. At this time, the working time of all workstations will exceed the pulsation cycle. Afterwards, the workstation continues to execute the genetic algorithm to find a better scheduling plan to reduce the completion time that does not exceed the pulsation cycle; once this goal is achieved, the resource input of the workstation will be reduced by one unit again, and the scheduling plan will continue to be optimized.

[0176] Iterate the above steps. If a better scheduling solution cannot be found, keep the resource input and scheduling solution of the previous iteration, and repeat the iteration until the iteration stop condition is met to obtain the optimal solution to the problem.

[0177] The above is the RReGA algorithm proposed in the present invention. Re-executing the genetic algorithm every time the resource allocation is updated will bring a huge computational burden, which seriously affects the efficiency. In fact, resource adjustments can be regarded as changes in the GA population environment. After the resources are reallocated, the existing GA population can still be used to represent the scheduling scheme. It is only necessary to re-evaluate the adaptability of the individuals in the new environment without restarting the GA from scratch, thus ensuring the continuity of evolution. In practice, the amount of resource adjustment for each iteration is small, and as the evolution proceeds, the resource allocation and the duration of each workstation will tend to stabilize. The degree of environmental change is becoming more and more subtle, thereby maintaining the adaptability of the population to the new environment.

[0178] In the resource transfer strategy, sufficient time must be reserved for the population evolution of each workstation after each resource adjustment. This ensures that the population can identify the individuals that are most adapted to the new environment, thereby accurately reflecting the performance of the workstation after the adjustment. In contrast, the resource recovery strategy has no fixed execution period. Once the performance of the group reaches the expected level, it can be adjusted. The flowchart of RReGA is shown in the figure below. Figure 6 shown.

[0179] T rt represents the execution period of the resource transfer strategy, which means that the population will experience T rt evolutionary iterations. During the resource transfer process, if no workstation contributes resources to the resource pool, it means that if resources are reallocated to the bottleneck workstation, all non-bottleneck workstations cannot maintain their status. This situation indicates that resource transfer should not be performed at this time. If this situation occurs s1 times in a row at the bottleneck workstation without any optimization in the pulsation cycle, the resource transfer strategy is stopped and the transition to the resource recovery phase is made. The termination condition of the resource recovery strategy is that s2 consecutive evolutions occur without recovering any resources. The RReGA algorithm ends when resource recovery stops or the number of evolutionary iterations reaches the maximum value.

[0180] In the case of extreme scarcity of total worker resources or special topological structure of a specific assembly task set (such as tasks are executed continuously), the resources allocated to other workstations may fall into the critical range. In this case, the decoding scheme using the original group in the new environment may fail. When the resource input is within the critical range, some individuals may fail to decode due to insufficient resources required by a specific task, which requires reinitializing the population of workstations to ensure feasibility and continue the subsequent iterations of the algorithm. Since this interval is derived from the resource requirements of a single task, the search space in this interval is greatly restricted, resulting in a significant reduction in the solution space, thereby improving the search efficiency. Therefore, even if the previous population is not inherited, the impact on efficiency is minimal.

[0181] Based on the above method, this embodiment provides the following examples to verify the effectiveness of the present invention.

[0182] 1. RReGA Performance Verification

[0183] To evaluate the efficacy of the RReGA algorithm, this embodiment uses multiple metrics to compare its performance with the GUROBI solver. In addition, extensive ablation experiments are performed to study the impact of resource transfer and recovery strategies on the RReGA algorithm.

[0184] 1. Comparison with GUROBI

[0185] This section introduces the experimental results of the GUROBI solver and the RReGA algorithm at different instance sizes, as shown in Table 3. The maximum solution time of GUROBI is limited to 3600 seconds. The average solution time of GUROBI and RReGA is t G and t R The number of feasible solutions found by GUROBI in each set of 20 experiments is denoted by #fea, while #opt is the number of optimal solutions. The gap is defined as follows:

[0186]

[0187] Table 3

[0188]

[0189]

[0190] In 20 experiments, the average difference between the solutions of the two methods in the pulsation period is expressed as GAP C Indicates. GAP R Refers to the average difference in resource consumption between two methods in the same pulse cycle. If the pulse cycles are different, the difference in pulse cycles is recorded, otherwise the difference in resource consumption is recorded. O Represents the average difference in optimization target values.

[0191] Efficiency is one of the most significant advantages of the RReGA algorithm, especially as the instance size increases. With 30 tasks, the average computation time of the RReGA algorithm is less than 300 seconds, while the GUROBI algorithm struggles to provide a feasible solution within an hour. In all other instances, the proposed algorithm reduces the computation time by half compared to GUROBI. Moreover, in instances of the same size, when resources are fully allocated, the computation time is longer than when resources are insufficiently allocated. The results of both methods show this pattern, although the root causes are different. For RReGA, the difference comes from the parameter T rtWhen resources are sufficient, multiple scheduling schemes can achieve the optimal pulse cycle, thus achieving more flexible resource allocation. Optimization mainly occurs in the subsequent resource recovery phase, so a smaller T is required. rt In contrast, when resources are insufficient, more precise and efficient resource allocation is required to achieve the optimal pulsation period, resulting in higher computational cost in the resource transfer phase. GUROBI uses branch-constraint method and various heuristic methods to solve integer linear programming problems (1). When resources are limited, the introduction of additional constraints leads to an increase in the number of branches. This complexity makes it more complicated to use heuristic methods to find feasible solutions, thereby increasing the computational time.

[0192] As shown in Table 3, as the number of tasks increases, it becomes increasingly difficult for GUROBI to obtain a feasible optimal solution within an hour. For cases where the number of tasks exceeds 20, GUROBI becomes inappropriate. The results in Table 3 show that for small-scale instances, the solution obtained by RReGA is close to the global optimal solution with an average error of less than 2%. For large-scale instances, the solution provided by the proposed algorithm is better than the GUROBI algorithm with an average error of negative value. C and GAP R The values ​​of reveal that the factors causing the gap vary depending on resource availability. When resources are sufficient, the gap between the solutions provided by RReGA and GUROBI is mainly reflected in resource consumption. RReGA achieves the optimal pulsation period, but the resource consumption is slightly higher than the global optimum. In contrast, when resources are insufficient, the gap between them is more related to the pulsation period. These findings confirm the previous discussion on different resource inputs to the same instance: sufficient resources are conducive to achieving the optimal pulsation period but may lead to resource waste; while insufficient resources lead to full utilization but difficulty in achieving optimal allocation, resulting in pulsation period deviation.

[0193] Figure 7 The comparison results of the objective values ​​of the two methods on different instances are shown. The top and bottom lines of each box plot represent the upper and lower quartiles of the objective function values, respectively. The center line in each box represents the median of the objective value. The whiskers extending from the box represent the maximum and minimum values. Outliers are marked with "o" and are removed from the statistical summary. The green dots represent the objective values, corresponding to the feasible solutions that GUROBI is able to give for large-scale instances within 3600 seconds. For small-scale instances containing 5, 7, and 10 tasks, the box plots of the two methods are almost identical, while for large-scale instances containing 15, 20, and 30 tasks, RReGA shows superior optimization results.

[0194] 2. Ablation experiment of RReGA

[0195] In order to further study the impact of resource transfer and recovery strategies on the performance of the RReGA algorithm, this embodiment performs an ablation experiment on the same 240 instances. This embodiment evaluates the performance of the RReGA algorithm after removing the resource transfer and recovery components. The algorithm that only retains the resource transfer part is named RTGA. RTGA transfers resources between workstations to optimize the pulsation cycle and balance the time span, while ignoring the recovery of redundant resources. The algorithm that removes the resource transfer strategy is called RRGA. After RRGA obtains the initial resource allocation based on the resource-span mapping, it performs span optimization and redundant resource recovery at each workstation. When no excellent offspring is found to reduce the makepan of the workstation for 20 consecutive generations, RRGA will transition from the pulsation cycle optimization stage to the resource recovery stage. RRGA and RTGA retain the other parameters of the RReGA algorithm.

[0196] Table 4 lists the various indicator results of the three methods in 12 groups of experiments. The CPU time represents the average duration of 20 experiments in each group. C and RPD R It represents the average relative deviation percentage of two targets (i.e., pulse period and resource consumption), and its definition is as follows.

[0197]

[0198] The cycle optimization results obtained by RTGA and RReGA with the resource transfer strategy are exactly the same, so only RPD R The main difference between RRGA and RReGA is the resource shift, which leads to significant differences in the pulsation cycle optimization results. C Can be used as an evaluation indicator.

[0199] Table 4

[0200]

[0201]

[0202] As shown in Table 4, in all instances, the computation time of RReGA is slightly higher than that of RTGA and RRGA. It is worth noting that in the case of insufficient resource input (e.g. Figure 8 (b) shows that the processing time of RReGA and RTGA is almost the same, which indicates that there are almost no redundant resources available for recovery.

[0203] Table 4 RPD CThe results show that the resource transfer strategy more effectively achieves the rationalization of resource allocation, thereby further optimizing the working time of the bottleneck workstation in most instances. However, for some instances with 10, 20, and 30 tasks, the pulsation period optimization effect of RReGA is worse than that of RRGA. This shows that when the initial resource allocation has reached the optimal level, directly optimizing the time span of each workstation can obtain a better pulsation period, while resource transfer may reduce the allocation effect. It is worth noting that when the initial resource allocation is optimal, performing resource transfer does not necessarily change the final allocation. Resources transferred between workstations may converge back to the initial optimal allocation, but optimizing the suboptimal allocation will incur a lot of cost compared to the RRGA algorithm without transfer, resulting in poor optimization effect. Figure 8 The box plot comparison of the two algorithms on the pulse period shows that RReGA outperforms RRGA on all instance sizes, and the difference is more obvious when resources are insufficient. This highlights the contribution of resource transfer strategy to effectively allocate resources and optimize the pulse period.

[0204] Resource recovery strategies aim to avoid redundant resource inputs, especially when total resources are excessive. Fig. 9 The results in (a) confirm this goal. When there are 20 and 30 tasks and sufficient resources, canceling resource recycling will lead to more than 30% resource waste. When the total amount of worker resources is insufficient, such as Fig. 9 As shown in (b), the algorithm without the resource recovery module performs roughly the same as the proposed algorithm in terms of resource consumption. Although this outstanding performance is closely related to the total amount of resources, it emphasizes the important contribution of the resource recovery strategy to reducing resource consumption.

[0205] 2. An Unbalanced Example

[0206] In actual production, it is challenging to ensure that an equal number of assembly tasks are assigned to each workstation. This imbalance often leads to idle time for some workstations, resulting in resource waste and project delays. Evaluating whether the completion time of all workstations in an unbalanced instance can be balanced by optimizing resource allocation is crucial for evaluating the robustness and adaptability of the algorithm. This example generates an unbalanced assembly task instance with the number of workstation tasks being {20, 20, 25, 25, 30}, the total number of workers being 120 and 50, the population size being 40, and other parameters remaining consistent.

[0207] Fig.10The whole resource reallocation process of the RReGA algorithm is described. The left side of the black dotted line represents the resource transfer process, and the right side represents the resource recovery process. During the resource transfer process, the resource input of the bottleneck workstation will increase, and this phenomenon is more obvious when the resources are sufficient. This shows that other non-bottleneck workstations continuously transfer resources to the bottleneck workstation to optimize the pulsation period C, which depends on the working time of the bottleneck workstation. In the resource recovery stage, due to insufficient resources, the workstation generally has no extra resources to recover. However, the bottleneck workstation will recover a large amount of resources when the resources are sufficient. Fig.11 The updates of the makespan for each workstation are shown. Fig.11 (a) The completion time update process of each workstation shows that the yellow line (the completion time of the bottleneck workstation) does not change after 30 iterations, while other workstations continue to transfer resources to it (e.g. Fig.10 (a), resulting in the accumulation of redundant resources at workstation w3. At the end of the resource transfer phase, the optimal resource allocation resulting from the updated resource-duration mapping allows the bottleneck workstation w3 to immediately unload a large amount of redundant resources.

[0208] like Fig.11 As shown in (b), in the second half of the resource transfer iteration, the workstation experiences more dramatic changes in time span when resources are insufficient. This occurs because as the resource input decreases, the work duration increases dramatically according to the resource-duration mapping. Under the two resource input conditions, the final survival periods of the workstations are {26, 27, 27, 27, 27} and {53, 50, 53, 53, 52}, respectively. This shows that in unbalanced assembly task instances, the RReGA algorithm can maintain a high degree of balance in the workstation time span while pursuing the optimal pulsation period C.

[0209] Fig.12 The blue curve assumes sufficient resource input and a short update cycle (T rt =5, each point represents a resource transfer update), the number of updates is small before converging to the optimal value. The yellow curve has a longer update cycle (T rt = 10), shows more updates before converging to the optimal value. This is because when resources are sufficient, multiple different coding solutions can reach the same validity period, while when resources are insufficient. This means that when the coding solution space is the same size, when resources are sufficient, there are more feasible coding solutions that can reach the optimal validity period. Therefore, this leads to a significant difference in search efficiency between sufficient and insufficient resources.

[0210] This example explores the lexicographic bi-objective optimization problem of resource allocation and task scheduling in an aircraft pulse assembly line, with the goal of minimizing the pulse cycle and resource consumption while balancing the workstation completion time. A lexicographic bi-objective optimization integer programming model is established, and an efficient RReGA method is proposed to achieve these goals. The core mechanism of the method is to adjust and optimize resource allocation through embedded resource reallocation. Resource reallocation includes two stages: resource transfer and resource recovery. The first stage distinguishes bottleneck workstations and non-bottleneck workstations according to the current optimal pulse cycle, thereby optimizing the pulse cycle, and follows a novel resource transfer principle to help bottleneck workstations. The second stage optimizes resource input by recovering redundant resources of workstations, ensuring that each workstation meets the pulse cycle constraint with minimal resource consumption. The resource recovery strategy is particularly effective when there is an overall surplus of workers. In addition, an updateable resource-duration mapping is developed based on a hybrid scheduling rule to provide initial allocation, guide resource transfer, and calculate the optimal resource allocation and pulse cycle.

[0211] A large number of experimental results verify the effect and performance of each algorithm module. In the mapping evaluation experiment, the efficiency and accuracy of the mapping estimation method are demonstrated. In the comparative experiment with GUROBI, the RReGA method shows higher efficiency and optimization results in terms of pulsation period and resource input. In order to fully demonstrate the effect and difference of resource transfer and resource recovery strategies, comprehensive ablation experiments are carried out. In the case of imbalance, the experimental results of highly consistent time spans of each workstation demonstrate the optimization ability of the RReGA method for unbalanced instances.

[0212] The flexibility of execution mode, the specialization of worker resources within a cycle, the simultaneous optimization of pulsation cycle and resource consumption, and the balance of workstation validity period considered in this invention are common complex features in the actual aircraft pulsation assembly line scenario. Future research can further consider the allocation of assembly tasks to make the entire production problem more systematic.

[0213] The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning, or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A joint optimization method for production scheduling and worker allocation for an aircraft pulse assembly line, characterized in that: The following steps are involved: Step 1) obtaining assembly task data and resource data of an aircraft pulse assembly line, wherein the resource is workers; Step 2) constructing a lexicographic order bi-objective optimization integer programming model to minimize the pulsation period and total resource consumption as the objective function; Step 3) Solve the lexicographic order bi-objective optimization integer programming model to achieve the joint optimization of worker allocation under the resource exclusive policy and multi-mode assembly tasks in the aircraft pulse assembly line: Step 31) estimating the resource-duration mapping of each workstation, and determining the initial resource allocation based on the estimated mapping; Step 32) Based on the determined initial resource allocation, a scheduling solution is searched using genetic operations; Step 33) Based on the scheduling scheme determined in step 32), a mapping update strategy is used to improve the mapping; Step 34) After a predetermined evolution cycle, resource reallocation is performed to adjust and optimize resource allocation, wherein the resource reallocation includes a resource transfer strategy and a resource recovery strategy, wherein the resource transfer strategy increases the resources of the bottleneck workstation by reallocating excess resources of the workstation with excess resources to optimize the pulsation cycle, and the resource recovery strategy recovers the remaining resources from each workstation after completing the pulsation cycle optimization to optimize the total resource consumption.

2. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The modeling assumptions of the lexicographically ordered bi-objective optimization integer programming model include: Each assembly task corresponds to a non-preemptive duration; After the assembly task starts, the execution mode cannot be changed; During the assembly cycle, resources cannot be transferred between workstations; Among them, consider an assembly line workshop with k workstations, denoted by W = {w1,w2,...,w k }, where k is the number of workstations, and each workstation w i Process a set of assembly tasks, using Indicates that, m i Indicates workstation w i The number of tasks to be processed, the assembly tasks at each workstation is predetermined, J i,j Denotes the assembly task set J i The jth task of the assembly task is j; the scheduling of tasks is subject to the priority constraint, that is, each task can only be started after all its predecessor tasks are completed; each assembly task J i,j Corresponding to multiple execution modes, each mode m∈M i,j Corresponding to a specific worker requirement r ijm and assembly time d ijm , where M i,j represents the assembly task J i,j All assembly mode sets of d ijm represents the assembly task J i,j The assembly time of mode m, r ijm represents the assembly task J i,j The number of assemblers in mode m.

3. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 2, characterized in that: The optimization objective of the lexicographic order bi-objective optimization integer programming model is expressed as: The constraints include: x ijm ∈{0,1} Where C represents the pulsation period, n i Indicates that it is assigned to workstation w i The number of resources, n represents the total number of resources, t i,j Represents task J i,j The start time, x ijm Represents the decision task J i,j Whether the assembly mode of is a 0-1 variable of m, A i,j Indicates that at time t i,j The task being assembled, Pre i,j Represents task J i,j All the preceding tasks, Represents a set of integers.

4. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The estimation of resource-duration mapping for each workstation specifically includes the following steps: Determine the resource interval, that is, the workstation w i Minimum resource investment and maximum resource investment in, It represents the minimum resources required for each workstation to complete the assembly task, which is equivalent to the maximum resources required for a single task in the assembly task set, and is executed in the mode with the least resource requirements. Among them, M i,j represents the assembly task J i,j The set of all assembly modes, r ijm represents the assembly task J i,j The number of assemblers in mode m, m i Indicates workstation w i The number of tasks to be processed; For a given assembly task set J i , maximum resource requirement Depends on the assembly task set J i Topology and assembly mode, when resource input exceeds the maximum resource demand The construction period will not be shortened. Represents the maximum resource demand J i All subsets of , where l is the total number of subsets, and each subset S i,p Contains assembly tasks that can be executed in parallel. Assume that S i,p All tasks in S are executed in the mode that requires the most resources, then all subsets S i,p The resource requirement R i,p The maximum value of To ensure that all workstations can complete their tasks, and assigned to workstation w i The number of workers is not less than Among them, n i Indicates that it is assigned to workstation w i The number of resources, n represents the total number of resources, is allocated to workstation w i The biggest resource It is expressed as: Then the maximum number of workers to be assigned to a workstation is calculated as follows: Based on the determined resource interval, a hybrid scheduling rule is established by considering both task selection and mode selection. Based on the hybrid scheduling rule, a task and mode list is generated for the assembly task of each workstation. The scheduling scheme is decoded by a serial scheduling generation method to determine the minimum duration and establish a resource-duration mapping. The scheduling rules for task selection include shortest assembly time priority, longest assembly time priority and maximum number of subsequent tasks priority, and the scheduling rules for mode selection include shortest assembly time mode priority and maximum resource utilization mode priority.

5. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 4, characterized in that: The determining of the initial resource allocation according to the estimated mapping is specifically as follows: According to the estimation mapping, the resource allocation problem is formulated as follows: minimizeC=max{f1(n1),f2(n2),...,f k (n k )} Among them, the function To map resource inputs to assembly task set J i The mapping function of the production cycle, f i (n i ) indicates workstation w i The working time, C represents the pulsation cycle; Solving the resource allocation problem obtains initial resource allocation.

6. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 5, characterized in that: The starting point of each constant completion time stage is defined as a critical point, and the resource allocation problem is efficiently solved by sampling the critical point.

7. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The step 32) is specifically as follows: the assembly task scheduling problem of each workstation is abstracted into a multi-mode resource-constrained project scheduling problem, wherein each workstation operates under a given resource allocation scheme, and the scheduling problems of each workstation are independent of each other after completing the worker resource allocation, and a genetic algorithm is simultaneously executed on each workstation to determine a scheduling scheme, which includes the following steps: Step 321) Encoding and decoding: Each individual's chromosome consists of two parts: (p, m), where represents a task list, subject to the priority constraint, p j represents the task index at position j, represents a list of task execution patterns subject to resource constraints, where each element specifies the number of workers and processing time required for the corresponding task, m i Indicates workstation w i The number of tasks to be processed; Step 322) Population initialization and parent selection: Generate an initial population POP0 of size N, where the chromosomes are randomly initialized: for the task list p, a task is selected only when all previous tasks have been scheduled; for the mode list m, determine the feasible modes of the task based on the available resources and randomly select a mode; Each individual in the initial population POP0 is a parent, and offspring are generated by randomly dividing into two sets of equal size (P1, P2). A pair of parents is drawn from each set in turn without replacement, generating N / 2 pairs of parents, and each pair of parents produces two offspring; Step 323) Crossover and mutation: The crossover operator is used with probability p c Applied to the task list p to generate a subtask list, and the corresponding pattern list m is interleaved in the same way; The mutation operator is p m The probability of is applied to the task list p and the pattern list m, where the mutation in the task list p is achieved by selecting a task p i And change the gene p i The position of the corresponding model gene m i The mutation in the pattern list m is achieved by selecting a task p i , and use a randomly selected feasible mode m i ′Replace its original pattern m i ; Step 324) Fitness calculation and population update: According to the individual fitness fit, individuals need to be evaluated before retaining or eliminating offspring: fit=C max =C i (p,m,n i ) Among them, C max represents the completion time solution, C i () indicates that the workstation w i The individual chromosomes (p,m) are decoded into functions of completion time, n i Indicates that it is assigned to workstation w i The number of resources; The offspring are stratified according to fitness, and the best 0.25N individuals out of a total of 2N individuals of the parents and offspring are retained. The remaining 1.75N individuals are divided into two sub-populations at a ratio of 2:1, and then 0.5N and 0.25N individuals are selected from them to form the next population POP. i+1 , repeat step 323) and step 324) to update the population.

8. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The step 33) is specifically as follows: For a given set of assembly tasks J i and resources i , when a better completion time solution C is obtained during the execution of the genetic algorithm max , by mapping assembly progress to resource investment in a non-increasing manner to improve the mapping f i :

9. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The resource transfer strategy is specifically as follows: According to the type and resource buffer status of each workstation, a corresponding resource transfer plan is determined for each workstation, wherein the workstation includes bottleneck workstations and non-bottleneck workstations. If its completion time is greater than the pulse cycle, it is defined as a bottleneck workstation, otherwise it is defined as a non-bottleneck workstation; a resource pool is defined to store resources provided by non-bottleneck workstations, and a resource buffer is defined as a temporary storage point for resources owned by each workstation; When a workstation is a non-bottleneck workstation and its resource buffer is empty, the workstation stores part of the resources into its resource buffer; When a workstation is a non-bottleneck workstation and its resource buffer is not empty, the workstation transfers all resources in its resource buffer to the resource pool and stores some resources transferred out in this round into the resource buffer; When a workstation is a bottleneck workstation and its resource buffer is empty, all resources in the resource pool are allocated to the workstation. If there are multiple such workstations, all resources are allocated to the workstation with the smallest index. When a workstation is a bottleneck workstation and its resource buffer is not empty, all resources are taken from the resource buffer and evenly distributed to all such workstations.

10. The method for joint optimization of production scheduling and worker allocation for an aircraft pulse assembly line according to claim 1, characterized in that: The resource recovery strategy is specifically as follows: After determining the best resource input of each workstation so far, the resource recovery strategy is implemented by reducing the resource input of each workstation by one unit. At this time, the working time of all workstations will exceed the pulse cycle. After that, the workstation continues to execute the genetic algorithm to find a better scheduling solution to reduce the completion time that does not exceed the pulse cycle. Iterate the above steps. If a better scheduling solution cannot be found, keep the resource input and scheduling solution of the previous iteration, and repeat the iteration until the iteration stop condition is met to obtain the optimal solution to the problem.

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