Method for joint optimization of production scheduling and worker allocation for aircraft pulsating assembly line

Abstract

The application relates to a production scheduling and worker allocation joint optimization method for an aircraft pulsating assembly line, which comprises the following steps: acquiring assembly task data and resource data of the aircraft pulsating assembly line, wherein the resource is a worker; constructing a dictionary order double-target optimization integer programming model, taking minimization of a pulsating period and total resource consumption as a target function; solving the dictionary order double-target optimization integer programming model, realizing joint optimization of worker allocation and multi-mode assembly task in the aircraft pulsating assembly line under a resource exclusive policy; estimating a resource-work period mapping of each workstation, and determining an initial resource allocation according to the estimated mapping; searching a scheduling scheme by using genetic operation; improving the mapping by using a mapping updating strategy; after a predetermined evolution period, performing resource reallocation to adjust and optimize the resource allocation. Compared with the prior art, the application has the advantages of high production efficiency of scheduling results, saved calculation overhead and the like.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of scheduling of aircraft pulsating assembly line, and particularly relates to a production scheduling and worker allocation joint optimization method for aircraft pulsating assembly line. BACKGROUND

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

[0003] The aircraft pulsating final assembly line is composed of a certain number of workstations, each of which has a set of assembly tasks that must comply with priority constraints. All assembly tasks must be completed within a predetermined cycle, and then the aircraft can move to the next workstation to complete the assembly of the remaining components. This requires that the completion time of all workstations be balanced. Worker allocation in aircraft pulsating final assembly line is one of the most important resource optimization problems, because more than 70% of the assembly tasks are performed manually. The flexibility of 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 multiple modes of assembly for the task, each corresponding to a certain worker input and processing time.

[0004] Existing research on scheduling of aircraft pulsating final assembly line reveals several shortcomings. First, few studies adequately address the necessary flexibility in resource allocation. In addition, since the two objectives of minimizing the pulsating cycle and minimizing the resource cost often conflict, few studies optimize both objectives simultaneously under limited resource constraints. Furthermore, the uniform pulsating cycle characteristic of aircraft pulsating final assembly line requires that the production cycle of all workstations be consistent to reduce resource waste caused by idle time. These gaps greatly limit the practical application of existing algorithms. SUMMARY

[0005] The purpose of the present application is to provide a production scheduling and worker allocation joint optimization method for aircraft pulsating assembly line, which realizes the joint optimization of worker allocation and production scheduling under resource exclusive policy by considering the characteristics of aircraft pulsating final assembly line, including limited resource constraints, balanced pulsating cycle and flexible resource allocation.

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

[0007] A production scheduling and worker allocation joint optimization method for aircraft pulsating assembly line, comprising the following steps:

[0008] Step 1) Obtain the assembly task data and resource data of the aircraft pulsating assembly line, the resource being workers;

[0009] Step 2) Construct a dictionary order double-objective optimization integer programming model to minimize the pulsating cycle and total resource consumption as the objective function;

[0010] Step 3) Solve the dictionary order double-objective optimization integer programming model to realize the joint optimization of worker allocation and multi-mode assembly task in the aircraft pulsating assembly line under the resource exclusive policy:

[0011] Step 31) Estimate the resource-duration mapping of each workstation, and determine the initial resource allocation according to the estimated mapping;

[0012] Step 32) Based on the determined initial resource allocation, search for a scheduling scheme using genetic operations;

[0013] Step 33) Based on the scheduling scheme determined in step 32), use the mapping update strategy to refine the mapping;

[0014] Step 34) After a predetermined evolution period, perform resource reallocation to adjust and optimize resource allocation, including resource transfer strategy and resource recycling strategy, wherein the resource transfer strategy increases the resources of the bottleneck workstation by reallocating the excess resources of the resource-excessive workstation to optimize the pulsating cycle, and the resource recycling strategy recovers the remaining resources from each workstation after completing the pulsating cycle optimization to optimize the total resource consumption.

[0015] The modeling assumptions of the dictionary order double-objective optimization integer programming model include:

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

[0017] The execution mode cannot be changed after the assembly task starts;

[0018] Resources cannot be transferred between workstations within the assembly cycle;

[0019] Wherein, considering an assembly line workshop with k workstations, denoted as W = {w1, w2,..., w k}, where k is the number of workstations, and each workstation w i Processes a set of assembly tasks, denoted as J i = {J i,0 , J i,2 ,..., J i,mi}, where m i represents the number of tasks to be processed by workstation w i , and the assembly tasks of each workstation are predetermined, J i,jdenotes the jth task of the set of assembly tasks J i The scheduling of tasks respects priority constraints, i.e. each task can start only after all its predecessor tasks have finished. Each assembly task J i,j corresponds to a plurality of execution modes, each mode m e M i,j corresponds to a specific worker requirement r ijm and an assembly time d ijm where M i,j denotes the set of all assembly modes of the assembly task J i,j d ijm denotes the assembly time of the assembly task J i,j in mode m, r ijm denotes the number of assembly workers of the assembly task J i,j in mode m.

[0020] The optimization objective of the dictionary-ordered bi-objective integer programming model is represented as:

[0021]

[0022] The constraints include:

[0023]

[0024] x ijm e {0, 1}

[0025]

[0026] where C denotes the pulsation period, n i denotes the number of resources allocated to workstation w i , n denotes the total number of resources, t i,j denotes the start time of task J i,j , x ijm denotes the 0-1 variable that determines whether the assembly mode of task J i,j is m, A i,j denotes the task being assembled at time t i,j , Pre i,j denotes all the immediate predecessor tasks of task J i,j , denotes the set of integers.

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

[0028] determining the resource interval, i.e. the minimum resource input and the maximum resource input of workstation w i wherein Indicates 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 a mode with the least resource requirements.

[0029]

[0030] Among them, M i,j Represents assembly task J i,j The set of all assembly modes, r ijm Represents assembly task J i,j The number of assembly workers 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 requirements 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, each subset S i,p Contains assembly tasks that can be executed in parallel, assuming 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 demand 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 the allocation to workstation w i The number of resources, n represents the total number of resources, is allocated to workstation w i The largest resource Expressed as:

[0034]

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

[0036]

[0037] Based on the determined resource interval, a hybrid scheduling rule is established considering task selection and mode selection, a task and mode list is generated for each workstation assembly task based on the hybrid scheduling rule, a serial scheduling generation scheme (SSGS) is used to decode the scheduling scheme to determine the minimum value of the duration, and a resource-duration mapping is established, wherein the scheduling rule of the task selection includes shortest assembly time first, longest assembly time first, and most number of tasks after priority, and the scheduling rule of the mode selection includes shortest assembly time mode first and maximum resource utilization mode first.

[0038] The initial resource allocation determined according to the estimation mapping is specifically:

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

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

[0041]

[0042] Wherein, the function f i ∶ is a mapping function for mapping resources to the production cycle of the assembly task set J i , f i (n i ) represents the working time of the workstation w i , and C represents the pulsation cycle;

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

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

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

[0046] Step 321) encoding and decoding:

[0047] The chromosome of each individual is composed of two parts: (p, m), wherein, represents a task list, which complies with priority constraints, p j represents the task index of the position j, denotes a list of task execution patterns, respecting the resource constraints, where each element specifies the number of workers and processing time required for the corresponding task, m i denotes the workstation w i the number of tasks to be processed;

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

[0049] An initial population POP0 of size N is generated, where the chromosomes are randomly initialized: for the task list p, only the tasks that have been scheduled before are selected; for the pattern list m, the feasible patterns for the tasks are determined according to the available resources, and a pattern is randomly selected;

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

[0051] Step 323) Crossover and mutation:

[0052] The crossover operator is applied to the task list p with a probability p c to generate a sub-task list, and the corresponding pattern list m is crossed in the same way;

[0053] The mutation operator is applied to the task list p and the pattern list m with a probability p m , where the mutation in the task list p is achieved by selecting a task p i and changing the position of the gene p i , while adjusting the position of the corresponding pattern gene m i , and the mutation in the pattern list m is achieved by selecting a task p i and replacing its original pattern m i with a randomly selected feasible pattern m i ;

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

[0055] Before the offspring are retained and eliminated, the individuals need to be evaluated according to their fitness fit:

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

[0057] where C max denotes the completion time solution, and C i () denotes the workstation w idecoding of the individual chromosomes (p, m) into a function of the completion time, n i representing the number of resources allocated to the workstation w i ;

[0058] The offspring are ranked according to fitness, the best 0.25N individuals of the total of 2N individuals of parents and offspring are kept, the remaining 1.75N individuals are divided into two sub-populations in a ratio of 2:1, and 0.5N and 0.25N individuals are selected from each of the two sub-populations to form the next population POP i+1 , steps 323) and 324) are repeated to update the population.

[0059] The step 33) is specifically:

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

[0061]

[0062] The resource transfer strategy is specifically:

[0063] According to the type and resource buffer state of each workstation, a corresponding resource transfer scheme is determined for each workstation, wherein the workstations include bottleneck workstations and non-bottleneck workstations, if the completion time is greater than the pulsation period, it is defined as a bottleneck workstation, otherwise it is defined as a non-bottleneck workstation; a resource pool is defined to store the resources given by the non-bottleneck workstations, and a resource buffer is defined as a temporary storage point for the resources owned by each workstation;

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

[0065] When the workstation is a non-bottleneck workstation and its resource buffer is not empty, all resources in the resource buffer of the workstation are transferred to the resource pool, and part of the resources released in this round are stored in the resource buffer;

[0066] When the 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 the workstation is a bottleneck workstation and its resource buffer is not empty, all resources in the resource buffer are taken out and evenly distributed to all such workstations.

[0068] The resource recycling strategy is specifically:

[0069] After determining the optimal resource input of each workstation so far, the resource recycling strategy is implemented by reducing the resource input of each workstation by one unit, at which time the working time of all workstations will exceed the pulse period, after which the workstations continue to execute the genetic algorithm to find a better scheduling scheme to reduce the completion time not exceeding the pulse period;

[0070] The above steps are iteratively executed, and if a better scheduling scheme cannot be found, the resource input and scheduling scheme of the last iteration are retained, and the iteration is repeated until the iteration stopping condition is met, and the optimal solution of the problem is obtained.

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

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

[0073] (2) The present application 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 working span of all workstations by transferring workers before execution, thereby optimizing the pulse period. After determining the pulse period, the resource recycling strategy is proposed to actively recycle the excess workers of each workstation, simplifying labor allocation while maintaining the optimal pulse period.

[0074] (3) The present application integrates resource reallocation into the evolutionary process, without the need for large computational overhead, saving computational resources. BRIEF DESCRIPTION OF DRAWINGS

[0075] Figure 1 is a flowchart of the method of the present application;

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

[0077] Figure 3 is a Gantt chart of a feasible scheduling solution in an embodiment;

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

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

[0080] Figure 6Flow chart for RReGA algorithm

[0081] Figure 7 Comparison results of the present application and GUROBI, wherein (a) is comparison under total worker sufficiency, and (b) is comparison under total worker insufficiency;

[0082] Figure 8 Comparison results of the present application and RRGA, wherein (a) is comparison under total worker sufficiency, and (b) is comparison under total worker insufficiency;

[0083] Figure 9 Comparison results of the present application and RTGA, wherein (a) is comparison under total worker sufficiency, and (b) is comparison under total worker insufficiency;

[0084] Figure 10 Adjustment process of resource reallocation, wherein (a) is adjustment process under total worker sufficiency, and (b) is adjustment process under total worker insufficiency;

[0085] Figure 11 Update process of completion time of each workstation, wherein (a) is update process under total worker sufficiency, and (b) is update process under total worker insufficiency;

[0086] Figure 12 Pulsation cycle optimization update process. DETAILED DESCRIPTION

[0087] The present application will be described in detail below with reference to the accompanying drawings and specific embodiments. The present embodiment is implemented on the premise of the technical solution of the present application, and detailed implementation and specific operation process are given, but the protection scope of the present application is not limited to the following embodiments.

[0088] Worker resource is critical in aircraft final assembly line (aircraft ripple final assembly line) which consists of a large number of manual assembly tasks. Aircraft ripple final assembly line has the characteristics of resource-constrained, station duration balancing and flexible resource allocation, which makes it more challenging than traditional scheduling problems. Therefore, this embodiment discusses the joint optimization problem of worker allocation and multi-mode task scheduling under resource-exclusive strategy in aircraft ripple final assembly line. The dual-objective of minimizing ripple period and total worker investment with lexicographic order is considered, and an integer programming model is established to express this problem. And a resource reallocation embedded genetic algorithm (RReGA) is proposed to effectively solve this optimization problem. First, hybrid dispatching rules (HDRs) are used to estimate the resource-completion time mapping of each workstation, so as to obtain a high-quality initial resource allocation scheme. Using these mappings, a resource reallocation method composed of resource transfer strategy and resource recovery strategy is embedded into the evolution process of genetic algorithm (GA) to search for assembly station scheduling solutions. The resource transfer strategy is responsible for dynamic resource transfer across workstations, which follows a novel transfer principle to optimize the ripple period; while the resource recovery strategy aims to meet the duration constraints with the least workers to minimize the cost. Through comprehensive comparison and ablation experiments and experiments of a non-balanced case, the effectiveness of the proposed algorithm is verified.

[0089] As Figure 1 shown, the embodiment provides a production scheduling and worker allocation joint optimization method for aircraft ripple assembly line, comprising the following steps:

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

[0091] Step 2) Build a lexicographic order dual-objective optimization integer programming model to minimize the ripple period and total resource consumption as the objective function.

[0092] This embodiment studies the worker allocation and multi-mode assembly task scheduling problem in an aircraft ripple assembly line, and considers the lexicographic order of the dual-objective optimization. Specifically, the primary objective is to minimize the workstation ripple period, 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 an aircraft ripple 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 incur a huge time cost, so it is not suitable to adopt the RS policy. Therefore, the worker allocation problem studied in this embodiment assumes the use of the 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 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 sets of assembly tasks. The problem studied in this embodiment, MRCMPSP under the resource exclusive strategy, can be considered as the combination of multiple coupled MRCPSPs. Therefore, it is also an NP-hard problem.

[0093] The secondary objective is to minimize the total investment in workers, aiming to reduce resource costs. Studies have shown that, considering the assembly tasks and the ripple period, minimizing the worker investment helps to balance the production cycle of the workstations. It expands the target space and has the potential to lead to local optimality. Existing researches mainly focus on resource allocation and scheduling under the resource exclusive policy in an aircraft ripple assembly line, but have not yet solved the optimization problem of the ripple period and resource cost at the same time, which are contradictory but often encountered in practice. Based on these challenges and deficiencies, this embodiment proposes the RReGA algorithm to solve this problem. The mapping-based resource reallocation method composed of the resource transfer strategy and the resource recycling strategy is embedded in the GA evolution process, thereby promoting the dynamic adjustment of worker allocation without increasing additional computational burden.

[0094] In this embodiment, "worker" and "resource" are synonymous, as this embodiment only considers worker resources. In other parts of this embodiment, the two words can be used interchangeably. The problem to be solved in this embodiment mainly includes two aspects: (1) allocating workers to workstations; (2) scheduling assembly tasks at each workstation.

[0095] Consider an assembly line workshop with k workstations, denoted as W = {w1, w2,..., wk}, where k is the number of workstations, and each workstation w k} processes a set of assembly tasks, denoted as T = {t1, t2,..., tm}, where m i} processes a set of assembly tasks, denoted as T = {t1, t2,..., tm}, where m i ​Workstations w i The number of tasks to be processed, the assembly tasks of each workstation are predetermined, J i,j Assembly task set J i The jth task; the scheduling of tasks complies with priority constraints, i.e. each task can only start after all its predecessor tasks have been completed. Each assembly task J i,j There can 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 Indicates the set of all assembly modes of assembly task J i,j d ijm Indicates the assembly time of assembly task J i,j In mode m, r ijm Indicates the number of assembly workers of assembly task J i,j In mode m. Generally speaking, the increase in the number of workers will result in the reduction of the processing time of a given task. The total number of available workers is n, and the difference in worker skills is not considered, assuming that the assembly efficiency of workers is constant. The number of workers allocated to each workstation and the start time and execution mode of each task need to be determined to optimize a lexicographic bi-objective. The primary objective is to minimize the pulsation period as much as possible, and the secondary objective is to minimize the worker input as much as possible.

[0096] The modeling assumptions of the problem include:

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

[0098] The execution mode cannot be changed after the assembly task starts;

[0099] Resources cannot be transferred between workstations within the assembly cycle;

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

[0101] Primary objective:

[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 the allocation 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 Is the assembly mode of m a 0-1 variable, 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 lexicographic order optimization idea 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 period. 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 followed. 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 lexicographically ordered 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, the present embodiment proposes a RReGA algorithm. First, a step is implemented to estimate the resource-completion time mapping of each workstation. Subsequently, the optimal resource allocation and pulse period are determined based on these mappings, using a method of sampling critical mapping points to speed up the computation. Subsequently, genetic operations are employed to search for a scheduling solution. Once an improved scheduling solution is determined for a given resource allocation, a mapping update strategy is employed to refine the mappings. After a predetermined evolution period, resource reallocation is performed to adjust and optimize the resource allocation.

[0114] Specifically comprising the following steps:

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

[0116] Function f i : Map the resource input to the production cycle of the assembly task set J i . These mappings {f1, f2,..., f k} help to obtain the optimal resource allocation and pulse period. However, it is impractical to achieve the optimal production cycle for all resource inputs due to the huge amount of computation. Therefore, a scheduling rule-based approach is usually employed to quickly obtain a feasible solution. However, a single scheduling rule is usually not universally applicable to all problems. To address this issue, the present embodiment employs a hybrid scheduling rule and serial scheduling generation scheme (SSGS) to speed up the estimation of these mappings. It must be acknowledged that this estimation deviates from the exact mapping and may mislead the optimization of resource allocation and pulse period. Therefore, resource reallocation and mapping update strategies are essential.

[0117] Step 311) Mapping estimation

[0118] Before estimation, the resource interval must be determined, i.e., the minimum resource input i and the maximum resource input of the workstation w represents the minimum resource required by each workstation to complete the assembly task, which is equivalent to the maximum resource required by a single task in the assembly task set, executed in the mode with the least resource demand, as shown in equation (14).

[0119]

[0120] For a given assembly task set J i , the maximum resource demand depends on the topology of the assembly task set J i and the assembly mode. When the resource input exceeds the maximum resource demand The project duration will not be shortened. represent the maximum resource demand J i , where l is the total number of subsets, each subset S i,p contains assembly tasks that can be executed in parallel, assuming that all tasks in S i,p are executed in the mode that requires the most resources, then the maximum resource demand R i,p of all subsets S i,p is the maximum value of R as shown in equation (15).

[0121]

[0122] To ensure that all workstations can complete the tasks, it is required that and the number of workers assigned to workstation w i is no less than The maximum resource i assigned to workstation w is represented as:

[0123]

[0124] The calculation method of the maximum number of workers assigned to the workstation is as follows:

[0125]

[0126] Given the difficulty of calculating is NP-hard, we use to estimate If This method may bring additional computational burden, but in most cases of aircraft pulsation assembly line, due to the lack of worker resources, this approximation method is still feasible.

[0127] Based on the determined resource interval, considering task selection and mode selection, a hybrid scheduling rule is established, based on which a task and mode list is generated for each workstation's assembly task, and a serial scheduling generation method (SSGS) is used to decode the scheduling scheme to determine the minimum value of the project duration, and a resource-duration mapping is established. Table 1 shows the detailed information of the hybrid scheduling rule.

[0128] Table 1

[0129]

[0130] Step 312) Determine the initial resource allocation according to the estimated mapping

[0131] According to the estimated mapping, the resource allocation problem can be described as follows:

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

[0133]

[0134] The optimization objective (18) is to minimize the cycle time, f i (n i ), of the workstation w i . The optimal solution, i.e. the optimal resource allocation, can be obtained by traversing all allocation schemes, but at a high computational cost. A characteristic of these mappings is that a significant number of iterations can be eliminated, thus improving the computational efficiency. For a given set of assembly tasks, the lead time does not immediately decrease with an increase in resource input. Instead, it exhibits periods of time where the curve remains constant. This phenomenon forms a distinct step-like pattern on the mapping curve, as shown in Figure 2 . This embodiment defines the starting point of each constant lead time phase as a critical point, marking the minimum resource required to achieve a certain lead time, as shown by the bold diamonds in Figure 2 . By sampling these critical points, an efficient solution to the problem (18) can be obtained, which only requires evaluating a few resource allocation schemes.

[0135] Step 32) Based on the determined initial resource allocation, search for a scheduling scheme using genetic operations.

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

[0137] Step 321) Encoding and decoding:

[0138] The chromosome of each individual is composed of two parts: (p, m), where, p j represents the task list, p represents the task execution mode list, 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 adhere to 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 demand of each task does not exceed the available resources.

[0139] After the solution represented by the task list p and the mode list m is determined, the decoding is performed by SSGS using the method in step 31) to determine the start processing time and the completion time of each task, and finally the duration is obtained. A simplified example is shown in Table 2, and the corresponding encoding and decoding results are shown in Table 3. Figure 3

[0140] Table 2

[0141] Task Possible assembly modes Immediately preceding task 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] An initial population POP0 of size N is generated, in which the chromosomes are randomly initialized: for the task list p, only after all the previous tasks are scheduled, the task is selected to ensure feasibility; for the mode list m, the feasible mode of the task is determined according to the available resources, and a mode is randomly selected; this method ensures that all solutions are essentially feasible without pre-processing before population evolution.

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

[0145] Step 323) crossover and mutation:

[0146] The crossover operator is applied to the task list p with a probability p c to generate a sub-task list, and the corresponding mode list m is crossed 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 the crossover does not directly operate on the mode list m, the coupling of p and m ensures the consistency of the genetic genes between the offspring.

[0147] The mutation operator is applied to the task list p and the mode list m with a probability p m , in which the mutation of the task list p is performed by selecting a task p i and changing the position of the gene p i within the allowed range, while adjusting the position of the corresponding mode gene m i , and the mutation of the mode list m is performed by selecting a task p i and replacing its original mode m i with a randomly selected feasible mode m i . Figure 4 The implementation of these operators is described in detail.

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

[0149] Before the offspring are discarded, the individuals need to be evaluated:

[0150] fit = C max = C i (p,m,n i ) (21)

[0151] where C max represents the makespan solution, C i () represents the function that decodes the individual chromosome (p,m) of workstation w i into makespan, n i represents the number of resources allocated to workstation w i ; as shown in equation (21), the fitness of an individual is determined by the makespan obtained after decoding. The lower the value, the better the quality of the individual.

[0152] After the offspring are generated from the parent POP i , the population size is increased to 2N. In order to improve the overall quality of the population while maintaining diversity, the offspring are stratified according to fitness, and the top 0.25N individuals are retained from the 2N individuals. The remaining 1.75N individuals are divided into two sub-populations in a 2:1 ratio, and 0.5N and 0.25N individuals are selected from each sub-population to form the next population POP i+1 , and steps 323) and 324) are repeated for population update.

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

[0154] For a given set of assembly tasks J i and resources n i , when a better makespan solution C max is obtained during the execution of the genetic algorithm, the mapping f i is refined through a non-decreasing mapping of assembly progress and resource investment, as shown in equation (22).

[0155]

[0156] For resource investment exceeding n i , this strategy replaces the previous C max with the newly determined optimal assembly period. After using the mapping update strategy, the mapping of each workstation will be closer to the global optimal value, making it possible to bring the resource allocation and pulsation period of problem (18) closer to the optimal value. This adjustment can guide subsequent resource reallocation.

[0157] Step 34) After a predetermined evolution period, resource reallocation is performed to adjust and optimize resource allocation, which includes resource transfer strategy and resource recovery strategy, wherein the resource transfer strategy increases the resources of the bottleneck workstations by reallocating the excess resources of the resource-excessive workstations to optimize the pulse period and achieve balance among the workstations, and the resource recovery strategy recovers the remaining resources from each workstation after the completion of the pulse period optimization to optimize the total resource consumption, thereby minimizing the resource investment without affecting the scheduling efficiency.

[0158] The deviation of the mixed scheduling rule in the mapping estimation means that the effectiveness of the initial resource allocation of the problem (18) cannot be guaranteed. Since the GA optimizes the span of the workstations, an inappropriate resource allocation scheme will exacerbate the imbalance of the workstation span and hinder the further optimization of the pulse period due to the lack of resources in the bottleneck workstations.

[0159] In order to strengthen the resource allocation and optimize the objective function, the embodiment proposes an efficient resource reallocation method composed of resource transfer strategy and resource recovery strategy. The resource transfer strategy includes increasing the resources of the bottleneck workstations by reallocating the excess resources of the resource-excessive workstations. This reallocation temporarily increases the working time of the non-bottleneck workstations and achieves the balance of the pulse period of the workstations. After the resource transfer, the pulse period optimization is completed, and the resource recovery strategy will recover the remaining resources from each workstation to manage the resource cost. This recovery strategy can ensure that each workstation maintains the normal working time of no more than the pulse period with the minimum resource investment.

[0160] 1. Resource transfer strategy

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

[0162] Definition (1): Bottleneck workstations and non-bottleneck workstations. 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. The resource given by the non-bottleneck workstation is stored.

[0164] Definition (3): Resource buffer. The temporary storage point of the resources owned by each workstation.

[0165] According to the type and resource buffer state of each workstation, principles to be followed by each workstation are formulated.

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

[0167] Type 2: If a workstation is not a bottleneck workstation and its resource buffer is not empty, it transfers all resources in its resource buffer to the resource pool and stores some of the resources it has 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, resources transferred out in this round can be stored in the resource buffer.

[0168] Type 3: When a workstation is a bottleneck and its resource buffer is empty, all resources in the resource pool are allocated to it. If there are multiple workstations of this type, all resources are allocated to the workstation with the lowest index. The empty resource buffer indicates that the workstation has exhausted its reserve resources. However, as a bottleneck workstation, it must acquire 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 from the resource buffer and distributed evenly among all workstations of this type. A non-empty buffer indicates that the workstation contributed resources in the previous round, and it is this resource transfer that turned it into a bottleneck workstation. Therefore, resources stored in the resource buffer must be promptly retrieved.

[0170] Obviously, only Type 3 workstations can utilize resources in the resource pool. Suppose, however, that during a round of resource transfer, there are no Type 3 workstations. In this case, resources in the resource pool will continue to accumulate, and the amount of resources available between workstations will decrease. To alleviate this problem, when there are no Type 3 workstations, Type 4 workstations are necessary due to the inevitability of bottleneck workstations. All resources in the resource pool are then evenly distributed among these Type 4 workstations. This strategy ensures that the resource pool is cleared after each round of resource transfer, preventing resource backlogs within the pool. Figure 5 The resource transfer strategies for different types of workstations are presented.

[0171] In the resource transfer process, all workers participate in the assembly task regardless of the minimization of resource cost. Each workstation has an upper limit of resource investment based on its assembly task topology, and any resource investment beyond this limit is futile. When a workstation reaches the upper limit but is still a bottleneck workstation, the resource transfer strategy will continue to allocate additional resources to it, especially in the case of abundant total resources. Although this approach does not affect the optimization of the pulse period, 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 pulse period, the temporary minimum resource requirement of the workstation is determined. At this stage, the resource allocation solution has been obtained for the primary objective, i.e., the optimization of the pulse period. After the optimization of the pulse period, the resource investment can be further reduced to optimize the secondary objective function.

[0174] During the entire resource transfer process, some workstations may always obtain new resources as bottleneck workstations. Therefore, their genetic algorithms are always executed in the case of abundant resources. However, in the case of limited resources, these workstations lack sufficient opportunities to update the mapping of the interval. In fact, they have the potential to achieve an interval not greater than the pulse period with reduced resource investment. Based on this core idea, the resource recovery strategy aims to enable all workstations to tap their potential to complete all assembly tasks within the pulse period with fewer resources than the current allocation.

[0175] Specifically, after determining the optimal resource investment of each workstation so far, the resource recovery strategy is implemented by reducing the resource investment of each workstation by one unit, at which time the working time of all workstations will exceed the pulse period, and thereafter, the workstations continue to execute the genetic algorithm to find a better scheduling solution to reduce the completion time not exceeding the pulse period; once this goal is achieved, the resource investment of the workstation will be reduced by one unit again, and the scheduling solution will continue to be optimized.

[0176] The above steps are iteratively executed, and if a better scheduling solution cannot be found, the resource input and scheduling solution of the last iteration are retained, and the iteration is repeated until the iteration stopping condition is met to obtain the optimal solution of the problem.

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

[0178] In the resource transfer strategy, enough time must be reserved for the evolution of the population of each workstation after each resource adjustment. This can ensure that the population can identify the individuals most adapted to the new environment, thereby accurately reflecting the performance of the workstations after adjustment. In contrast, the resource recovery strategy has no fixed execution period. Once the performance of the population reaches the expectation, adjustment can be made. The flowchart of RReGA is shown in Figure 6 .

[0179] T rt represents the execution period of the resource transfer strategy, which means that the population will undergo T rt evolution iterations. During the resource transfer process, if no workstation contributes resources to the resource pool, it indicates that if the resources are redistributed to the bottleneck workstations, all non-bottleneck workstations cannot maintain their state. This situation indicates that resource transfer should not be performed at this time. If the bottleneck workstation appears s1 times in succession without any optimization in the pulsation period, the resource transfer strategy is stopped, and the resource recovery phase is transitioned to. The termination condition of the resource recovery strategy is that s2 times of evolution occur successively without recovering any resources. When the resource recovery stops or the number of evolution iterations reaches the maximum value, the RReGA algorithm ends.

[0180] In the case of extreme scarcity of total worker resources or special topology of a specific set of assembly tasks (such as tasks are executed continuously), the resources allocated to other workstations may fall into the critical range. In this case, using the decoding scheme of the original population in the new environment may fail. When the resource input is in the critical range, some individuals may fail to decode due to insufficient resources required by specific tasks, which requires reinitialization of the population of the workstation to ensure feasibility and continue subsequent iterations of the algorithm. Since this interval is derived based on the resource requirements of individual tasks, the search space of this interval is greatly limited, resulting in a greatly reduced solution space and thus improved search efficiency. Therefore, even if the previous population is not inherited, the impact on efficiency is minimal.

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

[0182] I. RReGA performance verification

[0183] To evaluate the effectiveness of the RReGA algorithm, this embodiment compares its performance with the GUROBI solver using multiple indicators. In addition, extensive ablation experiments are conducted to study the impact of resource transfer and recovery strategies on the RReGA algorithm.

[0184] 1. Comparison with GUROBI

[0185] This section presents the experimental results of the GUROBI solver and the RReGA algorithm under different instance sizes, as shown in Table 3. The maximum solving time limit for GUROBI is 3600 seconds. The average solving time of GUROBI and RReGA is represented by t G and t R , respectively. The number of feasible solutions found by GUROBI in each group of 20 experiments is represented by #fea, while #opt represents the number of optimal solutions. The gap is defined as follows:

[0186]

[0187] Table 3

[0188]

[0189]

[0190] In 20 experiments, the average gap in solutions between the two methods over the pulsation period is represented by GAP C . GAP R refers to the average gap in resource consumption between the two methods within the same pulsation period. If the pulsation periods are different, the gap in the pulsation period is recorded, otherwise the gap in resource consumption is recorded. GAP O represents the average gap in the optimization objective value.

[0191] Efficiency is one of the most significant advantages of the RReGA algorithm, especially as the instance size increases. In the case of 30 tasks, the average computation time of the RReGA algorithm is less than 300 seconds, while the GUROBI algorithm is difficult to provide a feasible solution within an hour. In all other instances, the proposed algorithm is half the computation time of GUROBI. In addition, in instances of the same size, the computation time is longer when resource allocation is insufficient than when resource allocation is sufficient. Although the underlying reasons are different, the results of both methods exhibit this pattern. For RReGA, the difference comes from the parameter T rtchanges. When resources are abundant, multiple scheduling schemes can achieve the optimal pulsation period, thus enabling more flexible resource allocation. The optimization mainly occurs in the subsequent resource recovery phase, thus requiring a smaller T rt Conversely, when resources are scarce, more accurate and efficient resource allocation is required to achieve the optimal pulsation period, thus resulting in higher computational cost for the resource transfer phase. GUROBI employs branch-and-bound and various heuristics to solve the integer linear programming problem (1), and the introduction of additional constraints leads to an increase in the number of branches when resources are limited. This complexity makes it more difficult to find a feasible solution using heuristics, thus increasing the computation 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 one hour. For instances with more than 20 tasks, GUROBI becomes unsuitable. 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 proposed algorithm provides a better solution than the GUROBI algorithm, with an average error of negative value. The values of GAP C and GAP R reveal the factors that contribute to the gap, which vary with resource availability. When resources are abundant, 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. Conversely, when resources are scarce, the gap between them is more related to the pulsation period. These findings confirm the previous discussion on different resource inputs for the same instance: abundant resources facilitate the achievement of the optimal pulsation period, but may result in resource waste; while scarce resources lead to full utilization, but it is difficult to achieve optimal allocation, resulting in a deviation in the pulsation period.

[0193] Figure 7 The comparison of the objective values of the two methods in different instances is shown. The top and bottom lines of each box plot represent the upper and lower quartiles of the objective function values, respectively. The central line in each box represents the median of the objective values. The whiskers extending from the box represent the maximum and minimum values. Outliers are marked with "o" and excluded from the statistical summary. Green dots represent the objective values corresponding to the feasible solutions of large-scale instances that GUROBI can give 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 more superior optimization results.

[0194] 2. Ablation experiments of RReGA

[0195] To further investigate the impact of resource transfer and recovery strategies on the performance of RReGA, this embodiment conducted an ablation experiment on the same 240 instances. This embodiment evaluated the performance of RReGA with the removal of the resource transfer and recovery components. The algorithm that only retains the resource transfer component is named RTGA. RTGA transfers resources among workstations to optimize the pulse period and balance the time span, while ignoring the recovery of redundant resources. The algorithm that removes the resource transfer strategy is named RRGA. RRGA performs span optimization and redundant resource recycling at each workstation after obtaining the initial resource allocation from the resource-span mapping. When no superior offspring is found to reduce the makepan of the workstation for 20 consecutive generations, RRGA transitions from the pulse period optimization phase to the resource recycling phase. RRGA and RTGA retain other parameters of the RReGA algorithm.

[0196] Table 4 lists the results of various indicators of the three methods in 12 groups of experiments. CPU time represents the average duration of 20 experiments in each group. RPD C and RPD R represent the average relative deviation percentage of the two objectives (i.e., pulse period and resource consumption), which are defined as shown below.

[0197]

[0198] RTGA with resource transfer strategy is exactly the same as the period optimization result of RReGA, so only RPD R is compared. The main difference between RRGA and RReGA is resource transfer, which leads to significant differences in pulse period optimization results. Therefore, RPD C can be used as an evaluation indicator.

[0199] Table 4

[0200]

[0201]

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

[0203] 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 task numbers of 10, 20, and 30, the pulsation cycle optimization effect of RReGA is worse than that of RRGA. This indicates that when the initial resource allocation has reached the optimal, directly optimizing the time span of each workstation can obtain a better pulsation cycle, 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. The resources transferred between workstations may converge back to the initial optimal allocation, but optimizing the suboptimal allocation will produce a large cost compared to the RRGA algorithm without transfer, resulting in a poor optimization effect. Figure 8 The box plot comparison of the two algorithms on the pulsation cycle shows that RReGA is superior to RRGA in all instance sizes, and the difference is more pronounced when resources are insufficient. This highlights the contribution of the resource transfer strategy to effective resource allocation and optimization of the pulsation cycle.

[0204] The resource recycling strategy aims to avoid the input of excess resources, especially when the total resources are excessive. The results in Table 4 and Figure 9 (a) confirm this goal. In the case of 20 and 30 tasks and sufficient resources, canceling resource recycling leads to more than 30% of resource waste. When the total amount of worker resources is insufficient, as shown in Figure 9 (b), the algorithm without the resource recycling module performs similarly to 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 recycling strategy to reducing resource consumption.

[0205] II. An Unbalanced Instance

[0206] In actual production, it is challenging to ensure that each workstation is allocated an equal number of assembly tasks. This imbalance often leads to idle time for some workstations, causing resource waste and project delays. Evaluating whether the completion time of all workstations in an unbalanced instance can be balanced through optimized resource allocation is crucial for assessing the robustness and adaptability of the algorithm. This embodiment generates an unbalanced assembly task instance with workstation task numbers of {20, 20, 25, 25, 30}, a total number of workers of 120 and 50, a population size of 40, and other parameters remaining the same.

[0207] Figure 10The entire resource reallocation process of RReGA is described. The left side of the black dashed line represents the resource transfer process, and the right side represents the resource recovery process. In the resource transfer process, the resource input of the bottleneck workstation increases, which is more obvious in the case of sufficient resources. This indicates that other non-bottleneck workstations continuously transfer resources to the bottleneck workstation to optimize the pulse period C, which depends on the working time of the bottleneck workstation. In the resource recovery phase, the workstation generally has no excess resources to recover due to insufficient resources. However, the bottleneck workstation recovers a large amount of resources in the case of sufficient resources. Figure 11 The update process of the makespan of each workstation is shown. According to Figure 11 (a), the update process of the completion time of each workstation shows that the yellow dotted line (the completion time of the bottleneck workstation) no longer changes after 30 iterations, while other workstations continue to transfer resources to it (as shown in Figure 10 (a)), resulting in the accumulation of excess resources in workstation w3. At the end of the resource transfer phase, the optimal resource allocation derived from the updated resource-work period mapping allows the bottleneck workstation w3 to immediately unload a large amount of redundant resources.

[0208] As shown in Figure 11 (b), in the second half of the resource transfer iteration, the workstations experience more drastic changes in time span under the condition of insufficient resources. This occurs because as the resource input decreases, the working time increases sharply according to the resource-work period mapping. Under the two resource input conditions, the final survival time of the workstations is {26, 27, 27, 27, 27} and {53, 50, 53, 53, 52}, respectively. This indicates that in the unbalanced assembly task instance, the RReGA algorithm can maintain a high balance of workstation time span while pursuing the optimal pulse period C.

[0209] Figure 12 The update process of the primary objective is illustrated. The blue curve assumes sufficient resource input, with a shorter update period (T rt = 5, and each point represents one resource transfer update), and fewer updates before converging to the optimal value. The yellow curve, due to insufficient resource input, has a longer update period (T rt = 10), showing more updates before converging to the optimal value. This is because under the condition of sufficient resources, multiple different encoding schemes can achieve the same effective period, while under the condition of insufficient resources, this is not the case. This means that under the condition of the same size of encoding solution space, when resources are sufficient, there are more feasible encoding solutions that can achieve the best effective period. Therefore, this results in a significant difference in search efficiency between the cases of sufficient resources and insufficient resources.

[0210] This paper investigates the dictionary order bi-objective optimization problem of resource allocation and task scheduling in aircraft pulsating assembly lines, aiming to balance the workstation completion time while minimizing the pulsating cycle and resource consumption. A dictionary order bi-objective optimization integer programming model is established, and an efficient RReGA method is proposed to achieve these goals. The core mechanism of this method is to adjust and optimize resource allocation through embedded resource reallocation. Resource reallocation includes two stages: resource transfer and resource recycling. The first stage distinguishes between bottleneck workstations and non-bottleneck workstations according to the current best pulsating cycle, thereby optimizing the pulsating cycle and following a novel resource transfer principle to help bottleneck workstations. The second stage optimizes resource input by recycling redundant resources from workstations, ensuring that each workstation meets the pulsating cycle constraint with minimal resource consumption. In cases of overall worker surplus, the resource recycling strategy is particularly effective. In addition, an updatable resource-duration mapping is developed based on hybrid scheduling rules to provide initial allocation, guide resource transfer, and calculate the best resource allocation and pulsating cycle.

[0211] Extensive experimental results verify the effectiveness and performance of each algorithmic 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 pulsating cycle and resource input. To fully demonstrate the effectiveness and differences of the resource transfer and resource recycling strategies, comprehensive ablation experiments are conducted. In the unbalanced case, the experimental results show that the time span of each workstation is highly consistent, indicating the optimization ability of the RReGA method for unbalanced instances.

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

[0213] The above detailed the preferred embodiments of the invention. It should be understood that those skilled in the art can make many modifications and changes without creative labor based on the concept of the invention. Therefore, any technical solution obtained by logical analysis, reasoning, or limited experiments by those skilled in the art based on the concept of the invention and the prior art shall be within the protection scope determined by the claims.

Claims

1. A method for joint optimization of production scheduling and worker allocation for aircraft pulsating assembly lines, characterized in that, The method comprises the following steps: Step 1) obtaining assembly task data and resource data of an aircraft pulsating assembly line, the resource being a worker; Step 2) constructing a dictionary order double-target optimization integer programming model to minimize the pulsating cycle and total resource consumption as the objective function; The modeling assumptions of the dictionary order double-target optimization integer programming model comprise: Each assembly task corresponds to a non-preemptive duration; After the assembly task starts, the execution mode cannot be changed; Within the assembly cycle, resources cannot be transferred between workstations; Among them, consider a k The assembly line workshop of workstations is denoted as ,in, k is the number of workstations, each workstation Process a set of assembly tasks using Indicates that, Indicates a workstation The number of tasks to be processed, the assembly tasks at each workstation is predetermined, Represents an assembly task set No. j Tasks; Task scheduling follows priority constraints, i.e. each task can only start after all its predecessor tasks are completed; each assembly task Corresponding to multiple execution modes, each mode Corresponding to a specific worker requirement and assembly time ,in, Indicates assembly tasks All assembly mode sets of Indicates assembly tasks In mode m Assembly time, Indicates assembly tasks In mode m the number of assembly workers; The optimization objective of the dictionary order double-target optimization integer programming model is expressed as: minimize The constraint conditions comprise: wherein C denotes a pulsation period, denotes the number of resources assigned to the workstation , n denotes the total number of resources, denotes the start time of the task , denotes a 0-1 variable which decides whether the assembly pattern of the task is m, denotes the task being assembled at time , denotes all the preceding tasks of the task , denotes the set of integers; Step 3) solving the dictionary order double-target optimization integer programming model to realize joint optimization of worker allocation and multi-mode assembly task in the aircraft pulsating assembly line under the resource exclusive policy: Step 31) estimating the resource-duration mapping of each workstation to determine the initial resource allocation according to the estimated mapping; Step 32) searching for a scheduling scheme based on the determined initial resource allocation; Specifically, 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 the workstations are independent of each other after the worker resource allocation is completed, and a genetic algorithm is executed for each workstation to determine the scheduling scheme, which comprises the following steps: Step 321) encoding and decoding: Each individual's chromosome consists of two parts: wherein, represents a list of tasks, respecting priority constraints, represents a position of a task index, represents a list of task execution modes, respecting resource constraints, wherein each element specifies the number of workers and processing time required for the corresponding task, represents a workstation number of tasks to be processed; Step 322) population initialization and parent selection: Generate an initial population of size N where chromosomes are randomly initialized: for the task list , only after all previous tasks have been scheduled, a task is selected; for the mode list , the feasible modes for the task are determined based on available resources, and a mode is randomly selected; Initial population Each individual in the initial population is a parent, and offspring are produced by randomly dividing the parents into two equal sets and sequentially drawing a pair of parents from each set without replacement to produce N / 2 pairs of parents, each pair producing two offspring. Step 323) crossover and mutation: The crossover operator is applied with a probability to the task list to generate a sub-task list, and a corresponding mode list Crossover is performed in the same way; The mutation operator is applied to the task list and the schema list with a probability of where a mutation in the task list is performed by selecting a task and changing the position of the gene while adjusting the position of the corresponding schema gene and a mutation in the schema list is performed by selecting a task and replacing its original schema with a randomly selected feasible schema ;​ Step 324) fitness calculation and population update: According to individual fitness Preservation and culling of offspring requires evaluation of individuals prior to culling: wherein, denotes the completion time solution, () denotes a function that decodes the individual chromosomes of a work station into completion times, denotes the number of resources allocated to a work station ; The offspring are ranked according to fitness, the best 0.25N individuals are kept from the total of 2N individuals of the parent and offspring, the remaining 1.75N individuals are divided into two sub-populations in a ratio of 2:1, and 0.5N and 0.25N individuals are selected from each of the two sub-populations to form the next population Steps 323) and 324) are repeated to update the population. Step 33) based on the scheduling scheme determined in step 32), a mapping update strategy is used to refine the mapping; In particular: for a given set of assembly tasks and resources when a better completion time solution is obtained during the execution of the genetic algorithm the mapping is refined by a non-decreasing mapping of assembly progress and resource investment : Step 34) after a predetermined evolution period, resource reallocation is performed to adjust and optimize the resource allocation, the resource reallocation comprising a resource transfer strategy and a resource recovery strategy, wherein the resource transfer strategy increases the resources of the bottleneck workstations by reallocating the excess resources of the resource-excessive workstations to optimize the pulsating cycle, and the resource recovery strategy recovers the remaining resources from each workstation after the pulsating cycle optimization to optimize the total resource consumption.

2. The method of claim 1, wherein, The estimation of the resource-duration mapping of each workstation specifically comprises the following steps: determining a resource interval, i.e. the minimum resource input and the maximum resource input of a workstation wherein, min represents the minimum resource required for each workstation to complete the assembly task, which corresponds to the maximum resource required for a single task of the set of assembly tasks, performed in a mode with the least resource requirement, wherein, denotes the set of all assembly patterns of the assembly task , denotes the set of all assembly patterns of the assembly task , m denotes the number of assembly workers for the pattern , denotes the number of tasks to be processed at the workstation For a given set of assembly tasks , the maximum resource demand depends on the topology and assembly pattern, when the resource input exceeds the maximum resource demand , the duration will not be shortened, to represent all subsets of the set of assembly tasks , wherein is the total number of subsets, each subset contains assembly tasks that can be executed in parallel, assuming all tasks in are executed in the mode with the most required resources, the maximum of the resource demand of all subsets is : To ensure that all workstations can complete their tasks, it is required that , and the number of workers assigned to the workstations is not less than , where denotes the number of resources assigned to the workstations , n denotes the total number of resources, and the maximum number of resources assigned to the workstations is given by Then, the calculation method of the maximum number of worker allocation of the workstation is as follows: Based on the determined resource interval, hybrid scheduling rules are established considering task selection and mode selection, a task and mode list is generated for the assembly task of each workstation based on the hybrid scheduling rules, a scheduling scheme is decoded by a serial scheduling generation method to determine the minimum value of the duration, and a resource-duration mapping is established, wherein the scheduling rules of the task selection comprise shortest assembly time priority, longest assembly time priority and most number of tasks after priority, and the scheduling rules of the mode selection comprise shortest assembly time mode priority and maximum resource utilization mode priority.

3. The method of claim 2, wherein, The determination of the initial resource allocation according to the estimated mapping is specifically as follows: According to the estimated mapping, the resource allocation problem is expressed as follows: minimize Among them, the function To map resource inputs to assembly task sets The mapping function of the production cycle, Indicates a workstation The working time, C represents the pulsation cycle; Solving the resource allocation problem obtains the initial resource allocation.

4. The method of claim 3, wherein, 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 points.

5. The method of claim 1, wherein, The resource transfer strategy is specifically: According to the type and resource buffer state of each workstation, a corresponding resource transfer scheme is determined for each workstation, wherein the workstations include bottleneck workstations and non-bottleneck workstations, if the completion time thereof is greater than the pulse period, the workstation is defined as a bottleneck workstation, otherwise, the workstation is defined as a non-bottleneck workstation; a resource pool is defined to store the resources provided by the non-bottleneck workstations, and a resource buffer is defined as a temporary storage point of the resources owned by each workstation; When the workstation is a non-bottleneck workstation and the resource buffer thereof is empty, the workstation stores part of the resources in the resource buffer; When the workstation is a non-bottleneck workstation and the resource buffer thereof is not empty, all the resources in the resource buffer are transferred to the resource pool, and part of the resources discharged in this round are stored in the resource buffer; When the workstation is a bottleneck workstation and the resource buffer thereof is empty, all the resources in the resource pool are allocated to the workstation, if there are multiple such workstations, all the resources are allocated to the workstation with the smallest index; When the workstation is a bottleneck workstation and the resource buffer thereof is not empty, all the resources in the resource buffer are taken out and evenly allocated to all the workstations.

6. The method of claim 1, wherein, The resource recovery strategy is specifically: After determining the optimal 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 period, thereafter, the workstations continue to execute the genetic algorithm to find a better scheduling scheme to reduce the completion time not exceeding the pulse period; The above steps are iteratively executed, if a better scheduling scheme cannot be found, the resource input and scheduling scheme of the last iteration are retained, and the iteration is repeated until the iteration stopping condition is met, and the optimal solution of the problem is obtained.

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