Multiobjective optimization allocation of multi‐skilled workers considering the skill heterogeneity and time‐varying effects in unit brake production lines

The optimal allocation of multi‐skilled workers in labor‐intensive industries can improve production capacity and reduce production costs. In actual production, the efficiency of workers will change with time due to their proficiency, fatigue, and other effects. In this article, we attempt to solve the problem of multi‐skilled workers allocation in unit brake production lines considering the heterogeneity of skills and time‐varying effects. A nonlinear mixed‐integer programming model is established, which fully considers the impact of worker efficiency due to proficiency, fatigue, and multi‐task rest recovery. The product production cycle and worker cost are the two objectives of the optimization solution. An enhanced NSGA‐II algorithm that combines the improved NSGA‐II algorithm and the variable neighborhood search (VNS) algorithm is used to solve the multiobjective optimization problem. Finally, the weighted ideal point method is used to obtain the Pareto optimal solution. The application case of a unit brake production is considered to evaluate the proposed model. The results indicate that the time cost and salary cost of workers are reduced by 8.03% and 18.91% compared with the original scheduling. The scheduling model considering learning, fatigue and recovery factors is more suitable for the actual production situation, ensuring the completion time and reducing the labor cost.

7][8] These factors lead to temporal fluctuations in workers' productivity, significantly impacting overall production.Therefore, when scheduling workers, it is crucial to consider their efficiency in specific processes and the effects of time variation.
Workshop scheduling and personnel scheduling are intricate NP-hard problems. 91][12] However, due to the inherent complexity and distinct characteristics of scheduling optimization problems, there is no universal method suitable for solving various types of production scheduling problems.The existing solution methods can be categorized into three main types: 1. Methods based on operations research, such as the branch and bound method and enumeration method, 13,14 are widely utilized as representative approaches to address various scheduling problems.The branch-and-bound method can theoretically find the global optimal solution of the original problem by continuously tracing the upper bound and obtaining the lower bound through calculating the linear relaxation problem.Experimental results demonstrate that the branch-and-bound method is effective in solving small-scale problems.However, when dealing with large-scale problems, it may not be able to find the exact optimal solution within a feasible time frame. 15,16The reason for the inability to find the exact optimal solution in a feasible time for large-scale problems lies in the fact that the integer programming problem itself is NP-hard, and only exponential time-complexity optimization algorithms exist.2. Heuristics: Heuristics method is a general term for a class of methods that use prior knowledge and manual experience to solve the problem. 17,18These methods pre-design specific rules based on the problem's characteristics and quickly generate a feasible scheduling scheme using partial information from the production line.This kind of method usually originates from specific practical problems, allowing it to fully consider the characteristic constraints of complex industrial processes.It is easy to implement and capable of obtaining effective scheduling schemes.However, the effectiveness of heuristic rules is closely related to the choice of optimization targets and the specific problem's characteristics, resulting in poor generalizability.Moreover, heuristic rules typically lack a global search mechanism.Since the design of heuristic methods relies on the structure, characteristics, and experiential knowledge of specific problems, the solution quality may vary significantly for different problems.In practice, most heuristic algorithms cannot guarantee the quality of the solution, 19 and the same algorithm may not ensure consistent solution quality for different instances of the same problem.3. Metaheuristics: These methods generally involve randomly generating one or more vectors to represent potential solutions for the optimization problem.Through an iteration mechanism predefined by the algorithm, they evaluate the corresponding objective function values, generate new solutions based on the initial ones, and retain the better solutions by comparing the old and new solutions.This process improves with each iteration, aiming to find an approximate optimal solution to the problem.Common meta-heuristic methods include the genetic algorithm, 20 differential evolution algorithm, 21 variable neighborhood search (VNS) algorithm, 22 Tabu search algorithm, 23 simulated annealing algorithm, 24 artificial fish swarm algorithm, 25 etc.It is crucial to recognize that optimizing a single objective may not adequately fulfill the requirements of practical shop floor production.Currently, the most commonly employed computational intelligence methods in the investigation of multi-objective production scheduling are NSGA-II or its combination with some multi-objective algorithm. 26,27Lian et al. 4 solved a multi-skilled worker assignment problem using the NSGA-II in the context of seru production systems, in which differences in workers' skill sets and proficiency levels are considered.Tirkolaee et al. 28 developed two metaheuristics of NSGA II and multi-objective simulated annealing algorithm to solve the multi-objective multi-mode resource-constrained project scheduling problem with payment planning.Goli et al. 29 proposed a hybrid GA and a whale optimization algorithm to solve the cell formation problems including the scheduling of parts within cells in a cellular manufacturing system where several automated guided vehicles are in charge of transferring the exceptional parts.He et al. 30 proposed a multi-objective optimization framework based on the fitness evaluation mechanism.They also utilized a hybrid multi-objective genetic algorithm integrated into the framework to enhance the solving performance of the energy-efficient job-shop scheduling problem.The extensive literature reviewed above clearly indicates that meta-heuristic algorithms are well-suited for addressing complex industrial scheduling problems.Particularly, combining various intelligent methods can lead to complementary advantages, enhancing the search capability and efficiently achieving comprehensive optimization of scheduling problems.
In the problem of worker scheduling, the efficiency of workers is affected by the fatigue effect. 31Additionally, multi-skilled workers may have varying proficiency levels due to differences in training time and cognitive ability.Considering these factors in the scheduling treatments aligns better with actual production needs.As working hours increase, workers' efficiency gradually decreases, and over extended periods of work, their efficiency significantly deviates from the initial levels.Factors such as poor working environment, disturbed biological clock, and long operating hours are the leading causes of fatigue. 32Myszewski 33 and Michalos et al. 34 proposed utility function models to study the effects of fatigue on error rates, indicating that fatigue leads to an increase in error rates.Dong et al. 35 examined the influence of employee fatigue on service efficiency and developed a fitting model to describe the impact of fatigue factors on work efficiency.The model aims to minimize employee fatigue during the processing process and improve operational efficiency.In real production processes, workers typically take breaks after working for extended periods.During the rest process, workers' work efficiency can be significantly improved.Gentzler et al. 36 assumed that workers' efficiency is fully recovered during the rest interval and developed a mathematical model with multiple rest stages to maximize employees' output.Jaber and Neumann 37 quantified the relationship between physical fatigue and production output and proposed a "learning-forget-fatigue-recovery" model.In this model, the cumulative fatigue value after working time t is calculated with an initial fatigue value of 0. The residual fatigue of the worker after the rest will add to the fatigue accumulation of the next workpiece.Glock et al. 7 proposed an accumulation function of fatigue as an exponentially increasing function of time.Liu et al. 38 developed an agent-based simulation system to handle uncertainties in the worker fatigue model.They utilized a novel simulation-based optimization (SBO) framework that combines genetic algorithm (GA) and reinforcement learning (RL) to address the hybrid flow shop scheduling problem.In conclusion, studies on the fatigue-recovery effect aim to enhance employees' working efficiency and reduce production accidents.However, limited research has been conducted on the impact of fatigue on the dynamic efficiency of workers in a production line.
This article focuses on studying the scheduling problem of real-time changes in the working efficiency of multi-skilled workers, which arises from the production of unit brakes in locomotive components and is commonly encountered in worker scheduling for industrial production lines.The problem involves coordinating multiple tasks that require various skills, each worker possessing different skill levels.As workers utilize their skills, their proficiency improves with time, but fatigue also decreases their efficiency.Since each task requires different skills with varying durations, workers completing tasks can take specific rest periods, during which their skill levels increase.The main contributions of this article are outlined as follows: 1.The article employs the production of unit brakes as an application example and formulates a multi-objective nonlinear mixed integer programming model.This model incorporates worker skill heterogeneity and accounts for the impact of the learning-fatigue-recovery effect on work efficiency, addressing the scheduling optimization problem.2. To perform multi-objective optimization of completion time and worker salary cost, an improved NSGA-II algorithm combined with VNS is utilized.The algorithm is compared with other optimization methods regarding target optimization.3. The article provides management insights for enterprise managers based on the multi-skill and multi-project scheduling scenario of employees, taking into account the dynamic changes in employees' skill levels.

Problem description
This article focuses on addressing the multi-skilled worker allocation problem in the context of a unit brake production line.Unit brakes play a crucial role in locomotives as they contribute to reducing vehicle weight, ensuring even power distribution, and enhancing dynamic bogie performance.The production process of unit brakes involves two major steps: sub-assembly production and final assembly.This study presents a detailed description of the worker allocation problem for unit brake production and formulates a model to optimize the production cycle and labor cost by determining the optimal arrangement scheme for multi-skilled workers.The unit brake production project consists of K tasks that require S skills to complete.There are W multi-skilled workers available for scheduling in this project.Each task demands different skills, and the proficiency levels of multi-skilled workers vary for each skill.Let i, j, and k represent employees, skills, and tasks, respectively.The standard processing time is defined as the time required for a worker with proficiency level 1 to complete a specific task.Workers with proficiency levels greater than 1 can complete tasks faster than the standard processing time, while those with proficiency levels less than 1 require more time.During the specific production process, a worker's ability to perform a task improves over time, leading to a decrease in processing time.However, fatigue factors may also affect workers, reducing their efficiency.It is worth noting that the project comprises multiple tasks, and some tasks have a certain time sequence relationship.Therefore, after workers finish a task and before starting the next one, they can take a recovery period to restore their work efficiency.

Basic assumptions
The problem described above is subject to the following assumptions: 1.The number of workers remains constant throughout the entire project duration.
2. The initial skill proficiency and available time of each worker are known.
3. The standard processing times for each task are known and fixed.4. The construction period calculation does not consider time intervals or preparation time between task transitions.Therefore, the cost and time for task conversion are assumed to be zero.5.Each task cannot be interrupted and can only be assigned to one worker at a time.6.Each worker has the capacity to be assigned multiple tasks based on their individual abilities.

Notation
K is the number of tasks; S is the number of skills needed in the project; W is the number of workers; i is the index of workers t ijk represents the time when employee i uses skill j to participate in task k; t ij means the total time of worker i uses skill j in a certain working time; t sta jk is defined as the standard completion time of skill j required for task k.It represents the average level of skill j in the worker group; f ij represents the skill level of worker i on skill j, f ij = 1 means the standard level of worker i on skill j, f ij = 0 indicates that worker i does not possess skill j; f o ij indicates that worker i with the level of skill j at the beginning of the project.The value is determined based on workers' knowledge, ability and so on; f o jk is the skill level of worker i on skill j at the beginning of the project; Ft k is the finishing time of the task k; St k is the starting time of the task k; Ft k′ is the finish time of the preceding task; T k is the duration period of task k.It is also the maximum working time required by task k; E j k represents whether task k requires skill j, if yes, means the total salary of worker i in a certain working time.L ij is the learning coefficient of worker i on skill j. the formula is: where, l ij is the learning rate, a constant.F i is the fatigue coefficient of worker i. the formula is: where, ,  are the constant coefficient.R i is the recovery coefficient of worker i after a relaxing time.The formula is: where, ,  are the constant coefficient.
x ijk is the 0-1 decision making variable.x ijk = 1 represents that worker i participates in task k using skill j, otherwise, x ijk = 0.
y ijkt is the 0-1 auxiliary variable.y ijkt = 1 represent worker i uses skill j to participate in task k in time period t.

Mathematical model
Below is the established multi-objective scheduling model aimed at optimizing the project production cycle and labor cost.
The nonlinear mixed-integer programming model takes into account various factors such as worker skill heterogeneity, skill proficiency, and the impact of fatigue and recovery over time. min The objective functions ( 4) and ( 5) serve as the optimization goals in the mathematical model.Equation ( 4) aims to minimize the production period of the project, where the latest completion time represents the final project duration.Equation ( 5) aims to minimize the cost of workers for the project, with the total cost being the sum of workers' salaries for various tasks.Constraints (6) represent the time worker i spends using skill j to participate in task k.The more skilled the worker, the shorter the time taken to complete the task.Constraint (7) defines the duration of each task, which is the maximum time required for the skills involved in the task.Constraints (8) indicate that the completion time of task k is the sum of the start time of task k and the duration of the task.Constraints (9) determine that the start time of task k is the completion time of its primary tasks, while constraints (10) state that the completion time of the preceding task is the maximum time spent on the primary task.Constraint (11) comes into play when a task has no primary task, resulting in the completion time of the preceding task being empty.Constraint (12) represents the proficiency of worker i in skill j at the beginning of task k, considering the effects of learning, fatigue, and recovery.Constraint (13) ensures that each skill of each task can only be performed by one worker, while constraint (14) enforces the rule that a worker can participate in only one task at the same time.Constraint (15) represents the relationship between the decision-making variable x ijk and the auxiliary variable y ijk .Constraints ( 16) and ( 17) impose limits on the scope of the decision-making variable.

OPTIMIZATION ALGORITHM BASED ON AN ENHANCED NSGA-II ALGORITHM
The above content proposes a multi-objective optimization production scheduling model that considers the learning-fatigue-recovery factors of employees.The primary optimization goal is to minimize both production completion time and production costs.NSGA-II algorithm is proposed by Deb on the basis of a non-dominated sorting genetic algorithm, 39 which is satisfactory for solving multi-objective optimization problems with large search space and high complexity.While NSGA-II demonstrates strong global search ability in solving multi-objective optimization problems, its ability to explore individual solution neighborhoods is relatively weak.To address this limitation, this study introduces VNS in combination with NSGA-II to enhance the optimization capability of NSGA-II.VNS is an improved local search algorithm. 40It utilizes neighborhood structures formed by different actions to perform alternating searches, achieving a desirable balance between centrality and dispersity.The key to the VNS algorithm lies in its utilization of multiple different neighborhoods for exploration.The concept behind VNS-NSGA-II is to first utilize the NSGA-II algorithm for global search, and then apply the VNS algorithm during the evolution process to search for the optimal solution and achieve the best possible outcome.This property allows VNS to potentially discover better solutions that might be missed by NSGA-II's limited neighborhood search capabilities.By combining VNS with NSGA-II, the resulting VNS-NSGA-II approach aims to improve the overall optimization ability and achieve better solutions in multi-objective optimization problems.The VNS principle is straightforward yet effective, making it suitable for a variety of optimization problems.Its algorithmic structure remains independent of the specific problem, allowing for seamless integration into other algorithms.As a result, VNS has gained widespread usage in practical engineering scenarios.

Encoding
A double-layer coding method is employed to address the characteristics of the multi-skill staffing problem and to assign suitable personnel for each skill in the project.In the first layer of coding, both workers and skills are encoded using the natural number coding method.Specifically, W workers and S skills are coded accordingly.Moving on to the second layer, tasks are represented using a string of genes, with each gene representing a specific skill required for the corresponding task.The number of genes in the chromosome is equal to the total number of skills needed by all the tasks.Each chromosome in the second layer represents a potential solution to the model, where the combination of genes indicates the assignment of workers' skills to different tasks.An illustrative example is provided in Figure 1.Suppose the project involves four tasks, with task 1 requiring skills 1, 2, and 3.In the figure, "Layer 2" indicates that workers 2, 1, and 3 possess the skills required to complete task 1.Similar assignments are shown for the other tasks, and the overall encoding of the four tasks represents a solution to the model.The connections between workers and tasks are also depicted in Figure 1.

F I G U R E 1
Encoding method and links between workers and tasks.

Initialization
In the NSGA-II algorithm, the initialization strategy significantly influences the quality and convergence speed of the solutions.Given the algorithm's complexity and the limitation of the population size, we set the population size to 50 during the initialization stage.To ensure population diversity, a combination of random initialization and hybrid initialization is employed.This approach aims to enhance the convergence speed while also improving the global search capability.By using both random and hybrid initialization methods, the algorithm can efficiently explore the solution space and increase the likelihood of finding optimal solutions to the multi-objective optimization problem.

Operator selection
The objective of this work is to minimize the task completion time and labor cost.The fitness function of the algorithm is as follows: where, f 1 = T and f 2 = C, their minimization objective equations are evident in Equations ( 4) and ( 5), respectively.These two fitness indexes need to be balanced.A lower target value of task completion time corresponds to a higher individual fitness value, while a lower target value of cost results in a higher individual adaptation value.Based on the calculated fitness values, non-dominant ranking was performed for individuals, and the crowding degree of individuals at the same dominance level was computed.The chromosome with the highest fitness in the population was directly copied to the new population, and then other individuals were selected to form the new parent population using a dynamic crowding algorithm combined with the elite solution retention strategy.This approach aims to maintain diversity in the population while preserving promising solutions, which helps to explore the search space effectively and improve the quality of the obtained Pareto front.

Crossover and mutation
This article utilizes a binary league selection algorithm to prioritize the selection of individuals with lower ranks during the fast-non-dominated ranking process.In cases where the rank is the same, individuals with larger crowding distance are favored.For the crossover operation, a two-point crossover method is employed, while a single-point mutation mode is used for the mutation operation.It is important to note that the crossover and mutation operations are only applied to the first layer of chromosomes, as the alleles of the second layer are determined based on information from the first layer.The improved elite retention strategy is illustrated in Figure 2.With the parameter  being set, N ×  individuals are directly added to the new population, where N represents the total population size.The remaining N × (1 − ) individuals are randomly selected from the suboptimal front.This approach ensures that a certain percentage of the top-performing individuals, determined by the value of , are retained in the new population.Meanwhile, it also introduces diversity by randomly selecting a portion of individuals from the suboptimal front, which helps prevent premature convergence and allows for better exploration of the solution space.Furthermore, to prevent the solutions developed during population evolution from being trapped in local optima, a VNS algorithm is incorporated.The VNS algorithm generates feasible neighborhood solutions by altering specific gene loci of the current solution chromosome.This process aids in exploring alternative solutions and enhances the global search capability of the algorithm.In combination with various operators of the improved NSGA-II algorithm, two neighborhood search structures, namely the Inverse operator and the Swap operator, are introduced in the first layer of chromosome segments.The operators are described as follows: (1) Inverse operator: Two positions are randomly selected in the first layer of the chromosome, and the gene order between these positions is reversed.(2) Swap operator: Two positions are randomly selected in the first layer of the chromosome, and the genes at these positions are exchanged.

Optimal solution decision
The non-inferior solution set obtained through the fast non-dominated genetic algorithm provides a range of potential solutions for the problem.However, selecting the optimal solution requires further decision-making.In this study, two important factors, namely completion time and worker salary cost, need to be considered by a decision maker.Since the importance of these objectives may vary under different conditions, weight coefficients are introduced to aid in decision-making.The ideal point decision method is utilized to select the optimal solution from the non-inferior solution set by calculating the relative decision value.The relative decision value refers to the distance between each non-inferior solution and the positive and negative ideal points for each objective.The positive and negative ideal points are represented by the single objective optimal value and the worst value within the non-inferior solution set, respectively.The calculation formula for the relative decision value is as follows: where, D(p) means the relative decision-value of scheme p, superscript ⃛ + ⃜ and ⃛ − ⃜ mean the positive and negative ideal points, respectively, that is, the optimal and worst value of the single objective. and  are the weight coefficients of the completion time and worker salary cost,  +  = 1.
F I G U R E 2 Improved elite retention strategy.

F I G U R E 3
Calculation flow chart of the enhanced NSGA-II algorithm.

Algorithm procedure
This study utilizes an enhanced NSGA-II algorithm.The process involves applying a non-dominated sorting algorithm, a crowding algorithm, and an elite selection strategy to obtain a new population.Subsequently, the VNS algorithm is employed to construct a neighborhood structure set.The overall algorithm flowchart is illustrated in Figure 3.

Case description and parameter setting
The primary objective of this study is to address the allocation problem in the unit brake production lines, considering the presence of multi-skilled workers.The research is conducted at Qingdao City University's factory, and it takes into account various factors like learning, fatigue, and recovery that are relevant to actual production scenarios.To illustrate the problem, a specific unit brake production project consisting of nine tasks and requiring eight distinct skills is chosen as an example.These tasks exhibit a particular sequence relationship, and each task necessitates multiple skills to be completed.The time sequence relationship between tasks and their respective required skills is summarized in Table 1.
Each skill is considered independent and can be executed in parallel.Table 2 provides the minimum time required for each skill to complete each task.Currently, the factory employs a total of 30 multi-skilled workers who can be assigned and dispatched for various tasks.Table 3 presents the various types of skills possessed by each worker, along with their corresponding initial proficiency levels.It should be emphasized that the initial skill values of the workers are determined using real data collected from the factory's workforce.Notably, the standard time taken by workers with a specific skill to complete a particular task is considered as the benchmark time, and their skill coefficient is set to 1.For other workers possessing the same skill, their initial skill coefficients are calculated by comparing the time they take to complete the identical task.Table 4 provides an overview of the hourly salaries for each worker.Furthermore, the skill level of the workers is influenced by proficiency, fatigue, and recovery effects.To account for these effects, coefficients are assigned for each factor, as outlined in Table 5.These coefficients are derived from the factory's prior production experience.

TA B L E 2
Minimum time requirements for each skill in each task (unit: h).
Note: The dashed lines represent tasks that do not require a specific skill.

TA B L E 3
Worker heterogeneity and the initial skill proficiency.

Sensitivity analysis
To assess the impact of underestimation/overestimation of parameters in the NSGA-II-VNS algorithm on the objective function-specifically the completion time and the total salary of workers-and to examine the algorithm's stability, we conducted sensitivity analysis using the mentioned problems.During the analysis, one genetic algorithm parameter was replaced at a time, while keeping the other parameters unchanged.
In each case, we conducted 30 runs to identify the best implementation of the target.The analysis results are presented in Figure 4. From Figure 4A, it is evident that the minimum completion time and minimum worker's wage remain unchanged when the population size is 50 and above.Similarly, Figure 4B illustrates that the minimum completion time and minimum worker's wage no longer change when the mutation probability is greater than or equal to 0.3.Moving on to Figure 4C, we observe that the objective function values continuously optimize with increasing iteration times, indicating the algorithm's excellent convergence.After reaching 300 iterations, the objective function values tend to stabilize.Furthermore, Figure 4D demonstrates that the minimum completion time and minimum worker's wage do not change when the crossover probability is greater than or equal to 0.85.Based on the sensitivity analysis results presented in Figure 4, we can confidently conclude that the VNS-NSGA-II algorithm proposed in this article exhibits stability and delivers reliable results.

Numerical results and analysis
Different weights were assigned to each objective function to assess the practicality of the problem.Specifically, the weight values for completion time and production cost were set to 0.7 and 0.3, respectively.Figure 5 displays the Pareto front of the objective functions obtained through the improved NSGA-II algorithm.As evident from the data, completion time and salary cost of workers exhibit an inverse relationship.Decision makers can make informed decisions based on the specific requirements of the production process.Table 6 presents the top five schemes with the highest relative decision values, and scheme 1 is considered the optimal solution.The detailed worker scheduling solution is outlined in Table 7.
After normalizing each objective function, the optimal scheduling Gantt chart is generated, as depicted in Figure 6.In the context of multi-objective optimization, the organization can optimize workforce management by simultaneously considering multiple goals, such as minimizing project completion time, reducing worker wages, and maximizing the skill level of the participating employees.This approach allows for a comprehensive evaluation of trade-offs among these objectives and identifying a set of Pareto-optimal solutions that offer various combinations of objectives.For business managers, considering workforce skill levels becomes crucial in multi-objective optimization.Allocating employees with appropriate skill levels to projects can significantly impact project efficiency and quality.By matching the right skills to the tasks, the organization can minimize the learning curve and enhance overall productivity.In time-constrained projects, it is essential to optimize workforce scheduling to meet deadlines efficiently.Assigning tasks based on workers' skill levels, experience, and time availability can help ensure timely project completion.For projects with flexible timelines, the organization can use this advantage to optimize cost by choosing from a broader range of project schedules.
F I G U R E 5 Pareto front of the objective functions.Before implementing the optimal scheduling, the factory followed a principle of assigning the most skilled person to each skill.However, after adopting the optimal scheduling generated by the improved NSGA-II algorithm, a comparison of the results for each indicator is presented in Table 8.Data analysis from the table indicates that the time cost and salary cost of workers have both decreased by 8.03% and 18.91%, respectively, following the optimized scheduling.Moreover, the optimized scheduling has led to a reduction in the number of personnel involved in the scheduling process, contributing to improved efficiency and resource utilization in the unit brake production line.Regardless of the chosen decision approach, the number of workers involved in the projects and the expenditure of the worker salaries will be fewer than in non-optimized scenarios.By reducing the involvement of workers with shorter skill duration, the organization can focus on training and maintaining the skills of a select few, thereby maximizing talent utilization and enhancing project execution efficiency.

TA B L E 6
This article takes into account the important factors of fatigue and recovery effects in the mathematical model, which aligns well with the actual factory production scenario.While workers' skill proficiency increases with time when using their skills, the impact of fatigue on efficiency should not be disregarded.Additionally, since each task requires different skills, the completion time for each skill varies.Consequently, workers who complete tasks with their skills first can obtain some rest, leading to an increase in skill proficiency with additional rest time.Figure 7 displays the Pareto front of the objective function without considering the fatigue-recovery effect, along with the corresponding optimal scheduling Gantt chart.Upon considering the weight coefficients of completion time and labor cost, Table 9 presents the top three schemes with the highest relative decision values.Furthermore, Table 10 provides detailed solutions for the optimal worker schedules in the chosen scheme.A comparison between Figures 5 and 7 reveals that although the task completion time in the optimal solution set without considering the recovery factor does not change significantly, the number of workers assigned and the corresponding salary cost increase, with the salary cost showing a rise of 3.4%.This emphasizes the significance of considering the fatigue-recovery effect in worker scheduling, as it has a notable impact on labor costs and overall production efficiency.The scheduling model considering learning, fatigue and recovery factors is more suitable for the actual production situation, ensuring the completion time and reducing the labor cost.Implementing

TA B L E 9
Schemes with a high relative decision value (without considering the fatigue-recovery effect).

Scheme Completion time (h) Salary cost (RMB)
Relative decision value  appropriate recovery measures based on workforce considerations results in fewer employees participating in projects, leading to reduced expenditure on workforce wages and related costs.This cost-saving initiative can positively impact the company's financial performance and profitability.The organization should factor in employee recovery considerations while planning workloads, ensuring a balanced and supportive work environment to improve employee satisfaction and performance.

Performance comparison with NSGA-II
To demonstrate the superiority and reliability of the VNS-NSGA-II algorithm utilized in the article, we have conducted a comparison with the classical NSGA-II algorithm.Figure 8 displays a two-dimensional scatter plot of the solutions obtained by the NSGA-II and VNS-NSGA-II algorithms after running them for 10 iterations.The proximity of each point to the origin indicates the superiority level of the corresponding solution.As depicted in Figure 8, the red point is closer to the origin than the blue point, signifying that the Pareto rank achieved by the VNS-NSGA-II algorithm is higher.Furthermore, this study combines the solution sets obtained by both methods (VNS-NSGA-II and NSGA-II) to form a new solution set.Subsequently, the optimal Pareto solutions are reselected, and the proportion of optimal solutions obtained by each method is analyzed.The calculation reveals that the proportion of Pareto optimal solutions from NSGA-II is merely 2.25%.This finding further confirms that the VNS-NSGA-II algorithm yields higher Pareto grade solutions compared to the NSGA-II algorithm.

CONCLUSION
In this article, a nonlinear mixed-integer programming model was established to solve the problem of multi-skilled workers allocation in unit brake production line considering the skills heterogeneity and time-varying effects.Worker efficiency is a dynamic model with time, and the learning, fatigue and recovery effect that can affect worker efficiency were considered.The product production cycle and worker cost are the two conflicting objectives of the decision model.An enhanced NSGA-II algorithm that combines the improved NSGA-II algorithm and the VNS algorithm is used to solve the multiobjective optimization problem.The weighted ideal point method is used to help decision-makers obtain the Pareto optimal solution.The proposed model and algorithm were illustrated through a real industrial case of the unit brake production in locomotive manufacture.The results show that the model accords with the actual production situation.After the optimized scheduling, the time and salary costs are reduced by 8.03% and 18.91%.In addition, the model is compared with the model that does not consider the fatigue-recovery effect.The results show that the model can guarantee the completion time and save labor costs.Incorporating multi-objective optimization in workforce management allows the organization to strike a balance between project completion time, workforce wages, and skill utilization.By matching the right skill sets to projects, optimizing scheduling based on deadlines, and considering recovery factors for workforce well-being, the organization can achieve cost-efficiency and talent optimization.Future research can focus on the development of real-time adaptive scheduling algorithms that dynamically adjust worker assignments in response to changing conditions and unforeseen events in the production process.For instance, short-term training can be provided to employees who are not involved in the project to enhance their skills and enable them to participate in the scheduling of unfinished projects.This approach will result in a more agile and responsive production scheduling process.

4
Sensitivity analysis of different parameters on optimization algorithm.

F I G U R E 7
Pareto front of the objective function without considering the fatigue-recovery effect.

F I G U R E 8
Comparison of the solution for NSGA-II and VNS-NSGA-II.
Schemes with the high relative decision value.Gantt chart of the optimal solution.Comparison of results before and after optimized scheduling.
F I G U R E 6 TA B L E 8