## 3 A Collaborative Scheduling Framework for Human-Machine-Logistics Considering Workers' LFE
### Problem Description and Mathematical Model  
In the context of Industry 5.0, with the increasing prevalence of human-machine collaborative work scenarios, the role of humans as an indispensable part of the production system has become more prominent. The scope of resource scheduling should not only include processing and logistics equipment but also encompass workers as a critical factor. Due to the learning and forgetting effects experienced by workers during their tasks, worker efficiency exhibits dynamic variability, significantly impacting production efficiency and task allocation. Specifically, worker efficiency is not constant but gradually improves with accumulated work experience. However, once work is interrupted, efficiency begins to decline. This phenomenon is known as the learning and forgetting effect, which directly affects workers' ability to complete tasks over different time periods. Therefore, effectively quantifying and incorporating workers' learning and forgetting effects into human-machine-material collaborative scheduling is a critical issue that needs to be addressed. To address this issue, we developed a time-based mathematical model to quantify the impact of learning and forgetting on worker efficiency [7, 50], providing a more accurate description of how efficiency changes over time. \[p_{r}{{}^{\prime}}=p_{r}\Big{\{}max\Big{[}1+\Big{(}{\sum_{1}^{r-1}p_{[r-1]}}^{ r}\Big{)}^{a},\theta\Big{]}-1\Big{\}}e^{-bt}+p_{r} \tag{1}\]  
where \(p_{r}{{}^{\prime}}\) represents the actual working time, \(p_{r}\) represents the theoretical working time, \(\sum_{1}^{r-1}p_{[r-1]}{{}^{\prime}}\) represents the cumulative processing time of the worker, \(\theta\) is limiting parameter, \(0<\theta<1\), \(a\) is the learning factor, \(a<0\), \(b\) is the forgetting factor, \(0<b\), and \(t\) represents the interruption or break time between two consecutive operations. As can be seen from the model, this effect is influenced not only by work time but also by individual learning and forgetting factors. A worker's efficiency directly affects the production capacity of the equipment, which in turn impacts the collaborative operation of production-logistics units. Therefore, workshop scheduling must include workers in the scope, comprehensively considering the relationships between workers, machines, and logistics to deal with the uncertainties and complexities brought about by workers' learning and forgetting effects. This will achieve efficient collaborative scheduling. The scheduling problem can be described as follows: Given a set of jobs, each job has two time attributes--arrival time and deadline. If a job is completed after its due date, a penalty cost for lateness is generated. There is a set of AGVs (Automated Guided Vehicles) \(A=\{a_{i}|i=1,\,2,\,\dots,\,I\}\), a set of machines \(M=\{m_{j}|j=1,\,2,\,\dots,\,J\}\), and a set of workers \(H=\{h_{j}|j=1,\,2,\,\dots,\,J\}\). The objective of this method is to provide an adaptive task allocation strategy under the influence of workers' learning and forgetting effects, optimizing multiple objectives simultaneously. The assumptions for the problem are as follows:  
1. Jobs arrive randomly, and the deadlines vary. 2. Each machine can only process one job at a time, and each job can only be processed on one machine at a time. 3. Each AGV can only transport one job at a time, and the speed of AGVs is identical. 4. Jobs' arrival times and deadlines are only known upon their arrival. 5. Each machine has a buffer zone with a capacity of 10 jobs. 6. Each production unit provides loading and unloading areas for logistics units, and loading/unloading times are included in logistics time. 7. Each worker can only operate one machine at a time. 8. Each worker has a fixed location. 9. Each worker has different learning and forgetting effects, and their learning capabilities and forgetting speeds vary. 10. Workers' efficiency fluctuations conform to the learning-forgetting mathematical model.