## 3 Results of Simulation Experiments and Discussion
### Feasibility Analysis with Large-Scale Problems  
Finally, we examined the feasibility of our proposed approach by comparing the total time and energy consumption of three methods with large-scale problems (200-1000 tasks), as shown in Figure 9. Since the NBS-ISPS method cannot successfully generate solutions for large-scale problems (even with an allowed runtime of more than 1000 s), it is not included in this figure. Here, the total time and energy consumption (\(TC\)) are normalized following Equation (16):  
\[\text{normalized}\ TC=\frac{c-c_{min}}{c_{max}-c_{min}} \tag{16}\]  
where \(c\) is the time and energy consumed for transportation and processing, and \(c_{max}\) and \(c_{min}\) are the maximum and minimum sum of time or energy consumed for transporting and processing. Figure 9 shows that the total time and energy consumption grow with the number of tasks, and the RSS method performs better than the LSS method. Both RSS and LSS outperform the Baseline method for a varying number of robots and tasks. By comparing Figure 9a vs. Figures 9b and 9c vs. Figure 9d, we can see that when the Baseline method is adopted, increasing the number of robots will not significantly reduce the total consumption. For example, when using the Baseline method and \(n=1000\), the normalized \(TC\) values in Figure 9a,b are 0.923 and 0.911, respectively. However, the total consumption decreases markedly if the LSS or RSS method is used. When using LSS with \(n=1000\), the normalized \(TC\) values in Figure 9a,b are 0.819 and 0.761, respectively. One possible explanation is that when the processing time and energy consumption of  
Figure 8: The relationship between runtime and number of tasks (\(n\)) for our methods and the NBS-ISPS method with different numbers of robots (\(r\)): (**a,c**) \(r=5\); (**b,d**) \(r=10\). The time limit for the NBS-ISPS method is 100 s.  
machining centers are not considered (e.g., in warehousing and logistics scenarios), more robots will support the faster completion of tasks. However, when the processing time and energy consumption of machining centers are considered (e.g., in a smart factory environment), even if the number of robots increases and the tasks can be transported to machining centers faster, it still takes a certain amount of time for machining centers to process these tasks. This leads to the queuing of tasks and waiting of robots, and the total time consumption will not be significantly reduced. However, the LSS or RSS method can reorganize the processing sequence of tasks in real time, which can alleviate the queuing issue and reduce the waiting time for robots. Thus, the decrease in \(TC\) will be more obvious when the LSS or RSS method is adopted. In addition, by comparing Figure 9a vs. Figures 9c and 9b vs. Figure 9d, we can observe the influence of different weights of time and energy on the final \(TC\). For example, if using RSS with \(r=5,w_{t}=0.6,w_{e}=0.4,n=1000\), the normalized \(TC=0.732\), while when \(r=5,w_{t}=0.4,w_{e}=0.6,n=1000\), the normalized \(TC=0.881\). This result indicates that the advantage of RSS method is more pronounced when the weight of time (\(w_{t}\)) is larger. A possible explanation is that the search of the machining sequence is performed only after each determination of which machining center is assigned, i.e., the energy consumption does not vary with the insertion position since we only consider the time consumed by the machining center without considering the energy consumption during its idle time. Figure 10 shows the number of searching iterations and runtime of the proposed methods versus the number of tasks with large-scale problems. Figure 10a indicates that the number of search iterations increases along with the increasing number of tasks. The number of search iterations using the RSS method is greater than that using the LSS method, which can lead to slightly longer computation time, as shown in Figure 10b.