## 5 Experimental Results
### Performance of FFSP-AGV Parameters
#### 5.4.2 The Number of AGVs  
AGV quantity determination is one of the most prioritized key issues to be considered in the production activities of AGV participation scheduling, which is also the focus of this paper. For a production system driven only by AGV scheduling, if the number of AGVs is too small, the transportation of jobs will be slow, which will lead to a lot of time and resources being wasted, and the timeliness of the production tasks will be difficult to ensure. However, the number of AGVs should not be too large because too many AGVs are not only costly but may also increase energy consumption when traveling empty and waiting for tasks, which will reduce the efficiency of the whole production system. Therefore, investigating the most appropriate number of AGVs is one of the most important ways for manufacturing companies to reduce costs and increase efficiency. For this reason, in this section, instances J6-S2-M3- and J9-S2-M3- from Table 1 are used as benchmark data to investigate the performance of AGV count in the FFSP-AGV integration scheduling by varying the number of AGVs from one to eight. As mentioned earlier, in the MILP model constructed for the FFSP-AGV, the processing machines in each production stage are UPMs, which take different amounts of time in each parallel machine to complete specific operations. To be fair, the processing machines in each processing stage are also set as identical parallel machines (IPMs), and the processing time is averaged to cope with different production situations. The results of the test instances are recorded in Tables 3 and 4 and plotted in Figures 7 and 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Instances** & **NBV** & **NCV** & **NC** \\ \hline J2-S2-M3-A2 & 74 & 21 & 271 \\ J3-S2-M3-A2 & 138 & 31 & 553 \\ J4-S2-M3-A2 & 220 & 41 & 935 \\ J5-S2-M3-A2 & 320 & 51 & 1417 \\ J6-S2-M3-A2 & 438 & 61 & 1999 \\ J7-S2-M3-A2 & 574 & 71 & 2681 \\ J8-S2-M3-A2 & 728 & 81 & 3463 \\ J9-S2-M3-A2 & 900 & 91 & 4345 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results for the effect of change in the number of jobs on model complexity (time limit: 600 s). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test Instances** & **UPM** & **IPM** \\ \hline J9-S2-M3-A1 & 114 & 123 \\ J9-S2-M3-A2 & 77 & 105 \\ J9-S2-M3-A3 & 68 & 100 \\ J9-S2-M3-A4 & 68 & 99 \\ J9-S2-M3-A5 & 67 & 98 \\ J9-S2-M3-A6 & 67 & 99 \\ J9-S2-M3-A7 & 68 & 98 \\ J9-S2-M3-A8 & 66 & 98 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental results for the effect of AGV numbers on makespan under instance set J9-S2-M3 (time limit: 600 s). Figure 7: Effect of performance of AGV numbers on makespan under instance set J6-S2-M3. In Tables 3 and 4 and Figures 7 and 8, it can be observed that, whether in a UPM or an IPM environment, with the number of participating AGVs increasing, the optimization target makespan shows a decreasing trend. In other words, for a certain size of FFS, using more AGVs can effectively shorten the production cycle time and realize the improvement of whole production efficiency. At the same time, when the number of AGVs increases from one to two, the decrease in makespan is the largest. As the number of AGVs continues to increase, this decrease is gradually reduced. With the number of jobs being six, when the number of AGVs reaches two, the makespan no longer changes; meanwhile, in another group, with a job count of nine, when the number of AGVs reaches four, the makespan is not reduced, maintaining a near constant value. Analysis of the data shows that once the number of AGVs exceeds 60% of the capacity of parallel equipment for the same number of jobs, their incremental contribution to cycle time reduction becomes much smaller. Moreover, the two sets of data in Tables 3 and 4 conform to the law of marginal effect. For example, in Table 3, for the given production conditions in instance J6-S2-M3-, the optimal number of AGVs is three.