## 6 Experimental Comparison and Analysis
### Comparison of Metaheuristics  
The effectiveness of the proposed scheduling algorithms was assessed by comparing them with two well-established metaheuristic algorithms: Iterated Greedy (IG) and the Genetic Algorithm (GA). For the VND algorithm, no hyperparameters exist. For the IG algorithm and the GA, we used the best parameter settings suggested in the original paper for the parameters of the IG algorithm [30] and the GA [27]. Table 2 shows that the ARPD of the VND algorithm is significantly lower than that of the IG algorithm and the GA across all test configurations. For instance, when the number of workpieces \(n=20\), the ARPD of the VND algorithm is 0.07%, while the ARPDs for IG and the GA are 3.41% and 2.97%, respectively. As \(n\) increases, for instance, when \(n=50\), the ARPD of the VND algorithm drops to 0%, while the ARPDs of IG and the GA sharply increase to 74.01% and 65.55%, respectively. Additionally, as the number of machines \(m\) and AGVs \(a\) increases, the ARPD of the VND algorithm remains consistently low. For instance, when \(m=8\) and \(a=8\), the ARPD of the VND algorithm is 0.00% and 0.02%, respectively, whereas the corresponding values for IG and the GA are 37.42%, 32.11% and 31.06%, 26.62%. These trends indicate that the VND algorithm demonstrates greater robustness and adaptability when handling more complex scheduling tasks. In terms of CT, although the VND algorithm requires slightly more computation time in some configurations (for example, when \(n=30\), its CT is 11.74 min, higher than the 6.18 min for IG and 5.69 min for the GA), as the task size increases (for example, when \(n=50\)), the CT of the VND algorithm is only 11.64 min, while the CT of IG and the GA rises to 17.50 min and 21.83 min, respectively. This performance shows that the VND algorithm outperforms the other two algorithms in terms of computational efficiency when handling larger problems. A comparison of the performance of the three algorithms under different strategies is shown in Figure 4. The ARPD values of the VND algorithm are consistently close to \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{_Numerical Example_} & \multicolumn{2}{c}{**VND**} & \multicolumn{2}{c}{**IG**} & \multicolumn{2}{c}{**GA**} \\  & **ARPD (\%)** & **CT (min)** & **ARPD (\%)** & **CT (min)** & **ARPD (\%)** & **CT (min)** \\ \hline \multirow{3}{*}{\(n\)} & 20 & 0.07 & 3.02 & 3.41 & 18.78 & 2.97 & 18.00 \\  & 30 & 0.00 & 11.74 & 12.28 & 6.18 & 5.69 & 12.80 \\  & 50 & 0.00 & 11.64 & 74.01 & 17.50 & 65.55 & 21.83 \\ \hline \multirow{3}{*}{\(m\)} & 4 & 0.03 & 7.21 & 17.74 & 13.35 & 13.65 & 16.32 \\  & 6 & 0.01 & 8.41 & 27.92 & 13.80 & 22.67 & 17.28 \\  & 8 & 0.00 & 10.59 & 37.42 & 13.49 & 31.06 & 18.12 \\ \hline \multirow{3}{*}{\(a\)} & 4 & 0.03 & 7.84 & 26.52 & 14.06 & 21.67 & 17.03 \\  & 6 & 0.02 & 8.50 & 29.36 & 14.11 & 24.17 & 17.45 \\ \cline{1-1}  & 8 & 0.02 & 9.31 & 32.11 & 14.18 & 26.62 & 17.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The average RPD and CPU time of the VND, IG, and GA algorithms on the comparison instances. 0 across the nine strategies, indicating its significant superiority. In contrast, the ARPD values of IG and the GA fluctuate across strategies, with notably larger increases when strategy complexity is higher, suggesting that these algorithms have weaker adaptability to complex scheduling tasks. The VND algorithm outperforms both the IG algorithm and the GA in scheduling accuracy and computational efficiency, highlighting its effectiveness in solving the pulsating assembly flow shop scheduling problem. While the proposed VND algorithm demonstrates strong performance in terms of scheduling accuracy and computational efficiency, certain limitations exist. The computational time increases with the problem size, particularly when the number of jobs and machines grows significantly. Although VND outperforms IG and the GA in large-scale instances, its efficiency could be further improved through parallel computing or hybrid optimization approaches.