## 5 Application in Real-World HDS System
### Results of the Case Study  
The case study results from the methods of LH, A-star (\(l_{1}\)), BFS (\(b_{w}^{*}=0.7\)), and RRT-star (\(\eta=3\)) are displayed in orange, yellow, green and maroon, respectively, in Figure 12. The values of \(\overline{t}_{\text{search}}\), std\({}_{\text{search}}\), \(\overline{n}_{\text{search}}\), and std\({}_{\text{node}}\) are shown in Figure 12a-d to compare the computation cost. The average \(\overline{t}_{\text{ship}}\) and standard deviation std\({}_{\text{ship}}\) of the operational feasibility are displayed in Figure 12e.f. The value of \(r_{c}\) is displayed in Figure 12g, accounting for the performance robustness. In Figure 12, one can tell that with the incremental percentages of randomly settled obstacles, the completion rates from all four algorithms monotonically decrease, displaying \begin{table}
\begin{tabular}{l l} \hline \multicolumn{1}{c}{**Operations/Parameters**} & \multicolumn{1}{c}{**Time Consumption**} \\ \hline System initialization (bootup and warmup) & 0 s; assume no initialization required \\ Shipping speed with cargo loaded & 1.2 m/s \\ Shipping speed without cargo loaded & 0.8 m/s \\ Loading time & 1.2 s \\ Turning time & 0.9 s \\ Layout of squared single storage position & 1 square meter \\ Length/width of 4-way shuttle and conveyor & 0.8 m \\ Size of the fulfilment center’s layout & 22 by 20 grids, 22 m by 20 m \\ \hline \end{tabular}
\end{table} 
Table 2: The other necessary shipping operations with the parameters. Figure 12: The case study results; (**a**) The average searching time, (**b**) the standard deviation of the searching time, (**c**) the average traversed nodes, (**d**) the standard deviation of traversed nodes, (**e**) the average shipping time, (**f**) the standard deviation of shipping time, (**g**) the completion rate. the increased route searching challenges. The proposed LH method always returns the best \(r_{c}\) values, representing stronger robustness. After obtaining the planned shipping paths, the proposed LH method almost always searches faster while it traverses a smaller number of nodes than any of the benchmark methods, demonstrating superior computational performance. Although the average of the searching time of RRT-star (\(\eta=3\)) is slightly smaller in the scenario with 10% obstacle occupation (13.2 ms \(<\) 16.3 ms), it returns a much higher searching time standard deviation (15.9 ms \(\gg\) 1.2 ms). Therefore, our LH method is more stable and performs better overall than RRT-star (\(\eta=3\)). The values of the average and standard deviation of the searching time of LH and RRT-star (\(\eta=3\)) can be found in Figure 12a,b. Moreover, with all of the shipping operations and parameter settings under consideration, the confidence interval (CI) of the operational feasibility for industrial production in the 100-time simulation is calculated to compare the shipping time with the assigned 120 s time window. Given a 99% (\(z=2.58\)) successful rate, the CI of operational feasibility for industrial production is obtained through  
\[CI=\overline{I}_{\text{ship}}\pm 2.58\frac{\text{std}_{\text{ship}}}{\sqrt{100}}. \tag{14}\]  
Values of the average \(\overline{I}_{\text{ship}}\) and standard deviation \(\text{std}_{\text{ship}}\), as well as the 'Completion Rate' \(r_{c}\) and the results of the CI, are listed in Table 3. Upon returning a planned shipping path, all of the CI values from the two scenarios are within the 120 s time window, indicating that our is method feasible for industrial production.