## 4 Numerical Study
### Simulation Results
#### 4.3.3 Detailed Analysis of the Scenario with Random Obstacles  
We first evaluate the feasibility. With randomly settled obstacles, none of the four algorithms can guarantee a solution, and, therefore, the 'Completion Rate' \(r_{c}\) is compared in this scenario. As displayed in Figures 7e, 8e, and 9e, although \(r_{c}\) from our method decreases with the incremental percentage of obstacles, the completion values from our method are always larger than the values from the benchmark algorithms in the three scale maps, demonstrating the superior feasibility of our method.  
Next, we consider the optimality. With increased obstacle percentages, all four algorithms return increased values of \(\overline{t}_{\text{search}}\), \(\text{std}_{\text{search}}\), \(\overline{n}_{\text{search}}\), and \(\text{std}_{\text{node}}\), matching the theoretical analysis of computational complexity for a path planning algorithm. In comparison to the three benchmark algorithms, our method almost always provides a shipping route with the lowest average and standard deviation of the shipping time, as well as the lowest average and standard deviation of the number of traversed nodes, although RRT-star (\(\eta=3\)) may occasionally perform better. Therefore, our LH algorithm usually displays better optimality and better stability than the three benchmark methods.  
Moreover, we check the scalability. Because all of the completion rates are reduced with the increase in obstacle percentages given a larger map size, the worst case always happens in the map size of \(50~{}\times~{}50\) with 20% obstacles. In this case, our method performs much better than the algorithms A-star (\(l_{1}\)), BFS (\(b_{w}^{*}=0.7\)), and RRT-star (\(\eta=3\)), with completion rates of 69%, 53%, 23%, and 35%, respectively. Dynamically, by increasing obstacle percentages from 5% to 10% and 20%, the completion rate of our method is reduced by 7% (from 94% to 87%), 10% (89% to 79%), and 14% (83% to 69%) with map sizes of \(20~{}\times~{}20,30~{}\times~{}30\), and \(50~{}\times~{}50\), respectively. The reductions in A-star (\(l_{1}\)) are 11%, 16%, and 23%; in BFS (\(b_{w}^{*}=0.7\)), they are 24%, 21%, and 19%; and they are 14%, 17%, and 16% in RRT-star (\(\eta=3\)). Our method almost always returns a smaller reduction rate when adopted in a larger map. One can see these changes in both worst and dynamical cases by checking the slopes of the four curves in Figure 10, the enlarged graphs for the 'Completion Rate' of Figures 7-9, with the values displayed. Therefore, our method performs better than the three benchmark algorithms when scalability is concerned.  
After that, we analyze the robustness. Given the same map, we gradually increase the path planning difficulty by adding more obstacles, from 5% to 10% and 20%. By checking the orange curves in Figure 10, in a map of 20 \(\times\) 20, the reduction of the completion rate \(\Delta r_{c}\) in our method goes from 94% to 89%, 83%, and 11% (94% \(-\) 83%) in total. In a map of 30 \(\times\) 30, the reduction in our method is \(\Delta r_{c}\) = 12% (from 91% to 79%), and it is \(\Delta r_{c}\) = 18% (from 87% to 69%) in a 50 \(\times\) 50 map. The reductions from A-star (\(l_{1}\)) are 17%, 18%, and 29%; the reductions from BFS (\(b_{w}^{*}=0.7\)) are 44%, 42%, and 39%; and the reductions from RRT-star (\(\eta\) = 3) are 34%, 35% and 39%. The reductions of the completion rates for the three benchmark methods can be recognized from the slopes of the yellow, green, and maroon curves in Figure 10. Because the completion rates of our methods are always reduced slower, our method can plan the shipping path easier by adding more difficulties; therefore, we can conclude that our algorithm has a stronger optimal robustness.