## 4 Numerical Study
### Simulation Settings  
The numerical study is finished via MATLAB R2022a. The system configurations include CUP, Intel i7-12700H; GPU, NVIDIA RTX 3060; memory, 16 GB. The MATLAB function of tic-toc is adopted to record the computational time.  
Following a similar design to [41], we conduct three numerical studies at a large scale, medium scale, and small scale to simulate the real-world HDS systems in a regional logistical center, a fulfillment center, and a last-mile delivery station with map sizes of \(50\times 50\), \(30\times 30\), and \(20\times 20\) respectively. In all 3 scale maps, the start nodes are always located at far-right edges on the main tracks, keeping the greatest similarity to a real-world HDS system. The destinations are arbitrarily assigned on sub-tracks as the storage position, complying with the real operational rules of an HDS system. Only one four-way shuttle or one conveyor run in the map at the same time, and no multi-vehicle cooperations are considered.  
Two scenarios without obstacles and with random obstacles for each map size are conducted to fully demonstrate the optimality, scalability, and robustness of the proposed method. The obstacle percentages are set to 5%, 10%, and 20%. Obstacles are randomly located on both main and sub-tracks to simulate as many different kinds of route blockages as possible. Twelve map settings are under consideration in total. Samples of the 12 map settings are displayed in Figure 6. Each map setting and simulation is conducted 100 times with the same seeds to ensure that the numerical study is reproducible.  
Figure 6: Samples of the 12 map settings by using our topological map representation method. From left to right, they are without obstacles, with 5% obstacles, with 10% obstacles, and with 20% obstacles. From top to bottom, they are 20 \(\times\) 20, 30 \(\times\) 30, and 50 \(\times\) 50 map sizes.  
We define 'Completion Rate' \(r_{c}\) to check the feasibility (the successful rate) of an algorithm that returns a path from the start node to the destination. \(r_{c}\) is calculated as  
\[r_{c}=N_{\text{success}}/100, \tag{13}\]  
where \(N_{\text{success}}\) is the number of successful runs that can return a path. We also adopt the values of 'Searching Time' (in micro-second (ms), the operational time on the processor of an HDS system) and 'Number of Traversed Nodes' to measure the overall path planning cost from each algorithm. To comprehensively compare the path planning performances, and especially to compare the performance stability given increased randomness, we consider and compare the average and standard deviation, e.g., \(\bar{t}_{\text{search}}\), \(\text{std}_{\text{search}}\), \(\overline{n}_{\text{search}}\), and \(\text{std}_{\text{node}}\), of these two values from all of the four algorithms. Therefore, five metrics of \(r_{c}\), \(\bar{t}_{\text{search}}\), \(\text{std}_{\text{search}}\), \(\overline{n}_{\text{search}}\), and \(\text{std}_{\text{node}}\) are compared in our numerical study.  
It is worth mentioning that in the scenario without obstacles, randomness still exists from the arbitrarily settled destinations. Destinations are sampled from all of the generated nodes. Therefore, statistics of this sampling, e.g., \(\text{std}_{\text{search}}\), and \(\text{std}_{\text{node}}\) should be calculated and compared. Because all benchmark algorithms theoretically guarantee an optimal solution, the scenario without obstacles mainly checks the feasibility of our LH algorithm.