The overall design of our two-step LH method is presented in Figure 2.  
We apply the numerical and case studies to compare the performance of our algorithm to three benchmark algorithms, the renowned A-star algorithm, the BFS (Breadth-First Search) algorithm, and the RRT (Rapidly Exploring Random Tree) algorithm. All of the three benchmark algorithms are properly modified per requirements of an HDS system. Specifically, the A-star algorithm is assigned a heuristic estimation with Manhattan distance to keep a computational balance between traffic constraints and the searching distance. The BFS algorithm is given a breadth direction searching weight of 0.7 to enforce the goal-direction search. This value is given to help the planned path check sub-tracks but avoid certain blocked area and therefore return an optimal solution. RRT-star with an optimal searching step of three grids is used to replace the original RRT to smooth the searching route and to reach an asymptotic optimality via parent node selection and rewiring. In the numerical study, we design three sizes of maps, 20 \(\times\) 20, 30 \(\times\) 30, and 50 \(\times\) 50, to simulate the small-scale, medium-scale, and large-scale real-world HDS systems and to evaluate the optimality and scalability of our algorithm. In the case study, we adopt a local fulfillment center layout from Huade Co., Ltd., to validate the operational feasibility for industrial production of our algorithm. In both studies, we introduce two scenarios, without obstacles and with randomly settled obstacles of occupation percentages from 5% to 10% and 20%, to analyze the algorithm's stability and robustness. Both scenarios in the two studies from all of the four algorithms are conducted 100 times. The 'Completion Rate' is calculated to check the feasibility of the methods, given random route blockages. The average and standard deviations of 'Searching Time' and 'Number of Traversed Nodes' are calculated to evaluate the computational cost. In the case study, in consideration of all necessary shipping operations, the confidential interval (CI) for actual shipping time along the planned path is obtained to compare it with an assigned 120 s time window to analyze the algorithm's operational feasibility. The superiority of our method in all of the studies, in comparison to the three modified benchmark algo  
Figure 2: The overall design of the two-step LH method. rithms, demonstrates strong feasibility with optimality, proper scalability, rigid robustness, and suitable operability of the proposed method. The following parts of this paper are organized as follows. Related work on map representations and path planning algorithms for HDS systems in modern warehousing logistics are reviewed in Section 2. Our method of mesh-tree grid map representation and the LH algorithm are described in detail in Section 3. In Section 4, numerical studies on three scale maps with and without randomly located obstacles are simulated, and the results are compared to benchmark algorithms. In Section 5, the proposed method is applied to a real-world HDS system to show the usage of our algorithm in modern warehousing logistics. Section 6 summarizes this paper and points out potential future work.