Lattice-Hopping: A Novel Map-Representation-Based Path Planning Algorithm for a High-Density Storage System

Abstract

Optimal path planning algorithms offer substantial benefits in high-density storage (HDS) systems in modern smart manufacturing. However, traditional algorithms may encounter significant optimization challenges due to intricate architectural configurations and traffic constraints of the HDS system. This paper addresses these issues by introducing a two-step novel path planning method: (1) the mesh-tree grid map topological representation and the (2) Lattice-Hopping (LH) algorithm. The proposed method first converts the layout of an HDS system into a mesh-tree grid hierarchical structure by capturing and simplifying the spatial and geometrical information as well as the traffic constraints of the HDS system. Then, the LH algorithm is proposed to find optimal shipping path by leveraging the global connectivity of main tracks (main track priority) and the ‘jumping’ mechanism of sub-tracks. The main track priority and the ‘jumping’ mechanism work together to save computational complexity and enhance the feasibility and optimality of the proposed method. Numerical and case studies are performed to demonstrate the superiorities of our method to properly modified benchmark algorithms. Algorithm scalability, robustness, and operational feasibility for industrial production in modern smart manufacturing are also displayed and emphasized.

1. Introduction

With the accelerated progression of global economic integration and the exponential growth of e-commerce, warehousing logistics has emerged as a key component in smart manufacturing systems [1]. However, conventional storage methods are plagued by issues, such as suboptimal space utilization, inadequate operational efficiency, and exorbitant labor dependencies [2]. To solve these major challenges, high-density storage (HDS) systems have been introduced as a scientifically alternative solution offering robust frameworks to address the multifaceted issues inherent in contemporary warehousing logistics [3,4].

A typical HDS system, displayed in Figure 1, is mainly composed of main tracks, sub-tracks, automated transportation equipment, and other supporting components. The pictures in Figure 1 were taken from Nanjing Huade Storage Equipment Manufacturing Co., Ltd. (Huade Co., Ltd., Nanjing, China), the cooperator of our study. This is a leading Chinese supply chain company that provides warehousing solutions for modern smart manufacturing. In an HDS system, the main tracks serve as primary logistics arteries for four-way shuttles, the major transportation equipment, while sub-tracks arranged in arrays serve dual roles as storage locations and auxiliary paths. Supporting components, such as hoisters and conveyors, facilitate cargo loading, vertical movements, and lateral distribution [5]. Coordinated with sophisticated path planning algorithms and scheduling systems, HDS architectures are providing increasing production throughput and operational agility in modern smart manufacturing.

Despite their benefits, HDS systems also generate new difficulties, e.g., narrow transportation spaces, intricate device coordination, constrained path options due to blockages, etc., and therefore they impose stringent demands on intelligent and scalable path planning methods [6]. So far, designing an effective path planning algorithm for such a system remains an open challenge in modern warehousing logistics and smart manufacturing. We propose a two-step path planning algorithm in this paper to address this pragmatic challenge.

We first introduce a mesh-tree grid map topological representation to analog and simplify the HDS system layouts geometrically. In our representation, the main track is used for cargo transportation from the entrance to the storage area and plays a role as the backbone of a tree, while the sub-tracks are auxiliary paths to guide the shipping cargos into or from the identified storage position, as branches of a tree. In this way, an HDS layout is transferred structurally and geometrically into a mesh-tree grid topological network with categorized shipping paths and storage positions. This representation can significantly reduce computational costs in the subsequent path planning step.

Next, we introduce our Lattice-Hopping (LH) algorithm, where ‘Lattice’ represents the mesh-tree grid structure and ‘Hopping’ means ‘jumping’. Our algorithm always returns an optimal shipping path by leveraging the hierarchical structure of an HDS system with the design of main track priority and jumping mechanism, if such a path exists. During path planning, the search first moves along the main track to the storage area and then reaches the specific position via sub-tracks. This is the ‘main track priority’. Once obstacles are met, the jumping mechanism is triggered to enforce the path turning around from the obstacles and to reach the destination via another sub-track, if it is accessible. In this way, our method can provide superior performances by avoiding most unnecessary searches and return a shorter average searching time as well as a smaller number of traversed nodes. 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 × 20, 30 × 30, and 50 × 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 algorithms, 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.

2. Related Work

2.1. Map Representation

Map representations for warehousing logistics can provide a visualization of facility structures, pathways, obstacles, and storage locations and therefore facilitate enhanced spatial comprehension and path planning optimization. Mainstream methods include topological maps [7], which utilize node–edge structures to represent spatial connectivity and accessibility relationships, feature-based maps [8] that construct spatial representations via sensor that capture significant environmental characteristics, and symbolic maps [9] that employ abstract symbols to represent spatial information and structures. These map representations find primary application in advanced task planning and decision support systems but lack detection abilities for obstacles.

Other methods, e.g., vector maps [10], continuous space maps [11], three-dimensional maps [12], depth maps [13], etc., project detected geometric information and location specifications into coordinate systems and obtain accurate topological representations via sophisticated calculations. Although they display significant advantages in rapid obstacle detection, extra computation complexities should be considered, and advanced detection equipment is required. Therefore, utilization of these methods is limited.

SLAM (Simultaneous Localization and Mapping) [14] technology enables real-time positioning and environmental mapping without external sensing infrastructure. But, high computational costs and susceptibility to error accumulation in large-scale or featureless environments should be considered, and hence its utilization is restricted in HDS systems.

Besides these drawbacks, the existing map representations also struggle to accurately depict the hierarchical traffic constraints between main tracks and sub-tracks in the structured environment of an HDS systems. Traditional raster maps usually treat storage space as uniform grids, failing to account for the global connectivity of main tracks and the unidirectional dependency of sub-tracks. This limitation results in increased redundant node searches and frequent route conflicts during path planning. To overcome these limitations, we introduce the mesh-tree grid map topological representation to effectively distinguish track levels and enforce optimal path planning for HDS systems.

2.2. Path Planning

2.2.1. Traditional Path Planning Algorithm

The path planning algorithm is a computational method that determines an optimal or feasible trajectory for an agent (e.g., robot, vehicle) to navigate from a start point to a goal point while avoiding obstacles and satisfying constraints [15]. Path planning algorithm development constitutes a fundamental optimization challenge in modern warehousing logistics and smart manufacturing.

Classical graph-based algorithms include the BFS (Breadth-First Search) algorithm [16] and the Dijkstra algorithm [17], capable of determining the shortest paths between start and end points. Enhanced approaches, such as heuristic graph search algorithms [18] and the A-star algorithm [19], utilize heuristic functions for accelerated search processes. The incremental path planning method of the D-star algorithm [20] is developed for dynamic environments. Although these algorithms display their feasibilities in many problems, their finding efficiencies or results are constrained in high-dimensional complex environments, where path replanning is often required. The computational complexity increases exponentially when the problem scale becomes larger, and the generated paths typically consist of discrete connections that may not align with the actual shipping requirements.

To mitigate these drawbacks, sampling-based path planning algorithms, exemplified by the RRT (Rapidly Exploring Random Tree) algorithm, were developed [21]. RRT, with its asymptotic optimality, approaches constructing shipping paths through free space random sampling and demonstrates strong suitability in high-dimensional spaces. But, their dynamic environment adaptability remains limited, often requiring complete path regeneration.

Other methods include optimization approaches and learning approaches. The optimization approaches include gradient descent methods [22], energy consumption optimization [23], convex optimization [24], genetic algorithms [25], particle swarm optimization [26], ant colony optimization [27], etc. Their path planning efficiencies may be affected by complicated traffic constraints or for real-time implementation. These methods may also display high solution sensitivity when handling environmental-dependent problems and frequently converge to local optima in non-convex cases. Learning approaches, e.g., deep learning and reinforcement learning algorithms [28,29], transform path planning problems into learning problems, enabling them to autonomously learn and generate paths. But, these intelligent methods perform sensitively to parameter selection or under complicated applications.

2.2.2. Path Planning in High-Density Storage System

In the field of warehousing logistics and their management, scholars have explored diverse methodologies to identify optimal solutions. Gao et al. [30] proposed a bilevel optimization framework integrating job shop scheduling and collision-free route planning for high-density manufacturing systems. Fu et al. [31] introduced a path-net graph-based routing and scheduling algorithm for automated storage and retrieval systems (AS/RS), balancing path costs with trunk rail and pallet lifter utilization. Chen et al. [32] proposed a hierarchical framework that integrates centralized task assignment with a multi-agent reinforcement learning (MARL)-based distributed trajectory planning module to significantly improve task allocation efficiency and collision resolution in dynamic interactive scenarios. De Koster et al. [33] studied the scheduling optimization problem of the traditional AS/RS systems and proposed a storage location allocation method based on access frequencies. Although their work effectively reduces the average access time, its applicability is limited to single crane systems and cannot meet high concurrent requirements. Boysen et al. [34] proposed a modular 4D shuttle system that can operate simultaneously across multiple dimensions and greatly improve the system’s throughput. However, the task scheduling complexity of this system increases significantly when presenting challenges in multi-vehicle conflict detection. Azadeh et al. [35] analyzed the impact of vehicle scheduling strategies on system performance in multi-vehicle systems and proposed a distributed dispatching policy. Although decentralized control on an HDS system can improve the system’s robustness, path conflicts are inevitable under high-density scenarios, which affects the overall shipping efficiency. Konishi et al. [36] explored the application of AI methods in an HDS system by using a deep reinforcement learning algorithm to optimize shipping path selection, demonstrating strong adaptability in dynamic environments. However, the computational cost during the training phase is relatively high, limiting the real-time response of this method. All of the work mentioned above shows that although solutions exist, they are not tailored to topological representation or might be overly complex for HDS systems, rendering them impractical.

In summary, three major challenges exist in current path planning for HDS systems: (1) lack of universal map representation methods that can effectively describe facility structures and traffic constraints while recording accurate spatial information and geometric properties; (2) high computational complexity and high solution sensitivity when handling complex path-planning problems; and (3) limited application feasibility due to insufficient consideration of transportation kinematic constraints in HDS systems. Therefore, efficient path planning algorithmic approaches for HDS systems are urgently needed, and our mesh-tree-based LH algorithm is a potential solution.

3. Method

To address these limitations, we developed a novel two-step path planning method to find an optimal shipping trajectory with reasonable complexities in a smart HDS system. We first propose the mesh-tree grid map topological representation to physically describe the architectural characteristics and geometrical properties of an HDS system, and then we display our Lattice-Hopping (LH) algorithm in this map with the necessary details.

3.1. Mesh-Tree Grid Map Topological Representation

3.1.1. Topological Assumptions

Per geometrical, spatial characteristic, and traffic constraint requirements in an HDS system, and also by referring to the mapping methods in [37,38,39], we set four major assumptions for our mesh-tree grid map topological representation method.

• Assumptions for Geometric and Spatial Structures

• Uniform Grid Discretization

The warehouse’s layout is discretized into a uniform grid, where each cell corresponds to either a storage unit or a transportation space. Both main tracks and sub-tracks are discretized using the same grid unit, forming a structured mesh-tree topology.

• 2D Plan Modeling

Only the ground floor is considered, and vertical elements, such as hoisters, are ignored.

• Tree structure abstraction

Main tracks are modeled as the ‘backbone’ of a tree, while sub-tracks are considered ‘branches’. In this way, the whole layout is segmented into a tree structure to simplify the subsequent path planning process.

• Assumptions for Connectivity and Traffic Constraints

• Main Track Priority

All paths originate from a start point on main tracks. Navigation prioritizes movement along the main track until reaching an appropriate sub-track/storage area. Main track movement is constrained to lateral (horizontal) directions.

• Sub-Track Movement

Movements along sub-tracks are constrained to longitudinal (vertical) directions and generally from the main track to the sub-tracks. Movements can retreat from the sub-tracks to the main track when the path is blocked.

• Jumping Mechanism

To change from one sub-track to another, shipping equipment must return to the main track and then access the target via another sub-track. This is referred to as the ‘jumping’ mechanism. Direct switching between sub-tracks horizontally is not permitted.

• Restricted to Orthogonal Movement

Only four orthogonal direction (up, down, left, right) movements are allowed. Diagonal or non-grid-based transitions are excluded to maintain grid consistency and simplicity.

• Mathematical and Algorithmic Assumptions

• Uniform Node Cost

Assuming uniform distance and effort for movement between adjacent nodes, each grid node is assigned an equal transportation cost or computational cost in the algorithm.

• Random Obstacle Distribution

Obstacles are assumed to be randomly distributed across both main and sub-tracks, with an upper bound of 25% coverage [40] to simulate realistic congestion or blockage scenarios.

• Feasibility of Reachability

In the absence of obstacles, all positions are assumed to be reachable. If a destination is blocked by obstacles and all jumping attempts fail, the algorithm returns “No Solution.”

• Functional and Modeling Constraints

• Vehicle-to-Grid Matching

The size of four-way shuttles and conveyors is assumed to match the size of a single grid cell, which is also equal to the size of a standard storage unit.

• Single-Agent Assumption

The planning process assumes only one vehicle is active at a time. No inter-vehicle conflicts, collisions, or multi-vehicle corporations are considered.

3.1.2. Topological Representation

According to the assumptions listed above, our mesh-tree grid map topological representation converts the CAD drawing of an HDS system layout into a mesh-tree structure. Figure 3a is a layout CAD drawing, and Figure 3b is its simplified topological representation. In Figure 3a, the main track, represented by a yellow belt, is the major transportation route to ship the cargos from the start position to storage areas. Sub-tracks are grouped by the green boxes between two black lines, and they are also the storage areas. In Figure 3b, both main and sub-tracks are constructed of square grids with equal sizes. In Figure 3a, the two red boxes at the bottom are hoisters, which lift cargos from the input docks to the storage area. The crossed main tracks with the hoister on the side are a path start position. We do not consider hoisters in Figure 3b per our assumptions. No non-grid movement occurs during shipping, and all transportation is finished via four-way trucks or conveyers perpendicularly. The four-way trucks and conveyers and their geometric differences are neglected in Figure 3b. In this way, an HDS system layout is converted into a map representation.

3.2. LH Algorithm

• The Lattice-Hopping algorithm searches for a path from the start node to the destination in the represented map with the lowest overall cost. The objective cost function is

  $$\begin{array}{c}\mathrm{m}\mathrm{i}\mathrm{n} h\left(x\right)=m\left(x\right)+s\left(x\right)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o} x\in \mathcal{X}\end{array},$$

  (1)

  where h(∙) is the total cost function from the start node to node x, m(x) is the cost it spent along the main track, s(x) is the cost on sub-tracks, and X is the potential searching area in the represented map. Along the main track, we search for neighboring nodes toward the 4 orthogonal directions in the grid map (front, right, back, and left), Manhattan distance is used. In sub-tracks, we only search for the nodes ahead of the path. Therefore, $m\left(x\right)$ and $s\left(x\right)$ have different cost counting methods, revealing the design of the main track priority and the ‘jumping’ mechanism, respectively.

The algorithm flowchart is displayed in Figure 4, and the generated searching routes according to this flowchart are displayed in Figure 5. In Figure 5, node O is the original point. X is the main track search direction from node O to node I, and Y is the sub-track search direction from node O to node J. The start nodes are always set on the main track.

3.2.1. Feasibility Analysis

The feasibility of the proposed LH algorithm depends on obstacles. We analyze the algorithm’s feasibility via two cases: obstacles on sub-tracks and obstacles on the main track. The two cases are illustrated in Figure 5a and Figure 5b, respectively.

• Case One: obstacles on sub-tracks

In Figure 5a, the searching path starts along the main track from node O to position i_d and then turns from the main track to sub-track i_d. We search for neighboring nodes in the four orthogonal directions (front, right, back, and left) along the main track and the node ahead the route in sub-tracks. The traversed nodes are highlighted in yellow. If there are no obstacles, the path goes to (i_d,j_d) the blue box in Figure 5a, and returns to ‘Arrival’.

If an obstacle, a black box, is detected, the search is blocked and then retreats to the main track. The algorithm pushes the search one step back to position $i_d-1$ along the main track and checks the possibility to jump across the new sub-track $i_d-1$ to position $\left(i_d,j_d\right)$ from the opposite direction via another main track. If it is not possible to jump, the algorithm checks blockage along the new sub-track and continues the ‘retreat–one step back–check–jump–check’ loop to another sub-track until Arrival. If no steps can be pushed back, then no feasible path can be found, and the algorithm returns ‘No Solution’.

The searching process in Case One is displayed via the red curve in Figure 5a.

When the start node is on the right side of the destination $\left(i_d,j_d\right)$, the algorithm pushes the search back to $i_d+1$ after retreat. In other words, our algorithm always pushes the search one step back toward the start node. This process is shown via the purple curve.

• Case Two: obstacles on the main track

Once obstacles are on the main track, e.g., on position $i_d$, the searching route turns into the sub-track at node $i_d-1$ or $i_d+1$ and jumps to another main track, which is parallel to the current one. This is shown in Figure 5b in red and purple curves, respectively.

Therefore, the proposed algorithm can always find a feasible solution if it is possible.

3.2.2. Optimal Analysis

We compare our proposed algorithm with the famous A-star algorithm. According to the cost function of A-star, the total cost is the summation of the current travel cost and the heuristic estimation cost. In a mesh-tree grid topological representation, our travel cost should be no larger than the one in the A-star algorithm (using Manhattan distance for heuristic estimation in A-star) from the start node to the current node. But, we save more heuristic estimation via ‘jumping’ by only checking the node ahead in sub-tracks. In this way, we save more heuristic estimation cost and obtain a better optimality than A-star.

3.2.3. Complexity Estimation

Among all three benchmark methods, the A-star algorithm with Manhattan distance almost always has the best computational complexity for an HDS system; therefore, we compare the computational complexity of our method with the A-star algorithm.

(1)

Time complexity

We define $d$ as the length of the shortest path from the start node to the destination, and $b$ is the branching factor, the maximum nodes of successors for any given state. By using the two designs of main track priority and ‘jumping’ mechanisms on sub-tracks, the value of $b$ can be divided into two parts: (1) $b_1=4$ for nodes on the main track, where the Manhattan distance is adopted for the HDS system; and (2) $b_2=1$ on sub-tracks.

• Case One: best case—find the destination directly without any obstacles

The time complexity can be expressed as

$$T_{\mathrm{L}\mathrm{H}}^{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}=\mathcal{O}\left(4^{d_1}+d_2\right),$$

(2)

where $d_1$ is the number of nodes along the main track and d₂ is the number of nodes in the first sub-track until destination arrival.

• Case Two: worst case—find the destination after traversing all of the nodes in the map

The time complexity can be expressed as

$$T_{\mathrm{L}\mathrm{H}}^{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}=\mathcal{O}\left(4^{d_1}+{d_1\times d}_2\right).$$

(3)

Under the best case scenario, the time complexity of A-star is $\mathcal{O}\left(4^{d_1+d_2}\right)$, which is always larger than our best case time complexity of $\mathcal{O}\left(4^{d_1}+d_2\right)$. Under the worst case scenario, the time complexity of A-star is $\mathcal{O}\left(4^{d_1\times d_2}\right)$, which is also larger than $\mathcal{O}\left(4^{d_1}+{d_1\times d}_2\right)$, our time complexity.

Therefore, the time complexity of our algorithm performs better than the A-star algorithm. From Equations (2) and (3), one can also tell that the larger the map of an HDS system, the better the comparison result. This is displayed in both the numerical and case studies.

(2)

Space complexity

Not all traversed nodes during path planning need to be kept in memory. We reuse the nodes along the main track and reuse the majority of the nodes along the sub-tracks. In this way, data that are kept in memory are bounded by the map’s size, $\mathcal{O}\left({d_1\times d}_2\right)$. On the contrary, the worst space complexity in A-star is $\mathcal{O}\left(4^{d_1\times d_2}\right)$, which is much larger than our value. In a word, we have better computational complexity than the benchmark algorithms.

4. Numerical Study

In this section, we first modify the three well-known path planning algorithms of A-star, BFS, and RRT properly according to requirements in HDS system, and then we adopt our topological representation to construct the mesh-tree grid simulation networks for different scale maps of HDS layouts. Next, we utilize our LH algorithm to develop the optimal shipping paths and compare the performances with benchmark methods. In the end, we summarize the results of the numerical study and make necessary explanations.

4.1. Benchmark Algorithms

• A-star algorithm

In the A-star algorithm, the overall path planning cost function $f\left(n\right)$ can be defined as

$$f\left(n\right)=g\left(n\right)+h\left(n\right),$$

(4)

where $g\left(n\right)$ is the actual cost from the start node to node $n$ and $h\left(n\right)$ is the estimated cost from node $n$ to the destination.

To make the A-star algorithm intelligent, a proper heuristic estimation $h\left(n\right)$ should be selected. In our case of an HDS system, path searching is developed in two-dimensional space, and only 4-direction grid movement is allowed. Therefore, the $h\left(n\right)$ can be heuristically estimated by Manhattan distance as

$$h\left(n\right)=\left|x_1-x_2\right|+\left|y_1-y_2\right|,$$

(5)

where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are coordinates of two consecutive nodes along the developed path. In this way, we define our A-star as A-star ($l_1$), where $l_1$ represents the Manhattan distance heuristic estimation.

• BFS Algorithm

Graphically, the BFS (Breadth-First Search) algorithm searches nodes layer by layer, which guarantees the shortest distance from start to destination. All of the explored layers and nodes can be expressed as

$$\begin{gathered} L_0=\left\{s\right\} \\ {L}_{i+1}=\{v\in V\backslash {\bigcup }_{j=0}^i{L}_j|\exists u\in L_i,\left(u,v\right)\in E\}, \end{gathered}$$

(6)

where $V$ is the set of nodes, $E\subseteq V\times V$ is the set of edges, $s$ is the start node, $\left(u,v\right)$ is the edge between node $u$ and $v$, and $L_j, j=0,\cdots i+1$ are explored layers.

Adapting BFS to our mesh-tree grid network, we enhance the original BFS algorithm with the breadth direction weighted method. In our map representations, the depth and breadth directions are defined as the direction along the main track and the sub-tracks, respectively. Because all of the final storage locations are set in sub-tracks, we assign a positive breadth direction weight $b_w$ to enforce BFS searching always toward sub-tracks or final locations. The parameter $b_w$ is given by the cosine distance function as

$$b_w={\displaystyle \frac{d\left(u,v\right)\mathsf{\bullet }d\left(u,g\right)}{‖d\left(u,v\right)‖‖d\left(u,g\right)‖}},$$

(7)

given $\left(u,v\right)$ as the edge between node $u$ and $v$ and $\left(u,g\right)$ as the edge between node $u$ and the destination; $d$ is the direction vector, and $‖\mathsf{\bullet }‖$ is the distance calculation. Then, the explored nodes and layers can be converted into

$$\begin{gathered} L_0=\left\{s\right\} \\ {L}_{i+1}=\left\{v\in N_{b_w}\left(u\right)\backslash {\bigcup }_{j=0}^i{L}_j|\exists u\in L_i,\left(u,v\right)\in E\right\}, \end{gathered}$$

(8)

where $N_{b_w}\left(u\right)$ is the traversed nodes around $u$ according to the breadth direction weight $b_w$.

$b_w$ is between $-1$ and $1$, and $b_w>0$ encourages the breadth-direction search. After testing with different values of 0.3, 0.5, 0.7, and 0.9, we found that 0.7 returned the fastest searching speed and the least traversed nodes. Therefore, our BFS algorithm is assigned an optimal weight of 0.7 and represented as BFS ($b_w^{*}=0.7$).

• RRT-star Algorithm

Improved from the conventional RRT algorithm, RRT-star searches the optimal path asymptotically by selecting the optimal parent nodes among the new generated neighbor area and rewiring the route connectivity from this selected parent node. In RRT-star, a new generated node $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$ is calculated via the Steer function

$$x_{\mathrm{n}\mathrm{e}\mathrm{w}}=\left\{\begin{array}{c}\begin{array}{cc}{x}_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}},& ‖x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}-x_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}‖\le \eta \end{array}\\ \begin{array}{cc}{x}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}+\eta {\displaystyle \frac{x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}-x_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}}{‖x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}-x_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}‖}},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\end{array}\right.,$$

(9)

where $x_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}$ is the node randomly sampled from the free space ${\mathcal{X}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ (excluding obstacles) and $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$ is the nearest node to $x_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}$. Euclidean distance is adopted, $\eta$ is the searching step size, and $‖\mathsf{\bullet }‖$ is the distance calculation.

For the new generated node $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$, the optimal parent node $x_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}^{*}$ is optimally selected from the neighbor area $X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ by ensuring a minimum overall cost:

$$x_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}^{*}=\arg \underset{x_{\mathrm{near}}\in X_{\mathrm{neighbor}}}{\min }\left\{c\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\right)+c_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}},x_{\mathrm{n}\mathrm{e}\mathrm{w}}\right)\right\},$$

(10)

where $c\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\right)$ is the cost from the start point to a node $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\in X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ and $c_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}},x_{\mathrm{n}\mathrm{e}\mathrm{w}}\right)$ is the cost from node $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ to node $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$. $X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ is the circle sized neighbor area around $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$. The size of $X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ is determined by the total number of nodes $n$, the dimensionality $d$ of the searching space, the step size $\eta$, and a constant $\gamma$. The radius $r_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ of $X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ is calculated as

$$r_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}=\min \left(\gamma {\left({\displaystyle \frac{\log n}{n}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}},\eta \right).$$

(11)

Next, for each node $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\in X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$, if $c\left(x_{\mathrm{n}\mathrm{e}\mathrm{w}}\right)+c_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(x_{\mathrm{n}\mathrm{e}\mathrm{w}},x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\right)<c\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\right)$ holds and the path between $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$ and $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ is collision-free, the updated parent of $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ is set as

$$\mathrm{Parent}\left(x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\right)=x_{\mathrm{n}\mathrm{e}\mathrm{w}}.$$

(12)

In this way, rewiring from the generated node of $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$ to $x_{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ in the same neighbor area $X_{\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ is formed. This is the key update from conventional RRT to RRT-star.

From Equations (9)–(12), we can find that the step size $\eta$ plays multiple critical roles in RRT-star: (1) it controls searching expansion; (2) it ensures route connectivity; and (3) it updates the parent node for the next searching step. In an HDS system, $\eta$ is counted by the number of grids in our map representation. After careful comparison, the step size $\eta$ is set as 3 grids, and the RRT-star algorithm is presented as RRT-star ($\eta =3$). Therefore, the three properly selected and modified benchmark algorithms are the following:

(1)

A-star ($l_1$), the A-star algorithm with Manhattan distance heuristic estimation.

(2)

BFS ($b_w^{*}=0.7$), the BFS algorithm with a breadth direction weight of 0.7.

(3)

RRT-star ($\eta =3$), RRT-star with a searching step size $\eta$ of 3.

4.2. 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 × 50, 30 × 30, and 20 × 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.

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

$$r_c={\displaystyle \raisebox{1ex}{$N_{\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.},$$

(13)

where $N_{\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}$ 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., ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$, of these two values from all of the four algorithms. Therefore, five metrics of $r_c$, ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ 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., ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ 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.

4.3. Simulation Results

4.3.1. Simulation Results According to Map Sizes

The results for map scales of $20\times 20$, $30\times 30$, and $50\times 50$ without and with 5%, 10%, and 20% obstacles are displayed in Figure 7, Figure 8, and Figure 9, respectively. The five metrics of ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$, and $r_c$ are separated into (a), (b), (c), (d), and (e) in each figure. The results from the proposed LH algorithms are shown by the orange curve, and the results from the three benchmark algorithms of A-star (l₁), BFS ($b_w^{*}=0.7$), and RRT-star (η = 3) are shown by yellow, green, and maroon curves.

Illustrated in Figure 7, Figure 8 and Figure 9, the proposed LH algorithm almost always returns the best computational cost with the lowest average searching time and the smallest number of traversed nodes in all three map scales with or without randomly settled obstacles. The proposed LH algorithm also returns the best performance stability with the highest completion rate and the smallest standard deviation of the searching time, as well as the number of traversed nodes, if a solution/path can be returned/planned.

4.3.2. Detailed Analysis of the Scenario Without Obstacles

Presented in Figure 7, Figure 8 and Figure 9, all simulations can return a planned shipping path with no obstacles settled, demonstrating the feasibility of the proposed LH algorithm.

With the incremental map sizes, all of the metrics that evaluate the computation cost, e.g., the average and standard deviation of the ‘Searching Time’ and the ‘Number of Traversed Nodes’, increase for all four algorithms. But, the proposed method always returns a lower average value and a smaller standard deviation in comparison to the methods of A-star ($l_1$) and BFS ($b_w^{*}=0.7$), indicating better performance than these two methods. RRT-star ($\eta =3$), benefiting from its optimal parent node selection and rewiring mechanisms, exhibits the shortest average searching time ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$ in all three scale maps, shown in Figure 7a, Figure 8a, and Figure 9a, with a lower maroon curve. But, because this algorithm needs to check different nodes in the nearby neighborhood whenever a new node is generated, it returns a much larger value of ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ or poorer performance stability.

Therefore, the proposed LH algorithm can guarantee finding a conflict-free valid path and outperforms the three benchmark algorithms in the scenario without obstacles.

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 Figure 7e, Figure 8e, and Figure 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 ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$, 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 Figure 7, Figure 8 and Figure 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 × 20, the reduction of the completion rate ∆r_c in our method goes from 94% to 89%, 83%, and 11% (94% − 83%) in total. In a map of 30 × 30, the reduction in our method is ∆r_c = 12% (from 91% to 79%), and it is ∆r_c = 18% (from 87% to 69%) in a 50 × 50 map. The reductions from A-star (l₁) are 17%, 18%, and 29%; the reductions from BFS ($b_w^{*}=0.7$) are 44%, 42%, and 39%; and the reductions from RRT-star (η = 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.

4.4. Explanations

The superiority of our method can be explained by the properly designed main track priority and the sub-track jumping mechanism. The main track priority guarantees that the planned route utilizes the hierarchical structure of an HDS system and generates a shipping path along the main track first. The sub-track jumping mechanism pushes the search jump across the storage arrays though a nearby accessible sub-track once the current one is blocked and reaches the destination from the other side, if accessible. In this way, the obstacle can be bypassed, and the total number of traversed nodes can be reduced.

In comparison, A-star (l₁) performs the second best. This method cannot plan a path to reach the destination directly due to the fixed traffic constraints on sub-tracks, and therefore it always returns an alternative path after traversing more nodes. The BFS algorithm searches the storage areas when moving along the main tracks by setting the optimal breadth direction weight $b_w^{*}$ as 0.7. But, BFS ($b_w^{*}=0.7$) usually traverses more nodes than other methods and consumes almost the longest computation time. Although it can identify the destination, the generated path does not always comply with the traffic constraints and is thus infeasible, returning a lower completion rate. RRT-star (η = 3) is heavily influenced by optimal parent nodes’ identification and rewiring. Although this method may find the optimal path with the smallest searching time, result stability is low, and it sometimes has to traverse the largest number of nodes, increasing the overall computational complexity in the end.

In conclusion, through comparison to the properly modified benchmark algorithms of A-star, BFS, and RRT, per the requirements of the HDS system in both the scenario without obstacles and the scenario with an increased number of randomly settled obstacles, we demonstrate that our method is more feasible, optimal, scalable, and robust.

5. Application in Real-World HDS System

We conduct a case study to apply our proposed method to Huade Co., Ltd. to check the feasibility of our algorithm in a real-world HDS system. The conducted HDS system has a similar shape and size (22 × 20) as the fulfillment center, and it can be topologically represented in a medium-scale mesh-tree grid map. The topological representation of the conducted HDS system is displayed in Figure 11. Four four-way shuttles are allowed to operate in this HDS system. To simplify our case study and match the design of our algorithm, only one four-way shuttle and one conveyor are considered. The start position of shuttle D (bottom left corner) is the actual starting shipping position in our case study.

5.1. Settings of the Case Study

We implement our algorithm of the industrial microcontroller in Huade Co., Ltd. to plan a proper shipping path and then evaluate the operability of the planned path by executing the four-way shuttle with a conveyor to finish the whole shipping operation along the planned path. Settings of the industrial microcontroller are listed in

Table 1. The setting of the industrial microcontroller for the case study of Huade Co., Ltd.

Item	Specifications
CPU	NXP i.MX 6ULL ARM Cortex-A7 processor
Memory	512 MB RAM
Storage	8 GB onboard storage (eMMC/Flash)
Operations Temperature	−40 °C to +80 °C

. A time window of 120 s is assigned to cover the time it takes to transport the cargo along the planned path, as well as the extra time consumption from other necessary shipping operations. The other necessary shipping operations, as well as the parameters that may cause extra time consumption, are system initialization (microcontroller bootup and shipping equipment warm up), shipping speeds with and without cargo loaded, operations of load and turn, the size of shipping equipment (same length/width for four-way shuttles and conveyors), and the size of the storage position. The necessary shipping operations with the parameters are summarized in

Table 2. The other necessary shipping operations with the parameters.

Operations/Parameters	Time Consumption
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

.

Similarly to numerical studies, scenarios without obstacles and with randomly settled obstacles at 5%, 10%, and 20% occupations are considered. The average and standard deviation of ‘Searching Time’ and ‘Number of Traversed Nodes’ ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ are calculated to evaluate the optimality of the proposed algorithm. The ‘Completion Rate’ $r_c$ is adopted to analyze the feasibility and robustness of our method. The operational feasibility for industrial production is checked by executing the cargo transportation along the planned path, and the average and standard deviation of the shipping time, ${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$, are the two metrics. The three benchmark algorithms are A-star ($l_1$), BFS ($b_w^{*}=0.7$), and RRT-star ($\eta =3$). All of the simulations are executed 100 times with the same seeds.

5.2. 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 ${\overline{t}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, ${\overline{n}}_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$, and ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ are shown in Figure 12a–d to compare the computation cost. The average ${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ and standard deviation ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ 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 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 (η = 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 ≫ 1.2 ms). Therefore, our LH method is more stable and performs better overall than RRT-star (η = 3). The values of the average and standard deviation of the searching time of LH and RRT-star (η = 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

$$CI={\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}\pm 2.58{\displaystyle \frac{{\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}}{\sqrt{100}}}.$$

(14)

Values of the average ${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ and standard deviation ${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$, as well as the ‘Completion Rate’ $r_c$ and the results of the CI, are listed in

Table 3. The confidence interval for shipping time under the two scenarios.

HUADE Co., Ltd.	LH	$\mathbf{A}-\mathbf{Star} ({\mathit{l}}_1$)	$\mathbf{BFS} ({\mathit{b}}_{\mathit{w}}=0.7$)	$\mathbf{RRT}-\mathbf{Star} (\mathit{\eta }=3$)
Scenarios without obstacles
$r_c$	99%	99%	98%	99%
${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	11.6	11.6	11.6	14.3
${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	0.15	0.25	0.35	1.20
CI	$\left(11.56, 11.64\right)$	$\left(11.54, 11.66\right)$	$\left(11.51, 11.69\right)$	$\left(13.99, 14.61\right)$
Scenarios with 5% obstacles
$r_c$	92%	91%	87%	89%
${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	12.1	12.9	13.4	14.8
${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	0.2	0.4	0.6	1.8
CI	$\left(\mathrm{12.05,12.15}\right)$	$\left(\mathrm{12.8,13.0}\right)$	$\left(\mathrm{13.25,13.55}\right)$	$\left(\mathrm{14.34,15.26}\right)$
Scenarios with 10% obstacles
$r_c$	89%	87%	73%	78%
${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	13.7	14.2	15.1	16.7
${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	0.3	0.6	0.9	2.4
CI	$\left(\mathrm{13.62,13.78}\right)$	$\left(\mathrm{14.05,14.35}\right)$	$\left(\mathrm{14.87,15.33}\right)$	$\left(\mathrm{16.08,17.32}\right)$
Scenarios with 20% obstacles
$r_c$	84%	82%	66%	67%
${\overline{t}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	15.3	16.4	17.8	19.2
${\mathrm{s}\mathrm{t}\mathrm{d}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}$ (s)	0.4	0.9	1.3	3.1
CI	$\left(\mathrm{15.20,15.40}\right)$	$\left(\mathrm{16.17,16.63}\right)$	$\left(\mathrm{17.46,18.14}\right)$	$\left(\mathrm{18.40,20.00}\right)$

. 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.

It is also worth mentioning that although the proposed method can perform better than all three benchmark algorithms in both the numerical and case studies, Huade Co., Ltd., the cooperator in our study, has not applied our method in real production. An acceptable ‘Completion Rate’ for path planning and storage transportation in this company should always be higher than 95% for medium-size HDS systems, given two standard deviations. Therefore, as displayed in Table 3, only under the scenario without obstacles can the proposed algorithm be applied, but this ideal condition seldom happens. In a word, we will continuously improve our method for potential future industrial operations.

6. Conclusions

This paper presents a novel two-step path planning method for smart HDS systems in warehousing logistics. The proposed method first topologically represents the HDS layouts into mesh-tree grid maps and then introduces the Lattice (mesh-grid)-Hopping (jumping) algorithm for path planning. During representation, our method accurately captures the spatial and physical information, as well as the traffic constraints of the HDS layout, through a hierarchical ‘backbone–branch’ tree structure. The main track, the aisle where cargo is shipped along to the storage area from the entrance, is the backbone, and sub-tracks, where the shipping cargos are guided to the specific storage position, are branches. We designed a main track priority strategy and jump mechanism in our algorithm based on this ‘backbone-branch’ tree structure. By implementing these two designs, strong feasibility and optimality with low computational costs can be obtained without obstacles. Given obstacles in both the main track and the sub-tracks, the proposed algorithm still displays rigid stability and robustness. Both scenarios are theoretically proven.

Practically, numerical and case studies were undertaken to check the usability of our method. Three renowned benchmark algorithms, A-star ($l_1$), BFS ($b_w^{*}=0.7$), and RRT-star ($\eta =3$), after being properly modified according to the HDS system, were adopted for performance comparison. Two scenarios, without obstacles and with randomly settled obstacles, were considered. In the numerical study, three map scales were simulated to compare different types of HDS systems in modern warehousing logistics. The case study was conducted in a real warehousing environment from Huade Co., Ltd., our cooperator. A time window of 120 s was assigned to check the operational feasibility for industrial production.

In both the numerical and case studies, our proposed algorithm performs best, with the lowest computational cost and the highest feasibility when obstacles and time windows are given. These results exhibit the algorithm’s exceptional efficiency, stability, and robustness. Although the feasibility of the proposed algorithm is not sufficient to guarantee direct production deployment yet, our method still significantly enhances the likelihood of planning viable shipping routes under production settings when operational challenges cannot be neglected. In a word, our method displays outstanding performance and solid practical usage in modern warehousing logistics, and it could be a potential solution for difficulties in modern HDS systems.

For future work, our method can serve as a foundation for developing more efficient and robust algorithms in advanced HDS systems. Future algorithm development could integrate machine learning algorithms, e.g., reinforcement learning, to optimize time window allocation and enhance robustness when larger HDS systems are considered. With multiple shipping equipment in consideration, our method can be extended to design smart multi-vehicle cooperative transportation systems. Because lower computational cost usually implies low energy consumption, the proposed method can also be applied for sustainable equipment development for modern warehousing transportation.