The model uses a yard plane that is perpendicular to the quay shoreline, ensuring that no horizontal transportation equipment enters the box area and transfers operation with yards at both ends of the box area. Fig. 1 depicts the layout of the automated quay with 4 shore bridges and 12 yards built up. In this paper, a two-way single-lane path is chosen, and the AGV can travel to the nearby passable four nodes or wait in place. The grey obstacle designates the yard buffer region, which is regarded as an obstruction while the AGV conducts horizontal transportation operations.

The automated terminal’s AGV path planning system is a sophisticated system made up of numerous components, including shore bridges, yards, buffer zones, magnetic pegs, etc. The AGV horizontal transport area is taken into consideration, and the map is modeled using the grid method. Cells are used to represent the obstacle information of the AGV horizontal transport area, and their weights are Boolean variables where passable nodes are represented by 0 and impassable nodes by 1. The obstacle cells and the AGV cannot overlap.

A centralized control strategy is employed to discover a solution for all AGVs using a single central processing unit for horizontal transport operations at terminals with changeable surroundings. A wireless communication system, such as a communication interaction protocol between AGVs and the console, is required to implement communication between AGVs due to their constant movement and variety of horizontal transport tasks. The AGV sends the task starting point, the task deadline, and the following task receipt. Then, the AGV communicates the task start point, task finish point, and AGV cart number to the console, which then sends each cart the determined conflict-free path. This paper chooses user datagram protocol (UDP) wireless network communication, sets the network IP addresses of the AGV and console in the same IP network segment, and uses a wireless Wi-Fi signal to connect to the router to realize the communication between them while taking the cost of communication and the convenience of task operation into account. In the four corners and the center, a total of five WiFi access points (APs) are placed, using dual-band (2.4 and 5 GHz) APs, and installed at a height of 2–3 m above the ground to achieve complete WiFi coverage.

1. Intersection conflict;

The intersection conflict is produced, as seen in Fig. 2a, when AGVi,AGVj move vertically toward the intersecting grid Vm and reach the same grid point Vm at once at time point t. Add constraints (AGVi,Vm,t) for AGVi and (AGVj,Vm,t) for AGVj, respectively, to this conflict binary tree node and treat it as a point conflict (AGVi,AGVj,Vm,t).

2. Phase conflict;

A phase conflict develops when AGVi and AGVj travel to the same grid node Vn from opposing directions and arrive at the same grid point Vn at the same time at time point t, as shown in Fig. 2b. As with the intersection conflict, it is referred to as a point conflict (AGVi,AGVj,Vn,t) and constraints are added to AGVi(AGVi,Vn,t) and AGVj(AGVj,Vn,t) on the conflict binary tree nodes, respectively.

3. Exchange conflict;

As seen in Fig. 2c, an exchange conflict occurs when AGVi, AGVj move in opposite directions to exchange the positions of grid points Vm, Vn at time point t. Think of it as an edge conflict (AGVi,AGVj,Vm,Vn,t), and apply constraints toAGVi(AGVi,Vm,Vn,t) and AGVj(AGVj,Vm,Vn,t) on the conflict binary tree nodes, respectively.

4. AGV failure.

As seen in Fig. 2d, the equipment must wait while the faulty AGVi stops at the grid point Vm and the planned path AGVj crosses the fault point. This model does not take into account the possibility of an AGV conflict due to a fault.

On the automated terminal, both point conflicts and edge conflicts occur. After determining the shortest path for each AGV using the Manhattan distance-based A * algorithm, the search proceeds on the conflict binomial tree. AGV mobility is constrained by each node on the conflict tree, comprising a set of constraints (N.constraints), a solution (N.solution), and a total cost (N.cost).

Fig. 3 illustrates an instance of AGV conflict at the terminal. Each AGV must plan its entire route from the shore bridge to the stacking yard. In the diagram, and represent two shore bridges S1 and S2, while G1 and G2 denote two stacking areas. Both AGVs’ routes have lengths of 3: AGV1:S1,A1,D,G1;AGV2:S2,B1,D,G2. They simultaneously reach grid point D at time step t2, resulting in a conflict. The decision AGV1 is for to wait until a certain time point, and the AGV with the shortest total path length is designated to apply the restriction. Consequently, N.solution = 7 emerges as the optimal solution in this scenario.

Fig. 4 displays the related conflict binary tree. The initial paths of AGV1 and AGV2 from the shore bridge to the heap and the yard are calculated using the underlying A* algorithm as N.solution={AGV1:S1,A1,D,G1},{AGV2:S2,B1,D,G2}; this results in N.cost = 6. The root node of the conflict tree corresponds to an empty constraint set, denoted as N.constraints={ϕ}. The root node is where this data is kept. A conflict (AGV1,AGV2,D,t2) occurs when both AGVs arrive at the raster point D at the time t2 during the verification of the supplied solution. Hence, the target node is not the root node.

Two new child nodes are created to resolve the dispute. In contrast to the right child node, which adds constraints N.constraints={(AGV2,D,t2)}, the left child node adds constraints N.constraints={(AGV1,D,t2)}. To determine the best path while adhering to the new constraint, the left child node’s underlying A* search is invoked. To execute this, AGV1 must wait for a time point at A1 or S1, after which its new path becomes: {S1,A1,A1,D,G1}, whereas AGV2’s path in the left child node remains unchanged. The total cost for the left child node N.cost is 7.

Similarly, in creating the appropriate child node, the cost for Node N is 7. The OPEN node contains both child nodes. The left child node, which exhibits the lowest cost, is selected for expansion in the subsequent iteration of the while loop, confirming the underlying path. The left child node is identified as the target node because no conflict exists, making its solution N.solution the optimal one for this particular dock conflict case.

1. The shore bridge and yard are located in a fixed and well-known location, and there is one shore bridge for each section of the yard box;

2. Every AGV has an identical type definition;

3. The AGV runs at a steady pace, even while turning;

4. Without taking into account the impact of variables like failure, weather, and electricity while AGVs are being driven;

5. The idle AGV parking space is built up as a static obstruction in the buffer zone between the shore bridge and the yard, making it impossible for the AGV to move through when performing horizontal transportation operations;

6. AGVs can load and unload containers quickly between the yard and the beach bridge;

7. Every AGV grid route is accessible from both directions;

8. Several AGVs may be permitted to occupy the grid at the shore bridge, but only one AGV is permitted to occupy each grid at any time point.

In representing the path network of AGVs in the horizontal transport area of the automated terminal, the path network of AGVs is represented by the G=(V,E,ob) directed weighted graph. where V denotes the set of all node numbers of the AGV grid graph on the automated terminal, V=[(x1,y1),(x2,y2),…,(xn,yn)], where n×n denotes the number of grids, and E is the set of edges of V, which denotes the length corresponding to each V. obdenotes the set of obstacle coordinates ob=[(xm,y1),(xm,y2),…,(xm,yn)]. Other variables are set as shown in Table 1.

They should also be separated from the surrounding text by one space.

Objective function:

(1)min∑i=1NConi

Constraints:

(2)Xij={1,AGVa accesses node j after accessing node i0,Otherwise}

(3)∑i=1N∑j=1NXij=1

(4)T=Cov

(5)f(i)=g(i)+h(i)

(6)h(i)=|Pi[ t ][ 0 ]−gi[ 0 ]|+|Pi[ t ][ 1 ]−gi[ 1 ]|

(7)Td=∑i=1ktdi

(8)mov={1,Indicates that the AGV is waiting in place0,Otherwise}

(9)Co=∑i=1kPi−1

Eq. (1) indicates that the objective function of this model is the total cost minimization, Eqs. (2) and (3) indicates that each node is visited by AGV at most once at the same time, Eq. (4) indicates the time for all AGVs to complete the task, Eq. (5) indicates the A* algorithm used at the bottom, where f(i) is the estimation function, g(i) is the actual cost from the starting node to node i, h(i) is the estimated node i to the target node cost, Eq. (6) denotes the heuristic function using the Manhattan distance-based A* algorithm, Eq. (7) denotes the total waiting time, Eq. (8) denotes the choice of movement method for the AGVs, including waiting in place or moving one time step to a neighboring node, and Eq. (9) denotes the total cost of all AGVs to complete the task.

IBCBS(ωH,ωL) denote IBCBS uses ωH in high level and ωL in low level. IBCBS(ω,1) is a special case that focal search only used at a high level and IBCBS(1,ω) is focal search only used at a low level. The cost of IBCBS(ωH,ωL) is at most ωH∗ωL∗C∗. This also proves the boundedness of the algorithm.

The expansion cost of both IBCBS is no more than ω times higher than the ideal solution. Moreover, within OPEN, at least one CT node consistently aligns with every viable answer. Due to its systematic search, it eventually identifies all solutions. Consequently, IBCBS concludes.

Algorithm 1 presents the pseudo-code for a low level of IBCBS which uses a Manhattan distance-based A* algorithm with focal search (f,hc) for each single AGV path planning.

Algorithm 2 presents the pseudo-code for a high level of IBCBS which use focal-search (g,hc) to choose the cost of nodes in CT lower than ω∗g.

Set the number of AGVs as 10, 20, 30, 40, 50, and 60, and then randomly select the starting and endpoint based on Table 1. The shore bridge can be repeatedly selected. For AGV = 10, 20, 30, the starting point includes all shore bridges, and the endpoint is the yard. For AGV = 30, the endpoint includes all yards. However, for AGV = 40, 50, 60, 30 units move from the shore bridge to the yard, while the others move from the yard to the shore bridge. The starting point and endpoint remain fixed after random selection.

Set ω = 1.1. Four methods—IBCBS (ω, 1), IBCBS (1, ω), IBCBS (ω,ω) and CBS—are utilized for 100 experiments each. The effective time is set to 60 s, and if a solution cannot be obtained within this time, it is considered as unsolvable. Despite fixing the number of AGVs and algorithms, the computing time for each experiment differs while maintaining the same total path length. Thus, the computing time for 2400 experiments is depicted in Fig. 7.

It is evident from Fig. 7 that with a smaller AGV scale of 10, 20, 30, conflicts among AGVs are fewer. The effectiveness of focal search between the high-level and low-level in IBCBS is less pronounced. Additionally, the discrepancy between CBS and IBCBS (ω, 1), IBCBS (1, ω), IBCBS (ω,ω) is marginal. However, as the AGV scale increases to 40, 50, 60, AGV conflicts surge. IBCBS sacrifices optimality, leading to a slackening effect in both the high-level and low-level operations. Notably, CBS’s operational time is significantly longer compared to IBCBS (ω, 1), IBCBS (1, ω), IBCBS (ω,ω).

Fig. 8 illustrates that the effectiveness of bounded suboptimal search is not immediately evident with a low number of AGVs. Conflicts are minimal, below 30 units, resulting in comparable computing times for the six methods. However, as the AGV count exceeds 30 units, both point and edge conflicts rise, consequently escalating computing times for all six methods. The CBS algorithm consistently strives for optimal solutions, resulting in longer operational times. Due to convergence, GCBS algorithm operation times are extended, while BCBS operation times are marginally better than CBS and less than the other three IBCBS algorithms. The three IBCBS algorithms adopt focal search with different levels of permissive optimality constraints, substantially reducing runtime and enhancing computational efficiency. Among them, IBCBS (1.0488, 1.0488) uses focal search at both high and low levels, exhibiting the shortest operational time.

Fig. 9 indicates that CBS consistently yields the shortest total path length regardless of AGV count, with IBCBS following as the second shortest. At 10 AGVs, where path conflicts are absent, the total path length is identical for all four methods. Even with an increased AGV count, the total path length of the three IBCBS methods cannot exceed 1.1 times that of CBS due to CBS’s optimal nature, resulting in the shortest total path length. IBCBS is constrained by *cost, set at 1.1. By incorporating additional measures at the low level, potential conflicts can be averted. Consequently, IBCBS (1, 1) displays a shorter path length than IBCBS (1.0488, 1.0488) and IBCBS (1.1, 1).

Table 3 presents specific experimental data. The reduced time is calculated as (CBS’s average run time-IBCBS’s average run time)/CBS’s average run time, expressed as a percentage. Meanwhile, the over cost is determined by (IBCBS’s total path length-CBS’s total path length)/CBS’s total path length, also as a percentage. The results indicate that, in line with the analyses in Figs. 7 and 8, CBS exhibits the lowest operation time at AGV = 10, 20, 30; whereas, at AGV = 40, 50, 60, IBCBS (1.0488, 1.0488) demonstrates the lowest time, reduced by up to 64.06%. Regarding the total path length, as observed in Fig. 9, the sequence from shortest to longest is CBS, IBCBS (1, 1.1), IBCBS (1.0488, 1.0488), IBCBS (1.1, 1). All variants of IBCBS maintain an over the cost of no more than 1.2%.

Based on the analysis, when a low total path length is desired, the CBS algorithm is automatically preferred. For fewer AGVs (10, 20, or 30), CBS stands as the best choice due to its shorter computation time and total path length. On the other hand, when more AGVs are involved (40, 50, or 60), the IBCBS algorithm becomes the selection due to its shorter computation time and its ability to accept a more costly solution (1.0488, 1.0488).

AGV counts were set to 10, 20, 30, 40, 50, and 60, with starting and ending points chosen randomly from Table 1. The shore bridge can be selected repeatedly, without considering potential conflicts. For AGV = 10, 20, 30, the starting and ending points are selected from the shore bridge to the yard. For AGV = 40, 50, 60, 30 units are chosen from the shore bridge to the yard, while the remaining units are from the yard to the shore bridge.

In Fig. 10, a horizontal comparison highlights that as the number of AGVs increases, conflicts intensify, resulting in fewer valid points and declining success rates across all three algorithms. However, with the same AGV count, employing focal searches at the high level extends the search range and enhances the success rate, leading to an increased number of green points. These green points surpass the red and blue ones because only IBCBS (1, 1.1) and IBCBS (1.0488, 1.0488), which conduct focal search solely at the low level, might avoid potential conflicts and exhibit lower operation times.

Fig. 11, following the analysis from Fig. 10, displays the average run times of the three algorithms across 100 experiments. As the number of AGVs increases, all three algorithms take longer to compute. Notably, IBCBS (1.1, 1) takes significantly more time than IBCBS (1, 1.1) and IBCBS (1.0488, 1.0488). However, IBCBS (1.0488, 1.0488) exhibits the fastest operation time, employing focal search at both top and bottom levels, encompassing a wider search range.

Table 4 demonstrates specific success rates and average operation times using the three algorithms after randomly choosing starting and ending points. Based on the analysis, IBCBS (1.1, 1) attains the highest success rate among the three algorithms, while IBCBS (1.0488, 1.0488) records the shortest operation time. If the problem model prioritizes a higher success rate over operation time, selecting IBCBS (1.1, 1) is recommended, despite its longer operation time. Conversely, if a higher operation time is required, choosing IBCBS (1.0488, 1.0488) would be preferable.

Set the effective time limit at 60 s, beyond which the solution is considered invalid. Choose 60 AGVs, with half of them moving from the shore bridge to the yard and the other half from the yard to the shore bridge. Fix the values at 1.01, 1.05, 1.1, 1.5, and 2.0 for ω, and opt for IBCBS (ω, 1) with the highest success rate among the three methods. Conduct 100 experiments.

Fig. 12 depicts the computing time for 500 experiments, where times exceeding 60 s are deemed invalid. Experiment configurations, particularly the starting and endpoint selections, are random, occasionally resulting in computing times surpassing the limit. For instance, at ω = 1.01 and 1.05, some instances exceed 60 s, while other data cluster around 7.0 s. At ω = 1.01, the operation takes longer (illustrated by the green squares), with fewer instances indicating the lowest success rate compared to other data.

Fig. 13 displays the average operation time and success rate, excluding invalid data where operation times exceeded 60 s. With smaller ω values, the focal search scope decreases, making it challenging to find valid solutions through Focal_list traversal. Therefore, success rates rise and average run times decline as ω increases from 1.01 to 1.05 and 1.1. As ω grows larger, the Focal_list node count increases, enhancing the chance of the target node’s existence. Beyond ω = 1.1, the success rate plateaus, yet increasing ω widens the search range, enabling faster discovery of nodes with fewer conflicts, thereby enhancing algorithm efficiency.

Table 5 presents specific success rates and average operation time data for varying ω values in IBCBS (ω, 1). Based on the analysis, when dealing with 60 AGVs in the automated terminal, ω = 2.0 is preferable for high success rates and rapid computing times, whereas ω = 1.1 is preferable for maintaining a high success rate with excellent solution quality.