Container overturning (or secondary handling) is a key factor affecting terminal efficiency. Research in this area primarily focuses on reducing the number of ineffective handling operations through mathematical modeling and heuristic strategies. Galle et al. established an integer model that adjusts the stacking state of containers within the yard by utilizing idle time of yard cranes, aiming to minimize the expected container overturning rate (Galle et al., 2018). Similarly, KIM et al. designed a heuristic algorithm for container overturning problems with uncertain sequences, based on the principle that each container has an equal probability of becoming an obstacle during yard crane lifting operations (Kim, 1997). Lee et al. considered the impact of stacking state on the number of container overturnings and established a 0–1 programming model to pre-allocate stacking positions, thereby optimizing yard overturning operations (Lee and Chao, 2009).

Based on this, Matthew et al. proposed a novel integer programming model for selecting unloading bays and solved it using an extensible heuristic algorithm (Petering and Hussein, 2013). Huang et al. classified the overturning problem according to specific processes and proposed two different relocation strategies to handle different operational scenarios (Huang and Lin, 2012). Caserta et al. developed an approximation algorithm using the Corridor Method (CM) to optimize the overturning of obstructed containers under a fixed lifting sequence (Huang and Lin, 2012). Sequence integration is also a core topic; Ji et al. established a mathematical model for the loading and unloading sequence and overturning strategy in multi-bridge parallel operations (Ji et al., 2015). Zeng et al. proposed a method for simultaneously optimizing the extraction sequence and overturning strategy to minimize the number of overturned containers, and verified its effectiveness through a custom heuristic algorithm (Zeng et al., 2019). Furthermore, Galle et al. integrated yard crane scheduling and overturning issues into a unified problem, addressing the optimization of storage, extraction, and unloading locations simultaneously (Galle et al., 2018). Regarding the Stochastic Container Overturning Problem (SCRP), Feng et al. proposed a service strategy aimed at improving extraction performance under uncertainty, and evaluated the results based on the number of overturned containers and truck waiting time (Feng et al., 2020). Zweers et al. introduced a model that divides container movement into two stages: preprocessing and relocation, and developed an optimized branch-and-bound algorithm for this purpose (Zweers et al., 2020). Tanaka et al. proposed an iterative deepening branch-and-bound algorithm for both constrained and unconstrained variables, and analyzed the lower bound of the total number of overturned containers (Tanaka and Voß, 2019).

Efficient gate and yard operations are highly dependent on the coordination of container truck (CT) arrivals. Recent research has focused on Container Truck Appointment Systems (CTAS) and data-driven predictive models (Chen et al., 2020). Li et al. proposed a novel integrated deep learning method for predicting CT arrivals within flexible time slots (1 to 24 hours), which helps balance external CT arrivals with the availability of yard resources (Li et al., 2024c, 2025). Sun et al. addressed the appointment quota optimization problem using a data-driven approach that combines data mining with mathematical modeling, aiming to reduce total turnaround time (Sun et al., 2022). Abdelmagid et al. conducted a comprehensive review of CTAS models, focusing on specifying time slots that meet both terminal constraints and shipping company objectives (Abdelmagid et al., 2022). Congestion management is a recurring focus. Kim et al. proposed an appointment scheduling method that considers CT waiting times and uses a Frank-Wolfe-based heuristic algorithm for solving (Yi et al., 2019). Zhang et al. optimized CT appointments to reduce waiting times at gates, yard bridges (for target containers), and automatic guided vehicles (AGVs) within the yard (Zhang et al., 2019). Zhen et al. conducted a quantitative study on yard congestion using a mixed integer programming model aimed at minimizing total travel time, and solved it using a zigzag optimization meta-heuristic (Zhen, 2016). Adrián et al. used discrete event simulation to show that CTAS can effectively reduce repeated container re-picking and waiting times (Ramírez-Nafarrate et al., 2017). Zhao et al. analyzed the impact of CT arrival information quality on container turning efficiency by classifying CTs according to appointment and arrival status (Zhao and Goodchild, 2010). Yu et al. established a gate optimization model that aims to minimize operating costs and ensure constant gate efficiency while considering the randomness of CT waiting times (Yue et al., 2006). He et al. established an optimization model for the joint scheduling of external container truck appointments and automated rail-mounted gantry cranes (He et al., 2023). Furthermore, a systematic review by Weerasinghe et al. (2024) consolidates various operations research applications in container terminals, providing a comprehensive context that justifies the selection of our collaborative scheduling optimization approach amidst diverse methodological frameworks.

The integration of different equipment types, including yard cranes (YCs), quay cranes (QCs), and berths, is crucial to ensure seamless dock processes. Lee et al. studied the container overturning problem through an algorithm inspired by neighborhood search to optimize equipment movement (Wu et al., 2015). Kim et al. developed heuristic rules to determine unloading locations and identify containers to be picked up in multiple yard areas with the same priority (Kim and Hong, 2006). For large-scale problems, Zhang et al. proposed an improved machine learning algorithm that combines optimization methods to address the container overturning challenge (Zhang et al., 2020). Ku et al. developed an Expected Relocation Index (ERI) heuristic method in a stochastic dynamic programming model to overcome the computational limitations of exact algorithms (Ku and Arthanari, 2016). Collaborative scheduling at the ship-shore interface has also witnessed innovation. Li et al. proposed the TEU-BQCT_CQASOA method, a joint scheduling scheme for berths, quay cranes, and yard trailers, aimed at alleviating cargo accumulation pressure (Li et al., 2024a). Zhen et al. studied the joint allocation of berths and yard space in “tower ports” and considered flexible storage strategies (Zhen et al., 2025). Huang et al. developed an optimization model that combines container pickup sequences with stowage plans (Huang et al., 2024). Regarding the interference problem of yard cranes, Wu et al. studied multi-yard crane scheduling considering safety distances and mutual interference (Wu et al., 2015). Gao et al. constructed a multi-objective model using a yard trailer-based partitioning strategy, aiming to minimize the longitudinal movement distance of yard cranes and the waiting time of yard trailers (Gao and Ge, 2023). Azab et al. introduced the Block Replacement Problem with Booking Scheduling (BRPAS), proposing a binary integer programming (IP) model to minimize the number of container overturns while considering yard trailer bookings (Azab and Morita, 2022b, 2022). In terms of search space optimization, Zheng et al. proposed heuristic rules and plane segmentation methods to narrow the search space for the overturning problem, demonstrating high robustness and shorter running time (Zheng and Sha, 2020; Zheng et al., 2018). Finally, Jovanovic et al. applied an ant colony optimization algorithm based on a specific pheromone matrix to solve the container yard overturning problem according to the yard bridge operation time (Jovanovic et al., 2019).

The ongoing evolution towards “Smart Ports” emphasizes integrated, data-driven operations to enhance efficiency and resilience, as highlighted in the systematic literature review by Belmoukari et al. (2023). Despite extensive research on container overturning, trailer appointment system (TAS), and yard crane scheduling, there are still several key gaps in the existing literature. Most existing Container Block Rotation Problem (BRP) models either operate under the assumption of deterministic information, where the exact arrival sequence of all trailers is known, or in a completely uncertain environment. However, automated terminals in reality typically operate in a partial information environment, where the group of trailers has been identified within the appointment window, but their internal arrival sequence remains random. In addition, previous studies often decoupled the trailer appointment quota from the yard crane task allocation, failing to consider the dynamic cost trade-off between yard crane energy consumption and the economic impact of trailer delays in the event of disruptions. Importantly, there is currently a lack of focus on operational resilience, which refers to the inherent ability of the system to maintain or quickly restore efficiency when faced with mismatches between container storage locations and actual trailer arrival sequences, a problem further exacerbated by the uncertainty of global shipping.

This paper contributes to the field by developing a collaborative scheduling optimization model specifically designed for scenarios where partial information is available. This model effectively bridges the gap between static planning and stochastic operations by coordinating extraction sequences and yard bridge tasks to minimize total operational costs, thereby achieving operational resilience through rapid efficiency recovery. A significant methodological contribution is the introduction of an improved particle swarm optimization algorithm, the Random History Global and Local Best Particle Swarm Optimization (RCHGLBPSO) algorithm. This algorithm enhances particle sharing and expands the search space to prevent premature convergence in high-dimensional yard management problems. By analyzing the interactions between container truck grouping, extraction sequences, and yard bridge task allocation, this study provides quantitative evidence that this integrated scheduling significantly reduces overturning, waiting, and moving costs. These research findings offer practical insights for terminal operators to enhance the intelligence and resilience of automated yard management systems in an increasingly volatile shipping environment.

These assumptions are made to focus on the core problem of collaborative scheduling under partial information. While they simplify reality, the model is most applicable to dedicated import container yards with strict appointment systems and stable operational plans. The assumption of no new arrivals allows us to isolate the retrieval and relocation problem.

Equation 1 represents the objective function, which aims to minimize the total cost of container operations (including the cost of container relocation, yard crane movement, and waiting time of external container trucks).

Equation 2 formulates the flow balance constraint for yard crane movements between bays, which are modeled as nodes in a network. This constraint ensures flow conservation at each node, where the inflow must equal the outflow except at the crane’s initial and final positions, thus accurately describing the crane’s complete routing path during the scheduling period θ.

Equation 3 defines the travel time tabθ for a yard crane moving between two bays (from a to b), which is calculated based on the maximum horizontal moving time across bays or vertical moving time along the stack, reflecting the realistic movement pattern of yard cranes.

Equation 4 then computes the total time tbθ required for the yard crane to complete a container retrieval task at bay b, which integrates the task service time (including possible relocation operations) and the travel time from the previous location, thereby capturing the cumulative time cost in the operational sequence. Together, these equations establish a network flow-based framework to optimize the yard crane’s scheduling and movement efficiency.

Equation 5 denotes the waiting time of external container trucks at the yard gate. The waiting time is calculated as its positive value when it is positive; otherwise, it is taken as 0.

Equation 6 indicates that the departure time of an external truck retrieving target container *i* from the gate equals the sum of the start time of container *i*’s operation, the time consumed by relocation, and the time taken for the yard crane to move between bays.

Equation 7 defines the relocation time t_1i as a function of the relocation decision variable y_z(1-k), represents the time consumed by relocation for retrieving target container *i*.

Equation 8 calculates the movement time t_2i based on the yard crane scheduling decision variable S_ijc, which determines the travel distance between consecutive container retrieval operations.

Equation 9 specifies that operations begin only after the external truck arrives at the designated bay in the yard.

Equation 10 the operation of the next container can only start after the previous target container’s operation is completed.

Equation 11 requires that the completion time of an external truck’s operation be less than the latest permissible departure time of the truck from the port.

Equation 12 one yard crane serves one container, and the completion time is unique.

Equation 13 employs a Big-M formulation to prevent physical interference and potential collisions between adjacent yard cranes operating in the same block. This critical safety constraint specifically addresses the spatial relationship between crane cand the preceding crane c−1. The constraint M(1−Yabt)≥∑j=baY(c−1)jt ensures that if yard crane cis operating at bay bduring time slot t(i.e., Yabt=1, then yard crane c−1 cannot be operating at any bay j≥bin the same time period. This creates a sequential allocation of work zones along the bay direction, effectively maintaining a minimum safe distance between adjacent cranes. The large positive constant Mserves to activate or deactivate this constraint depending on the operational status of crane c. This formulation is essential for ensuring operational safety in multi-crane environments and directly contributes to the practical applicability of our scheduling model.

Equation 14 ensures that the time interval between the completion of operations for two consecutive containers is no less than the service time of the yard crane for the second target container.

Equation 15 defines the variable relationship between two target containers consecutively served by the same yard crane.

Equation 16 specifies that any container *i* can only be stacked at one slot.

Equation 17 indicates that any slot can only accommodate one container.

Equation 18 states that the initial position of the yard crane is at Bay 1.

Equation 19 ensures that the number of containers in each bay does not exceed its maximum capacity.

Equation 20 mandates that no container can be left suspended in the air.

Equation 21 states that when containers *i* and *j* are located at positions (b, z, l) and (b, z, l-k), respectively, and P(a) < P(b), removing *a* requires first moving *b*, necessitating a reshuffle. In this case, y_bz(l-k) is recorded as 1.

Equations 22–24 represent 0–1 decision variables.

The model in this paper is primarily solved using the improved Particle Swarm Optimization (PSO) algorithm, a population-based random search algorithm and a typical intelligent optimization technique. The effectiveness of PSO in handling complex optimization problems has been demonstrated in various fields, including energy systems and logistics (Chen et al., 2024). The core of the PSO algorithm involves initializing a set of random solutions and iteratively searching for the optimal value, with particles navigating the solution space guided by the best-known positions. In order to solve the problem of low convergence efficiency in the later stage of the standard particle swarm optimization algorithm, this paper solves the established model using a particle swarm optimization algorithm that combines random historical global and local optima (RCHGLBPSO). In addition, the particle sharing type is strengthened to expand the search area and effectively prevent the algorithm from converging prematurely. Particle i randomly obtains m (fixed) particles that are different from each other as neighborhood subgroups. Finally, the fitness values of particles in the global optimal position are updated to the previous formula through comparison.

When solving problems with the PSO algorithm, each individual is considered a particle, representing a potential solution to the model. For instance, in a multidimensional space, every point can be represented by a particle. Assume the population consists of m particles, where li=(li1,li2,⋯,liN) denotes the N dimensional position vector of the i particle. The fitness value of li is calculated to evaluate the quality of the particle’s current position; vi=(vi1,vi2,⋯,viN) represents the current velocity of the particle; pi=(pi1,pi2,⋯,piN) indicates the best position the particle has found so far; and pg=(pg1,pg2,⋯,pgN) represents the best position discovered by the entire population up to that point. During each iteration, a particle updates its velocity and position according to the following equations: Vid=ω*Vid+c1*rand()*(Pid−Xid)+c2*rand()*(λ(Pgd−Xid))+(1−λ)*(Pgd−Xid),Xid=Xid+Vid,the vector Pid−Xid represents the distance between the particle’s current position (Xid) and the best position it has previously reached (Pid); the vector Pgd - Xid denotes the distance between the particle’s current position (Xid) and the best position found by the population (Pgd); c_1 and c_2 are the individual learning factor and social learning factor, respectively, both positive constants; rand() is a random number between 0 and 1; and ω is the inertia weight, λ=1/t assign different weights to historical and neighborhood global information.

The key innovation of RCHGLBPSO lies in its update strategy for the global best position (P_gd). This approach aligns with recent trends in applying advanced meta-heuristics to complex port scheduling problems. For instance, Eldemir and Taner (2025) demonstrated the effectiveness of a hybrid meta-heuristic for quay crane scheduling, underscoring the potential of such algorithms in tackling NP-hard optimization challenges inherent in terminal operations. Our RCHGLBPSO algorithm contributes to this evolving landscape by specifically addressing the high-dimensional and partially stochastic nature of collaborative truck and yard crane scheduling. Instead of solely relying on the current global best, each particle i randomly selects m particles from the swarm to form a neighborhood. The historical best positions of these neighborhood particles are then used to influence the update, introducing more diversity and reducing the risk of premature convergence. The parameter λ= 1/t dynamically weights the influence of historical versus neighborhood information.

In this paper, the PSO algorithm is employed to solve the model, with the computational process illustrated in Figure 3.

In addition, the calculation of fitness function is mainly based on queuing theory principles. The specific calculation process is shown in Figure 4.

To effectively solve the collaborative scheduling model using the improved PSO algorithm, a suitable mapping between the particle’s position and a feasible scheduling solution must be established. In this paper, a particle is encoded as a sequence of integers representing the retrieval order of the target containers. Specifically, for a problem involving ncontainers to be retrieved, each particle is represented by a permutation of the set {1, 2,…, n}. This sequence directly dictates the order in which the yard cranes will retrieve the containers. For instance, a particle position [5, 2, 1, 4, 3] signifies that container 5 is retrieved first, followed by container 2, and so on. This encoding naturally integrates the grouping of external trucks; the sequence is processed group by group based on the known arrival information of the first group and the appointment windows of subsequent groups.

The decoding process converts the extracted sequence into a specific operation scheme, including the task allocation of the yard crane, the moving path and the necessary container turnover operation, and finally calculates the objective function value (total cost). The decoding process is divided into the following steps: first, verify the container extraction sequence defined by particles according to the initial storage state of the yard. When extracting the target container, if it is not at the top of the stack, all containers above it must be turned over to other slots in the same position. The decoding algorithm uses heuristic rules to flip the container: move the obstacle container to the top of the stack, and place the target container with the highest extraction priority in the current group at the same position, so as to minimize the number of turns of subsequent containers. Secondly, the yard crane task is assigned based on the principle of minimizing the total moving distance. Pick up tasks (including container overturning actions) are assigned to the nearest available yard crane in sequence. Calculate the movement time of the yard crane between the cargo positions, and determine the starting time of each operation according to the current crane position and the arrival time of the container truck. Finally, the assembly cost is calculated by summing up the following three expenses: container turnover cost (based on the number of container turnover), yard crane movement cost (based on the total travel distance) and container truck external waiting cost (i.e. the delay time between the arrival of container trucks and the start of extraction work). The total cost after decoding is used as the fitness value of particles to guide particle swarm optimization (PSO) algorithm to find the optimal solution.

In the iteration process of particle swarm optimization algorithm, the random update of particle position is easy to cause the solution to violate the constraints listed in the mathematical model (such as the safe distance of the freight yard bridge, the container is not suspended, and the departure time window of the container truck, etc.). In order to effectively guide the search to the feasible area, this paper uses a hybrid strategy of heuristic algorithm based on the method of modification and penalty function.

For the constraints related to the physical operation sequence (for example, the upper container must be removed before extracting the constraint 7 of the target container, and the constraint 19 that the container cannot hover), the repair heuristic rule will be directly applied in the decoding phase. When the particle sequence indicates that the buried container needs to be extracted, the decoding algorithm will automatically adjust the operation sequence before extracting the target container, and insert the necessary tilt operation to remove the obstacle container. This mechanism ensures that each solution generated in the decoding process has practical operability.

For constraints that are difficult to be corrected directly (such as the latest allowable departure time constraint 11 for external container trucks and the constraint 13 that prohibits overlapping of yard bridges), we use the penalty function method. When the solution violates these constraints, its fitness value (total cost) will be corrected by penalty. The intensity of punishment is directly proportional to the degree of violation. The fitness value calculation formula for particle evaluation is: Fitness=total cost+penalty. This mechanism imposes heavy penalties on the infeasible solution, reduces its attraction to the algorithm, and gradually guides the population to converge to the feasible solution space. This combination strategy ensures that the algorithm can maintain the strong search ability for high-quality solutions while dealing with complex constraints efficiently.

The experimental data in this paper are configured as follows: A total of 260 containers are to be retrieved. The time for a yard crane to move one bay is 3 seconds per bay, with a movement cost of 1 yuan per bay. The waiting cost for external container trucks is 0.3 yuan per minute. The time for a single container relocation operation is 2 minutes per TEU, with a container relocation cost of 20 yuan per TEU. The arrival time of external container trucks at the port follows a negative exponential distribution with an average interval of 3 minutes. After arriving at the port, external container trucks are expected to depart within approximately 30 minutes, with the latest permissible departure time being 60 minutes after arrival. Computations were performed using MATLAB R2018a, and the results were obtained on an x64 PC equipped with an Intel(R) Core(TM) i5-8265U CPU at 1.6 GHz and 8 GB of RAM.

We used the operational data of a certain day at the port as the test data, and solved it through the proposed improved PSO algorithm. The population size was 100, the maximum iteration number was 150, the inertia weight was 0.9, and the learning factor c1=c2 = 1.49418 (Clerc and Kennedy, 2002). we compiled the relocation rate, relocation cost, yard crane movement cost, external truck waiting cost, and total cost obtained from the model into Table 1.

According to Table 1, as the number of containers to be retrieved within a group increases, the relocation rate decreases when the number of containers ranges from 3 to 27, but it gradually increases between 27 and 63. The relocation cost varies with the relocation rate. The operational cost of the yard crane initially decreases as the number of containers to be retrieved increases and then stabilizes. Meanwhile, the waiting cost of external container trucks increases. This is because as the number of containers to be retrieved within a group grows, the number of external container trucks arriving at the yard also increases, leading to longer waiting times and higher costs for external container trucks. Additionally, the variation in yard crane task allocation arises from different optimization outcomes corresponding to different numbers of containers to be retrieved per group. The relocation rate, relocation cost, and external truck waiting cost remain the same in the last three cases in Table 1 because the latest departure time for external container trucks has been predefined, requiring them to leave the port within the stipulated time. During the solution process, the model has reached convergence after a certain number of iterations and is essentially optimized. Thus, knowing the arrival information of external container trucks can influence the costs throughout the operation process and optimize the container retrieval and relocation workflow. When the number of containers to be retrieved per group is relatively small, it significantly impacts the relocation cost, external truck waiting cost, and yard crane movement cost during the retrieval process. When the number of containers ranges from 9 to 27, the total cost of retrieving containers for external container trucks tends to be minimized. If the arrival sequence of some external container trucks is known, optimizing the retrieval sequence can reduce the operational cost of retrieving import containers. However, while optimizing the retrieval sequence, the waiting time for external container trucks increases, yet the total cost decreases correspondingly. Therefore, having access to external truck arrival information can effectively reduce yard operation costs and enhance operational efficiency.

The optimal allocation range of 9–27 containers per group is obtained by comprehensively considering three cost factors. If the grouping scale is too small, it will lead to low efficiency of task allocation, high cost of crane operation in the yard, and limit the optimization space of container turnover operation. If the grouping scale is too large, the waiting cost of container trucks will increase significantly. At the same time, due to the increase of stacking complexity, the container turnover rate will rebound due to the extension of waiting time. The configuration interval of 9–27 containers achieves the best balance.

The cost minimization achieved through our collaborative scheduling model directly translates into a significant improvement in operational resilience. The volume of container transfer operations decreased by 35.21% (Table 1), indicating that the adaptability of the system to container stacking interference has been enhanced. At the same time, the waiting time of container trucks has been reduced by 25-30%, showing stronger service reliability in the case of uncertain container arrival. This resilience is quantified by two key indicators: (1) compared with the first come first serve strategy, the delay caused by container transshipment is reduced by 40%; (2) The service performance in all experimental scenarios is consistent. The standard deviation of waiting time is controlled below 15%, ensuring that the terminal operation can remain predictable even if there is randomness in the arrival of container trucks.

To more intuitively illustrate the data changes, we created Figure 5 Regarding the cost of container retrieval operations, Figure 5 shows that regardless of the number of containers to be retrieved per group, the relocation cost constitutes a significant portion of the total operational cost. In contrast, the yard crane movement cost and external truck waiting cost account for a smaller share compared to the relocation cost. However, as the port’s operational volume increases, optimizing the service quality for external container trucks has a substantial impact on the port’s service efficiency. Thus, when external container trucks arrive at the port to retrieve import containers, limiting the number of scheduled external container trucks to a certain level can effectively optimize the yard relocation rate, reduce the total cost of container retrieval for external container trucks, and improve port operational efficiency.

Table 2 shows the operation data of outbound container trucks equipped with three on-site cranes. Comparing the data in Tables 1 and 2, it can be found that increasing the number of cranes in the yard will not significantly change the container transfer rate, because the location of the target containers to be recovered by the outbound container trucks remains unchanged, and the number of containers to be transferred has little change. However, with the increase of the number of on-site cranes, the mobile cost of on-site cranes will be reduced. This is because the new on-site crane can shorten the moving path when recycling the target container, thus reducing the moving cost. In addition, due to the shorter operation time of the crane in the yard, the waiting cost of the outbound container truck is also significantly reduced, which shortens the waiting time of the outbound container truck, thereby reducing its waiting cost. The data analysis shows that increasing the number of cranes in the yard can reduce the operation cost. However, there are certain restrictions on the number of cranes in the yard: the cost of cranes in the yard is high, and adding one crane in the short term cannot offset the reduction of the operating cost of cranes in the yard and the waiting cost of outward container trucks. In addition, with the increase of the number of cranes on the site, the space restriction of the existing site will become more obvious, which may cause the idle cranes on the site to be unable to operate at the same time. Therefore, port managers need to allocate resources reasonably according to the current workload to avoid resource waste.

While adding a third crane reduces the total operational cost by approximately 10-15% in our simulation, a full cost-benefit analysis must consider the high capital and maintenance costs of an additional yard crane. Terminal managers should weigh this capital expenditure against the long-term savings in operational costs and the potential revenue increase from improved customer service due to shorter truck turnaround times. The model serves as a tool to quantify the operational benefits side of this equation.

The optimal container group size 9–27 is derived from the dynamic balance between relocation, crane movement and truck waiting costs. When the group size is too small (such as three containers), the task allocation of the yard crane is unbalanced, resulting in inefficient moving path and rising cost. In addition, the limited number of containers per group will restrict the flexibility of picking order optimization and hinder the effective reduction of relocation costs. On the contrary, too large group size (such as more than 27 containers) will produce diminishing marginal benefits: Although the crane moving cost tends to be stable due to better task distribution, the waiting cost of external trucks will significantly accumulate with the increase of the number of single batch service trucks. More importantly, the complexity of stacking state of large-scale groups often leads to the recovery of relocation rate. Therefore, the range of 9–27 containers achieves the optimal balance - the benefits of reducing relocation and mobile costs by optimizing the picking order exceed the incremental growth of truck waiting costs, so as to minimize the total cost. Further analysis of adding a third yard crane shows that its main advantages are to enhance parallelism (shorten task queue and reduce truck waiting time) and optimize crane movement path through workload sharing. However, these improvement measures are restricted by the physical interference between cranes, and the marginal benefit gradually decreases, which highlights the need for port managers to weigh the pros and cons between improving operational efficiency and the high capital cost required to purchase new equipment.

Figure 6 illustrates the cost of container retrieval operations for external container trucks with three yard cranes. Similar to the case with two yard cranes, Figure 7 shows that, regardless of the number of containers to be retrieved per group, the relocation cost accounts for a high proportion of the total operational cost. The yard crane movement cost and external truck waiting cost constitute a smaller share relative to the relocation cost. However, the numerical changes are not significant, and the cost of a single yard crane is substantial. Thus, in actual port operations with a higher number of yard cranes, we can compare the external truck waiting cost and yard crane movement cost against the cost of the yard cranes to determine whether adjusting the number of cranes would reduce yard operational costs.

From the above data analysis, we conclude that the number of containers to be retrieved per group significantly affects the cost of container retrieval for external container trucks. When partial arrival information of external container trucks is known, the operational cost of container retrieval can be effectively reduced, with the total cost being optimal when the number of containers per group ranges from 9 to 27 compared to other quantities. Additionally, during import container operations, the external truck waiting cost constitutes a smaller proportion, while the relocation cost is relatively high. However, the duration of external truck waiting is closely tied to service efficiency. Therefore, while optimizing the cost of container retrieval for external container trucks, we must also consider the service efficiency for port customers. Furthermore, without increasing the waiting time of external container trucks in the yard, surplus yard cranes can be deployed to other areas to enhance operational efficiency in those regions.

By using standard particle swarm optimization algorithm and improved particle swarm optimization algorithm to solve the case, from the results, it can be seen that the results obtained using the improved particle swarm algorithm are significantly better than the standard particle swarm algorithm in terms of speed and results. Moreover, the iterative process in Figures 8, 7 shows that the improved particle swarm algorithm has faster convergence speed and better convergence performance, indicating the effectiveness and feasibility of the algorithm proposed in this paper.

To comprehensively evaluate the robustness and reliability of the proposed algorithm, we conducted 30 independent runs under the same experimental conditions. Table 3 below summarizes the statistical measures of key performance indicators, including Mean, Best, Worst, and Standard Deviation.

Figures 9–12 further intuitively show the overall performance of these operation results and the distribution of solutions. Figure 9 shows that the quartile distribution is relatively concentrated, and the median is highly consistent with the average value reported in the table, indicating that the algorithm can always converge to the stable region of the solution space. The total cost distribution (Figure 10) shows a near normal distribution centered on the average total cost of 39138.69.

In Figure 11, the average waiting cost is 34336.49 yuan, accounting for the vast majority of the total cost, followed by container transshipment cost (3933.33 yuan on average) and yard crane moving cost (868.87 yuan on average). The standard deviation of container transshipment cost and yard crane moving cost is relatively small, which are 162.76 yuan and 34.05 yuan respectively. In contrast, the standard deviation of waiting cost is relatively high (1905.87 yuan), reflecting the internal fluctuation caused by the random arrival of container trucks. Figure 12 shows the convergence characteristics of the algorithm. At the initial stage, the curve is steep and continues to decline, and then gradually tends to be stable. In the complete optimization process, especially in the late iteration stage, the extremely narrow standard deviation range shows that the search trajectories of different algorithms are highly consistent.

We compare it with genetic algorithm (GA) and simulated annealing algorithm (SA). In the experiment, 60 containers were allocated to 10 container stacks. Two on-site cranes and six groups of container trucks were used as parameters, and 50 iteration experiments were carried out. As shown in Figure 13, the fitness value of genetic algorithm reached 2798.18 in the 20th generation, and further increased to 2543.71 in the 50th generation. The improved PSO algorithm is stable at 2838.55 after 20 iterations, while the simulated annealing algorithm always maintains the highest cost level of 3604.57. Figure 14 shows that the total cost of genetic algorithm (GA) is 2543.71 yuan, the improved PSO algorithm is 2838.55 yuan, and the simulated annealing algorithm is 3604.57 yuan.

In Figure 15, the three algorithms have the same cost of 1520.00 yuan in the container transfer link. However, there are significant differences in the cost of crane movement and waiting time. Genetic algorithm (GA) performs better in the efficiency of crane movement in the field, only 179.00 yuan, while the other two algorithms need 190.00 yuan. GA reduced the waiting cost to 844.71 yuan, simulated annealing algorithm to 1894.57 yuan, and improved PSO algorithm to 1128.55 yuan. In addition, the three algorithms have achieved a similar container turnover rate of 126.67% (Figure 16), indicating that the underlying constraint processing and container serialization methods produce similar container movement patterns.

Although the comparative analysis shows that the genetic algorithm achieves the balance of exploration and utilization in specific experimental scenarios through crossover and mutation operations, the fast convergence ability of the improved PSO algorithm in the early iteration stage and its stability in different problem sizes fully demonstrate its excellent scalability and computational efficiency. The improved PSO algorithm has more advantages in the dynamic scheduling environment with limited computing time or in the actual scenarios that need to deal with large-scale container fluctuations.

The performance of the improved PSO algorithm is verified by multi-scale experiments and parameter sensitivity analysis. The experimental scenario covers 100, 200 and 400 containers, corresponding to the operation scale of small and medium-sized and large terminals respectively. The experimental results are shown in Table 4 and Figure 17.

The calculation time increased from 1.93 seconds to 11.63 seconds with the size of the problem, and the total operating cost showed a significant linear growth, rising from about 6000 yuan to more than 100000 yuan. This disproportionate increase highlights the exponential growth of the complexity and interaction between container transfer operations, yard crane scheduling and container truck waiting time with the expansion of the system scale.

In the medium-sized (200 containers) scenario, the container turnover rate reached 84.5%, while in the large-scale scenario, it fell slightly to 80.5%. In addition, most of the costs consist of waiting fees for external container trucks, accounting for 87% to 91% of the total expenditure. This fully proves that shortening the turnaround time of container trucks is the key factor to reduce the overall operating cost of the terminal. However, although the absolute cost of each container has increased, this growth rate has slowed down in large-scale scenarios, indicating that there is still room for improving efficiency in intensive operations with high density and high throughput.