Joint Optimization of Berths and Quay Cranes Considering Carbon Emissions: A Case Study of a Container Terminal in China

Abstract

The International Maritime Organization (IMO) aims for net zero emissions in shipping by 2050. Ports, key links in the supply chain, are embracing green innovation, focusing on efficient berth and quay crane scheduling to support green port development amid limited resources. Additionally, the energy consumption and carbon emissions from the port shipping industry contribute significantly to environmental challenges and the sustainable development of ports. Therefore, reducing carbon emissions, particularly those generated during vessel berthing, has become a pressing task for the industry. The increasing complexity of berth allocation now requires compliance to vessel service standards while controlling carbon emissions. This study presents an integrated model that incorporates tidal factors into the joint optimization of berth and quay crane operations, addressing both service standards and emissions during port stays and crane activities, and further designs a PSO-GA hybrid algorithm, combining particle swarm optimization (PSO) with crossover and mutation operators from a genetic algorithm (GA), to enhance optimization accuracy and efficiency. Numerical experiments using actual data from a container terminal demonstrate the effectiveness and superiority of the PSO-GA algorithm compared to the traditional GA and PSO. The results show a reduction in total operational costs by 24.1% and carbon emissions by 15.3%, highlighting significant potential savings and environmental benefits for port operators. Furthermore, the findings reveal the critical role of tidal factors in improving berth and quay crane scheduling. The results provide decision-making support for the efficient operation and carbon emission control of green ports.

1. Introduction

The global shipping industry consumes a significant amount of energy and contributes substantially to carbon emissions. It is estimated that the global shipping industry emits between 800 million and 900 million tons of carbon dioxide annually, accounting for 2% to 3% of total global carbon emissions [1]. The rational allocation of berths and quay cranes directly impacts carbon emissions: on one hand, it can reduce the time ships spend in port, thereby lowering their carbon emissions; on the other hand, it can improve the loading and unloading efficiency of quay cranes, reducing carbon emissions during their operation. Therefore, to achieve a “carbon peak” and “carbon neutrality”, it is essential to study the joint allocation of berths and quay cranes with consideration of carbon emissions [2]. Against this backdrop, the issue of port carbon emissions has become a focal point of research for scholars both domestically and internationally.

By analyzing the various workflows of container terminals [3], it is evident that carbon emissions primarily stem from two sources: first, the fuel consumption of ships and terminal equipment, and second, the indirect carbon emissions generated from the use of electrical equipment.

Ships consume energy and produce significant carbon emissions during the following stages: (i) When ships slow down to enter the port area from a certain distance offshore, the carbon emissions at this stage are several times higher than when cruising at sea. Both excessively high or low speeds can lead to increased energy consumption and carbon emissions, making it crucial to maintain an optimal speed when entering the port area for energy conservation [4]. (ii) When ships are anchored at berths, the main engine used for propulsion is typically shut down, but auxiliary engines continue to run to meet the daily living needs of the crew. The longer a ship stays in port, the higher its carbon emissions. This necessitates efficient scheduling, the rapid completion of loading and unloading operations, and minimizing ship queuing time to reduce the overall energy consumption and carbon emissions of the port [5].

Various production activities at container terminals consume a significant amount of electricity. Currently, the main source of power generation in China is still thermal power, meaning that electricity consumption corresponds to a certain level of carbon emissions. The port’s electrical equipment includes quay cranes, lighthouses, and a range of monitoring equipment, with quay cranes being the largest energy consumers. This study focuses on how to efficiently and rationally allocate berths and quay cranes under the premise of low carbon emissions, thus only considering the energy consumption of quay cranes. The energy consumption of quay cranes is proportional to their working hours. Therefore, the scheduling method aims to complete as many container loading and unloading operations as possible within the same working time of the quay cranes [6].

1.1. Literature Review

In recent years, as the importance of sustainable development in maritime logistics has increased, research on berth allocation optimization has gradually focused on reducing fuel consumption and carbon emissions. Venturini et al. (2017) introduced a multi-port berth allocation problem incorporating speed optimization and emission considerations. Their research demonstrated that optimizing vessel speeds on sailing legs between ports can significantly reduce fuel consumption and emissions, achieving up to a 42% reduction in emissions across an entire shipping network [7]. Genetic algorithms have become a common tool in this field due to their effectiveness in multi-objective optimization problems. For example, Wang et al. (2019) proposed an optimization method that integrates berth allocation with quay crane–yard truck joint scheduling, emphasizing the minimization of carbon emissions in port areas. The study showed that optimizing berth and quay crane allocation strategies using genetic algorithms can effectively reduce carbon emissions and improve port efficiency [8]. Yu et al. (2022) studied the continuous berth allocation problem considering carbon emissions and uncertainty, using a multi-objective genetic algorithm to optimize berth allocation strategies under different operational scenarios, highlighting the need for adaptive strategies in dynamic port environments [9]. Additionally, Jiang et al. (2022) proposed an integrated scheduling model for vessel scheduling and berth allocation with carbon emissions as a consideration. By using an adaptive dual-population multi-objective genetic algorithm (NSGA-II-DP), they not only reduced carbon emissions but also improved port scheduling efficiency and service quality [10]. Sharifi et al. (2023) studied berth allocation and quay crane scheduling at Rajaee Port in Iran, incorporating speed optimization and air emission considerations. By using advanced algorithms like NSGA-II and MOSA, their model provided Pareto optimal solutions, highlighting the trade-offs between economic and environmental objectives [11]. Yu et al. (2022) also developed a discrete berth allocation method under uncertainty, emphasizing the use of robust low-carbon discrete berth allocation methods to address environmental challenges in port operations [12]. Finally, Fan et al. (2019) investigated a multi-type tramp ship routing and speed optimization model aimed at reducing fuel consumption and carbon emissions. This study demonstrated the broad application potential of the genetic simulated annealing algorithm in enhancing overall maritime logistics optimization [13].

Berths and quay cranes are both limited resources in a port, and their allocation is interdependent. The efficiency of quay crane operations depends on the rational allocation of berths, while the effective turnover of berths relies on the appropriate configuration of quay cranes. Therefore, academic research often focuses on the joint optimization of these two aspects. Park and Kim (2003) developed a BAP + QCAP cost model for quay crane operations, analyzing the relationship between ship port time, the number of quay cranes, and overall scheduling costs [14]. Their study found that within a certain range, an increase in the number of quay cranes and a decrease in loading and unloading times led to a reduction in the ship’s port time. Thus, measures need to be taken in both areas to reduce quay crane operation costs. Imai et al. (2008) investigated the discrete berth allocation and quay crane allocation problem under dynamic ship arrival conditions, analyzing the BAP + QCAP (quantity and specificity) problem with the goal of minimizing the ship’s port time (waiting and loading/unloading times) [15]. Xu and Shi (2023) studied the integrated problem of continuous berth allocation and quay crane scheduling, proposing a solution based on a particle swarm optimization algorithm. Their research demonstrated that this algorithm performed exceptionally well in improving port operational efficiency and significantly reduced ship mooring times [16]. Nazri et al. (2024) developed a mixed-integer linear programming (MILP) model to optimize berth allocation in Malaysian container terminals, focusing on reducing the vessel turnaround time by 38.54%. Their approach utilized Pyomo and Google Colab to enhance computational efficiency while addressing environmental concerns [17]. Dai et al. (2023) utilized a mixed-integer model incorporating quay crane setup times and cargo types, combining offline greedy insertion algorithms with online reinforcement learning algorithms for berth allocation optimization. Their results showed that this model performed more efficiently and flexibly in handling complex real-world scenarios compared to traditional methods [18]. Tang et al. (2023) proposed a distributionally robust optimization model that accounts for uncertainties in quay crane operational efficiency. The results indicated that despite these uncertainties, the model still provided more stable scheduling results, especially in handling fluctuations in real operations [19]. Fatemi-Anaraki et al. (2020) developed a new mathematical heuristic method for scheduling waterways, berths, and quay cranes simultaneously. Considering various maritime constraints, their study showed that this method significantly improved overall maritime operation efficiency, especially in complex port environments [20]. Chargui et al. (2023) proposed a robust optimization model addressing berth and quay crane allocation while incorporating renewable energy uncertainty. This study demonstrated how balancing renewable energy usage and operational costs could improve sustainability in container terminals [21]. Li et al. (2022) investigated a bi-objective optimization model that, for the first time, incorporated the preventive maintenance of quay cranes into the integrated berth and quay crane allocation model. The experimental results indicated that this model effectively reduced ship turnaround time while lowering the penalty costs associated with maintenance activities [22]. Yin et al. (2022) proposed a two-layer model to optimize quay crane and berth allocation in U-shaped ports using Bayesian network analysis and mixed-integer programming. Their implementation of an improved search genetic algorithm in MATLAB successfully reduced port operation costs and showed an excellent performance in terms of solution quality [23]. Chargui et al. (2023) further explored energy efficiency in berth and quay crane allocation by integrating energy price variations into the optimization process. Their robust decomposition algorithm effectively handled uncertainties in quay crane processing times and energy costs, ensuring significant operational savings while maintaining environmental compliance [24]. Wang et al. (2024) introduced an optimization method based on an active–reactive strategy for solving continuous berth allocation and quay crane allocation problems with mixed uncertainties. Their research demonstrated the practicality of this method in enhancing port scheduling efficiency, particularly in dealing with uncertain factors [25]. Cao et al. (2022) proposed a new integrated berth–quay crane allocation model using the chaotic quantum sparrow search algorithm (CQSSA). This algorithm significantly improved port congestion issues and optimized allocations based on economic factors [26]. Ji et al. (2023) developed an enhanced non-dominated sorting genetic algorithm (ENSGA-II) to address berth and quay crane allocation problems with stochastic arrival times. Their research indicated that ENSGA-II had significant advantages in handling multi-objective optimization problems, particularly under various constraints, compared to other multi-objective and single-objective methods [27]. These studies not only provide theoretical support for improving port operational efficiency but also demonstrate the application potential of advanced algorithms in solving practical operational problems.

Existing research has made significant progress in the joint scheduling of berths and quay cranes, particularly in the use of intelligent algorithms such as genetic algorithms (GAs) and particle swarm optimization (PSO) for multi-objective optimization. These studies have effectively reduced the time costs and carbon emissions of port operations, providing theoretical support for the sustainable development of green ports. For instance, some studies have optimized ship departure delays and energy consumption through the establishment of dual-objective models, while others have incorporated quay crane maintenance activities into the models, further improving scheduling efficiency and economic viability. However, there are still some limitations in the existing research. Firstly, most studies have oversimplified the uncertainties in practical operations, such as tidal effects and the dynamic arrival times of ships, resulting in the limited applicability of the models. Secondly, the berth allocation for ultra-large container ships and the consideration of tidal factors have been insufficiently addressed, failing to fully reflect the complexity of actual port operations.

Synthesizing the above research, it is evident that under a low-carbon framework, the current focus of studies is how to balance ship service satisfaction with the minimization of port carbon emission costs. Additionally, some studies, in an effort to simplify models, have used related constraints that are too ideal. In recent years, there has been an increase in the number of large container ships. Due to the depth limitations of ports, these large vessels often need to wait for high tide to enter or leave, leading to increased waiting times at anchorages. This not only raises terminal operation costs but also indirectly increases the system’s carbon emissions. Therefore, to develop an effective and feasible port operation scheme, it is essential to fully consider all these factors. In response to this, this paper incorporates tidal factors to improve the constraint conditions. By taking into account both the carbon emissions of the port area and the service level of ships, a mixed-integer programming model has been established. The objective function of this model is to minimize the combined cost of port area carbon emissions and the cost of ships in port.

1.2. The Contributions of This Paper

This paper addresses the joint allocation problem of berths and quay cranes in green port operations, with a particular focus on the consideration of tidal factors and carbon emissions—elements that have been largely overlooked in the existing literature. It compares and summarizes previous models and methods, identifying their limitations, especially the insufficient consideration of dynamic operational constraints such as tides. By incorporating tidal factors, carbon emissions, and ship service satisfaction into the constraints, the study improves upon these models, thereby establishing a more realistic and comprehensive single-objective mixed-integer programming model. The objective of this model is to minimize the combined costs of ships in port and port carbon emissions. To solve this model, the study employs the CPLEX solver (Version 12.10.0) and introduces a hybrid PSO-GA (particle swarm optimization–genetic algorithm) approach. This hybrid method aims to provide relatively optimal solutions for berth and quay crane allocation, as well as ship port duration, outperforming traditional genetic algorithms and particle swarm optimization methods. Additionally, a sensitivity analysis of key parameters is conducted to assess the robustness of the proposed model under varying conditions.

2. Problem Description and Model Formulations

This section first presents the problem by giving a qualitative description in Section 2.1. Subsequently, we formulate a linear programming (ILP) method to deal with the problem of combining berth and yard allocation with carbon emissions.

2.1. Problem Description

In the berth and quay crane joint allocation problem, vessels arrive at the port over a period of time, and port operators formulate berth and quay crane schedules based on the information provided by shipping lines and forecasts of container capacity. Once a vessel enters the port area, it communicates relevant information (including the time of entry into the port area, vessel speed, and the volume of cargo to be loaded or unloaded) to the port. Subsequently, port planners determine the vessel’s berthing time and location, allocate the appropriate number of quay cranes (QCs), and provide feedback on the recommended sailing speed to the vessel to enhance service quality and reduce carbon emissions. Following berthing, loading and unloading operations commence and continue until the vessel departs.

Figure 1 provides a two-dimensional representation of the berth and quay crane joint allocation problem. In the figure, the time unit is set to 1 h, and the berth coordinate unit is set to one meter. Each vessel is represented by a rectangle, with the length and width corresponding to the vessel’s length and the duration of container loading and unloading operations, respectively. The numbers in parentheses indicate the number of quay cranes assigned to each vessel by the port. During the planning period, it is assumed that the port will handle three types of container ships: feeder ships (V1), which do not require consideration of navigational tidal depth differences; handy-size ships (V2); and Panama ships (V3), which require consideration of tidal depth differences. Due to the greater draft depth of V3 vessels, they may be unable to berth safely when approaching the port, necessitating berthing during high-tide periods.

Figure 2 shows that carbon emissions in the port area mainly come from two sources: vessels and port service equipment. Emissions from vessels are mainly due to fuel consumption, which is significantly influenced by the vessel’s sailing speed and dwell time in the port. Emissions from port equipment are primarily affected by service efficiency. A well-planned berth and quay crane joint allocation scheme is crucial for reducing carbon emissions. The berth allocation plan must address three interrelated aspects: berthing time, berthing position, and departure time. A vessel’s sailing speed affects its arrival time and berthing time; conversely, an inefficient berthing plan may lead to delays in the vessel’s departure. The allocation of quay cranes determines the vessel’s handling time in the port, which directly impacts its departure time. The berthing position affects the vessel’s operational efficiency, further influencing carbon emissions within the port area. Thus, the vessel, berth, and quay cranes (QCs) form an integrated system, requiring the development of a joint resource allocation model to effectively address the berth and quay crane joint allocation problem.

2.2. Model Formulations

In the process of berth and quay crane allocation, port carbon emissions mainly consist of two parts: the carbon emissions produced by ships during their port stay and those generated by quay crane operations (excluding the carbon emissions from quay crane movement). However, in addressing berth and quay crane allocation issues, carbon emissions cannot be the sole consideration; the service level of the ships must also be comprehensively considered. Thus, this paper introduces three types of costs to measure the service level of ships: the departure delay cost, anchorage waiting cost, and cost deviation from preferred berths.

For the sake of simplicity and clarity of the model, the following key assumptions are made (Park and Kim, 2003): [14]

(1)

All quay cranes operate on the same track and have uniform loading and unloading efficiencies;

(2)

The length of each ship includes the actual length plus the necessary gap required between docked ships;

(3)

Once a ship has docked, it cannot move until its loading and unloading operations are fully completed;

(4)

The energy consumption per unit time is consistent across all quay cranes and ships;

(5)

All ships arriving at the port must undergo loading and unloading operations within the constraints of their allocated resources;

(6)

Each ship has predefined minimum and maximum limits for the number of quay cranes it can be allocated, with the actual allocation falling within this range;

(7)

The effects of ship departure speed and quay crane movement on carbon emissions are not explicitly considered in the model.

2.3. ILP Model

The notations used in the ILP model are explained in

Table 1. Notations of the ILP model.

Indices:
$V$	set of ships, $V=\left\{1,2,3\cdots ,N\right\},i\in V$;
$V^S$	set of small ships;
$V^m$	set of medium-sized ships;
$V^h$	set of large ships;
$k$	A tidal sequence, each sequence includes one high tide and one low tide;
$K$	set of tidal sequences;
$S$	duration of high tide;
$T$	set of time periods, $T=\left\{1,2,3\cdots ,t\right\},t\in T$;
$Q$	set of quay cranes, $Q=\left\{1,2,3\cdots ,q\right\},q\in Q$, $\left|Q\right|$ is the total number of quay cranes;
$L$	length of the container terminal berth shoreline
$l_i$	length of the ship’s hull;
$t_{ai}$	ship’s estimated time of arrival (ETA);
$t_{di}^0$	ship’s estimated time of departure (ETD);
$q_i^{max}$	maximum number of quay cranes that can be allocated to the ship;
$q_i^{min}$	minimum number of quay cranes that can be allocated to the ship;
$a_i$	The volume of containers to be loaded/unloaded for the ship;
$\mathsf{\eta }$	loading/unloading efficiency of the quay crane;
${\alpha }_1$	carbon emission factor of the ship during its stay in the port;
$P{O}_i$	auxiliary engine’s rated power for the ship;
$LF$	ship’s load factor;
$E{N}_i$	number of auxiliary engines on the ship;
${\alpha }_2$	carbon emission factor of the quay crane;
$P$	carbon tax;
$\mathsf{\omega }$	energy consumption per unit time of each quay crane;
$c_0$	cost coefficient for ships waiting at anchor to dock;
$c_1$	cost coefficient for ships with delayed departures;
$c_2$	cost coefficient for ships deviating from their preferred berth;
$b_i^0$	ship’s preferred berth (measured from the bow);
$M$	a very large positive number.
Decision variables:
$X_{ij}$	value of 1 when a ship docks in front of another ship; otherwise, it is 0.
$Y_{ij}$	value of 1 when a ship docks after another ship has departed; otherwise, it is 0.
$b_i$	actual docking position of the ship.
$t_{bi}$	actual docking time of the ship.
$t_{di}$	actual departure time of the ship.
$qhea{d}_i$	number of the foremost quay cranes allocated to the ship.
$qtai{l}_i$	number of the rearmost quay cranes allocated to the ship.
$s{s}_{iq}$	Indicates whether the allocated number of quay cranes can meet the loading and unloading requirements of the ship. If not, it takes a value of 1; otherwise, it is 0.
${\lambda }_{ik}$	value of 1 if the ship enters the channel and docks during the nth high tide period; otherwise, it is 0.
${\beta }_{ik}$	value of 1 if the ship enters the channel and departs during the nth high tide period; otherwise, it is 0.
Auxiliary variables:
$\Delta b_i$	deviation between the ship’s actual docking position and its preferred berth position;
$\Delta t_{di}$	delay in the ship’s departure time.

.

To ensure the practicality of the model proposed in this paper, the model must not only allocate berths and quay cranes in a logical and efficient manner but also simultaneously consider both the carbon emissions of the port and the total costs incurred by ships while in port. Therefore, following the aforementioned principles, the objective function is established as follows:

$$\mathrm{m}\mathrm{i}\mathrm{n}Z={\sum }_{i\in V}\left({\mathrm{F}}_i^1+{\mathrm{F}}_i^2+{\mathrm{f}}_i^1+{\mathrm{f}}_i^2+{\mathrm{f}}_i^3\right),$$

(1)

Carbon emission cost incurred by ships during their stay in port:

$$F_i^1={\mathsf{\alpha }}_1\cdot P{O}_i\cdot E{N}_i\cdot LF\cdot \left(t_{di}-t_{ai}\right)\cdot P,$$

(2)

Carbon emission cost from quay crane operations:

$$F_i^2={\alpha }_2\cdot \omega \cdot P\cdot a_i/\eta ,$$

(3)

Cost of ships waiting at anchorage:

$$f_i^1=c_0\cdot \left(t_{bi}-t_{ai}\right),$$

(4)

Cost of departure delays:

$$f_i^2=c_1\cdot {\left(t_{di}-t_{di}^0\right)}^{+},$$

(5)

Cost of deviating from preferred berth positions:

$$f_i^3=c_2\cdot \left|b_i-b_i^0\right|,$$

(6)

The model is subject to the following constraints:

$$b_i+l_i\le b_j+M\times \left(1-X_{ij}\right);\forall i,j\in V,i\ne j,$$

(7)

$$t_{di}\le t_{bj}+M\times \left(1-Y_{ij}\right);\forall i,j\in V,i\ne j,$$

(8)

$$1\le X_{ij}+X_{ji}+Y_{ij}+Y_{ji}\le 2;\forall i,j\in V,i\ne j,$$

(9)

$$t_{ai}\le t_{bi};\forall i\in V,$$

(10)

$$b_i+l_i\le L;\forall i\in V,$$

(11)

$$q_i^{min}\le qtai{l}_i-qhea{d}_i+1\le q_i^{max};\forall i\in V,\forall t\in T,$$

(12)

$$\Delta b_i\ge b_i-b_i^0;\forall i\in V,$$

(13)

$$\Delta b_i\ge b_i^0-b_i;\forall i\in V,$$

(14)

$$\Delta t_{di}\ge t_{di}-t_{di}^0;\forall i\in V,$$

(15)

$$\Delta t_{di}\ge 0;\forall i\in V,$$

(16)

$${qhead}_j\ge {qtail}_i+1+M\times {(X}_{ij}-1);\forall i,j\in V,i\ne j,$$

(17)

$$\left(t_{ai}-t_{bi}\right)\cdot \left(qtai{l}_i-qhea{d}_i+1\right)+M\cdot s{s}_{iq}\ge a_i/\eta ;\forall q\in Q,\forall i\in V,$$

(18)

$$\left(t_{di}-t_{bi}\right)\cdot q+M\cdot s{s}_{iq}\ge a_i/\eta ;\forall q\in Q,\forall i\in V,$$

(19)

$$qtai{l}_i-qhea{d}_i\ge q\cdot s{s}_{iq};\forall q\in Q,\forall i\in V,$$

(20)

$${\sum }_{k\in K}{\lambda }_{ik}=1,\forall i\in V^h,$$

(21)

$$\begin{array}{ll}& 2\left(k-1\right)S-M\left(1-{\lambda }_{ik}\right)\le t_{bi}\le \left(2k-1\right)S+M\left(1-{\lambda }_{ik}\right)-1;\\ & \forall i\in V^h,\forall k\in K\end{array},$$

(22)

$${\sum }_{k\in K}{X}_{ik}=1,\forall i\in V^h,$$

(23)

$$\begin{array}{ll}& 2\left(k-1\right)S-M\left(1-X_{ik}\right)\le t_{ik}\le \left(2k-1\right)S+M\left(1-X_{ik}\right)-1;\\ & \forall i\in V^h,\forall k\in K\end{array},$$

(24)

$$b_i,t_{bi},t_{di},q_{it},qhea{d}_i,qtai{l}_i,k,S\ge 0;\forall i\in V,\forall t\in T,$$

(25)

$$s{s}_{iq},X_{ij},Z_{ij},{\lambda }_{ik},X_{ik}\in \left\{\mathrm{0,1}\right\};\forall i,j\in V,\forall q\in Q,$$

(26)

Constraints (7)–(9) ensure that no two ships overlap in the space–time graph. Constraint (10) specifies that a ship’s docking time must occur after its arrival time (berths may not be in a free state). Constraint (11) mandates that all ships must dock within the terminal’s shoreline. Constraint (12) ensures that the number of quay cranes allocated to a ship at any given time falls within the permissible range for that ship. The deviation of berths is determined by Constraints (13)–(14). The delay in ship departure times is established by Constraints (15)–(16). Constraint (17) guarantees that quay cranes allocated to two simultaneously docked ships do not intersect and also ensures that if cranes at both ends are working, those in the middle are considered to be working as well. Constraint (18) ensures the completion of workloads. Constraints (19) and (20) linearize Constraint (18). Constraints (21) and (22) state that large ships can only enter the channel and dock during high-tide periods. Constraints (23) and (24) state that large ships can only enter the channel and depart during high-tide periods. Constraints (25) and (26) define the range of values for the decision variables.

3. Heuristic Approach

According to Imai et al. (2008) [15], considering that this study addresses an NP-Hard problem, such problems are often approached with swarm intelligence algorithms, with genetic algorithms (GAs) and particle swarm optimization (PSO) being the most commonly used. However, both have their drawbacks and limitations: compared to genetic algorithms, particle swarm optimization exhibits a higher convergence speed. Nonetheless, a significant drawback of PSO is its tendency to get trapped in local optima in high-dimensional search spaces, leading to ineffective computational results. Similarly, the slow convergence rate of genetic algorithms also affects their practicality. Therefore, this study employs a hybrid metaheuristic algorithm to address the aforementioned issues. The flow of this algorithm is illustrated in Figure 3:

3.1. Encoding and Decoding

In alignment with the characteristics of the mathematical model presented in this paper, the following encoding method is selected for the solution process using the hybrid algorithm. The following decision variables are set: the sequence in which each ship is scheduled and the number of quay cranes allocated to each ship. For these decision variables, floating-point encoding is used for both the ship scheduling order and the number of quay cranes allocated to each ship, meaning random numbers between 0 and 1 are generated to represent N1 (the ship scheduling order) and N2 (the number of quay cranes allocated to each ship). The first and second tiers represent the ship’s scheduling order and the allocation of quay cranes, respectively.

Initially, the ships’ scheduling order is determined based on the magnitude of the N1 floating-point numbers. Figure 4 shows how the scheduling order for ten ships is determined by comparing floating-point numbers.

Subsequently, the floating-point numbers of N2 are each multiplied by a specific factor $\left(q_i^{max}-q_i^{min}\right)$ and added to $q_i^{min}$, then rounded up to the nearest whole number to determine the consecutive number of quay cranes allocated to each ship. Figure 5 shows how the number of quay cranes for each of the ten ships is determined using the described method.

3.2. Population Initialization

The strategy for generating the initial population is as follows:

The particles in this study consist of two components: the scheduling order of ships and the allocation of quay cranes. Population initialization for these particles is conducted based on actual conditions.

The first layer of genes represents the scheduling order of ships, in which the values within this segment of the particle must be unique and are randomly generated within the range of the total number of ships.

The second layer of genes corresponds to the number of quay cranes allocated to each ship, with their values randomly generated within a specified range ($q_i^{min},q_i^{max}$).

3.3. Fitness Function

The fitness function of the algorithm in this paper is as follows:

$$Fit\left(x\right)=1/\left(1+Z\right),$$

(27)

Additionally, this paper employs a roulette wheel selection mechanism and an elitism retention strategy, in which individuals with a higher level of fitness are directly copied to the next generation.

3.4. Particle Update

In each iteration, the velocity and position of each particle are updated according to Equations (28) and (29), respectively.

$$V_i=w^{*}{V}_i+c_1^{*}rand{\left(\right)}^{*}\left(Pbes{t}_i-X_i\right)+c_2^{*}rand{\left(\right)}^{*}\left(Gbes{t}_i-x_i\right),$$

(28)

$$X_i=X_i+V_i,$$

(29)

After updating the positions, the algorithm assesses the fitness of each particle according to the scheduling scheme obtained. If the current fitness value of a particle is superior to that of its best individual record, its optimum is updated accordingly. Thus, the following is obtained:

$$Pbes{t}_i=\left\{\begin{array}{l}{P}_i,fitness\left(P_i\right)<fitness\left(Pbes{t}_i\right)\\ Pbes{t}_i,otherwise\end{array}\right.,$$

(30)

3.5. Crossover and Mutation

3.5.1. Crossover

The crossover method employed in the algorithm of this study uses two-point crossover. This involves randomly selecting two points on the chromosome and using the gene segment between these two points as the crossover fragment. Then, crossover operations are conducted on two paired individuals, with the specific crossover operation illustrated in Figure 6, which involves randomly selecting two points on the chromosome and using the gene segment between these points as the crossover fragment. Crossover operations are then conducted between two paired individuals, with the operation aiming to improve the diversity of solutions and accelerate convergence. The figure provides a detailed example of this operation, showcasing how the gene segments are swapped between paired chromosomes.

3.5.2. Mutation

Given the use of floating-point encoding and the presence of two layers of chromosomes, different mutation strategies are applied to these two layers. The specific mutation operations are as follows:

(1) For the first layer of chromosomes (the scheduling order of ships), swap mutation is employed, as illustrated in Figure 7, which demonstrates the swap mutation process applied to the first layer of chromosomes, which represents the scheduling order of ships. In this operation, two genes are randomly selected, and their positions are swapped. This mutation method enhances the exploration capability of the algorithm by introducing new potential solutions into the population, thereby reducing the likelihood of premature convergence.

(2) For the second layer of chromosomes (quay crane allocation), insertion mutation is used, as illustrated in Figure 8, which depicts the insertion mutation method used for the second layer of chromosomes, which corresponds to the quay crane allocation. In this operation, a randomly selected gene is removed from its original position and inserted into a new, randomly selected position. This operation allows the algorithm to explore different quay crane allocation scenarios, improving its ability to identify optimal resource configurations.

4. Case Study

4.1. Cases and Solutions

This study utilizes empirical data from a specific container terminal as a case example. The terminal features a quay length of 1600 m, equipped with 20 quay cranes distributed along the quay front for cargo handling operations. The operational window considered encompasses 24 h, with the quay cranes exhibiting a handling efficiency of 35 TEU per hour. Additionally, the cost associated with deviating from the preferred berth is quantified, with a cost coefficient of 0.0014 USD per meter of deviation per container.

Table 2. Model parameter settings.

Parameter	Meaning	Value
${\mathsf{\alpha }}_1$	Carbon Emission Factor During Vessel Stay	$0.683\mathrm{kg}/$kWh
${\mathsf{\alpha }}_2$	Quay Crane Carbon Emission Factor	1.0935 kg/kWh
$\mathsf{\omega }$	Energy Consumption per Quay Crane per Unit Time	93.1 kWh/h
$\mathrm{E}\mathrm{N}$	Number of Auxiliary Engines per Vessel	4
$\mathrm{P}\mathrm{O}$	Rated Power of Vessel Auxiliary Engines	250 kW
$\mathrm{L}\mathrm{F}$	Vessel Load Rate	0.5
${\mathrm{c}}_1$	Departure Delay Cost	17.08 USD/h
${\mathrm{c}}_0$	Anchorage Waiting Cost	5.74 USD/h
$\mathrm{P}$	Carbon Tax	0.035 USD/kg

outlines additional parameters used in this study, which considers two large vessels, two medium-sized vessels, and one small vessel. All cost calculations in this study are presented in USD, the currency used for international standardization. For the convenience of readers, a currency conversion table has been provided in Appendix A, showing equivalent values in major currencies such as CNY and EUR based on the exchange rates during the study period.

The start of the planning period is set at the 0 h mark, coinciding with low tide, when the minimum water depth is eight meters. The maximum water depth, occurring at high tide, is 18 m, with the time interval between low tide and high tide being 6 h. Figure 9 illustrates the tidal pattern at this port over a 24 h period.

Utilizing the PSO-GA hybrid algorithm designed in the previous sections, a small-scale case study was conducted using the data provided in

Table 3. Detailed data of arriving vessels (small).

Vessel ID	Vessel Length ${\mathbf{l}}_{\mathbf{i}}$/m	Preferred Berth ${\mathbf{b}}_{\mathbf{i}}^0$/m	Arrival Time ${\mathbf{t}}_{\mathbf{a}\mathbf{i}}$/h	Expected Departure Time ${\mathbf{t}}_{\mathbf{d}\mathbf{i}}^0$/h	Handling Volume (TEU) ${\mathbf{a}}_{\mathbf{i}}$/TEU	Minimum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	Maximum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$	Deviation Cost ${\mathbf{c}}_2$/(USD/m)
1	285	100	1:00	10:30	2360	2	5	3.30
2	130	300	12:30	14:57	520	1	3	0.73
3	326	0	10:10	21:24	2100	2	6	2.94
4	136	200	13:10	16:59	735	1	3	1.03
5	81	300	18:30	20:21	280	1	2	0.39

and

Table 4. Optimal allocation scheme (small).

Vessel ID	Berth Position	Berthing Time	Departure Time	Quay Crane Allocation
1	327	2:00	16:00	5
2	748	13:00	18:00	3
3	1	13:00	24:00	6
4	612	14:00	21:00	3
5	327	19:00	23:00	2

. Table 3 presents detailed information on the arriving vessels, including their lengths, preferred berth positions, arrival and expected departure times, handling volumes, and quay crane requirements. For instance, Vessel 1, with a length of 285 m and a handling volume of 2360 TEU, requires between two and five quay cranes and has a deviation cost of 3.30 USD per meter if it deviates from its preferred berth position. Similarly, other vessels exhibit varied requirements and handling volumes, reflecting the diverse operational demands at the port. Table 4 shows the optimal allocation scheme as determined by the hybrid PSO-GA algorithm. This table details the assigned berth positions, berthing and departure times, as well as the quay crane allocations for each vessel. For example, Vessel 1 is assigned to berth position 327, with a berthing time of 2:00 and a departure time of 16:00, utilizing five quay cranes. The optimal allocation scheme reflects the algorithm’s ability to efficiently allocate port resources, minimizing total operational costs while meeting the service requirements of each vessel. The comparison between the expected and actual allocation outcomes underscores the efficiency of the proposed hybrid algorithm in handling the complexities of berth and quay crane scheduling.

Figure 10 provides the results of small-scale instances solved by the proposed algorithm, which reveals the cost values produced during each iteration. The minimum total cost output at the termination of the algorithm is 2501.56 USD, representing the optimal allocation solution. Table 4 details the specific berthing positions, times, departure times, and assigned quay cranes for the five vessels.

Figure 11 visually presents the berth and quay crane allocation scheme optimized by the algorithm. In the figure, the horizontal and vertical axes represent the berth length and berthing time, respectively. The rectangles represent the specific vessels, with the numbers before and within the parentheses indicating the vessel identification numbers and the corresponding number of assigned quay cranes. The blue bars denote the high-tide windows.

To further validate the algorithm’s effectiveness, the number of arriving vessels was increased to 15 for a medium-scale case study, with a time period extended to 48 h. All other parameters remained consistent with those provided in Table 2.

Table 5. Detailed data of arriving vessels (medium).

Vessel ID	Vessel Length ${\mathbf{l}}_{\mathbf{i}}$/m	Preferred Berth ${\mathbf{b}}_{\mathbf{i}}^0$/m	Arrival Time ${\mathbf{t}}_{\mathbf{a}\mathbf{i}}$/h	Expected Departure Time ${\mathbf{t}}_{\mathbf{d}\mathbf{i}}^0$/h	Handling Volume (TEU) ${\mathbf{a}}_{\mathbf{i}}$/TEU	Minimum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	Maximum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$	Deviation Cost ${\mathbf{c}}_2$/USD/m)
1	120	25	1:00	16:00	366	1	3	0.51
2	130	570	2:00	14:00	1050	1	5	1.47
3	140	30	4:00	16:00	203	2	3	0.28
4	180	25	3:00	10:00	525	2	4	0.74
5	120	330	5:00	17:00	280	1	3	0.39
6	140	320	10:00	21:00	735	1	3	1.03
7	100	610	7:00	13:00	210	1	2	0.29
8	180	340	13:0	23:00	735	2	4	1.03
9	100	630	17:00	23:00	150	1	2	0.21
10	140	60	18:00	22:00	180	1	3	0.25
11	96	211	12:00	15:33	630	3	4	0.88
12	310	60	1:00	20:00	2200	2	5	3.08
13	326	675	10:10	21:24	2000	2	6	2.80
14	290	910	2:00	20:00	2300	3	6	3.22
15	320	989	0:00	23:58	2500	3	7	3.50

presents detailed data for the arriving vessels, including four large, nine medium-sized, and two small vessels.

Figure 12 illustrates the iterative curve of the algorithm output after the program’s execution. Figure 12 reveals that the algorithm stabilizes in a convergent state after approximately 30 iterations. The minimum total cost output after the algorithm’s termination is 3608.29 USD.

Table 6. Optimal allocation scheme (medium).

Vessel ID	Berth Position	Berthing Time	Departure Time	Quay Crane Allocation
1	621	2	6	3
2	491	2	17	2
3	1	9	11	3
4	1	3	8	3
5	621	7	11	2
6	321	16	23	3
7	741	8	11	2
8	1	14	25	2
9	461	18	21	2
10	181	18	20	3
11	181	21	27	3
12	181	2	15	5
13	621	13	28	4
14	947	2	13	6
15	947	14	26	6

details the berthing and departure times, berthing positions, and number of assigned quay cranes for the 15 vessels. Figure 13 shows a two-dimensional spatio-temporal simulation of the joint berth and quay crane allocation based on the optimized algorithm results. In the figure, for Panamax vessels V12, V13, V14, and V15, the channel depth only meets the berthing requirements during high tide, necessitating that these vessels enter and leave the port during the high-tide windows. For example, vessel V12 arrives at the port at time 1, which falls within the high-tide window. After navigating the port channel, it berths at time 2 to commence loading and unloading operations, and departs at time 15, again within the high-tide window. Vessel V13, on the other hand, arrives at the port outside of the high-tide window and must wait at the anchorage until reaching an appropriate time to enter the channel, conduct cargo operations, and leave the port. From this analysis, it is evident that if the berth and quay crane allocation scheme were developed without considering tidal influences, the resulting total time the vessels spend in port could significantly exceed the theoretically calculated time. Therefore, in container terminal operational planning, it is crucial to fully account for tidal factors to ensure that the plan is more scientific and better aligned with actual conditions.

Finally, the number of arriving vessels was increased to 20 for a large-scale case study using the same algorithm, with a time period of 48 h. All other parameters are consistent with those provided in Table 2, with the numbers of large, medium-sized, and small vessels being five, thirteen, and two, respectively. The detailed data of arriving vessels are shown in

Table 7. Detailed data of arriving vessels (large).

Vessel ID	Vessel Length ${\mathbf{l}}_{\mathbf{i}}$/m	Preferred Berth ${\mathbf{b}}_{\mathbf{i}}^0$/m	Arrival Time ${\mathbf{t}}_{\mathbf{a}\mathbf{i}}$/h	Expected Departure Time ${\mathbf{t}}_{\mathbf{d}\mathbf{i}}^0$/h	Handling Volume (TEU) ${\mathbf{a}}_{\mathbf{i}}$/TEU	Minimum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	Maximum Quay Cranes ${\mathbf{q}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$	Deviation Cost ${\mathbf{c}}_2$/USD/m)
1	120	25	1:00	16:00	366	1	3	0.51
2	130	570	2:00	14:00	1050	1	5	1.47
3	140	30	4:00	16:00	203	2	3	0.28
4	180	320	5:00	9:00	294	1	3	0.41
5	180	25	3:00	10:00	525	2	4	0.74
6	120	330	5:00	17:00	280	1	3	0.39
7	140	320	10:00	21:00	735	1	3	1.03
8	180	340	10:00	22:00	840	2	5	1.18
9	100	610	7:00	13:00	210	1	2	0.29
10	180	340	13:00	23:00	735	2	4	1.03
11	180	570	12:00	23:00	635	1	3	0.89
12	100	630	17:00	23:00	150	1	2	0.21
13	140	1270	13:00	20:00	190	1	3	0.27
14	140	60	18:00	22:00	180	1	3	0.25
15	96	211	12:00	15:33	630	3	4	0.88
16	285	300	1:00	10:30	2360	5	7	3.30
17	310	60	1:00	20:00	2200	2	5	3.08
18	326	675	1:10	10:24	1710	5	8	2.39
19	290	910	2:00	20:00	2300	2	5	3.22
20	320	989	0:00	23:58	2500	3	5	3.50

below.

Figure 14 presents the iterative curve of the algorithm output after the program’s execution. According to Figure 14, the algorithm reaches a convergent state after approximately 47 iterations. Upon termination, the algorithm outputs a minimum total cost of 7649.77 USD.

Table 8. Optimal allocation scheme (large).

Vessel ID	Berth Position	Berthing Time	Departure Time	Quay Crane Allocation
1	1	2	6	3
2	121	2	17	2
3	251	5	7	3
4	1002	7	10	3
5	237	21	25	4
6	1	8	16	1
7	251	11	18	3
8	311	26	31	5
9	1182	8	11	2
10	491	14	25	3
11	491	28	35	2
12	391	17	20	2
13	961	14	17	3
14	1	18	20	3
15	141	19	25	3
16	391	2	12	7
17	1	26	39	5
18	676	2	12	5
19	671	13	27	5
20	961	18	36	5

details the berthing and departure times, berthing positions, quay crane allocations, and respective cost values for all 20 vessels.

Figure 15 presents a two-dimensional spatio-temporal simulation of the joint berth and quay crane allocation based on the optimized algorithm results. In this figure, the berth length is represented on the horizontal axis, while time is depicted on the vertical axis. The numbers within the rectangles represent the vessel identification numbers, and the numbers in parentheses indicate the number of quay cranes allocated to each vessel. The blue bars represent the high-tide windows.

4.2. Algorithm Performance Analysis

In the algorithm performance verification stage, the PSO-GA algorithm was used to output the optimal solution for a small-scale example, which was then compared with the optimal solution obtained by CPLEX. The results are shown in

Table 9. Results of CPLEX and PSO-GA algorithms.

Solution	${\mathbf{F}}_{\mathbf{i}}^1$ Cost (USD)	${\mathbf{F}}_{\mathbf{i}}^2$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^1$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^2$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^3$ Cost (USD)	Carbon Emission Cost (USD)	Vessel In-Port Cost (USD)	Total Cost (USD)
CPLEX	559.83	586.14	29.66	287.23	1035.12	1145.97	1352.00	2497.97
PSO-GA	561.80	586.14	29.73	290.31	1035.55	1147.94	1355.59	2503.52

.

Figure 16 uses a bar chart to visually compare the various costs. By comparing the two schemes, it can be observed that the optimized allocation scheme from the algorithm differs from the CPLEX allocation scheme by approximately 0.3% in vessel in-port costs, 0.17% in carbon emission costs, and 0.22% in total costs. These data indicate that the PSO-GA algorithm designed in this paper has a very low difference compared to the results obtained by the CPLEX solver, demonstrating that the algorithm has high levels of accuracy and feasibility.

Table 10. Cost of solving each algorithm.

Solution	${\mathbf{F}}_{\mathbf{i}}^1$ Cost (USD)	${\mathbf{F}}_{\mathbf{i}}^2$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^1$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^2$ Cost (USD)	${\mathbf{f}}_{\mathbf{i}}^3$ Cost (USD)	Carbon Emission Cost (USD)	Vessel In-Port Cost (USD)	Total Cost (USD)
GA	1468.17	527.35	705.07	1854.61	4890.12	1995.52	7449.79	9445.3
PSO	1563.79	509.49	750.99	1820.45	3065.4	2073.32	5636.82	7710.14
PSO-GA	1480.12	516.66	710.81	1581.33	3360.87	1996.78	5652.99	7649.77

compares the optimal solution obtained by the PSO-GA hybrid algorithm with traditional genetic and particle swarm optimization algorithms.

The iteration diagrams of the three algorithms are shown in Figure 17.

Figure 18 presents a bar chart comparing the solutions obtained by each algorithm. By comparing the schemes and iteration diagrams output by the three algorithms, the conclusion is that the optimized algorithm used in this paper yields allocation schemes with lower total costs compared to the traditional genetic algorithm and particle swarm optimization algorithm, with reductions of 24.1% and 0.78%, respectively. It is also evident that the optimized algorithm converges faster and has a higher solution accuracy compared to traditional heuristic algorithms.

Based on the above results, it can be demonstrated that the model and algorithm used in this study are superior and can be applied as practical guidance for berth and quay crane allocation at container terminals. Therefore, this study is of significant importance. Using the algorithm proposed in this paper for berth and quay crane allocation can not only substantially reduces the actual costs of the terminal but also alleviate pressure on carbon emissions in port areas.

4.3. Sensitivity Analysis of Quay Crane Handling Efficiency

In this study, the quay crane handling efficiency is set at 35 TEU/h, based on actual operations at a container terminal. According to relevant sources, the theoretical maximum efficiency of dual-trolley twin-lift quay cranes is approximately 80–100 TEU/h for the sensitivity experiments. Then, the quay crane efficiency values are set at 35, 50, 65, 80, and 95 TEU/h. It is assumed that the increase in quay crane handling efficiency is accompanied by a proportional increase in the power consumption per unit time of the quay cranes. The results of the model calculation under these assumptions are presented in Table 9. Figure 19 presents line graphs that illustrate the fluctuations in various indicators with changes in quay crane handling efficiency. The time spent in port, being relatively minor, is represented on the secondary y-axis (on the right side of the Y-axis). As shown in the figure, the enhancement of the quay crane handling efficiency effectively reduces various cost indicators, including carbon emission costs and vessel port costs, while also significantly decreasing the time vessels spend in port.

4.4. Comprehensive Impact Analysis of Efficiency Improvement on Operational Costs and Carbon Emissions

In the previous case studies and algorithm performance validation, this paper optimized the allocation of berths and quay cranes using the hybrid PSO-GA algorithm, significantly reducing port operational costs and carbon emissions.

By comparing carbon emission data under different efficiency levels, it was observed that as the quay crane handling efficiency improves, the time each vessel spends in port is significantly reduced, thereby directly decreasing the carbon emissions generated during the vessel’s stay. For instance, when the quay crane handling efficiency increased from 35 TEU/h to 95 TEU/h, the reduction in carbon emissions exceeded 15%. It indicates that enhancing quay crane efficiency not only accelerates cargo handling processes but also effectively reduces the carbon emissions associated with vessel berthing. In terms of costs, as the handling efficiency improves, both the in-port costs of vessels and the costs associated with deviating from preferred berths decrease significantly. For example, when the quay crane handling efficiency reaches 95 TEU/h, the total operational costs decrease by approximately 20% compared to the initial level. This result suggests that efficient quay crane operations can not only reduce the time vessels spend in port but optimize berth allocation, thereby reducing additional costs incurred due to deviations from preferred berth positions. There is a certain correlation between costs and carbon emissions. When the operational efficiency improves, although the energy consumption of quay cranes may increase, the overall reduction in operational time leads to a significant decrease in carbon emissions per unit of time. This result highlights the necessity of considering both costs and carbon emissions in port scheduling, and the hybrid PSO-GA algorithm performs exceptionally well in this regard.

Therefore, by optimizing berth and quay crane allocation strategies, significant reductions in operational costs were achieved, while effectively controlling carbon emissions. These findings will be the basis for future efforts to demonstrate the potential of the model and algorithm proposed in this paper for practical applications. Future research can build on this foundation to explore more operational factors and constraints, further optimizing the management of green port operations.

5. Conclusions

This paper conducts an in-depth study on the joint scheduling problem of continuous berths and quay cranes, particularly incorporating tidal influence factors into the model to make it more consistent with the actual conditions of terminal production organization. Considering the port’s carbon emissions and vessel service levels, a joint optimization model for berth and quay crane allocation is established. In the process of solving the model, CPLEX was first utilized to solve small-scale problems, and the results showed that the model performed well with a satisfactory solution quality and a short computation time for small-scale problems. Building on this, the study further applied a hybrid PSO-GA algorithm to solve large-scale cases, and the results indicated that this algorithm not only outperformed traditional methods in terms of solution quality but also demonstrated excellent computational efficiency. The model provides optimal berthing times, berthing positions, and berthing sequences for vessels within a given time period while also allocating quay cranes efficiently. The case study demonstrates that the established model is feasible and effective, making it a valuable reference for quay crane and berth planning at container terminals. The core advantage of this study lies in its ability to significantly enhance port operational efficiency and effectively reduce carbon emissions through optimized scheduling strategies.

However, the study has several limitations that should be acknowledged. First, the model assumes simplified operational conditions and does not fully account for factors such as extreme weather, equipment failures, or vessel draft limitations, which may impact real-world applicability. Second, the influence of peripheral equipment, such as yard cranes and trucks, is not included, which could affect terminal operations. Third, quay crane availability is treated as static, without considering dynamic constraints such as temporary downtimes or operational disruptions. These limitations may affect the practical application of the model in real-world port scenarios.

Future research can be further developed in the following aspects: (1) incorporating the specific requirements of different vessel types and the updates to quay crane equipment into the model to improve its applicability and accuracy; (2) considering more complex tidal and weather conditions to develop more robust scheduling schemes; and (3) exploring the integration of real-time data with the model to enable dynamic scheduling optimization, thereby further enhancing port operational efficiency and environmental benefits.