1. Introduction
Because about 80% of global cargo transportation is by sea, maritime container transportation has become the most important type of transportation for international trade. In recent years, the low rate of container vessels arriving on schedule due to uncertainties, such as epidemics and wars, has had a significant impact on container terminal operations, leading to severe congestion in ports around the world. Berth and quay cranes are important operational resources for terminals. The key to ensuring the terminal’s efficient operation is a suitable berth allocation and quay crane assignment scheme. The integrated rescheduling problem of the berth allocation and quay crane assignment problem with uncertainty is investigated in this paper to enhance the operational efficiency of container terminals by coping with the disturbance of terminal operations caused by events such as vessel delays and unscheduled vessel arrivals.
As the key resource of the container terminal at sea, the berth allocation and quay crane assignment problem (BAQCAP) has been a hot topic of concern for all parties concerned [
1]. The integrated scheduling of berths and quay cranes has attracted the attention of scholars, and a wealth of research results have been achieved [
2,
3,
4,
5,
6,
7]. At present, most of the literature takes the premise that the vessels arrive at the scheduled time and the number of arriving vessels remains unchanged. It also takes all the vessels expected to arrive at a container terminal in a single planning period as the optimization objectives, such as the minimum total waiting time for all vessels or the earliest end time for the last vessel, and finally gives the berth position, handling time, and the number of allocated quay cranes for each expected arriving vessel, which has certain guidance and reference values for the daily operations of the terminal. However, uncertain events such as vessel delays and unscheduled vessel arrivals often occur in operation. The above uncertain events may lead terminals into such predicaments as berth resource constraints and abnormal blockage during some hours. The study of integrated rescheduling of berths and quay cranes under the influence of such events is valuable to the application of terminals and has received much attention.
In earlier studies, scholars set buffers to deal with the disturbance of uncertain events before terminal operations. For example, Rodriguez-Molins et al. [
8] established multi-objective robust optimization by introducing buffer times to maximize the absorption of possible uncertainty events and to solve them by a genetic algorithm. Zhen and Chang [
9] proposed a bi-objective optimization model with the objectives of minimizing operation costs and maximizing operation plan robustness by considering the uncertainties (e.g., vessel arrival time and handling time), and designed a heuristic algorithm to solve the large-scale problem. Zhen et al. [
10] considered the uncertainty of vessel arrival time and operation time to establish a stochastic model, considered a robust model with limited information on probability distribution, analyzed the relationship between the two models, and proposed a variety of heuristic algorithms for the solution. Xu et al. [
11] considered the uncertainty of vessel delay and handling time, inserted a time buffer between two vessels at the same berth to improve the robustness of the scheme, established a model intending to minimize the total delay time of all vessels and maximize the robustness of the scheme, and designed a robust berth scheduling algorithm (RBSA) solution combining simulated annealing and branch delimitation algorithms.
Following this research line, Du [
12] considered, for the first time, the specific impact of uncertainty scenarios in the plan implementation process, introduced a feedback mechanism into the berth allocation problem for vessel delays, and emphasized the use of real-time operational progress to guide the time buffer settings for different phases. Zhen et al. [
13] studied the berth allocation problem with uncertainty in vessel arrival time and handling time, considered a certain degree of uncertainty in the baseline plan, proposed a proactive recovery strategy in the plan execution, and established a two-stage model under uncertainty. Liu et al. [
14] developed a baseline plan cost minimization decision model, including delay cost and offset preference berth cost, by using discrete scenarios to explicitly deal with uncertainty, considering uncertainty in vessel arrival time and handling time. Xiang et al. [
15] studied the BAP problem with uncertainty in vessel arrival time and handling time, dealt with the uncertainty linearly using a discretized scenario, developed a bi-objective robust optimization model focusing on economic efficiency and customer satisfaction, and designed an adaptive gray wolf algorithm to solve it. Sheikholeslami and Ilat [
16] considered the effect of tides on the port berth allocation scheme and the uncertainty of vessel arrival time. They also proposed a rigorous mathematical model that presented the sample average approximation method to generate an efficient berth allocation scheme. Liu et al. [
17] used the uncertainty set to describe the possible scenarios without using probability distributions and designed a two-stage robust model that gave the baseline plan first and the rescheduling plan after the uncertainty scenario was explicit. Rodrigues and Agra [
18] considered the berth allocation and quay cranes assignment problem with uncertain vessel arrival time and developed a two-stage robust mixed-integer programming model where berth allocation occurred before the exact arrival time and the adjusted crane assignment operations, according to the arrival time. Xiang and Liu [
19] considered the integrated berth allocation and quay crane assignment problem with uncertain vessel delays and container quantities, developed a robust model to minimize the total cost, and proposed a decomposition method to solve the problem.
Both of the above two types of studies set aside a time buffer or resource buffer in the plan, and once the uncertainty event has not occurred during the operation, the reserved resources are wasted, so the operation scheme given by the above studies is relatively conservative. Therefore, some scholars have proposed a reactive approach in recent years, which first gives a baseline plan to minimize costs or vessel handling time, and then adjusts the baseline plan when an uncertainty event occurs. Liu et al. [
20] focused on the problem of rescheduling quay cranes in the event of a breakdown during plan execution and rescheduling the system to minimize negative deviations from the baseline plan. Xiang et al. [
21] studied the problem of container terminals in uncertain berth allocation and quay crane assignments and developed a mixed-integer programming model considering practical constraints to obtain a baseline plan to minimize the baseline cost. Considering disruptions (e.g., vessel arrival time deviation, vessel handling time deviation, unscheduled vessel call, and quay crane breakdown) when executing the baseline plan, a response strategy with the baseline plan as a reference and the objective of minimizing the recovery cost was proposed. Li et al. [
22] studied the disruption recovery optimization problem in the integrated berth allocation and quay crane assignment problem of container terminals and developed a multi-objective programming model to maximize the service quality while minimizing the recovery cost. They also designed a heuristic method based on squeaky wheel optimization to solve the model. Al-Refaie and Abedalqader [
23] studied the berth allocation problem under unscheduled vessel arrivals, proposed an optimization method for container ports when an unscheduled vessel arrives, and developed three sequential models to maximize the number of unscheduled vessels, served with minimal disruption to the regular vessel service schedule. Kim [
24] proposed a method of berth and quay crane rescheduling caused by the updated information on vessel arrival time to solve the problem of frequent fluctuations in vessel arrival time at container terminals, established a mixed-integer linear programming model for the berth allocation and quay crane allocation problem, and solved it using a rolling-horizon method.
From an application point of view, the reactive approach is divided into two steps in the application process. The first step is the identification of uncertain events. Xiang et al. [
21] proposed a reactive strategy (a rolling-horizon heuristic method), which takes the baseline schedule as a reference and aims to minimize the recovery cost. The second step is to design the algorithm to solve the model and obtain the rescheduling scheme. Tan et al. [
25] developed a two-stage metaheuristic framework based on GA to solve this problem. Li et al. [
26] proposed an efficient optimization algorithm that is a hybrid of the iterated greedy and simulated annealing algorithms to solve the flexible job shop scheduling problem with crane transportation processes. Rodrigues et al. [
7] counted the algorithms used in all relevant studies from 2006 to 2021, with metaheuristic algorithms accounting for 55% and genetic algorithms for 20% of all studies. It shows that GA is an effective algorithm for solving the BAQCAP, especially for continuous berth. Some scholars have also started to use machine-learning methods to solve such problems. Du et al. [
27] proposed a deep Q-network model to solve a multi-objective flexible job shop scheduling problem with crane transportation and setup times, and in the following research, a flexible job shop scheduling problem with a time-of-use electricity price constraint is considered [
28].
In summary, most of the existing literature on the BAQCAP for coping with the effects of various uncertain events initially used robust optimization or two-stage disturbance recovery models. Due to the high number and irregularity of various uncertain events in recent years and the high impact leading to the near impossibility of restoring the baseline plan, there has been a gradual increase in the literature on constructing rescheduling models [
29,
30,
31,
32], which can provide a reference for terminals when developing contingency plans. Additionally, there are still the following problems: the above article only considers some uncertain events in vessel delay, handling time uncertainty, and unscheduled vessels and does not include all three types of uncertain events. The time of rescheduling is determined in the reactive strategy, but the vessels that operate in the port are not considered to be moved to improve the utilization of berth.
2. Problem Formulation
During the planning period, vessels inform the terminal of their arrival information in advance, and the terminal develops a baseline plan based on the information and the terminal’s conditions. However, some vessels are affected by uncertain events during terminal operations that can cause vessel delays. At the same time, there are be unscheduled vessel arrivals during the planning period. The baseline plan needs to be adjusted after vessel delays and unplanned vessel arrivals to avoid disorderly terminal operations, resulting in wasted resources and reduced operational efficiency.
In this study, the berth type refers to the continuous berth, where the vessel can berth at any position as long as the space allows. Quay cranes interfere with each other in operation, resulting in a decrease in operational efficiency. The marginal productivity loss index of shore bridges is denoted by α. In addition, the operational efficiency is also affected by the transport time of land-side trucks. The closer a vessel berth is to its yard, the shorter the transport time of trucks, and these locations are usually the best berthing position for vessels.
A specific description is given in a two-dimensional coordinate diagram in
Figure 1 to visualize the disruptions to the baseline plan caused by vessel delays and unplanned vessel arrivals. The horizontal axis represents the length of the shoreline, the vertical axis represents the time, the rectangles represent the vessels, and the length and width of the rectangle represent the length (e.g.,
l1 of vessel 1) and the handling time (e.g.,
h1 of vessel 1) of the vessel, respectively. The distribution of the quay cranes on the same track can be panned but not crossed. The number of quay cranes that can be allocated at the same time varies from vessel to vessel, and the longer the vessel, the higher the number of quay cranes that can be allocated at the same time (e.g., quay cranes [
8,
9] for vessel 1, quay cranes [
5,
6,
7] for vessel 2). Because the vessels must ensure spatial and temporal independence between them, i.e., there can be no overlap between the two rectangles, when a vessel is delaying, it might overlap with others (e.g., vessel 4 and delayed vessel 3 overlapped). By the same token, unscheduled vessels (e.g., vessel 6) can cause overlap between vessels. Both of these situations can lead to confusion in the baseline plan and therefore require timely adjustments to the baseline plan. The optimization objective of rescheduling is to minimize the delay between the actual departure time of the original vessels and the expected departure time in the baseline plan, as well as the waiting time of the unscheduled vessels.
The rescheduling strategy includes moving the vessels operating in the terminal, increasing the number of quay cranes, adjusting the berthing time of vessels, and reallocating berth positions, as shown in
Figure 2. The red rectangle in the figure shows the vessels in the baseline plan and the black rectangle shows the vessels in the rescheduling plan. In
Figure 2, vessel 1 and vessel 2, which are operating, are shifted to the sides so that space can be freed up for the unscheduled vessel 9 to berth. Because vessel 3 is delayed, the quay cranes are added to vessel 3 to reduce the handling time and avoid overlapping with vessel 4, which is berthed later. However, not all vessels can add quay cranes in the baseline plan, so vessels that need to berth in the back order adjust their berthing time to avoid overlap (e.g., vessel 6 adjusts its berthing time backward to avoid overlapping with vessel 5). If overlap cannot be avoided by adding quay cranes and adjusting berthing times, the berths need to be reallocated (e.g., vessel 7 is delayed so the berthing positions of vessel 7 and vessel 8 are adjusted). Again, the above adjustment strategies are all applicable to overlaps caused by unscheduled vessels, and a combination of rescheduling strategies can be implemented for the same vessel.