Ship–Infrastructure Cooperation: Survey on Infrastructure Scheduling for Waterborne Transportation Systems

Abstract

Ship–infrastructure cooperation, i.e., infrastructure scheduling, is significant for optimizing the utilization of spatial-temporal resources of infrastructures and improving the efficiency and safety of waterborne transportation systems. This paper carries out a systematic review of the scheduling problems of the infrastructures in waterborne transportation systems, including locks, terminals, berths, and waterway intersections. The infrastructure scheduling problems are linked to the classical optimization problems, and a generalized infrastructure scheduling problem is formulated. For lock scheduling, the ship placement sub-problem aims at minimizing the number of lockages, which is a kind of classic 2D bin packing problem; the lockage scheduling sub-problem deals with chamber assignment and lockage operation planning, which is modeled as a single or parallel machine scheduling problem. For berth and terminal scheduling, the idea of queuing theory (for discrete terminal) and 2D bin packing (for continuous terminal) are usually applied. Most research aims at minimizing the waiting time of ships and focuses on the continuous dynamic terminal scheduling problems. As a special infrastructure, the waterway intersection receives little attention. Most research focuses on traffic conflicts and capacity problems. Future research directions are provided based on the review results and problems of infrastructure scheduling in practice.

1. Introduction

The waterborne transportation system is a giant system. Various infrastructures, such as locks, terminals, berths, etc., provide appropriate places for navigation, berthing, and the operation of ships. However, with the trend of increasing demand for freight transport, and larger ships, it is increasingly obvious that the infrastructures become the bottlenecks of inland shipping. Taking the Three Gorges Dam of the Yangtze River as an example, during the flood season, the unilateral backlog can reach 700–800 ships, and the waiting time for the lock can reach up to ten days [1]. Construction of new infrastructures is time- and money-consuming. Thus, it is important to make full use of existing infrastructure resources (i.e., spatial-temporal resources) through proper scheduling.

The existing review on waterborne transportation infrastructures mainly focuses on a specific type of infrastructure. As the main bottleneck, the lock scheduling problem has attracted much attention. The methods used in the lock scheduling problem are reviewed and compared in [2]. Moreover, the review on berth allocation, quay crane scheduling, and integrated planning is provided in [3]. In fact, the process of ships passing through various infrastructures, such as locks, berths, and terminals, can be regarded as the occupation of spatial-temporal resources. Thus, the core of scheduling problems of different infrastructures is the same, i.e., the allocation of spatial-temporal resources. The problem can be expressed as classic scheduling problems, such as single machine scheduling problems, parallel machine scheduling problems, or job shop scheduling problems.

Therefore, this paper carries out a systematic review on the scheduling problems of the infrastructures in waterborne transportation systems, including locks, terminals, berths, and waterway intersections. The purpose is to analyze the current research status of the scheduling of waterborne transportation infrastructures and to correspond the scheduling problem to the classical optimization problem. The present research trends and gaps are identified, and suggestions for future development are put forward. The main contributions of this article are as follows:

(1)

The existing infrastructure scheduling methods for waterborne transportation systems are reviewed and linked to the classical optimization problems.

(2)

A generalized form of infrastructure scheduling problem is formulated.

The remainder of this paper is organized as follows. Section 2 provides the process of data collection and review criteria for the subsequent analysis. In Section 3, the existing literature on the scheduling of locks, terminals, berths, and waterway intersections is analyzed. In Section 4, the generalized infrastructure scheduling problem is formulated, and the research gaps and future research directions are provided. The conclusions of this article are provided in Section 5.

2. Research Methodology

2.1. Collection of the Literature

The analysis presented in this paper is based on the literature database established with the steps shown in Figure 1.

• Search: First, related journal papers in the Web of Science (WOS) published from January 2010 to November 2022, with topics including “ship” or “ship” or “surface vehicle” or its synonyms, or “ship lock” or “(lock)” or “lockage” or “berth” or “port” or “terminal” or “intersection”, and “scheduling” or “planning” or “optimiz*” or “allocation” or “collaboration” or “collaboration” or “organization”. Then, the database is supplemented through the snowball search by the reference list and authors.
• Selection: Papers beyond the scope of this survey are removed from the database after reading:

  (a)

  Articles that only focus on theoretical analysis of generalized intelligent scheduling systems;

  (b)

  Articles that focus on resource optimization between infrastructures, such as quayside or port quay cranes and yards;

  (c)

  Articles that focus on the optimization of the structure of the infrastructure;

  (d)

  Articles that only focus on economic analysis.

Seventy-six pieces of literature are collected through the above steps. The word cloud of the keywords is shown in Figure 2. Ship is the core of the waterborne transportation system scheduling problem. The berth allocation and the lock scheduling have attracted much attention. As the main terminal type, container terminal is the research focus in the berth allocation problems. Environmental-related objectives, such as energy-saving, gain increasing concern. Besides the spatial-temporal constraints, tidal constraints, weather conditions, uncertainty, and service priority have gradually been the hotspots in recent research.

From the perspective of the time dimension, the ship lock is the earliest infrastructure to attract attention. Large locks, such as the Three Gorges dam, are the hotspot. Due to the different calculation performances, some heuristic algorithms, such as the Bee Colony Optimization (BCO) and Evolutionary Algorithm (EA), were the main solution method before 2018. Later, other heuristic algorithms, such as the Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), were increasingly introduced to solve large-scale optimization problems and find a balance between solution speed and quality. As a result, the exact method has been widely adopted. The lock scheduling problem, which can be seen as a variant of the job shop scheduling, has long been modeled as mixed integer linear models, and the goal of sustainable development and cooperative optimization has gained the latest attention.

2.2. Review Criteria

According to the collected data, this paper classifies the existing research from the following aspects: system characteristics, model characteristics, and solution methods, as shown in

Table 1. Model evaluation and classification criteria.

		Abbreviation	Description
System characteristics	Spatial-scale	Dis	Discrete
		Con	Continuous
		Mix	Mixed model
	Temporal-scale	St	Static
		Dy	Dynamic
Model characteristics	Objective function	min T	Minimize average waiting time
		max U	Maximize area utilization of infrastructures
	Constraints	T	Temporal constraint
		Sp	Spatial constraint
		Ti	Tidal constraint
		C	Cargo adaptation constraint
		Pr	Priority constraint
Solution methods		PA	Exact algorithm
		MA	Heuristic algorithm

.

The core of infrastructure scheduling is the allocation of spatial-temporal resources. Therefore, the system characteristics are reflected in spatial and temporal scales. The spatial scale describes the spatial resource characteristics of infrastructures. For example, in the berth allocation problem, the spatial scale characteristics mainly refer to the distribution of berths. The discrete type means that berths are separated. That is, ships can only operate in a designated berth, and the same berth can only provide services for exactly one ship simultaneously. Then, the continuous type means that there is no clear boundary between berths, and the ship can occupy one or more berths simultaneously [4]. Furthermore, the mixed type is the scheduling problem that a port includes both the discrete and continuous types. Regarding temporal scales, scheduling problems can be divided into static and dynamic. The static scheduling problem refers to the ideal situation where all the scheduled ships have already arrived or the arrival time of the ships are known in the scheduling period. However, in the dynamic scheduling problem, ships arrive one after another and the arrival of ships is uncertain, which can be disturbed by weather and other external factors.

The infrastructure scheduling problem is generally established as an optimization model. The commonest optimization objectives are to minimize ships’ waiting time and maximize the utilization of infrastructure areas. For each optimization model, due to the different optimization objects, optimization objectives, and application scenarios, various influencing factors need to be considered in the modeling process, such as the external natural environment, infrastructure conditions, scheduling rules, etc. Such factors are usually regarded as constraints, which can be divided into five categories: temporal, spatial, tidal, cargo adaptation, and priority. Among them, temporal constraints include restrictions on the order of ships’ arrival time and infrastructure idle and operation time. Spatial constraints include requirements for the scale of both ships and infrastructures, as well as restrictions on the occupancy of the infrastructure. Tidal constraints restrict the passage time of ships in need of the tide. As for the infrastructures that ships pass through and operate on, the cargo adaptation constraints refer to the requirements on the cargos, and priority constraints can be the service rules of infrastructures or preferences of ships and controllers, such as First Come First Serve (FCFS).

The exact and heuristic algorithms are used to solve infrastructure scheduling problems. The exact algorithm refers to the algorithm that can find the optimal solution. However, it has limitations in solving large-scale complex problems. In related papers, solvers such as CPLEX are often used to obtain the optimal solution. Further, the heuristic algorithm is based on intuition or experience, which can give a feasible solution to a large-scale multi-objective optimization problem [5], including GA, Ant Colony Optimization (ACO), PSO, Simulated Annealing (SA), and Tabu Search, etc. To prevent falling into local optimum, improvements of heuristic algorithms are proposed, such as the combination of two and more meta-heuristics [6].

Moreover, the information provided by ships is the basis of the scheduling decision-making process. Hence, the following four classes of information involved in the scheduling process are provided in the following analysis:

(a)

Static information includes the length, width, tonnage, type, and other information about ships preset in construction or a voyage.

(b)

Dynamic information is updated in real-time with speed and heading of the ship, such as the position and navigation status of the ship.

(c)

The information related to the voyage refers to the data associated with the specific voyage of the ship, including the draft, the estimated time of arrival, the type of dangerous goods, the port of departure and the port of destination, etc.

(d)

The information related to the use of the infrastructure refers to the information that needs to be provided when passing through a specific infrastructure, such as the ship service priority in the ship lock scheduling problem.

3. Infrastructure Scheduling for Waterborne Transportation Systems

3.1. Ship Lock

A ship lock is a navigation construction composed of lock chambers, the head of upper and lower locks, gates, a water conveyance system, approach navigation channel, and related equipment. The water conveyance system adjusts the water level in the lock chamber so ships and fleets can pass through smoothly.

The process of ships passing through the lock is shown in Figure 3. The lock scheduling problem aims to provide the time, the chamber, and the position in the lock chamber for each ship passing through the lock. The lock scheduling problem includes two sub-problems: the ship placement sub-problem and the lockage scheduling sub-problem. The purpose of the ship placement sub-problem is to put ships of different sizes and types into lock chambers according to particular orders and rules. On this basis, making full use of the limited spatial resources of the lock chamber, shortening the time of entering the lock, and reducing the lockage numbers are all considered in the problem, which is usually described as the classical 2D bin packing problem. The lockage scheduling sub-problem aims to determine the time for ships to pass through each chamber and the running time of each lock, which is often modeled as a single or parallel machine scheduling problem.

3.1.1. Lock Scheduling Problem

The ship lock scheduling problem is an optimization problem to reasonably allocate the spatial-temporal resources of the lock chamber. Thus, spatial and temporal constraints need to be considered. The lock scheduling problem is a multi-constraint and multi-objective combinatorial optimization coupled in spatial and temporal dimensions. Equations (1)–(8) are the general expressions for ship lock scheduling problems summarized in the literature. The model is based on the following three assumptions:

(a)

All ships and their safety distance areas are uniformly regarded as rectangular, with the bow facing forward, which is arranged and distributed parallel to the upstream and downstream directions and cannot be changed.

(b)

The lock chamber is rectangular, and the influence of drought, wind and other hydrology, waterway, meteorological climate, and priority order on ship lock scheduling is ignored.

(c)

The waiting time only considers the time needed for ships to wait from entering the security inspection qualified area to entering the lock chamber.

Lock Scheduling Problem (LSP):

$$\mathrm{minimize} t^{\mathrm{avg}}=\frac{{\displaystyle \sum _{k=1}^{\mathrm{K}}{\displaystyle \sum _{m=1}^{\mathrm{M}}{\displaystyle \sum _{i=1}^{\mathrm{N}}(t_{im}^{\mathrm{en}}-t_i^{\mathrm{de}})x_{im}^k}}}}{\mathrm{N}}$$

(1)

$$\mathrm{subject} \mathrm{to} \forall i,j=1,2,\dots ,\mathrm{N}, i\ne j, m=1,2,\dots ,\mathrm{M}; k=1,2,\dots ,\mathrm{K}$$

$$0<L_i<L_m$$

(2)

$$0<W_i<W_m$$

(3)

$${\displaystyle \sum _{i=1}^{\mathrm{N}}{F}_i}{x}_{im}^k\le F_m$$

(4)

$$\begin{array}{l}\mathsf{\mu }(x_{im}^k·a_{im}^k-x_{jm}^k·a_{jm}^k-x_{jm}^k·L_j)+\mathsf{\mu }(x_{jm}^k·a_{jm}^k-x_{im}^k·a_{im}^k-x_{im}^k·L_i)+\\ \mathsf{\mu }(x_{im}^k·b_{im}^k-x_{jm}^k·b_{jm}^k-x_{jm}^k·W_j)+\mathsf{\mu }(x_{jm}^k·b_{jm}^k-x_{im}^k·b_{im}^k-x_{im}^k·W_i)\ge 1\end{array}$$

(5)

$$x_{im}^k·t_i^{\mathrm{de}}\le t_{im}^{\mathrm{en}}$$

(6)

$$(t_{im}^{\mathrm{en}}-t_i^{\mathrm{de}})·x_{im}^k\le t^{\mathrm{wait},\max }$$

(7)

$${\displaystyle \sum _{k=1}^{\mathrm{K}}{\displaystyle \sum _{m=1}^{\mathrm{M}}{x}_{im}^k\le 1}}$$

(8)

where, $i,j$ is the number of the ship, $m,k$ represents the number of the locks and the order in which the ships are served. $t^{avg}$ is the average waiting time of ships. $t_i^{\mathrm{de}}$ and $t_{im}^{\mathrm{en}}$ is the time when ship $i$ applies for passing through the lock and finally enters the lock chamber. $F_i$ and $F_m$ represent the area of ship $i$ and lock $m$, respectively. $\mathsf{\mu }(x)$ is a step function that indicates whether $x>0$ (1) or $x\le 0$ (0). $a_{im}^k$ and $b_{im}^k$ represent the abscissa and ordinate of ship $i$ in lockage $k$ of lock chamber $m$, and $t^{\mathrm{wait},\max }$ indicates the maximum waiting time.

Minimizing the average total waiting time of ships is the objective (Equation (1)), in which the waiting time refers to the duration between the time when the ship applies for passing through the lock and when the ship enters the lock chamber. Constraints (2)–(4) are the boundary constraints of the lock chamber. Constraint (2) ensures that the length of the ship is smaller than the length of the lock chamber, and Constraint (3) ensures that the width of the ship is smaller than the width of the lock chamber. Constraint (4) means that the total area of ships in the same lock chamber is not larger than the lock chamber area. Constraint (5) means ships in the same lockage do not overlap. Constraint (6) ensures that the time the ship applies to pass through the lock is earlier than when it enters. To improve the service quality, the ship lock management department generally sets a maximum waiting time. For example, in the linkage control scheme of ships passing through the lock on the Yangtze River, there are the overall control objectives that the average waiting time of ships cannot exceed a threshold. Thus, Constraint (7) ensures that the waiting time of each ship is not larger than the specified maximum waiting time. Constraint (8) guarantees that a ship can enter one lock chamber of one lockage.

3.1.2. Ship Lock Communication

Table 2. Information in the lock scheduling problems.

	Static Information	Dynamic Information	Navigation-Related Information	Lock-Crossing-Related Information
Single- chamber	Length of the ship; Width of the ship; Ship tonnage; Empty weight; Deadweight; Type of ship; Ship’s actual load.	Speed of ship	Ship arrival time; Estimated arrival time.	Total number of ships; Maximum allowable waiting time; Ship delay time; Costs of ship navigation; Number of tugs used by ships; Safety distance between ships; The direction of ship navigation.
Multi- chamber	Length of the ship; Width of the ship; Ship tonnage; Type of ship.	-	Ship arrival time.	Total number of ships; Ships that need to follow FCFS rules; Ship delay time; The direction of navigation (crossing locks); Maximum waiting time; Safety distance between ships; Ships that need to ride the tide.

summarizes the information involved in the literature related to the lock scheduling problem, including four types: static information, dynamic information, navigation-related information, and lock-crossing-related information. Among them, static information is involved in both single-chamber and multi-chamber lock scheduling research. Dynamic information mainly consists of the navigation speed of ships in single-lock chamber research, but has not been considered in multi-chamber studies. Navigation-related information has been mentioned in single and multi-chamber lock research, such as the ships’ arrival and estimated arrival times. Then, the lock-crossing-related information includes the following: total number of ships, the maximum allowable waiting time, ship delay time, costs of ship navigation, number of tugs used by ships, safety distance between ships, and direction of ship navigation, etc. In some studies of multi-chamber lock scheduling in tidal river reach, the relevant information on ship tide riding is also included in the decision information.

However, the existing literature all assumes that the required information is known, without mentioning how to obtain the information. It is worth noting that in the current research, ships generally provide relevant information to the ship lock manager and can only passively accept the scheduling results. When the ship cannot arrive within the specified time window, it can only wait for the next opportunity, resulting in an unnecessarily long waiting time.

3.1.3. Research Status of Ship Lock Scheduling

The model characteristics and solution methods of the collected literature related to ship lock scheduling are summarized in

Table 3. Overview of research on ship lock scheduling problems.

No.	Published Time	Model Characteristics												Solution Methods 3
		Objective Function 1							Constraints 2
		Min T	Min C	Min Z	Max Q	Max Pr	Max U	Oth	T	Sp	Ti	Pr	Oth	PA	MA
[11]	2022		√						√	√				CPLEX	Local-Search Heuristic
[12]	2022		√						√	√		√	√	√	√
[10]	2021	√			√				√	√		√			LNS Heuristics
[13]	2021	√		√					√	√				Gurobi
[14]	2021	√							√	√		√	√	Gurobi
[15]	2021	√		√					√	√			√	Gurobi
[16]	2021		√						√	√				√
[17]	2020	√							√	√			√	√	Adaptive Large Neighborhood Search Algorithm
[7]	2020	√					√		√	√		√			Binary Quantum-Inspired Gravitational Search Algorithm; Multi-Order Best-Fit Tabu Search Algorithm
[18]	2019	√		√					√	√		√			Quantum-Inspired Binary Gravitational Search Algorithm andModified Moth-Flame Optimization
[19]	2019	√		√					√	√		√			Modified Binary Nondominated Sorting Genetic Algorithm II
[20]	2019	√	√												Greedy Particle Swarm Optimization Algorithm
[21]	2017	√					√		√	√		√		√
[22]	2016	√		√					√	√					Genetic Operators-Based Artificial Bee Colony Algorithm
[9]	2015	√		√					√	√		√		√
[23]	2014	√							√	√			√	√
[24]	2014						√	√	√	√		√			Hooke–Jeeves Algorithm
[25]	2014	√							√	√				√
[26]	2013							√	√	√	√	√	√	Gurobi
[27]	2013							√		√		√	√		Multi-Order Best-Fit Heuristic
[28]	2013			√					√	√					Ant Colony Optimization Algorithm
[29]	2013	√		√					√	√				√
[30]	2010			√					√	√					Ant Colony Optimization Algorithm

¹ Objective function: min T—minimize the time of ship passing through locks; min C—minimize ship carbon emissions; min Z—minimize costs (including ship navigation costs and water consumption costs, etc.); max Q—maximize lock throughput; max Pr—maximize ships’ priority in locks; max U—maximize lock chamber utilization; Oth—others. ² Constraints: T—temporal constraint; Sp—spatial constraint; Ti—tidal constraint; Pr—priority constraint; Oth—others. ³ Solution methods: PA—exact algorithm; MA—heuristic algorithm.

.

Regarding the ship placement sub-problem, the service rules are determined first, and specific scheduling rules and priorities are adopted to ensure the rationality of the scheduling plan. The most widely accepted rules are FCFS, the shortest processing time first (SPF), etc. FCFS is currently the most popular rule for ship locks. However, FCFS can easily lead to low utilization of lock chamber areas [7]. Moreover, the total waiting time when adopting the SPF is less than that of the FCFS, by analyzing and comparing the different service strategies of multiple locks on the Mississippi River [8]. The priorities of ships are considered by assigning weights to ships, such as in [9,10].

Moreover, the ship placement sub-problem is usually described as a classic 2D bin packing problem. The process of putting ships into lock chambers is regarded as the process of filling large rectangles with small rectangles, which satisfies the constraint that the sum of the areas of ships arranged into each lock chamber does not exceed the lock’s area. In general, additional constraints are considered, such as the effects of adding mooring constraints and safety constraints to ships [27]. Ref. [15] also considers mooring constraints, but it deals with the serial-lock scheduling problem from the view of a multi-commodity network.

The study of the lockage scheduling sub-problem deals with chamber assignment and lockage operation planning. This problem is often modeled as a parallel machine scheduling problem where chambers map to machines and the lockage to jobs. In [21], the lockage scheduling problem is divided into two parts: arranging the sequence of every lock chamber and predicting lockage time by calculating the flexible lockage time of every lock time, which improves lockage efficiency. In [18], the co-scheduling problem of cascaded locks with multiple chambers is investigated, focusing on determining the lockage number, the lockage direction combination, and timetable optimization.

In recent years, the issue of ship carbon emissions has gained great attention. Especially for the ship lock of the Three Gorges dam, an improved mixed-integer nonlinear programming model is proposed focusing on the fuel consumption and exhaust emission pollution caused by ships sailing and waiting for a long time in the ship lock scheduling, especially for the ship lock process [20]. The obtained scheduling scheme can significantly reduce the total carbon emissions of all ships passing through the lock without compromising efficiency. Considering the uncertainties in the lock processing time, a speed optimization model is put forward in [12] to minimize ships’ fuel consumption.

While solving the lock scheduling model, different methods are often chosen based on the trade-off between solving time and solving accuracy for problems of different scales and solving requirements. The exact algorithm performs better in solving some small-scale cases, such as the method based on the combinatorial benders’ decomposition proposed in [23], which is proved to perform well in lock scheduling problems with a single large lock chamber with up to 90 ships However, compared with the proposed exact algorithm, the heuristic can reduce the maximal computation time from 527 to 0.5 s with a maximal optimality gap of 2.93%. Similarly, a polynomial algorithm based on Forbidden Zones is put forward in [25] to solve the NP-hard problem of ship lock scheduling, which can solve the problem that only one ship can pass through a single ship lock at one time. Using a graph problem to improve the existing algorithm is highly recommended. Taking the aggregated fuel consumption by ships into consideration, an iterative payment scheme algorithm is proposed in [16] to decide the speed and payment of ships, which can not only guarantee a Nash equilibrium but also minimize the fuel consumption. In conclusion, compared with the heuristic algorithm, the exact algorithm is widely used because it can always obtain the optimal solution of the scheduling problem; although the computation time is longer. In future work, the characteristics of these exact algorithms should be considered in the heuristic algorithm, so as to improve the accuracy of the solution of the heuristic algorithm.

Due to the capability of solving large-scale scheduling problems, the heuristic algorithm has received tremendous attention in the research of ship lock scheduling. Regarding the optimization model of the co-scheduling of two-line ship locks based on nonlinear objective programming, the Hooke–Jeeves Algorithm is proposed in [24] to solve it. Then in [28], the detailed ship scheduling plan is obtained using the Ant Colony Optimization (ACO) to achieve the optimization goal of minimizing the operating cost within the planning period, and the convergence is analyzed. Some combinatorial optimization approaches are then mentioned in future research, such as applying the neural network to solve the time scheduling problem or integrating neural networks with ACO. The adaptive large neighborhood search heuristic is proposed to obtain high-quality scheduling solutions relatively quickly [17], especially for instances without FCFS restriction or with small inter-arrival times. Therefore, nowadays, some effective heuristic algorithms have been designed and proposed to solve the medium- and large-scale instances with a small optimality gap of the solutions, such as the adaptive large neighborhood search heuristic.

In order to choose the most appropriate solution method, some studies have paid attention to comparing the exact algorithms and heuristic algorithms in lock scheduling problems. In [27], the ship placement problem is studied, and three algorithms are designed and compared. The first one is an exact approach, which has a good performance in dealing with the small-scale problems (referring to the problems that the number of ships that fit in one lockage is less than 18), and the results also show that a feasible solution can be generated more easily when solving several single-lockage ship placement problems than one multi-lockage instance. The second and third approaches are left-right-left-back heuristic and multi-order best fit heuristic, respectively. For the large-scale instances with up to 1000 ships, the solution can be obtained within 0.1 s using the multi-order best fit heuristic, which is much more efficient than the other two approaches. Similarly, in [11], the local search heuristic is compared with the exact approach based on MILP. Results show that the MILP solver can only find a feasible solution for instances within 40 ships, and the local search heuristic can produce a schedule for nearly 118 ships in 17.62 s. Thus, heuristic algorithms are proved to be more suitable for real-life instances. To conclude, the exact algorithms are more suitable to solve small instances within approximately 18 ships, while heuristic algorithms have better performance in large scale instances even up to 1000 ships.

The ship lock scheduling problem is an optimization problem to reasonably allocate the spatial-temporal resources of the lock chamber. Thus, spatial and temporal constraints need to be considered. In the water–land transshipment model of the ship lock established in [22], the time ships go into the chamber, the opening time of the adjacent ship locks, etc., are constrained. At the same time, the spatial conditions are restricted, considering the lock chamber capacity. A one-way single ship lock model is established in [30], considering the arrival time, scheduling starting time, and maximum waiting time of the ship. Further, the ship is deemed as a rectangle to consider the spatial constraints through the 2D bin packing problem. To deal with the cascaded lock scheduling problem, the time a ship arrives at a lock is closely related to its completion time at the previous lockage and the traveling time between locks [13,14].

If the traffic volume in the upstream and downstream directions of the lock is relatively balanced, the two-way operation mode of alternating upstream and downstream is generally adopted. When the demand for passing through the lock in a certain direction is greater, the one-way operation mode of continuous passage in the same direction (referred to as an inverted gate) can be adopted. In such conditions, there is an additional water filling (discharging) process between the adjacent two lockages. The number of inverted gates is considered in the objective function of the scheduling model to avoid waste of resources in [29]. In reality, the multiple cascaded continuous locks usually adopt a one-way operation mode. For example, the normal working mode of the Three Gorges lock is that the north ship lock goes upstream and the south ship lock goes downstream. When there is an asymmetric backlog of ships above and below the dam, the mode will be all upstream or all downstream. Still, studies on the timing of switching modes and the scheduling during switching are rarely seen in the relevant literature. In addition, specific constraints, such as priority constraints, ship mooring constraints, and tidal limitations, are considered in [26] to make it closer to practice. However, these constraints complicate the problem and limit the model’s applications.

3.2. Terminal and Berth

The terminal scheduling problem aims to balance the efficient operation of ships and the scientific utilization of terminal resources, such as berths. A berth is a fundamental unit of ports and terminals and is the essential place for ships to load and unload. Its size and quantity are important factors in measuring the scale of ports and terminals. The terminal scheduling problem includes both single-berth and multi-berth allocation. Accordingly, the allocation of berths directly impacts the operational efficiency of the entire terminal and port.

Single-berth allocation is a process in which a single or multiple ships arrive at a particular berth and formulate their berthing and operation plans before berthing. Similar to the queueing problem, the single-berth allocation problem can be regarded as the ship’s arrival following a specific law, and the service time of the berthing system obeys a particular distribution. Based on these rules, the berth allocation system is updated to meet the needs of the ship and realize the goal of maximizing economic benefits. In the current literature, there are few studies on single-berth allocation. A hybrid model is established in [31] to determine the probability model of the arrival time of different types of ships at a single berth. In [32], ships are regarded as “customers”, while “service stations” are represented by the berths of the port. It concluded that the expected queuing time can be reduced by increasing the loading and unloading speed with specialized berths.

According to the berth layout, the multi-berth allocation problem can be divided into discrete and continuous berths, as shown in Figure 4. The discrete berth allocation problem is a variant of the queuing problem. That is, the ship arrives according to specific rules, the occupancy of the service infrastructures, and the service time of the terminal system obeys a particular distribution. Contrarily, the continuous berth allocation problem can be solved using the 2D bin packing problem. That is, taking the length of the terminal shoreline and operation time as two solving dimensions to determine the ship’s birthing position and berthing time to achieve the optimization objectives of the shortest waiting time and the maximum utilization of the terminal shoreline.

3.2.1. Terminal Scheduling Problem

The terminal scheduling problem aims to minimize the time of the ship in port, which is defined as the duration from the beginning of the ship’s berthing to the end of the ship’s final departure. The general models of discrete and continuous berth scheduling are constructed based on the following assumptions:

(a)

Ship shifting between berths is ignored.

(b)

Each ship has and only has one berthing opportunity.

(c)

Emergencies, such as severe sea conditions and equipment failures, are not considered.

The discrete berth allocation is shown in Equations (9)–(14), where Constraint (10) ensures that the approved berthing capacity of the berth is larger than the tonnage of the corresponding ship. Constraint (11) ensures that the berth length is longer than that of the corresponding ship. Constraint (12) is the time constraint between ships operating at the same berth. Constraint (13) requires that the ship’s berthing time is later than its arrival time. Constraint (14) guarantees that a ship is served at only one berth.

Discrete Berth Allocation Problem (DBAP):

$$\mathrm{minimize} t^{\mathrm{avg}}=\frac{{\displaystyle \sum _{i=1}^{\mathrm{N}}{\displaystyle \sum _{m=1}^{\mathrm{M}}(t_i^{\mathrm{mo}}-t_{im}^{\mathrm{arr}})y_{im}}}}{\mathrm{N}}$$

(9)

$$\mathrm{subject} \mathrm{to} \forall i,j=1,2,\dots ,\mathrm{N}, i\ne j, m=1,2,\dots ,\mathrm{M}$$

$$(A_m-A_i^{\mathrm{ton}})y_{im}\ge 0$$

(10)

$$(L_m-L_i)y_{im}\ge 0$$

(11)

$$\left\{\begin{array}{l}{t}_i^{\mathrm{mo}}+t_{im}^{\mathrm{op}}+\mathrm{H}(y_{im}-1)\le \\ t_j^{\mathrm{mo}}+\mathrm{H}(1-y_{jm})+\mathrm{H}(1-x_{ij})\\ t_j^{\mathrm{mo}}+t_{jm}^{\mathrm{op}}+\mathrm{H}(y_{jm}-1)\le \\ t_i^{\mathrm{mo}}+\mathrm{H}(1-y_{im})+\mathrm{H}·x_{ij}\end{array}\right.$$

(12)

$$t_{im}^{\mathrm{arr}}\le t_i^{\mathrm{mo}}$$

(13)

$${\displaystyle \sum _{m=1}^{\mathrm{M}}{y}_{im}=1}$$

(14)

where, $i,j$ is the number of the ship, $m$ represents the number of the berths, $A_m$ indicates the approved berthing capacity of berth $m$, $A_i^{\mathrm{ton}}$ is the tonnage of the ship $i$, $y_{im}$ is a binary variable that denotes whether the ship $i$ berths at berth $m$ (1) or not (0).

The continuous scheduling model also takes the objective function of minimizing the total time of ships in port. Constraints (16)–(20) ensure that any two ships do not overlap in spatial-temporal dimension. Constraints (21) and (22) require the ship to be berthed within the shoreline of the terminal. Constraint (23) requires the ship to berth after its arrival. Constraint (24) is the time relationship that the ship needs to meet for berthing operations at the terminal, which means the ship can leave the berth after completing the berthing, unberthing, and loading and unloading operations.

Continuous Berth Allocation Problem (CBAP):

$$\mathrm{minimize} t^{\mathrm{po}}={\displaystyle \sum _{i=1}^{\mathrm{N}}{\displaystyle \sum _{m=1}^{\mathrm{M}}(t_{im}^{\mathrm{un}}-t_{im}^{\mathrm{arr}})}}$$

(15)

$$\mathrm{subject} \mathrm{to} \forall i,j=1,2,\dots ,\mathrm{N}, i\ne j, m=1,2,\dots ,\mathrm{M}$$

$$R_i+\frac{L_i+L_j}{2}\le R_j+L_m(1-{\lambda }_{ij}^{\mathrm{Le}})$$

(16)

$$t_{im}^{\mathrm{mo}}+t_{im}^{\mathrm{mo},\mathrm{op}}+t_{im}^{\mathrm{op}}+t_{im}^{\mathrm{un},\mathrm{op}}\le t_{jm}^{\mathrm{mo}}+\mathrm{H}(1-{\lambda }_{ij}^{\mathrm{be}})$$

(17)

$${\lambda }_{ij}^{\mathrm{Le}}+{\lambda }_{ji}^{\mathrm{Le}}+{\lambda }_{ij}^{\mathrm{be}}+{\lambda }_{ji}^{\mathrm{be}}\ge 1$$

(18)

$${\lambda }_{ij}^{\mathrm{Le}}+{\lambda }_{ji}^{\mathrm{Le}}\le 1$$

(19)

$${\lambda }_{ij}^{\mathrm{be}}+{\lambda }_{ji}^{\mathrm{be}}\le 1$$

(20)

$$R_i-\frac{L_i}{2}\ge 0$$

(21)

$$R_i+\frac{L_i}{2}\le L_m$$

(22)

$$t_{im}^{\mathrm{mo}}\ge t_{im}^{\mathrm{arr}}$$

(23)

$$t_{im}^{\mathrm{un}}=t_{im}^{\mathrm{mo}}+t_{im}^{\mathrm{op}}+t_{im}^{\mathrm{mo},\mathrm{op}}+t_{im}^{\mathrm{un},\mathrm{op}}$$

(24)

where, $i,j$ is the number of the ship, $m$ represents the number of berths, $t^{\mathrm{po}}$ is the total time of all ships in port, $R_i$ indicates the actual berthing position of ship $i$, referring to the berth position corresponding to the center of the ship, ${\lambda }_{ij}^{\mathrm{Le}}$ is a binary variable that denotes whether ship $i$ is on the left side of ship $j$ in the two-dimensional map of berth allocation (1) or not (0), $t_{im}^{\mathrm{mo},\mathrm{op}}$ and $t_{im}^{\mathrm{un},\mathrm{op}}$ represent the berthing and unberthing time of ship $i$, ${\lambda }_{ij}^{\mathrm{be}}$ is a binary variable that denotes whether ship $i$ is below the ship $j$ in the two-dimensional map of berth allocation (1) or not (0).

3.2.2. Ship Berth Communication

Table 4. Information in the terminal/berth scheduling problems.

	Static Information	Dynamic Information	Navigation-Related Information	Terminal/Berth-Related Information
Terminal	Length of the ship; Width of the ship	-	Draft of the ship; Ship arrival time; Estimated arrival time; The ship’s required departure time	Cargoes loaded and unloaded; Ship’s preferred berth; Unit penalty cost caused by ship departure delay; Ship loading and unloading time; Time of berthing and unberthing operation; The time when ships begin to berth; Ship’s maximum and minimum number of loading and unloading quay cranes; Priority of ship; Total number of ships; Maximum and minimum deadweight tonnage or container of all ships; Maximum waiting time of ship.
Berth	Length of the ship; Width of the ship; Type of ship	-	Draft of the ship	Estimated number of ships arriving; Estimated operating time per ship; Ship arrival and departure time; Arrangement of the pilot; Shipborne cargo information.

summarizes the information involved in terminal scheduling and berth allocation problems. In addition to the dynamic, static, and navigation information related to ships, the information used in terminal scheduling also includes cargoes information, berth capacity, and berth operation.

3.2.3. Research Status of Terminal Scheduling

Table 5. Overview of research on terminal/berth scheduling problems.

Ref.	Year	System Characteristics					Model Characteristics								Solution Methods 5
		Spatial-Scale 1			Temporal-Scale 2		Objective Function 3				Constraints 4
		Dis	Con	Mix	St	Dy	Min T	Min Z	Min F	Oth	T	Sp	Ti	Pr	PA	MA
[31]	2017									√	√				Generalized Queuing Models
[32]	2019						√				√	√			Queuing Models
[33]	2020		√			√		√			√	√			CPLEX	Particle Swarm Optimization
[34]	2019		√			√		√			√	√				Improved Shuffled Frog-Leaping Algorithm
[35]	2017		√			√	√	√			√	√				Simulated Annealing Algorithms
[36]	2022		√			√	√	√			√	√			Gurobi
[37]	2022		√			√	√				√	√	√			Iterated Local Search
[38]	2022		√			√		√			√	√			CPLEX	Variable Neighborhood Search
[39]	2022	√					√	√			√	√				AEMOHCO Algorithm
[40]	2022	√					√				√	√			√
[41]	2022	√					√			√	√	√	√		OR-Tools
[42]	2022	√				√	√				√	√		√		Scatter Search Metaheuristic
[43]	2022		√		√		√				√	√		√		Genetic Algorithm
[44]	2022						√	√	√		√	√				Genetic Algorithm
[45]	2022	√						√			√	√			CPLEX	Genetic Algorithm
[46]	2022	√					√				√	√		√	√	Benders Cut Strengthening Heuristic
[47]	2021	√						√			√	√			Queueing Theory
[48]	2021							√			√	√			√
[49]	2021				√			√	√		√	√				Multi-Objective Particle Swarm Optimization Algorithm
[50]	2021		√			√		√			√	√				Improved Particle Swarm Optimization
[51]	2021	√					√				√	√	√		Column Generation
[52]	2021	√					√				√	√	√			Adaptive Large Neighborhood Search Algorithm
[53]	2021		√					√			√	√				Particle Swarm Optimization Algorithm + Machine Learning
[54]	2020						√				√	√		√		Greedy Algorithms
[55]	2020					√	√				√	√		√	√
[56]	2020							√			√	√				Squeaky Wheel Optimization Heuristic
[57]	2020					√	√	√			√	√		√		Genetic Algorithm
[58]	2020		√			√	√		√		√	√			√
[59]	2020			√		√	√	√			√	√				Chemical Reaction Optimization
[60]	2020		√			√	√				√	√	√			Reduced VNS-Based Approach
[61]	2020	√				√	√				√	√				Bee Colony Optimization
[62]	2020	√				√	√			√	√	√				New Hybrid Genetic Algorithms
[63]	2020		√			√	√	√		√	√	√		√	LINGO
[64]	2020		√			√	√				√	√		√		Swarm-Based Artificial Bee Colony Optimization
[65]	2019	√				√		√			√	√				Evolutionary Algorithm
[66]	2019	√				√		√			√	√		√		Adaptive Island Evolutionary
[67]	2019			√		√	√				√	√				Modified Sailfish Optimizer
[68]	2019	√				√		√			√	√			CPLEX
[69]	2019	√				√	√				√	√	√		CPLEX
[70]	2019	√				√	√				√	√				Chaotic Genetic Algorithm-Based Method
[71]	2017		√			√	√				√	√	√		CPLEX
[72]	2017		√				√				√	√				Modified NSGA-II Algorithm
[73]	2016			√		√		√			√	√				Smart Greedy Algorithm
[74]	2016							√	√		√	√				Memetic Algorithm; Genetic Algorithm; Simulated Annealing Algorithm
[75]	2016							√			√	√				Adaptive Large Neighborhood Search Heuristic
[76]	2016		√			√	√				√	√			GAMS
[77]	2013							√	√		√	√			CPLEX
[78]	2012	√				√	√					√				Artificial Intelligence Hybrid Heuristic Based on Tabu Search
[79]	2010			√	√		√			√	√	√		√		Multi-Objective Evolutionary Algorithm

¹ Spatial-scale: Dis—discrete; Con—continuous; Mix—mixed model. ² Temporal-scale: St—static; Dy—dynamic. ³ Objective functions: min T—minimize average waiting time; min Z—minimize costs; min F—minimize total fuel consumption; Oth—others. ⁴ Constraints: T—temporal constraint; Sp—spatial constraint; Ti—tidal constraint; Pr—priority constraint. ⁵ Solution methods: PA—exact algorithm; MA—heuristic algorithm.

summarizes the system characteristics, model characteristics, and solution methods in the existing research on the terminal scheduling and berth allocation problem.

From the analysis of berth layout and time characteristics, most current papers pay attention to the dynamic continuous berth allocation problem. To minimize the total operation cost, the continuous berth scheduling with the “Double-Line Ship Mooring” mode is illustrated in [33]. Then, both exact and heuristic methods are proposed to solve this model. The dynamic continuous berth allocation problem is also studied in two stages in [34]. In the first stage, an initial solution to the problem is generated based on the FCFS. The continuous berth allocation problem is regarded as a 2D bin packing problem in spatial and temporal dimensions. In [35], a mixed-integer programming model for the berth allocation problem of the dynamic continuous terminal is established, aiming at minimizing the total weighted service time and the cost of deviating from the preferred berth position. Two SA-based Algorithms are proposed to determine the berthing position of the ship. Considering the uncertainties from the external environment, a dynamic continuous berth allocation problem is modeled and solved in [36], which can minimize the ships’ actual waiting times and achieve enhanced service quality.

For the discrete terminal scheduling problem, the mixed-integer programming model considering the priority of ship service is established in [66] to minimize the total weighted cost of ship service, including ship handling operation cost, ship waiting operation cost, and delay departure cost. Due to the discrete layout of berths, each can only serve one ship simultaneously in the model. In the case of the dynamic arrivals, the mixed-integer linear programming model of discrete berth allocation is established in [65] and solved by EA. This method shows good computational superiority and stability compared to the other nine heuristic algorithms. In [46], a terminal with two sides of symmetrically distributed berths is considered, and an exact method based on logic-based Benders decomposition is proposed to minimize the total tardiness of all ships. The algorithm’s performance is then evaluated based on experiments with real data from China. To investigate the discrete berth allocation problem of multiple liner carriers, a three-stage method is put forward in [47] to minimize the number of berths to be assigned for all considered liner carriers.

However, few studies focus on the hybrid terminal scheduling problem. In [67], the terminal scheduling problem of a variable hybrid layout is studied. The small ships share the same berth for operation, and an improved Sail Fish Optimization Algorithm is proposed for berth allocation. The simulation shows that this method can significantly improve terminal utilization. In [79], a more general situation is adopted where the whole berthing space is separated into many discrete sections, and the space is continuous within each other, which leads to a scheme of a hybrid terminal schedule. A Multi-Objective Evolutionary Algorithm is then proposed to solve this multi-objective problem, including minimizing the makespan, waiting time, and priority deviation of ships. Similarly, in [59], the terminal is separated into several sets of segments. A ship can occupy some of the segments or several ships are allowed to berth in one segment simultaneously. A chemical reaction optimization algorithm is proposed to solve the large-scale problem, which is proved to have better exploration capability in searching effectively and attaining promising near-optimal solutions, compared with GA and PSO.

In the research of terminal scheduling, the primary goals are to reduce the delay time of ships and time in port, as well as reduce the cost of sailing operation of ships and the operation cost of terminals, and to maximize the benefits under limited resources. Considering the uncertainties in the arrival time and handling time of ships, a dynamic hybrid berth allocation problem is modeled in [73] to minimize the total realized cost of the updated berthing plan. Furthermore, to reduce the total time that ships stay in the port, an artificial intelligence hybrid heuristic algorithm based on Tabu search is proposed in [78], which has a good performance in large, medium, and small instances. An ore terminal in a Brazilian port is discussed in [75] to minimize the delay cost and dispatch cost of ships due to exceeding or ahead of loading and unloading time, which is conducive to maximizing the broad interests of ship owners and port management agencies. Furthermore, most studies consider the weighting of multiple objectives. In [62], the terminal scheduling scheme of minimizing the total ship service time and maximizing the terminal occupancy is sought, and the berth preference of ships is also considered.

Under the development trend of green ports, energy conservation and emission reduction have also begun to be reflected in berth allocation. In [74], a mixed-integer programming model is established to minimize the departure delay and energy consumption of all ships, and the effects and contributions of the “energy-saving strategy” and “time-saving strategy” in environmental protection are then analyzed. Ref. [77] considers fuel consumption and ship emissions and proposes a mixed-integer linear programming model to minimize delay and fuel costs. Then, two quadratic outer approximation approaches are presented and proved to be able to handle this model efficiently. To balance the port efficiency and environment protection, berth allocation is studied in [44] from energy saving and emission reduction perspectives. Research with similar optimization goals can be found in [49,58].

Terminal scheduling usually involves large- or medium-scale mixed-integer planning. It is challenging to obtain the optimal solution in a short time using an exact algorithm. Therefore, commercial solvers, such as CPLEX and GAMS, are preferred. In [68], three mixed-integer linear programming models for the berth allocation problem of container terminals are solved by CPLEX with high computational efficiency. Furthermore, it is proved that the proposed models can maintain superior computational performance with less optimality gap with more than 110 ships in the instances. Ref. [71] considers the influence of tide in the berth allocation for dry bulk terminals with CPLEX, which concludes that some valid inequalities can significantly improve the solution quality and the computation time of the MILP model. Ref. [76] considers different berth lengths and service requirements, and the GAMS is used. To conclude, using commercial solvers to solve MILP is widely accepted.

Designing heuristic algorithms to solve the established terminal scheduling problem is the current mainstream. In [61], a meta-heuristic algorithm based on BCO is used to solve the berth allocation problem of the large-scale composite terminal. The real data of the Livorno port in Italy are used, and compared with the CPLEX solver and Ant Colony Algorithm, the results of the BCO are proved to be near-optimal (average gap of 0.12%) and are calculated in a reasonably low computation time (on average 10.3 s). Furthermore, a chaos GA is proposed in [70] to solve the berth allocation problem with physical constraints and ship service priority constraints. The results are compared with GA, which proves that the algorithm proposed has a better performance on balancing the service priority and terminal productivity. In [50], a core operator is designed to improve the PSO algorithm used in the studied robust berth allocation problem. Moreover, the smart update strategy outperforms the traditional ones in 40 test instances with solution improvements of 0.90%. Moreover, a Squeaky Wheel Optimization heuristic is adopted in [56] to deal with unforeseen incidents. Compared with CPLEX, the proposed method can more successfully give high-quality solutions to the scales up to 8 mother vessels, 20 feeder vessels, and 20 transshipment flows. Ref. [53] proposed a PSO algorithm embedded with a machine learning approach to deal with unpredictable weather conditions with two-stage optimization. Results show that for small-scale and medium-scale instances, the average deviation of the proposed PSO algorithm does not exceed 1% compared with CPLEX, and PSO can also find solutions quickly for large-scale instances. In conclusion, more and more heuristics are used and improved to solve BAP and proved to have good performance, such as BCO, chaos GA, and the PSO with a core operator. Future research can pay attention to the unavoidable optimality solution gap, especially solving the large-scale instances.

Some research also compares the performance of commercial solvers and heuristic algorithms. A MILP formulation, a scheduling heuristic, and an Iterated Local Search are presented to solve the tidal bulk berth allocation problem in [37], and the MILP is solved by CPLEX. Results show that CPLEX can perform well with instances within 40 ships, while these heuristic methods can provide better solutions starting from instances with 40 ships, and can prove that they could provide a better alternative to commercial software when dealing with medium- and large-scale instances. Furthermore, in [38], a MILP formulation and a Generalized Variable Neighborhood Search solution approach (GVNS) are developed to solve the berth allocation problem with continuous layout. The instances in the experiments are divided into three groups: small-scale (18–30 ships), medium-scale (40–60 ships), and large-scale (80–120 ships). The results show that the MILP’s solution is better than the GVNS when solving the small-scale and medium-scale instances. As for the large-scale instances, GVNS provides better solutions than MILP.

In terms of the constraints, the terminal scheduling models mainly consider temporal constraints, such as the arrival time of the ship, the service time, the loading and unloading time, and the operation time, as well as spatial constraints, such as the occupation of the continuous shoreline of the terminal [60,63,64,69,72]. In [39], the first ship is stipulated to be served when it arrives, and ships can be accommodated in their selected berth. Furthermore, in [40], an exact approach that combines Benders decomposition and column generation is proposed to solve the berth allocation problem, with the constraints of the cargo handling time and the mooring limits of ships. Similar temporal and spatial constraints are found in [45,48].

The tidal influences are considered in large seaports. For example, the dynamic continuous berth allocation problem under tidal constraints is studied in [60]. Ref. [69] takes into account the fact that ships with large drafts need to wait for anchoring at low tide when designing the scheduling algorithm. In the case study, the delay is effectively reduced by 46%. An integrated optimization method is proposed in [41] for cyclic berth planning accounting for tidal windows. Furthermore, in [51,52], tidal factors are modeled and systematically studied in the scheduling models.

Furthermore, in practice, terminal operators decide the berthing order according to specific priority rules. Four different experimental scenarios based on single-queue and multi-queue models are combined with FCFS rules to study the influence of ship priority on terminal scheduling problems in [64]. In addition, the scheduling model of emergency ships arriving at the port is established in [63]. Emergency ships, such as ships transporting disaster relief materials, have absolute berthing priority compared to regular ships. However, the adjustment should not make significant changes to the current service schedule but consider the priority rules and fairness. Several priority control procedures are presented in [42], such as the dedicated terminal configuration, FCFS, and ships with the shortest processing time first, etc. The results show that choosing different priority rules for different situations helps improve efficiency. To minimize the total delay, a mixed integer programming model is given in [43] to solve the integrated continuous berth allocation problem with the service priority. The priority rules are also considered in the study in [54,55,57].

3.3. Waterway Intersection

The waterway intersection refers to the waterway that the ship’s flow crosses through due to the staggering of the shipping routes. Different from the hydraulic construction infrastructures, such as ship locks and terminals, the waterway intersections often exist in the areas where the waterways intersect, such as ports and waterway networks, and are the nodes of the waterway network. In the waterway intersections, ships often come from different directions. With the increase in the number of ships and the expansion of the channel scale, the traffic flow in the intersections becomes more complex, and the encounters between ships increases, which makes it difficult to guarantee the safety and efficiency of ship navigation. Existing studies on the waterway intersections usually focus on expanding the width and depth of the waterway or adopting the traffic separation scheme to organize and manage ship traffic in the waterway. The waterway intersections can also be a kind of special navigation infrastructure. Making full use of ship navigation information and scheduling research on the conflict areas in the waterway intersections can effectively improve the utilization of spatial-temporal resources and ship traffic efficiency in the waterway intersections to ensure the safety of navigation.

The waterway intersection scheduling problem can be regarded as a variant of the job shop scheduling problem (JSP). Ships pass through the intersections divided into blocks, and each ship passes through each block in a specific order, as shown in Figure 5. To avoid conflicts, only one ship can pass through each block at a time. Unlike the JSP problem, the ship does not stay in the waterway intersections and does not need to cross all the blocks.

3.3.1. Intersection Scheduling Problem

Intersection scheduling aims to achieve the goal of efficient passage by rationally arranging the time and location of ships passing through the waterway intersections. The intersection scheduling problem (ISP) is shown in Equations (25)–(33). The model’s objective is to minimize the total time required for all ships to pass through the waterway intersections. Constraint (26) ensures that the entire time needed by all ships is larger or equal to the sum of the arrival time of ships and the time required to pass through the intersections. Constraint (27) is the earliest arrival time of each ship. Constraint (28) gives the time and range necessary for ship i to pass through block m. Constraint (29) is for the sequential and no-wait constraints. Constraint (30) gives the range of time required for ship i to sail from block m to block n. Constraint (31) indicates that the arrival time of the latter ship should not be earlier than the time when the former ship leaves the block. Constraint (32) ensures that the arrival time interval of ships in the same block should be larger than the predefined safe time interval. Constraint (33) ensures that the departure time interval of ships in the same block should be larger than the predefined safe time interval.

Intersection scheduling problem (ISP):

$$\mathrm{minimize}{T}_{\max }$$

(25)

$$\mathrm{subject} \mathrm{to} \forall i,j=1,2,\dots ,\mathrm{N}, m,n=1,2,\dots ,\mathrm{M}, n=m+1$$

$$T_{\max }\ge {\displaystyle \sum _{i=1}^{\mathrm{N}}(t_{im}^{\mathrm{arr}}+t_{im}^{\mathrm{op}})}$$

(26)

$$t_{im}^{\mathrm{arr}}\ge E_i$$

(27)

$$\frac{l_{im}}{v_{i,\max }}\le t_{im}^{\mathrm{op}}\le \frac{l_{im}}{v_{i,\min }}$$

(28)

$$t_{in}^{\mathrm{arr}}=t_{im}^{\mathrm{arr}}+t_{im}^{\mathrm{op}}+T_{i,m\to n}$$

(29)

$$\frac{d_{i,m\to n}}{v_{i,\max }}\le t_{i,m\to n}\le \frac{d_{i,m\to n}}{v_{i,\min }}$$

(30)

$$\left\{\begin{array}{l}{t}_{im}^{\mathrm{arr}}+t_{im}^{\mathrm{op}}\le t_{jm}^{\mathrm{arr}}+\mathrm{H}(1-x_{ij,m})\\ t_{jm}^{\mathrm{arr}}+t_{jm}^{\mathrm{op}}\le t_{im}^{\mathrm{arr}}+\mathrm{H}·x_{ij,m}\end{array}\right.$$

(31)

$$\left\{\begin{array}{l}{t}_{im}^{\mathrm{arr}}+t_{i,\mathrm{safe}}\le t_{jm}^{\mathrm{arr}}+\mathrm{H}(1-x_{ij,m})\\ t_{jm}^{\mathrm{arr}}+t_{i,\mathrm{safe}}\le t_{im}^{\mathrm{arr}}+\mathrm{H}·x_{ij,m}\end{array}\right.$$

(32)

$$\left\{\begin{array}{l}{t}_{im}^{\mathrm{arr}}+t_{im}^{\mathrm{op}}+t_{j,\mathrm{safe}}\le t_{jm}^{\mathrm{arr}}+t_{jm}^{\mathrm{op}}+\mathrm{H}(1-x_{ij,m})\\ t_{jm}^{\mathrm{arr}}+t_{jm}^{\mathrm{op}}+t_{i,\mathrm{safe}}\le t_{im}^{\mathrm{arr}}+t_{im}^{\mathrm{op}}+\mathrm{H}·x_{ij,m}\end{array}\right.$$

(33)

where, $i,j$ is the number of the ship, $m,n$ represents the number of the infrastructures, $T_{\max }$ is the total sailing time, $E_i$ is the earliest arrival time of the ship $i$, $l_{im}$ is the length of the path that ship $i$ travels through block $m$, $v_{i,\max }$ and $v_{i,\min }$ represent the maximum and minimum speed limits of ship $i$, respectively, $d_{i,m\to n}$ and $t_{i,m\to n}$ represent the distance and time ship $i$ sailed from block $m$ to block $n$, $x_{ij,m}$ is a binary variable that denotes whether ship $i$ passes block $m$ before ship $j$(1) or not (0), $t_{i,\mathrm{safe}}$ is the predefined safe time interval for ship $i$.

3.3.2. Communication in Intersection Scheduling Problems

Table 6. Information in the intersection scheduling problems.

Static Information	Dynamic Information	Waterway Intersection-Related Information
Length of the ship; Width of the ship; Ship tonnage	Speed of ships; Acceleration of ships; Courses of ships; Average speed;	Safety distance between ships; Rules of the ships’ arrival; Total number of ships or ship flows; Time of ships passing through the intersections.

summarizes the information in the waterway intersection scheduling problem. Unlike the scheduling of locks, terminals, and berths, waterway intersections are generally aimed at a short period. Therefore, there is a high demand for dynamic information containing the ship’s speed, acceleration, and course.

3.3.3. Research Status of Intersection Scheduling

Table 7. Overview of research on intersection scheduling problems.

No.	Published Time	Objective Function 1					Constraints 2
		Min T	Max Sa	Max Tr	Max U	Oth	T	Sp
[81]	2019				√		√	√
[82]	2018	√	√	√			√	√
[80]	2015		√				√	√
[83]	2011					√	√	√

¹ Objective function: min T—minimize the time of ship passing through the waterway intersections; max Sa—maximize ship navigation safety (the lowest probability of traffic conflict); max Tr—maximize the smoothness of ship trajectory; max U—maximize space utilization; Oth—others. ² Constraints: T—temporal constraint; Sp—spatial constraint.

summarizes the relevant literature on the waterway intersection scheduling problem. Nowadays, few studies are related to the scheduling of waterway intersections. The existing research mainly analyzes the traffic conflict rate and calculates the passing capacity. In [80], a navigational traffic conflict point is described as where traffic flows cross, merge, or diverge. The traffic conflict technique is introduced in the quantitative analysis of the influence degree of traffic flow, considering the similarity between the road intersections and the ship routing precautionary. In [81], conflict technology is applied to study the capacity estimation of waterway intersections to improve the ship traffic rules and obtain the maximum number of ships passing through waterway intersections per unit of time.

Nowadays, the authorities mainly adopt ship routing schemes or implement other specific navigation aides to ensure the safety of navigation in the waterway intersections. However, with the increasing number of ships, these measures have limited effects. To avoid collision, ships often maintain a large distance due to the lack of effective communication between ships. As a result, the spatial and temporal resources in the intersection waters cannot be effectively utilized. In [82], the multi-ship coordination system composed of ship formation and scheduling of the waterway intersections is proposed. The ADMM negotiation framework is used to realize the cooperation between controllers. This study shows that the scheduling method can effectively reduce the total time of ships passing through the waterway intersections. At the same time, the ships cooperate with each other, and there is no conflict in the waterway intersections. Thus, the ships can have a smoother trajectory than in the non-cooperative case.

4. Discussion

4.1. Generalized Model for Infrastructure Scheduling Problem

Table 8. Comparison of research on infrastructure scheduling problems.

Infrastructure Scheduling Problem	Model Characteristics										Solution Methods
	Objective Function						Constraints
	Min T	Min Z	Min F/Min C	Max U	Max Sa	Max Tr	T	Sp	Ti	Pr	PA	MA
Ship lock	●	◐	○	○			●	●	○	◐	◐	●
Terminal	●	◐	○				●	●	○	○	○	●
Berth	◐						●	◐			●
Waterway intersection	○			○	◐	○	●	●

●—The amount of literature that uses the characteristics or methods is more than 60%. ◐—The amount of literature that uses the characteristics or methods is more than 30% and less than 60%. ○—The amount of literature that uses the characteristics or methods is less than 30%.

summarizes the characteristics of four infrastructure scheduling problems in the relevant literature. Most literature focuses on the spatial and temporal dimension in the objective function and constraint, where the objective function is minimizing the time for ships passing through the infrastructures. In addition, more than 60% of the literature for all infrastructures consider time constraints, and more than 60% of the literature for infrastructures, except single berth scheduling, consider space constraints, which reflects the essence of infrastructure scheduling problem, that is, the allocation of spatial-temporal resources. Furthermore, the heuristic algorithm is more popular in the lock scheduling problem and terminal scheduling problem, while the exact algorithm is preferred in the berth scheduling problem.

Through the analysis of the existing literature, the following conclusions are drawn:

(a)

The process of ships passing through various infrastructures can be regarded as the occupation of spatial-temporal resources. Therefore, the main problem of infrastructure scheduling is the allocation of spatial-temporal resources, which can be formulated as the classical optimization problem. The ship placement sub-problem in ship lock scheduling is a classical 2D bin packing problem. The lockage scheduling sub-problem is a single machine (a lock with a single chamber) or a parallel machine scheduling problem (a lock with multiple lock chambers). The terminal scheduling problem includes both the scheduling of a single berth and multiple berths at the same time. The single berth and discrete multiple berth scheduling problems can be regarded as a variant of the queuing theory problem. At the same time, the continuous multiple berth scheduling problem can be solved by two dimensions, i.e., the terminal shoreline length and the ship navigation operation time, with the idea of a 2D bin packing problem. Furthermore, though it is rarely mentioned in the literature, the waterway intersections can be regarded as a variant of the JSP problem.

(b)

From the perspective of model establishment, the scheduling of locks, terminals, berths, and waterway intersections is a multi-constrained and multi-objective combinatorial optimization problem coupled in spatial-temporal dimensions. The objective function and constraints are mostly related to spatial-temporal dimensions. Temporal constraints are expressed by inequality constraints, while spatial constraints are represented by binary variables (0–1). The constraints such as tides and priority levels of special cases are rarely considered in some studies. Carbon emissions, energy conservation, and emission reduction targets and constraints have gradually become concerns.

(c)

As for the solution to the infrastructure scheduling problem, most studies propose or improve various heuristic algorithms or use solvers, such as CPLEX and GAMS. Moreover, the effectiveness of the model and method is verified by simulation.

(d)

As the research object, the existing research mainly focuses on locks and terminals. However, in key areas, such as ports and intersections where ship traffic flows converge, the interaction between ships increases, which is also a vital node worthy of attention.

To conclude, the main problem that needs to be solved for waterborne transportation infrastructures can be expressed as the following Generalized Infrastructure Scheduling Problem (GISP). The objective function is to minimize the total cost, which generally refers to the weighted sum of spatial-temporal cost and other related expenses. Constraint (35) is a temporal constraint, which means that the arrival time of the latter ship cannot be earlier than the time when the former ship leaves the area. Constraints (36)–(39) are spatial constraints, ensuring that the number of ships arriving/operating simultaneously does not exceed the maximum capacity of the infrastructure and that the size of ships is smaller than the maximum size allowed. Constraints (40)–(43) are tidal constraints, which mean that the ship that needs to ride the tide should enter/leave the port within the tidal period. Constraints (40)–(41) and (42)–(43) represent the constraints of tidal periods of ships entering and leaving the port, respectively. Constraint (44) is a cargo adaptation constraint, ensuring that the ship or its cargo type should match the type of infrastructure. Constraint (45) is a priority constraint.

Generalized infrastructure scheduling problem (GISP):

$$\mathrm{minimize}Z$$

(34)

$$\mathrm{subject} \mathrm{to} \forall i,j=1,2,\dots ,\mathrm{N}, i<j , m=1,2,\dots ,\mathrm{M};k=1,2,\dots ,\mathrm{K}$$

$$\mathrm{subject} \mathrm{to}$$

(35)

$${\displaystyle \sum _{i=1}^{\mathrm{N}}{x}_{im}^k\le U_m}$$

(36)

$$(L_m-L_i)x_{im}^k\ge 0$$

(37)

$$(D_m-D_i)x_{im}^k\ge 0$$

(38)

$$(W_m-W_i)x_{im}^k\ge 0$$

(39)

$$\mathrm{H}\times (1-e_i^p)+t_{im}^{\mathrm{mo}}\ge t_{\mathrm{s}}^{\mathrm{Ti}}+(P-1)\times {\mathrm{t}}^{\mathrm{inter}}$$

(40)

$$\mathrm{H}\times (e_i^p-1)+t_{im}^{\mathrm{mo}}\le t_e^{\mathrm{Ti}}+(P-1)\times {\mathrm{t}}^{\mathrm{inter}}$$

(41)

$$\mathrm{H}\times (1-d_i^p)+t_{im}^{\mathrm{un}}\ge t_{\mathrm{s}}^{\mathrm{Ti}}+(P-1)\times {\mathrm{t}}^{\mathrm{inter}}$$

(42)

$$\mathrm{H}\times (d_i^p-1)+t_{im}^{\mathrm{un}}\le t_{\mathrm{e}}^{\mathrm{Ti}}+(P-1)\times {\mathrm{t}}^{\mathrm{inter}}$$

(43)

$$B_m-g_i+\mathrm{H}\times (1-x_{im}^k)\ge 0$$

(44)

$$t_{im}^{\mathrm{s}}\le t_{(i+1)m}^{\mathrm{s}}$$

(45)

where, $i,j$ is the number of ships, $m,k$ represents the number of infrastructures and the order in which the ships are served, $Z$ is the comprehensive cost, which can be one or more weights of spatial and temporal related variables, $\mathrm{N},\mathrm{M}$ is the total number ships and the total number of infrastructures, respectively, $\mathrm{K}$ is the total number of services for all ships, $t_{im}^{\mathrm{arr}}$ is the time when ship $i$ arrive at infrastructure or area $m$, $t_{im}^{\mathrm{s}}$ and $t_{im}^{\mathrm{op}}$ is the start time and the duration of the operation, $x_{im}^k$ is a binary variable that denotes whether ship $i$ is assigned to the $k\mathrm{th}$ order of infrastructure or area $m$ (1) or not (0), $L_m$, $W_m$, $D_m$ and $U_m$ are the length, width, depth, and maximum capacity of the infrastructure $m$ (in units of ships), respectively, $L_i$, $W_i$ and $D_i$ are the length, width, and draft of the ship $i$, $\mathrm{H}$ is an arbitrarily large number, $P$ is the tidal period, $e_i^p$ is a binary variable, $e_i^p=1$ if and only if ship $i$ enters the port during the tidal period $P$, $t_{im}^{\mathrm{mo}}$ is the berthing time of ship $i$, $t_{\mathrm{s}}^{\mathrm{Ti}}$ and $t_{\mathrm{e}}^{\mathrm{Ti}}$ is the start and end time of the first tidal period, ${\mathrm{t}}^{\mathrm{inter}}$ is the tidal interval, $d_i^p$ is a binary variable, $d_i^p=1$ if and only if ship $i$ leaves the port during the tidal period $P$, $t_{im}^{\mathrm{un}}$ is the departure time of ship $i$, $B_m$ is the type of infrastructure , and $g_i$ is the type of ship $i$.

4.2. Research Gaps

In recent years, more and more scholars and related management agencies have devoted themselves to optimizing the operation efficiency of the waterborne transportation system, among which the problem of infrastructure scheduling is the core problem to be concerned. Relevant research results have relieved the operating pressure of waterborne infrastructures, such as terminals, to a certain extent. However, the following issues in the actual scheduling process are rarely mentioned in existing research:

(a)

Problems of “lack of connection” and “poor coordination” between infrastructures and ships

In the actual infrastructure scheduling process and existing research, ships can only passively accept the scheduling results of ship lock. Still, they have no idea of the specific situation of lock operation, and thus ships cannot prepare in advance. Similarly, the infrastructure management cannot accurately and in a timely manner grasp the status of ships. When the ship fails to arrive as planned, it often needs to wait for the next window, resulting in a long waiting time. The problems of “lack of connection” and “poor coordination” between the above infrastructures and ships have varying degrees of affecting the ship’s passing efficiency.

(b)

Problem of information isolated island

At present, the ways for infrastructure managers to obtain information mainly rely on ships’ reports. Although in recent years, the informatization of waterborne transportation organization and management has developed by leaps and bounds, the systems in the same jurisdiction and different jurisdictions are relatively independent. Data exchange and information sharing involve the interaction between different technical systems, which is still tricky.

(c)

Adaptability problem in the process of operation control

Many scholars have researched ship lock and terminal scheduling problems in-depth and proposed various algorithms. However, the berth allocation and ship lock scheduling decisions still rely on managers’ experience. On the one hand, the scheme designed in the study has been set in advance, which limits its application scenarios, and the results may be inconsistent with managers’ experience and preferences. On the other hand, uncertain events, such as ship delays and bad weather, significantly impact infrastructure schedules. Thus, it is always tricky for managers to re-establish a berth scheduling plan quickly to ensure economy and stakeholders’ satisfaction. The existing algorithms are also complicated to take these factors into account.

(d)

Lack of coordination of the interconnected infrastructures

Currently, most research is aimed at an individual infrastructure. The cooperative scheduling of ship locks and berths is studied in [19,22,84,85]. However, research on the coordination scheduling of different types of infrastructures and the cooperation of the whole transportation system is still lacking. In fact, as a huge system, the interconnected infrastructures in a waterborne transport system are interdependent. Improving the traffic situation at one piece of infrastructure may lead to congestion in other places. Thus, it is necessary to research cooperative scheduling of the interconnected infrastructures.

4.3. Recommendation

Based on the analysis of the research status of waterborne transportation infrastructure scheduling, this paper summarizes the key problems to be solved in future development and puts forward the following suggestions:

(a)

Human–machine collaborative decision-making: At present, the scheduling scheme of locks, waterway intersections, and other infrastructures still depends on the experience of managers. In the future, it is necessary to enhance managers’ trust in the algorithm and improve its flexibility for emergencies through mutual feedback, learning, and collaboration between waterborne transportation infrastructures and scheduling algorithms. For example, the uncertain operation time and different weather conditions are considered in the scheduling model in [48,53], which improved the practicability of the model to a certain extent and enhance managers’ trust in the model and algorithm.

(b)

Ship–infrastructure coordination: Most of the existing infrastructure scheduling methods are from the perspective of infrastructures, and the feedback of ships to scheduling results should be further considered. A reasonable coordination mechanism should be designed to receive the feedback in time. The scheduling plan can be updated to realize real-time closed-loop coordination between ships and infrastructures. The feedbacks of ships are considered in [73] to realize the real-time recovery of the berth scheduling plan, in case of the ships’ uncertain arrival time. Furthermore, in the waterway intersection scheduling problem proposed in [86], an ADMM-based negotiation framework is proposed for the cooperation among both the intersection controllers and vessel controllers.

(c)

Infrastructure–infrastructure coordination: Existing research is aimed at one infrastructure, but the waterborne transportation system is a whole, including various waterborne transportation infrastructures. Each infrastructure affects the others through the ships sailing in it. Focusing only on local efficiency may lead to the deterioration of global efficiency. Therefore, it is necessary to consider the coupling relationship between infrastructures from the global perspective and improve the operation efficiency of the whole system through coordination. Different levels of coordination have been gradually applied to infrastructure scheduling, such as the lock scheduling problem with a central coordinator [10].

(d)

Communication methods: Information is the basis of coordination. How to realize the information interaction between ships and ships, ships and infrastructures, and how to apply interactive information to the scheduling decision of waterborne transportation infrastructures are significant to improve the efficiency of waterborne transportation scheduling and need future efforts. Besides conventional communication methods such as VHF, Ship Reporting Systems, etc., novel communication methods can be adopted. Moreover, the limitations, such as bandwidth, delays, and package loss, should also be considered.

(e)

Energy conservation and emission reduction: The impacts of the shipping industry on the global environment cannot be ignored. Ship emissions will also increase accordingly in ports, locks, and other frequently manipulated areas [87,88]. Thus, energy conservation and emission reduction should also be paid more attention to when formulating infrastructure scheduling decisions.

5. Conclusions

This paper reviews the research on the scheduling for waterborne transportation infrastructures in the past ten years. The research status of four typical infrastructures, including ship locks, terminals, berths, and waterway intersections, are analyzed from system characteristics, model characteristics, and solution methods. Then, each infrastructure scheduling problem corresponds to the classical optimization problem: the ship placement sub-problem in ship lock scheduling is a classical 2D bin packing problem; the lockage scheduling sub-problem is a single machine (a lock with a single chamber) or a parallel machine scheduling problem (a lock with multiple lock chambers); the single berth and discrete multiple berth scheduling problems can be regarded as a variant of the queuing theory problem; the continuous multiple berth scheduling problem can be solved with the idea of a 2D bin packing problem; the waterway intersections can be regarded as a variant of the JSP problem. A general model of the infrastructure schedule is then constructed.

The main research gaps of the scheduling of waterborne transport infrastructures are identified, including lack of connection and poor coordination between infrastructures and ships, information isolated island, lack of adaptability, and lack of coordination of the interconnected infrastructures. The following research directions are recommended: human–machine collaborative decision-making; ship–infrastructure and infrastructure–infrastructure coordination; communication methods and constraints; energy conservation; and emission reduction.