Optimizing Pilotage Efficiency with Autonomous Surface Vehicle Assistance

Abstract

Efficient pilotage planning is essential, particularly due to the increasing demand for skilled pilots amid frequent vessel traffic. Addressing pilot shortages and ensuring navigational safety, this study presents an innovative pilot-ASV scheduling strategy. This approach utilizes autonomous surface vehicles (ASVs) to assist or replace junior pilots in specific tasks, thereby alleviating pilot resource constraints and upholding safety standards. We develop a comprehensive mathematical model that accommodates pilot work time windows, various pilot levels, and ASV battery limitations. An improved artificial bee colony algorithm is proposed to solve this model effectively, integrating breadth-first and depth-first search strategies to enhance solution quality and efficiency uniquely. Extensive numerical experiments corroborate the model’s effectiveness, showing that our integrated optimization approach decreases vessel waiting times by an average of 9.18% compared to traditional methods without ASV integration. The findings underscore the potential of pilot-ASV scheduling to significantly improve both the efficiency and safety of vessel pilotages.

1. Introduction

Port operations are a critical component of the global supply chain, significantly impacting the efficiency and safety of maritime transport, according to the International Maritime Organization [1]. Furthermore, pilot scheduling constitutes a vital part of port operations [2]. Pilots play a crucial role in port operations, safely guiding vessels into and out of berths. Their task is challenging in the dynamic maritime environment, requiring strict adherence to safety protocols to prevent accidents and ensure smooth port operations [3,4]. This results in skilled pilots becoming a scarce resource in ports. With the high frequency of domestic vessel traffic, the shortage of pilots has become a bottleneck in port operations [5]. To address this problem, some ports, such as Tianjin Port in China, implement a pilot scheduling approach whereby domestic vessels require pilot assistance for initial port entries and departures, but not necessarily for subsequent visits. While this approach helps alleviate the pressure on pilot resources, it increases the risk associated with vessel navigation. Specifically, vessels without pilot assistance are prone to maritime accidents, leading to frequent tugboat emergencies and resulting in port scheduling chaos. To address this issue, ports are seeking technological solutions that can provide pilot-like assistance to captains when vessels make subsequent entries to the port. With technological advancements, autonomous surface vehicles (ASVs) are now capable of effectively monitoring underwater currents and environmental conditions, providing essential data for safe maritime navigation [6,7]. With advances in artificial intelligence, ASVs can now learn local regulations and port channel conditions, offering navigation guidance to captains that meets port requirements, similar to local pilots [8]. Therefore, this paper proposes a pilot-ASV scheduling approach to alleviate the pressure on pilot resources. In this pilot-ASV scheduling strategy, vessels arriving at the port for the first time must still be piloted by pilots. However, for subsequent visits to the port, ASVs can be utilized to provide pilot assistance for vessels shorter than 180 m and not carrying hazardous materials. This approach not only eases the strain on pilot resources but also reduces the navigational risks associated with vessels entering and leaving the port. In light of this, the research problem addressed in this paper can be described as developing a pilotage planning in seaports with pilot-ASV scheduling. The problem takes into account various factors such as the work time windows of pilots, the different levels of pilots, and the battery limitations of ASVs, aiming to reduce the stay time of vessels.

The rest of the paper is structured as follows. Section 2 describes the related works for the pilotage planning with pilot-ASV scheduling and emphasizes our contribution. Section 3 formulates the problem with a mathematical model. Section 4 details a heuristic algorithm for the problem-solving. Numerical experiments are conducted in Section 5. Finally, conclusions are made in Section 6.

2. Literature Review

Pilotage planning encompasses various port operations, including vessel entries and departures, as well as pilot scheduling. The optimization of vessel traffic sequencing is a critical aspect of pilotage planning, aiming to minimize port stay time, reduce congestion, and enhance overall port throughput. Agussurja et al. initially formulated the maritime traffic management problem as a variant of the resource-constrained project scheduling problem, conducting numerical experiments based on a real case study of Singapore waters [9]. Then, Zhang et al. developed a mathematical model aiming to minimize the total waiting time, taking into account the scheduling order, travel direction, and distance to the berth [10]. To address the channel constraints impacted by tidal conditions, Li et al. proposed a multi-objective optimization model for vessel traffic scheduling in multi-harbor basins with restricted channels [11]. Regarding navigational safety, Xia et al. developed a model for precomputing key collision avoidance points [12]. This model assessed the safety level of each vessel scheduling plan and its resulting efficiency in the unidirectional, multi-junction waterways of port areas. In addition to these studies, some research efforts incorporated pilots as a critical constraint in vessel traffic sequencing to enhance port efficiency. Jia et al. considered integrating pilot scheduling decisions into vessel traffic management to mitigate congestion and enhance vessel services [13]. On this basis, Abou Kasm et al. initially considered both pilotage and tugging requirements in their study, creating a vessel schedule that incorporates these constraints within berthing operations, accounting for channel limitations at seaports [14]. From their research, it is evident that pilot scheduling plays a crucial role in pilotage planning. Research in this area typically concentrates on optimizing pilot schedules to minimize vessel waiting times and enhance overall operational efficiency. Wu et al. explored a vessel service planning problem in seaports, tackling both berth allocation and pilotage planning concurrently [15]. Shahpanah et al. solved the pilotage scheduling and the queuing problem at ship tugging operation to reduce vessel waiting times at berthing areas of a container terminal [16]. They indirectly highlighted the critical role of efficient pilotage scheduling in mitigating berthing delays and enhancing overall port operational efficiency. Jia et al. considered the number of tugboats as a constrained resource, constructing a mathematical model to solve the pilotage scheduling problem [17]. Their model incorporated the vessel berthing schedule, integrating it into the optimization of pilot assignments. Building on this foundation, Xiao et al. conducted a more complex study on integrated scheduling optimization [18]. They introduced an innovative integer programming model that integrates essential resources, including berths, quay cranes, and maritime pilots, and took into account channel capacity constraints, aiming to enhance the overall efficiency of the logistical process. These studies aim to address how to effectively allocate multiple pilots to meet the growing demand for vessel services. However, they overlooked the working conditions of pilots, particularly the work time window. Gregory et al. developed an evaluation of fatigue factors in maritime pilot work scheduling [19]. They considered the time window in pilot work scheduling and introduced concepts such as maximum continuous working hours and maximum weekly working hours for pilots into their model. Wermus and Pope devised a straightforward heuristic for harbor pilots, creating schedules that are both fair and equitable while meeting the required off-duty periods [20]. Furthermore, Lorenzo-Espejo et al. introduced a mixed-integer linear programming models for scheduling days-on and days-off, designed to arrange extended breaks for each staff member [21]. To improve pilots’ job satisfaction, Xiao et al. studied a maritime pilot scheduling problem with working hour regulations [22]. Their study prominently reduced the working time and working timespan of pilots. However, it becomes particularly problematic in seaports with frequent domestic vessel traffic, where existing pilot scheduling is unable to meet the high demands of port operations.

Simultaneously, unmanned technology has advanced rapidly. Williams et al. developed ASV systems that accurately capture time-series benthic imagery at designated reference stations [23]. These data serve as foundational ecological information, enabling researchers to quantitatively assess the long-term impacts of climate change and human activities on benthic environments. Ciaccio and Troisi conducted a comprehensive review of global scientific literature concerning the use of ASVs in environmental applications [24]. Yuan et al. compared ASVs with satellite remote sensing [25]. Furthermore, Glaviano et al. examined from an ecological perspective the most striking innovations applied by ASVs and analyzed their advantages and limits to depict scenarios of monitoring activities made possible for the next decade [26]. Additionally, the literature indicates that ASVs are capable of monitoring oceanographic data such as currents underwater, providing navigational support for vessels [27,28]. Therefore, this paper proposes a joint pilot-ASV scheduling to perform vessel pilotage planning in seaport, alleviating the strain on pilot resources. Although there is currently no literature specifically on maritime pilot-ASV scheduling, analogous research on truck-UAV scheduling has already developed substantially. Long et al. proposed a dynamic truck-UAV collaboration strategy to enhance the efficiency and resilience of urban emergency response systems [29]. Sajid et al. devised a joint optimization framework to tackle UAV-route scheduling issues within the truck-UAV delivery system [30].

Pilotage planning is a typical NP-hard problem. Consequently, research on solving methods has mainly focused on metaheuristic approaches. Among them, researchers have extensively explored the Artificial Bee Colony (ABC) algorithm, aiming to boost its performance through various modifications. Zhang et al. [31] incorporated genetic algorithm concepts to develop a hybrid ABC algorithm, which accelerated the convergence rate of the algorithm. Ng et al. [32] introduced a multiple-colony ABC algorithm, improving the global search capability through the cooperation of multiple populations. Wu et al. [33] developed an ABC algorithm with an adaptive strategy that effectively prevents the algorithm from getting stuck in local optima. Jiang et al. [34] proposed a dominant ABC algorithm that enhances the balance between global and local search capabilities. Building on foundational studies in ABC, this paper focuses on the distinct roles played by different bee groups in the optimization process, particularly designing strategies and operators based on Breadth-First Search (BFS) and Depth-First Search (DFS) during the employed and onlooker bee phases, respectively, to enhance both the global and local search capabilities of the algorithm.

In summary, optimizing pilot scheduling is crucial for enhancing the efficiency of port pilotage operations. However, the frequent arrivals and departures of domestic vessels far exceed the workload that the available number of pilots can handle. Additionally, with the rapid development of unmanned technologies, ASVs are now capable of providing essential subsea data to guide maritime navigation. Therefore, this paper proposes a pilot-ASV scheduling strategy to effectively manage the pilotage planning in seaport. The contributions of this study include:

• This study first integrates pilot-ASV scheduling into pilotage planning, including considerations of pilot work time window and the battery limitation of ASVs.
• We develop an optimization mathematical model, factoring in multiple practical constraints. Compared to existing research, our model considers more realistic constraints, such as pilot levels and the operational scope of ASVs.
• Our study conducts extensive numerical experiments of varying scales. The results demonstrate that our solution method can achieve optimal or near-optimal solutions for real-world cases.

3. Problem Formulation

3.1. Problem Description

Within a seaport, a pilot station is tasked with deploying $\left|P\right|$ pilots and $\left|U\right|$ ASVs to manage the entry and departure navigation operations for $\left|\mathrm{V}\right|$ vessels over a planning period $\left|T\right|$. Pilots $(p\in P)$ are classified into two levels, each qualified to navigate vessels of varying sizes, with specifics provided in

Table 1. Relation between pilots’ level and vessels’ information.

Size of Vessels	Type of Vessels	Capacity of Pilots
		Junior Pilot	Chief Pilot
<180 m	Non-hazardous cargo vessel	✓	✓
	Hazardous cargo vessel		✓
≥180 m	Non-hazardous cargo vessel		✓
	Hazardous cargo vessel		✓

[35]. Additionally, ASVs have battery constraints, necessitating that their total round-trip duration does not exceed $\overline{e}$. In our research, vessel $i\in V$ requires the assistance of pilots or ASVs to reach its designated berth after arriving at the port at time ${TA}_i$. After completing cargo loading and unloading operations lasting $t_k^{car}$, the vessel still needs the assistance of pilots or ASVs to navigate through the channel and finally sails to the open sea. If vessel $i$ arrives at the port for the first time, it must complete its pilotage tasks with the assistance of pilots. If a vessel requires a chief pilot, the use of ASVs for pilotage is not permissible. Furthermore, any pilotage tasks conducted by a pilot must occur within the assigned pilot’s available work time window.

Therefore, the research problem of this paper can be described as follows: within a port, $\left|P\right|$ pilots and $\left|U\right|$ ASVs are tasked with managing the entry and departure operations of $\left|V\right|$ vessels within a planning period $\left|T\right|$. The goal is to minimize total vessels’ stay time, while considering the work time windows of pilots, the different levels of pilots, and the battery limitations of ASVs.

3.2. Assumptions and Notations

To optimize the pilot-ASV scheduling, and to maintain the model’s scientific integrity, the following assumptions are made:

• The planning horizon is discretized into a sequence of time points: This means that the time period under consideration is divided into specific points in time.
• Pilots of the same level are homogeneous and proficient in performing all tasks; this implies that all pilots possess similar skill levels and can execute any required task equally well.
• Sufficient numbers of pilot boats and helicopters are available: There are an ample amount of both pilot boats and helicopters accessible for the tasks at hand.
• The time and cost associated with each non-through-channel transport operation are negligible and assumed to be zero: This indicates that the time taken and the expenses incurred for transportation operations not involving direct routes are considered insignificant and are treated as zero for the purpose of analysis.

We have followed the assumptions of existing research and augmented the second assumptions to align with our research question [15]. Furthermore,

Table 2. Notations and descriptions.

Notation	Description
Inputs
$V$	The set of vessels entering and leaving the harbor during the planning period, i, j are vessel indexes.
$P$	The set of pilots, p is the pilot index.
$O$	The set of nodes such as pilot bases, anchorages, and berths.
$U$	The set of ASVs, u is the ASV index.
$Q$	The number of trips in one cycle of the pilot (ASV), q is the trip index.
$T$	The operational cycle, t is the time index.
$c_i$	The minimum rank of the pilot that satisfies the i.
$C_p$	The rank of the pilot p.
$\left[\underset{\_}{w_p^n},\overline{w_p^n}\right]$	The time window in which the pilot p can perform pilotage operations, n is the time window index.
$\overline{e}$	Maximum duration of the ASV battery.
${T1}_{ab}$	The time required for the pilot to perform the pilot mission from a to b.
${{T1}_{ab}}^{\prime }$	The time required for the pilot to travel from a to b without task.
${T2}_{ab}$	The time required for the ASV to perform the pilot mission from a to b.
${{T2}_{ab}}^{\prime }$	The time required for the ASV to sail from a to b without tasks.
$T{A}_i$	The expected pilotage start time of the vessel i.
$x_{ab}^i$	1 if vessel i needs to sail from a to b, 0 otherwise.
${\varphi }_i$	1 if vessel i arrives at the port for the first time, 0 otherwise.
$M$	A large positive number.
Decision variables
${\theta }_{iq}$	1 if vessel i in trip q, 0 otherwise.
${\mu }_{pq}$	1 if pilot p is in charge of trip q, 0 otherwise.
${\mu }_{uq}$	1 if ASV u is in charge of trip q, 0 otherwise.
${\chi }_{qik}$	1 if vessel i is the kth task of trip q, 0 otherwise.
${\gamma }_{pi}$	1 if pilot p is responsible for vessel i, 0 otherwise.
${\gamma }_{ui}$	1 if ASV u is responsible for vessel i, 0 otherwise.
$z_{kij}^q$	1 if the vessel i is the kth task of trip q and the vessel j is the k + 1th task of trip q, 0 otherwise.
$T{B}_{pk}^q$	The time when pilot p starts the kth task in trip q.
$T{C}_{pk}^q$	The time when pilot p completes the kth task in trip q.
$T{B}_{uk}^q$	The time when ASV u starts the kth task in trip q.
$T{C}_{uk}^q$	The time when ASV u completes the kth task in trip q.

lists the notations used to formulate the model.

3.3. Model Formulation

Based on the notation defined in Table 2, the objective function of the integrated model can be expressed as follows:

$$F=\mathit{min}\sum _{i\in V}\sum _{q\in Q}{\chi }_{pik}\left(T{B}_{pk}^q-T{A}_i\right)+{\chi }_{uik}\left(T{B}_{uk}^q-T{A}_i\right)$$

(1)

Equation (1) is the objective function that represents minimizing the waiting time of all vessels waiting for a pilot. Vessels, pilots (ASVs), and pilotage tasks are bound by ${\chi }_{pik}$ and ${\chi }_{uik}$.

$$T{A}_i\le {TB}_{pk}^q+M(1-{\chi }_{qik}),\forall i\in V,\forall q\in Q,\forall k$$

(2)

$${TC}_{pk}^q={TB}_{bk}^q+\sum _{i\in V}{\chi }_{qik}\sum _{a\in O}{x}_{ab}^i{T1}_{ab},\forall p\in P,\forall q\in Q,\forall k,\forall a,b\in O$$

(3)

$${TC}_{pk}^q+z_{kij}^q{T1}_{bc}^{\prime }(x_{ab}^i+x_{cd}^j-1)\le {TB}_{b(k=1)}^q,\forall i,j\in V,\forall p\in P,\forall q\in Q,\forall k,\forall a,b,c,d\in O$$

(4)

$$T{A}_i\le {TB}_{uk}^q+M(1-{\chi }_{qik}),\forall i\in V,\forall q\in Q,\forall u\in U,\forall k$$

(5)

$${TC}_{uk}^q={TB}_{uk}^q+\sum _{i\in V}{\chi }_{qik}\sum _{a\in O}{x}_{ab}^i{T2}_{ab},\forall q\in Q,\forall u\in U,\forall k,\forall a,b\in O$$

(6)

$${TC}_{uk}^q+z_{kij}^q{T2}_{bc}^{\prime }\left(x_{ab}^i+x_{cd}^j-1\right)\le {TB}_{u\left(k=1\right)}^q,\forall i,j\in V,\forall q\in Q,\forall u\in U,\forall k,\forall a,b,c,d\in O$$

(7)

Constraints (2)–(6) specify time-related requirements, with Constraints (2)–(4) focusing on the pilot’s operational timings. Specifically, Constraint (2) stipulates that a pilot may only begin piloting duties after receiving a formal request from the vessel. Constraint (3) defines the duration required for pilot p to accomplish the kth task within trip q. Following this, Constraint (4) denotes the time needed for the pilot to reach the start of the subsequent task after completing the kth task, in preparation for initiating the k + 1th task. Similarly, Constraints (5)–(7) outline the operational timings for ASVs.

$$T{B}_{p1}^q\ge \overline{w_p^n},\forall p\in P,\forall q\in Q,\forall n\in N$$

(8)

$$T{C}_{p\left|k\right|}^q\le \underset{\_}{w_p^n},\forall p\in P,\forall q\in Q,\forall n\in N$$

(9)

$${TC}_{u\left|k\right|}^q-{TB}_{u1}^q\le \overline{e},\forall u\in U,\forall k$$

(10)

Constraints (8)–(10) pertain to the time windows for pilot operations and the battery limitations of ASVs. Constraints (8) and (9) specify that pilots must operate within predetermined time windows. On the other hand, Constraint (10) ensures that the sailing duration of the ASV stays within its battery limitations.

$$\sum _{q\in Q}{\theta }_{iq}=1,i\in V$$

(11)

$$\sum _{p\in P}{\mu }_{pq}+{\mu }_{uq}(1-{\phi }_i)=1,q\in Q,i\in V$$

(12)

$$({\mu }_{pq}+{\theta }_{iq}-1)\le {\gamma }_{pi},q\in Q,\forall i\in V,p\in P$$

(13)

$${\mu }_{pq}+{\theta }_{iq}>{\gamma }_{pi},q\in Q,\forall i\in V,p\in P$$

(14)

$$({\mu }_{uq}+{\theta }_{iq}-1)\le {\gamma }_{pi},q\in Q,\forall i\in V,p\in P$$

(15)

$${\mu }_{uq}+{\theta }_{iq}>{\gamma }_{ui},q\in Q,\forall i\in V,p\in P$$

(16)

$$\sum _{i\in V}{\chi }_{qik}\ge \sum _{i\in V}{\chi }_{qi(k+1)},\forall i\in V,\forall p\in P$$

(17)

Constraint (11) ensures that each vessel is assigned to a specific trip. Constraint (12) specifies that each trip must be managed by a pilot/ASV, while vessels making their initial port call require pilotage. Constraints (13) and (14) establish the relationship between vessels and pilots: if vessel i is in trip q, and trip q is managed by pilot p, then pilot p is responsible for piloting vessel i. Similarly, the ASV within each trip also has the same constraints, replacing ${\mu }_{pq},{\gamma }_{pi},$ in Constraints (15) and (16) with ${\mu }_{uq},{\gamma }_{ui}$, respectively. Lastly, Constraint (17) guarantees that tasks are consecutive, with the commencement of the next task immediately following the completion of the kth task.

$$z_{kij}^q={\chi }_{qik}\ast {\chi }_{qi(k+1)},\forall i\in V,\forall p\in P,\forall k$$

(18)

Constraint (18) represents the relationship between two consecutive tasks. Since it is a nonlinear constraint that the solver cannot directly solve, it needs to be linearized.

$$z_{kij}^q\le {\chi }_{qik},\forall i,j\in V,\forall p\in P,\forall k$$

(19)

$$z_{kij}^q\le {\chi }_{qi(k+1)},\forall i,j\in V,\forall p\in P,\forall k$$

(20)

$$z_{kij}^q\ge {\chi }_{qik}+{\chi }_{qi(k+1)}-1,\forall i,j\in V,\forall p\in P,\forall k$$

(21)

Constraints (19)–(21) linearize Constraint (18). After linearization, the meaning of the constraint expression remains unchanged, but there is no multiplication of variables.

$${\gamma }_{pi}{C}_p\ge c_i,\forall i\in V,\forall p\in P$$

(22)

Constraint (22) indicates that the level of pilot conducting the pilotage task must meet the minimum level limits for vessel pilotage.

$${\theta }_{iq},{\mu }_{pq},{\chi }_{qik},{\gamma }_{pi},z_{kij}^q,{\mu }_{pq},{\gamma }_{ui}\in \left\{\mathrm{0,1}\right\}$$

(23)

$${TB}_{pk}^q,{TB}_{uk}^q\ge 0$$

(24)

Constraints (23) and (24) specify the range of values for the variables.

4. Solution Method

4.1. Improved Artificial Bee Colony Algorithm

The problem studied in this article is an NP-hard problem that requires assigning tasks to pilots/ASVs under multiple constraints to ensure the safety and efficiency of vessel navigation. The difficulty lies in considering the pilots’ qualification matching, working hours, and time windows. The Artificial Bee Colony algorithm (ABC) can efficiently search for the optimal scheduling solution by simulating the intelligent foraging behavior of bees. It can also flexibly adapt to changing conditions and avoid falling into local optima, thus performing well in solving complex pilot scheduling problems. To further enhance the performance of the ABC, this paper proposes an improved version named the Improved Artificial Bee Colony algorithm (IABC), which builds upon the standard algorithm. IABC incorporates operators based on breadth-first and depth-first search strategies in the employed bee and follower bee phases, respectively. This enhancement effectively strengthens both global and local search capabilities. The algorithm’s flow is depicted in Figure 1.

4.2. Decoding and Calculating Fitness

We use natural number coding to represent the matching relationship between pilots/ASVs and pilotage tasks. Assuming that there are 15 pilotage tasks (11 ordinary tasks and 4 advanced tasks) and a total of 2 pilots (1 ordinary pilot and 1 advanced pilot) currently on duty, in addition to two ASVs, the solution to the problem in this case can be represented by the sequence shown in Figure 2. Here, the value 0 represents the pilot station, and the non-zero numbers represent the pilot tasks. The sequence between two values of 0 represents a single trip of one pilot/ASV, and all trips of two pilots and two ASVs constitute one solution to the problem.

The first row represents the ordinary pilot’s first trip to pilot vessel 1 and vessel 4, the second trip to pilot vessel 5 and vessel 6. The third row represents an ASV’s first trip to pilot vessel 10 and vessel 11, and the second trip to pilot vessel 9 and vessel 12.

In IABC, the fitness value, denoted as Z, is directly utilized in the evaluation of the solution, as expressed in Equation (25), where F represents the objective function value of the solution.

$$Z={\displaystyle \frac{1}{F}}$$

(25)

4.3. Initial Population Generation

After determining how the solution is encoded, the initial population is generated using the steps described below:

Step 1: A set of sequences (Sequence 1) containing all vessels is randomly exchanged several times to obtain Sequence 2.

Step 2: Randomly select a pilot/ASV and assign it an initial position in the empty Sequence 3 (empty trip), adding a value of 0 to the center of the pilot station. Exclude selection if the pilot’s time window does not meet demand or if the ASV’s duration is not sufficient for the mission.

Step 3: Evaluate vessel demand for Sequence 2. If the vessel fits the pilot’s operation sequence’s time window and the pilot’s rating allows piloting the vessel, it is considered successfully added. Otherwise, skip the vessel. For ASVs, if the sum of operation times on the current trip does not exceed the upper bound of one trip’s continuation, the vessel is considered successfully added.

Step 4: Vessels are judged and added to Sequence 3 sequentially backward. If the pilot’s operating time window constraint is not satisfied or the upper bound of the ASV’s range is reached, then the current trip ends, and a value of 0 is added at the end of the sequence to close it out.

Step 5: Repeat steps 2 to 4 to generate all trips.

4.4. Employed Bee Stage

The Hired Bee phase draws on the idea of Breadth-First Search (BFS), which emphasizes extensive exploration of the search space to expand the solution search space. To emphasize the role played by this search strategy, the search operators are divided into two categories [32]: Category 1 operators, which induce significant changes in individuals before and after updates, and Category 2 operators, which induce minor changes in individuals before and after updates. In the Hired Bee phase, Category 1 operators are employed for neighborhood search for each individual in the population.

Category 1 operators involve vessel movements between different trips or cross movements between different individuals in the population. These operators result in substantial differences in individuals before and after iterations, enabling the Hired Bee phase to achieve extensive search in the search space. There are four operators in Category 1, namely exchange, insertion, reversal, and crossover operators. The specific operations are illustrated in Figure 3. The light blue and light green in the figure represent unchanged encoding, while the dark blue and dark green represent encoding involved in neighborhood search.

The exchange operator randomly selects 1 to 3 vessels to be exchanged within different sub-paths. The insertion operator randomly selects 1 to 3 customer points within one sub-path to insert into another sub-path. The reversal operator randomly selects a range of vessels and then reverses their order. The crossover operator randomly selects 1 to 3 vessels for crossover between two pilots of equal class, or between a pilot and an ASV.

4.5. Following Bee Stage

The Following Bee phase draws inspiration from the concept of Depth-First Search (DFS) and focuses on delving deeply into the search space to accelerate the algorithm’s convergence. Initially, the Following Bee phase selects an individual based on the roulette principle and subsequently conducts a neighborhood search using Type 2 operators. Type 2 operators primarily facilitate intra-trip customer point adjustments. These operators resemble the three operations of exchange, insertion, and reversal employed in the Hired Bee stage. However, in the Type 2 operators, only one customer point is selected in the exchange and insertion operations, and the operations are confined to a single trip of an individual. Consequently, the disparity between an individual before and after iteration remains minor. Additionally, in this phase, after each iteration of searching using the operators, if there is an enhancement in the individual’s performance, the search within that phase is reiterated; otherwise, it proceeds to the subsequent phase.

4.6. Reconnaissance Bee Stage

The employed bee and following bee phases involve comparing the fitness value of the new individual with that of the original individual. If the newly generated individual has a smaller fitness value, the original individual remains unchanged, and a consecutive counter N, tracking the number of times it is not improved, is incremented by one. Conversely, if the new individual exhibits better fitness, it replaces the original individual, and the counter N resets to zero.

In the detection bee phase, the algorithm initially identifies the individual with the largest N value in the population. Subsequently, it compares the N value of this individual with the threshold value L. If N exceeds L, the algorithm regenerates a new individual to replace the selected individual based on the initial solution generation method.

5. Computational Experiments

To validate the effectiveness of our proposed model and solution, this paper presents a series of experiments programmed in Python on a PC with an Intel (R) Core (TM) i5-12600KF CPU @ 3.70GHz processor and 32 GB of RAM. The experimental framework is divided into three key parts: (i) experiments involving four pilots (2 junior pilots, 2 chief pilots), 4 ASVs and 20 vessels; (ii) analysis of the effectiveness of our proposed solutions; and (iii) evaluation of the algorithmic solution quality.

5.1. 4-Pilots-4-ASVs-20-Vessels Experiment

In generating this arithmetic example, we first considered the arrival time and loading/unloading time of 20 vessels. To simulate the real situation in the port, we used a random number generator to generate the arrival times of the vessels. These arrival times are distributed over the 24 h of the day and follow a uniform distribution. Vessel loading and unloading operations are simulated using random numbers between 4 and 10 h. Secondly, we considered the type of vessel. Vessels are categorized into two types: large vessels $(>180\mathrm{m})$ and small vessels $(\le 180\mathrm{m})$. Large vessels require a chief pilot, while small vessels require a pilot with any level or an ASV. The berths at which the vessels are berthed are randomized, with a total of eight berths available for random selection.

Two pilot stations are set up, each with two types of pilots: ordinary pilots and senior pilots. Each station has two pilots (one junior pilot and one chief pilot) and two ASVs. The pilots have only one operating window of eight hours in a 24 h period, while each ASV has an endurance time of four hours.

The aforementioned data are input into the algorithm, which is run for 300 iterations. The results are illustrated in Figure 4. The rectangles with the same color and number in the figure represent the same ship.

The total waiting time for the vessel is 64.23 h. The program ensures a match between pilot and vessel type, satisfying the pilot time window constraints. Additionally, the ASV’s single voyage meets the endurance requirements.

5.2. Scheme Effectiveness: Comparison with Existing Schemes

To prove that the scheme presented in this paper is more reasonable and effective compared to other schemes, it is compared with the first-come-first-served (FCFS) scheme and the scheme without ASVs. When the port business volume is low, the number of arriving vessels is fewer than 15. When the port business volume is moderate, the number of vessels ranges from 15 to 20. In a busy commercial environment, the number of vessels also ranges from 20 to 25 [13]. In extreme cases, the maximum number of vessels is 36 [15]. Therefore, the examples in this paper set the number of vessels awaiting pilotage to 20, 25, 30, and 35. These vessels adhere to the generation rules described in Section 5.1. In the FCFS scheme, we continue to employ four pilots and two ASVs, yet without the application of our model optimization. Conversely, in the no-ASV scheme, six pilots are utilized to service the vessels, with the solution being derived based on our model. The following Figure 5 illustrates the comparison results of the objective function values.

As shown in Figure 5, this paper’s scheme outperforms the FCFS scheme by an average of 17.12%. The optimization rate of this scheme exceeds 18% when there are 35 vessels, demonstrating its scalability. Compared to the no-ASV scheme, this paper’s scheme shows an average improvement of 9.18%, with an optimization rate exceeding 10% for 35 vessels. The advantage of this scheme lies in the efficient deployment of pilots and ASVs: longer pilotage tasks are performed by the ASV, while the pilot handles several shorter tasks within a single trip.

$$GAP\left(a\right)={\displaystyle \frac{FCFS\left(Obj\right)-Ourscheme\left(Obj\right)}{Ourscheme\left(Obj\right)}}\times 100\%$$

(26)

$$GAP\left(b\right)={\displaystyle \frac{WithoutASV\left(Obj\right)-Ourscheme\left(Obj\right)}{Ourscheme\left(Obj\right)}}\times 100\%$$

(27)

The above Equations (26) and (27) are the gap calculation formulas for Figure 5a,b, respectively.

5.3. Solution Quality: Comparison with Existing Methods

To illustrate the superior performance of this paper’s algorithm compared to other algorithms, we compared IABC with Gurobi (https://www.gurobi.com/ (accessed on 31 July 2024)), traditional ABC algorithms [31], Genetic Algorithm (GA), and Simulated Annealing Algorithm (SA). The algorithm follows the same rules as described above, varying the number of vessels waiting to be piloted under different pilot and ASV configurations, with 20, 25, 30, and 35 vessels, respectively. Solve the same example five times and use the best result. The objective function values obtained by the logging algorithm are shown in

Table 3. Comparison of algorithmic solutions.

Number of Pilots	Number of Vessels	Objective Value (h)					Gap (%)
		IABC	GUROBI	ABC	GA	SA	GUROBI	ABC	GA	SA
6	20	74.56	74.56	74.56	74.56	74.56	0%	0%	0%	0%
6	25	86.83	86.83	86.83	86.83	86.83	0%	0%	0%	0%
6	30	101.74	-	108.95	107.4	110.82	-	7.09%	5.56%	8.92%
6	35	123.85	-	140.63	136.86	147.28	-	13.55%	10.50%	18.92%
8	20	60.42	60.42	60.42	60.42	60.42	0%	0%	0%	0%
8	25	78.51	78.51	78.51	78.51	78.51	0%	0%	0%	0%
8	30	87.85	-	96.68	92.28	99.13	-	10.05%	5.04%	12.84%
8	35	106.47	-	116.85	113.70	118.47	-	9.75%	6.79%	11.27%
Average		-	-	-			-	5.05%	3.49%	6.49%

below, and the solution times are shown in

Table 4. Comparison of algorithm solving times.

Number of Pilots	Number of Vessels	Solving Time (s)					Gap (%)
		GUROBI	IABC	ABC	GA	SA	(ABC-IABC)/IABC	(GA-IABC)/IABC	(SA-IABC)/IABC
6	20	409.29	7.46	10.23	7.74	9.98	37.13%	3.75%	33.78%
6	25	1498.02	23.28	32.90	26.74	31.65	41.32%	14.86%	35.95%
6	30	3600 s +	445.08	560.87	447.29	547.29	26.02%	0.50%	22.96%
6	35	3600 s +	654.90	800.18	672.5	824.7	22.18%	2.69%	25.93%
8	20	512.04	9.08	11.34	9.32	10.76	24.89%	2.64%	18.50%
8	25	1892.59	30.86	35.29	32.26	37.92	14.36%	4.54%	22.88%
8	30	3600 s +	472.95	582.73	488.2	600.29	23.21%	3.22%	26.92%
8	35	3600 s +	760.61	872.36	778.22	914.61	14.69%	2.32%	20.25%
Average		-	-	-	-	-	25.48%	4.31%	25.90%

. Set the upper limit of the solving time to 3600 s. If it exceeds this limit, denote it as 3600 s +. The convergence results of the different algorithms for the example with 35 vessels are shown in Figure 6.

From Table 3, it can be seen that all algorithms provide the optimal solution when the number of vessels does not exceed 25. Once the number of vessels exceeds 25, Gurobi cannot deliver an effective solution within 3600 s, while heuristic algorithms can still provide feasible solutions. Our algorithm’s average optimization rates compared to the other three heuristic algorithms are 5.05%, 3.49%, and 6.49%, respectively. Table 4 compares the solving times of the solver and the algorithm. Compared to the other three heuristic algorithms, our algorithm demonstrates higher solving efficiency, with average improvements of 25.48%, 4.31%, and 25.90% in solving time, respectively. Figure 6 shows that the traditional ABC algorithm, due to its limited search operator performance, cannot find a better solution when the number of ships exceeds 25. In contrast, the algorithm proposed in this paper has stronger neighborhood search capability, faster convergence speed, and higher solving efficiency.

A comprehensive comparison verifies that the algorithm proposed in this paper has better solution quality and efficiency.

6. Conclusions

The primary goal of this study is to enhance port efficiency and ensure navigation safety by a pilot-ASV scheduling within pilotage planning. We develop an innovative mathematical model that considers pilot work time windows, pilot levels, and ASV battery constraints. Our main contributions are the introduction of a “pilot-ASV” strategy, the development of this model, and the creation of IABC tailored for this complex optimization challenge. This algorithm incorporates innovative search strategies that significantly improve the solution quality and efficiency. Compared to the traditional FCFS method without optimization, our approach reduces the average vessel waiting time by 17.12%. In extensive numerical experiments, our optimization algorithm demonstrates stable solving capabilities, surpassing common heuristic algorithms in both solving speed and quality. Furthermore, we compared our pilot-ASV strategy against the no-AUV strategy. The results show that substituting junior pilots with ASVs can reduce vessel waiting times by an average of 9.18%, further validating the effectiveness and practicality of our proposed pilot-ASV scheduling approach.

Despite its strengths, the study has limitations that could affect its application in real-world scenarios. We assumed a fixed pilot scheduling protocol without considering the potential variability in pilot availability or the dynamic nature of port conditions, such as weather impacts and operational disruptions, which could influence pilotage and vessel sequencing. These factors are vital as they can markedly affect port operations. Therefore, future research will extend the model to consider additional constraints, such as weather effect, different work policies of pilots, and multiple types of ASVs. Such improvements could make the model more flexible and applicable to various port environments, potentially leading to broader adoption and better optimization of port operations globally.

In conclusion, this study offers valuable insights into the pilotage planning, proposing a novel “pilot-ASV” scheduling approach that promises to enhance port operational efficiency and ensure navigation safety. While recognizing its limitations, this research lays a solid foundation for future developments and encourages further exploration into comprehensive port management strategies.