Logic-Based Benders Decomposition for Unmanned Electric Tugboat Scheduling Considering Battery-Swapping Operations

Abstract

As the electrification reform accelerates in ports worldwide, the application of electric tugboats is becoming more widely applied, posing a challenge in the balance between working arrangement and energy replenishment, especially when the shore energy replenishment facilities are limited. Aligning with the emerging trends of port electrification, unmanned operations, and intelligentization, this paper investigates unmanned electric tugboat scheduling considering battery-swapping operations that combine the assignment of tasks to the working periods of tugboats, the allocation of battery-swapping operations to the shore battery-swapping stations, and the sequencing of operations at each station. The problem is formulated into a mixed-integer linear programming to minimize the total completion time of the battery-swapping operations. A logic-based Benders decomposition method is proposed that decomposes the problem into a master problem and a subproblem. The master problem relaxes the sequencing constraints and solves the assignment of tasks to tugboats and the allocation of battery-swapping operations to stations. The SP, based on the solution to the master problem, determines the sequencing of battery-swapping operations at each station. Considering the interdependence of swapping operations of each tugboat that might be allocated to different stations, a dispatching heuristic is designed to efficiently obtain high-quality sequences for the stations. Numerical experiments are conducted based on 80 randomly-generated instances with up to 100 tasks, ten tugboats, and six battery-swapping stations. The results demonstrate that LBBD is capable of solving all 80 instances, whereas the commercial solver CPLEX fails to solve those with 80 or more tasks. Moreover, the average computational time of CPLEX on the instances it can solve is 241.32 s, nearly 32 times that of LBBD (7.57 s). This clearly indicates that LBBD significantly outperforms CPLEX in terms of both computational capacity and efficiency. Further analyses show that the increase in the number of tugboats will significantly shorten the makespan and make ETSBS easier to solve, while the increase in the number of battery-swapping stations makes the problem more challenging with longer computational time.

1. Introduction

Shipping is crucial to global trade, as 85% global trade volumes are carried by ships [1]. In 2024, the global shipping industry gradually emerged from the decline caused by the COVID-19 pandemic, economic sluggishness, and other factors, and it has started to show a positive trend of steady recovery and improvement. According to Clarksons Research Services Ltd. (London, UK), global new building orders in 2024 surpassed 65.81 million compensated gross tons—the highest figure recorded since 2008 [2]. One notable characteristic is that the proportion of new ship orders with low-carbon and zero-carbon power has significantly increased to 48% of the total orders, which fully reflects the steady progress of the green development of the shipping industry. In the past few years, the maritime industry, especially the ports worldwide, has witnessed a significant revolution in working boats towards electrification, unmanned operations, and intelligentization. Recent research conducted by [3] has found that to reduce carbon emissions, numerous ports and companies, including the Port of San Diego, the Port of Seattle, the Port of Vancouver, Washington State Ferries, and San Francisco Bay, have either already adopted or are on the verge of adopting electric vessels and tugboats for their operations.

The introduction and promotion of electric tugboats in port operations, compared with traditional fossil fuel-powered counterparts, offer numerous advantages, including reduced carbon emissions, enhanced operational efficiency, lower operating costs, and richer electrification and intelligent functions. However, their power form determines that they must return to the port for energy replenishment to ensure continuous working ability. At present, the battery-swapping mode is one of the main forms for electric vessels, including tugboats, to replenish electricity and restore working capacity. The advantage of battery-swapping for ship include faster speed, reduced time and cost consumption, and lower initial investment pressure [4,5]. Also, battery-swapping stations can stock a large inventory of standardized battery packs. This centralized storage can schedule batteries to be charged during off-peak hours or whenever low-cost periods occur to relieve peak-load pressure on the local distribution network and to reduce operating expenses [5]. When the electric tugboats under battery-swapping mode return to the shore battery-swapping station, the depleted battery is replaced with a fully charged battery that has been prepared in advance, and then it immediately returns to the task area assisting vessel berthing or unberthing. This mode allows the electric tugboat to fully restore power and return to work in a relatively short time.

In the operation of electric tugboats, enhancing the utilization of the working period between two consecutive battery-swapping operations is a key concern for decision-makers. On one hand, it is essential to ensure that all the tasks assigned to the same working period can be completed before the battery is depleted. On the other hand, to maximize battery usage, the remaining energy upon returning to the shore station for battery swapping is supposed to be as low as possible. In this context, a growing body of research has focused on electric tugboat scheduling to minimize total operation cost/time, or to maximize profits. Some of these studies have considered the energy-replenishing requirements of electric tugboats. However, most of them assume that there are enough facilities to support the replenishment. This assumption ignores the potential conflict between the high demand for battery-swapping operations and the limited availability of battery-swapping stations and pays little attention to the need to investigate recharging-stations scheduling problems.

In fact, the scheduling of electric tugboat tasks and the scheduling of battery-swapping stations are interrelated combinatorial optimization problems. If only the scheduling of tugboat tasks is considered, it may lead to a large number of tugboats returning to the port for battery swapping at the same time, causing a large amount of waiting time and significantly reducing the availability of tugboats. Focusing solely on the scheduling of battery-swapping stations may compel tugboats to return to the stations while their battery levels are still relatively high, potentially resulting in inefficient electricity usage and reduced operational hours. Therefore, investigating the scheduling of electric tugboats while simultaneously considering battery-swapping operations is conducive to optimizing tugboat management from a global perspective, thereby obtaining a globally optimal plan for tugboats completing tasks and replenishing energy.

To fill this gap, this study addresses the electric tugboat scheduling considering battery-swapping (ETSBS) operations in a more comprehensive way. ETSBS includes the assignment of tasks to the working periods of tugboats, the allocation of tugboat battery-swapping operations to shore battery-swapping stations, and the sequencing of multiple battery-swapping operations at each station simultaneously. We develop a mixed-integer linear programming (MILP) model to accurately describe this complex problem. The model aims to minimize the makespan of all the battery-swapping operations. Considering the complexity of the model, we apply a logic-based Benders decomposition (LBBD) algorithm to solve it efficiently. This algorithm takes advantage of the unique structural features of the MILP model to improve computational performance. We conduct extensive computational experiments with large-scale instances to validate our model and algorithm. The results show that the methods proposed are effective in solving ETSBS.

This study makes three key contributions. First, we propose a novel logic-based Benders decomposition method that integrates electric tugboat scheduling with battery-swapping operations, addressing limitations of prior work. Second, our method achieves significantly faster computational times than commercial solvers like CPLEX, making it highly practical for real-world applications. Third, we highlight the substantial environmental benefits of electric tugboats, providing a case study for their adoption in port operations.

The remainder of this paper is structured as follows. Section 2 reviews the relevant literature, emphasizing the research gap. Section 3 formally defines the ETSBS and formulates it as a MILP model. The LBBD algorithm is detailed in Section 4. Section 5 conducts a case study based on data from Port, while Section 6 presents comprehensive numerical experiments on randomly-generated instances to verify the effectiveness of the LBBD algorithm. Finally, Section 7 concludes the paper and proposes future research directions.

2. Literature Review

The ETSBS investigated in this paper encompasses two primary research domains: electric tugboat scheduling with recharging considerations (eTug-SR) and the berth allocation problem (BAP). In the following section, we provide a concise review of the relevant literature on these two areas.

2.1. Tugboat Scheduling Problem with Recharging Consideration

Extensive research has been devoted to tugboat scheduling problems (Tug-SP), yielding effective methods tailored to diverse scenarios. These include adaptive large neighborhood search under the service pattern of multiple services at multiple points [6], extended Markov decision processes for dynamic Tug-SP [7,8], and a Stackelberg game-based fuzzy model and seagull optimization algorithm [9] to address dynamic task arrivals and imprecise operation times. Additionally, an improved gray wolf optimizer has been proposed for multi-objective Tug-SP [10]. Nevertheless, studies on eTug-SR, i.e., Tug-SP incorporating electric propulsion and recharging constraints, remain scarce.

Given the high frequency of berthing or unberthing assistance tasks in ports, conventional fossil fuel-powered tugboats face significant environmental and operational challenges due to their substantial contribution to port pollution. To meet the net-zero emissions goal set by the International Maritime Organization [11], many ports around the world have accelerated the electrification of working boats, including tugboats. For example, the first all-electric tugboat in the United States, the eWolf, was put into operation at the Port of San Diego in 2024. It is estimated to save 178 tons of nitrogen oxides, 2.5 tons of diesel particulate matter, and 3100 metric tons of carbon in 10 years’ time [12]. Studies have demonstrated that employing electric tugboats can reduce CO₂ emissions by 38% compared with their conventional counterparts [13].

With the rapid global adoption of electric tugboats, several key issues have come to the fore in recent discussions. These include the advantages of electric tugboats over their conventional counterparts, their operational methods, and the optimization of their usage, especially considering their recharging requirements. Acomi et al. [14] conducted a systematic review to assess the feasibility of electric tugboats as a sustainable alternative. The authors synthesized data from 42 studies and validated their findings through a focus group of industry experts. The review identified key advantages of electric tugboats, including reduced greenhouse gas emissions, lower operational costs, and enhanced energy efficiency. However, significant challenges such as high initial investment costs, limited battery range, and infrastructure requirements for port electrification were also highlighted. The study concluded that while electric tugboats offer substantial environmental and economic benefits, widespread adoption is hindered by technological and financial barriers. The authors recommended strategic investments in infrastructure, advancements in battery technology, and policy support to facilitate the transition to greener maritime operations. Another review by [15] compared the long-term economic performance of conventional, hybrid, and electric tugboats, finding that electric tugboats offer significant operational cost savings despite higher initial investments. A conclusion was finally made that electric tugboats are a promising alternative for sustainable maritime operations, but further investigation into deployment obstacles is needed.

With the advancement of technology, some researchers have explored the concept of controlling unmanned electric tugboats for vessel assistance. Koznowski and Łebkowski [13] explored energy consumption reduction in unmanned electric tugboats using a multi-agent control system for formations. They analyzed four identical electric tugboats in eight formation configurations during a simulated port entry and exit scenario. The study found that a linear formation achieved a 57.6% reduction in energy consumption compared with independent operation. Choi et al. [16] conducted a comprehensive review of autonomous tugboat operations for efficient and safe ship berthing. By analyzing various projects related to autonomous tugboats as well as the technologies required, the authors highlighted the need for further research on communication between the mothership and tugboats, real-world verification of control methods beyond simulations, and the development of algorithms for automatic task assignment and reassignment within the tugboat swarm. The study concluded that achieving fully autonomous ship berthing using unmanned tugboats requires addressing these limitations and integrating advanced perception and decision-making technologies.

Once unmanned electric tugboats are put into use, they will offer several advantages, such as precise and timely control, long availability without the need to consider crew rest and shift changes, nearly zero carbon emissions, and low operating costs. However, their effectiveness will heavily rely on the implementation of scheduling models and algorithms, as well as battery recharging management. To the best of our knowledge, there are currently no research results on scheduling models and algorithms specifically for unmanned electric tugboats, with only Ma et al. [17] addressing the eTug-SR at container terminals to minimize the total weighted tardiness of all container ships. They proposed a MILP model and developed two algorithms: a matheuristic method and an adaptive large neighborhood search (ALNS) algorithm. Computational experiments on randomly generated instances and Zhenhai Port data showed that the matheuristic method outperformed ALNS. However, for large-scale instances where the matheuristic failed to find a feasible solution, ALNS remained effective.

Despite these contributions, Ma et al.’s work leaves room for further research. First, they assumed a fixed recharging mode for tugboats, with each recharging operation taking 90 or 120 min, which is relatively long compared with the processing times $p_j$ (60 to 180 min), potentially limiting tugboat utilization. Second, they did not account for limitations in recharging facilities, assuming immediate recharging once a tugboat’s energy was depleted. This assumption may not hold in busy, large ports with numerous tugboats, where recharging facilities are limited.

In summary, although prior work has examined the environmental and economic benefits of electric tugboats, research on eTug-SR models and their associated algorithms remains scarce, Moreover, joint optimization of tugboat scheduling and recharging, especially in congested ports where berths equipped with recharging infrastructure are outnumbered by tugboats, warrants further attention. Future studies should develop integrated frameworks that concurrently address scheduling and charging constraints to guarantee efficient operation of electric tugboats.

2.2. Berth Allocation Problem with Electric Power Management

Electric tugboats must return to shore frequently to replenish their batteries. To streamline this process, some ports have equipped certain berths with battery-swapping stations. In large, high-traffic ports where many tugboats service simultaneously, the surge in battery-swapping requests can overwhelm the limited number of swapping-enabled berths, creating a classic BAP. Traditional BAP seeks optimal berthing times and positions for various vessels to minimize operating time and costs, maximize total profit, or achieve other strategic objectives. Failure to do so may result in a large number of vessels crowding and queuing near the berths, causing waste of operational resources and further reducing the overall profits. Plenty of studies have been devoted to typical BAP in the efficient management of marine container terminals [18,19] and proposed effective models and algorithms for different varieties, including multi-port BAP [20,21], dynamic discrete BAP [22], and BAP under disruptions [23]. Recently, the electrification of maritime fleets has shifted attention to BAP variants that embed electric-energy supply. Mao et al. [24] pioneered this direction by unifying berth scheduling with on-shore power management, estimating an annual savings of $1.013 million at the Port of Houston. Subsequent studies have elevated emission reduction to a primary objective alongside cost and efficiency and proposed multi-objective models. Yue et al. [25] developed a multi-objective model for BAP integrating shore-side electricity to save total costs and reduce emissions. An improved NSGA-III algorithm was proposed, which was validated to outperform classic heuristics, with a roughly 30% improvement in effectiveness. Zhang et al. [26] jointly optimized shore-power allocation and berth scheduling to minimize total cost, ship emissions, and queuing time, again employing NSGA-III algorithm. Peng et al. [27] cast the problem as a cooperative multi-objective model and solved it with the particle swarm optimization method. Experiments on a ten-berth bulk terminal under two carbon-pricing scenarios showed that the Pareto front dominated the no-shore-power baseline, cutting pollutants ≥ 3.8% while reducing total cost up to 35.9%, or eliminating emissions at 183% cost increase.

While the BAP has been extensively studied, its integration with electric energy supply remains underexplored. Most existing work centers on shore-side power supply, leaving the specific challenges of battery-swapping tugboats unaddressed. This variant can be recast as a parallel-machine scheduling problem in which battery-swapping operations become jobs, and berths equipped with swap stations act as machines. Moreover, by simultaneously optimizing tugboat scheduling, the BAP can be integrated with a more complicated combinatorial problem, yielding a globally optimal solution for the entire tugboat fleet management.

To conclude, based on the above review of eTug-SR and BAP, our analysis uncovers several gaps that remain to be addressed. First, the majority of existing studies assume that energy-replenishment facilities are sufficient, thereby neglecting the necessity to optimally determine the allocation of berths equipped with such facilities. However, in some busy ports, a contradiction does exist between the recharging demand and the limited number of energy-replenishment facilities. If not properly managed, this contradiction will significantly reduce the utilization rate of tugboats and ultimately impact the overall operation of the port. Second, despite the tight interdependence between berth allocation and tugboat scheduling, few studies have simultaneously optimized these two issues and provided globally optimal solutions for tugboats.

In sum, the literature review on eTug-SR and BAP exposes two critical gaps. First, prevailing studies presume abundant energy-replenishment facilities, ignoring the need to optimize the scarce berths that house them; in congested ports, this oversight creates a mismatch between recharging demand and limited facilities, eroding tugboat utilization and port efficiency. Second, the tight coupling between berth allocation and tugboat scheduling remains largely unaddressed, with few works offering joint, globally optimal solutions for the entire tugboat fleet.

To address these gaps, this paper integrates eTug-SR and BAP and investigates the ETSBS. The ETSBS involves decisions such as the assignments of berthing or unberthing jobs to the working periods of each tugboat, the allocation of berths to the recharging demands of each tugboat, and the starting and completion times of each recharging operation for the tugboats. Compared with the work by Ma et al. [17], we apply a fast battery-swapping mode to allow tugboats to be fully re-energized in less than 30 min, significantly improving the availability of tugboats and contributing to the efficient operation of busy ports. Moreover, we incorporate the limitation of recharging facilities into our model, integrating the scheduling of battery-swapping stations with tugboat scheduling. This integration is more realistic, considering the potential conflict between the increasing number of tugboats (and their recharging demands) and the limited resources available.

To deal with the ETSBS problem, we formulate it into a MILP model to minimize the makespan of recharging, i.e., the completion time of the last recharging operation in all the berths. Considering the complexity of the problem and the structure of the model, we develop an efficient LBBD method and verify its effectiveness through extensive numerical experiments. Next, we formally describe the ETSBS and build the MILP model.

3. Problem Description and Mathematical Formulation

In electrified ports with a large number of electric tugboats, the scheduling of tugboats to assist with the berthing/unberthing operations of vessels and the allocation of tugboats to energy replenishment berths are two crucial issues. These issues are essential to ensure the readiness of tugboats and the overall efficiency of the ports. This problem integrates both aspects and addresses all the decisions simultaneously. However, several challenges arise. First, each tugboat needs to recharge multiple times. Each pair of adjacent recharging operations forms a working period during which the tugboat can perform berthing or unberthing assistance. The recharging operations have inherent time-sequencing constraints, meaning that the $(r+1)$-th recharging operation can only start after the r-th operation is completed. This adds complexity to the berth allocation and makespan calculation. Second, tugboats in a port are typically heterogeneous in terms of size, horsepower, and battery capacity. This heterogeneity results in varying eligibility and lengths of working periods among the tugboats.

3.1. Problem Description

The ETSBS problem investigated in this paper is illustrated in Figure 1. In the figure, vessels are assisted by unmanned electric tugboats to perform berthing and unberthing operations. Each tugboat can handle multiple tasks until its battery is depleted. Before the battery runs out, the tugboat travels to a battery-swapping station to replace the depleted battery with a fully charged one and then returns to service. Due to the limited number of battery-swapping stations, tugboats may need to wait if all stations are occupied until one becomes available.

Under the above scenario, we consider an electrified port that is equipped with a set K of heterogeneous unmanned electric tugboats under battery-swapping mode to serve large vessels. Up to the end of the last planning horizon, the command center has received a set J of berthing/unberthing tasks from various types of vessels. Each task $j\in J$ is associated with a processing time $t_j$ and an energy consumption $e_j$. To meet the energy-replenishment demands of the tugboats, the port has constructed a set H of battery-swapping stations. These stations are capable of swiftly replacing the depleted battery of a tugboat with a fully charged one within a fixed battery-swapping time t. Each tugboat $k\in K$ is characterized by a heterogeneous battery capacity $b_k$. Considering the working eligibility, an auxiliary binary parameter ${\alpha }_{j}k$ is defined, which takes the value one if task j is served by tugboat k, and zero otherwise. Each tugboat k can consecutively perform multiple tasks until its remaining battery power is insufficient to undertake the next task. At this juncture, a battery-swapping operation $r\in R$ arises, prompting the tugboat to return to the shore station, swap its battery, and promptly resume work with full energy. The tugboat can be assigned to any station to non-preemptively swap its battery. The travel time among the stations and vessels (tasks) is considered negligible compared with the processing time $p_j$ and battery-swapping time t.

The decisions to be made in this ETSBS include the assignment of tasks to the working period after the rth battery-swapping, the execution or not of the rth battery-swapping, and the allocation to a shore station m, the sequencing of different battery-swapping operations at each station, and the starting and completion times of each battery-swapping. The objective function of the ETSBS, from the battery-swapping perspective, is to minimize the makespan of the stations, i.e., the completion time of the last battery-swapping operation across all stations. Next, we formulate the above described ETSBS into a MILP model.

3.2. Mathematical Formulation

Based on the description above, this section formulates the problem into a MILP model. At first, for the sake of manageability and model simplicity, the following assumptions are made:

(1)

The newly swapped battery is fully charged, enabling the tugboat to restore its battery capacity to full.

(2)

Despite the varying battery capacities among the tugboats, they can all complete the battery-swapping process at any station.

(3)

The travel time of tugboats between the stations and tasks is neglected.

(4)

Accidents are not considered in the system, including those involving the tasks, the tugboats, the batteries, and the stations.

We next define the sets, parameters, and decision variables for the ETSBS, all of which are summarized in

Table 1. Definition of sets, parameters, and decision variables.

Sets
J	set of berthing/unberthing tasks, indexed by $i,j$;
K	set of tugboats, indexed by k;
R	set of battery-swapping operations, start from 0, indexed by r;
H	set of battery-swapping station, indexed by m;
Parameters
$b_k$	the battery capacity of tugboat k;
$e_j$	the amount of energy required for task $j\in J$;
$p_j$	the processing time of task j;
t	the fixed battery-swapping time;
${\alpha }_{jk}$	equal to 1 if task j is served by tugboat k, and 0 otherwise;
M	a large enough positive value;
Decision variables:
$C_{max}$	the maximum completion time, i.e., the makespan;
$C_{kr}$	the completion time of the rth battery-swapping operation of tugboat k;
$y_{krm}$	equal to 1 if the rth battery-swapping is performed by tugboat k at station m, and 0 otherwise;
$v_{jkr}$	equal to 1 if task j is served after the rth battery-swapping of tugboat k, and 0 otherwise;
$x_{kr{k}^{\prime }{r}^{\prime }}$	equal to 1 if the rth battery-swapping operation of tugboat k is performed before the $r^{\prime }$th
	battery-swapping operation of tugboat $k^{\prime }$, and 0 otherwise;
$w_{kr}$	equal to 1 if the rth battery-swapping of tugboat k is performed, and 0 otherwise.

.

The MILP model (denoted as model $\mathcal{P}$) is formulated as follows.

$$\begin{array}{c}min{C}_{max}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill & C_{max}\ge C_{kr}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(2)

$$\begin{array}{ccc}\hfill & C_{kr}\ge C_{k,r-1}+t{w}_{kr}+{\displaystyle \sum _{j\in J}}{p}_j{v}_{jk,r-1}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(3)

$$\begin{array}{ccc}\hfill & C_{k^{\prime }{r}^{\prime }}\ge C_{kr}+t{w}_{k^{\prime }{r}^{\prime }}-M(3-x_{kr{k}^{\prime }{r}^{\prime }}-y_{krm}-y_{k^{\prime }{r}^{\prime }m})\hfill & \hfill \forall k,k^{\prime }\in K,k\ne k^{\prime },r,r^{\prime }\in R,m\in H\end{array}$$

(4)

$$\begin{array}{ccc}\hfill & C_{kr}\ge C_{k^{\prime }{r}^{\prime }}+t{w}_{kr}-M(2+x_{kr{k}^{\prime }{r}^{\prime }}-y_{krm}-y_{k^{\prime }{r}^{\prime }m})\hfill & \hfill \forall k,k^{\prime }\in K,k\ne k^{\prime },r,r^{\prime }\in R,m\in H\end{array}$$

(5)

$$\begin{array}{ccc}\hfill & w_{kr}={\displaystyle \sum _{m\in H}}{y}_{krm}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(6)

$$\begin{array}{ccc}\hfill & w_{kr}\le {\displaystyle \sum _{j\in J}}{v}_{jkr}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(7)

$$\begin{array}{ccc}\hfill & v_{jkr}\le w_{kr}\hfill & \hfill \forall j\in J,k\in K,r\in R\end{array}$$

(8)

$$\begin{array}{ccc}\hfill & w_{kr}\le w_{k,r-1}\hfill & \hfill \forall k\in K,r\in R\\ \hfill & v_{jkr}\le {\displaystyle \sum _{h=0}^r}{v}_{ikh}\hfill & \hfill \forall k\in K,r\in R,i,j\in J,i<j\end{array}$$

(9)

$$\begin{array}{ccc}\hfill & \hfill & \hfill {\alpha }_{ik},{\alpha }_{jk}=1\end{array}$$

(10)

$$\begin{array}{ccc}\hfill & {\displaystyle \sum _{j\in J}}{e}_j{v}_{jkr}\le b_k\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(11)

$$\begin{array}{ccc}\hfill & {\displaystyle \sum _{r\in R}}{v}_{jkr}={\alpha }_{jk}\hfill & \hfill \forall j\in J,k\in K\end{array}$$

(12)

$$\begin{array}{ccc}\hfill & w_{k,0}=1\hfill & \hfill \forall k\in K\end{array}$$

(13)

$$\begin{array}{ccc}\hfill & C_{k,0}=0\hfill & \hfill \forall k\in K\end{array}$$

(14)

$$\begin{array}{ccc}\hfill & C_{kr}\ge 0\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(15)

$$\begin{array}{ccc}\hfill & w_{kr}\in \{0,1\}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(16)

$$\begin{array}{ccc}\hfill & y_{krm}\in \{0,1\}\hfill & \hfill \forall k\in K,r\in R,m\in H\end{array}$$

(17)

$$\begin{array}{ccc}\hfill & v_{jkr}\in \{0,1\}\hfill & \hfill \forall j\in J_k,k\in K,r\in R,m\in H\end{array}$$

(18)

$$\begin{array}{ccc}\hfill & x_{kr{k}^{\prime }{r}^{\prime }}\in \{0,1\}\hfill & \hfill \forall k,k^{\prime }\in K,r,r^{\prime }\in R\end{array}$$

(19)

The objective function of this problem minimizes the makespan, i.e., the maximum completion time of the last battery-swapping operation, which is shown in (1). Constraints (2) make sure that the makespan is no less than the completion time of any battery-swapping. Constraints (3) make sure that for tugboat k, the completion time of the rth battery-swapping is at least that of the ($r-1$)th plus the battery-swapping time t and the total processing time of the tasks that are assigned to the working period after the ($r-1$)th battery-swapping. Constraints (4) and (5) show the time sequencing constraints between two battery-swapping at the same berth. If two battery-swapping operations, for example, the rth of tugboat k and the $r^{\prime }$th of tugboat $k^{\prime }$ ($k\ne k^{\prime }$), are both assigned to station m and tugboat k is served earlier than tugboat $k^{\prime }$ (i.e., $x_{kr{k}^{\prime }{r}^{\prime }}=1$), then the completion time $C_{k^{\prime }{r}^{\prime }}$ of tugboat $k^{\prime }$ is at least the completion time of $C_{kr}$ of tugboat k plus the fixed battery-swapping time t. Otherwise, given M being a large enough positive value, $C_{k^{\prime }{r}^{\prime }}$ is no earlier than a negative value, which stands all the time. Constraints (6) enforce that only if a battery-swapping operation is performed can it be assigned to a battery-swapping station. Similarly, constraints (7) and (8) give the relationship between variables $v_{jk}$ and $y_{kr}$. Namely, only when a battery-swapping operation is performed can a task be served after it, and at the same time, at least one task must be assigned when a certain battery-swapping operation happens. Constraints (9) require that for any tugboat k, the rth battery-swapping operation can be performed only if the former one is completed. Constraints (10) means that if task j is assigned to the working period after the rth battery-swapping of tugboat k, than the former tasks, e.g., task i ($i<j$), must have been assigned. Constraints (11) enforce that the total energy consumption of tasks assigned to the same working period of a tugboat cannot exceed its battery capacity. Constraints (12) highlight the tugboat eligibility constraints, i.e., tasks can only be assigned to tugboats that are capable of providing the services (${\alpha }_{jk}=1$). Constraints (13) tell that a battery-swapping operation must be performed before the tugboats start to work, making sure that all the tugboats are fully charged at the very beginning. Constraints (14) denote the completion time of the 0th battery swapping as the start of the planning horizon. Constraints (15)–(19) define the domains of the decision variables.

4. Solution Method

The ETSBS investigated in this paper simultaneously optimizes the assignment of berthing or unberthing tasks to the working periods of tugboats, the allocation of battery-swapping operations of tugboats to battery-swapping stations, and the sequencing of the demands at each station. The ETSBS is highly complicated since its simplified version, i.e., ignoring the assignment of tasks to tugboat working periods, is the difficult parallel machine scheduling problem. Preliminary experiments show that commercial solver CPLEX spends a long computational time solving medium-sized instances and fails to provide feasible solutions for practical instances with 80 tasks. Therefore, by observing the structure of the problem, we propose a logic-based Benders decomposition (LBBD) algorithm. The LBBD algorithm, firstly proposed by Hooker and Ottosson [28], is extended from the classic Benders decomposition (CBD) algorithm. It decomposes the complicated problem into a master problem (MP) and a series of subproblems (SP). The MP relaxes some difficult variables and determines the remaining parts to obtain a lower bound (LB) of the original problem. The solution to MP is then passed to the SP as input parameters to solve the remaining variables. If the SP is infeasible, a feasibility Benders cut is generated; else, the SP is solved, and a complete solution, for which the objective function value is the upper bound (UB). Also, an optimality Benders cut arises after the SP is solved. The two kinds of Benders cuts are added to the MP in the following iterations to avoid the same solution being obtained again and to improve the LB of MP, respectively. With the continuation of the iteration between MP and SP, the LB and the UB gradually reconcile, and a globally optimal solution is found when they are equal. The LBBD terminates when the optimal solution is obtained or when some other conditions are met, such as the computational time limit and the maximum number of iterations. LBBD allows the SP to be any form of optimization problem instead of being limited to linear programming in the CBD, which has greatly expanded its applicability. In the past decade, the LBBD algorithm has raised remarkable attention from researchers and has achieved great performances in solving complicated combinatorial optimization problems such as integrated scheduling and location [29], doctor–patient matching and scheduling [30], railway timetable planning [31]. Readers are recommended to read the book by [32] that systematically introduces the theory and application of LBBD and provides a comprehensive guide for LBBD users.

The general structure of LBBD for our ETSBS problem is illustrated in Figure 2. The primary objective of the LBBD algorithm for ETSBS is to minimize the total completion time of battery-swapping operations while efficiently scheduling electric tugboats and battery-swapping stations. This is achieved by decomposing the problem into an MP and an SP, as step 1 in the figure. Specifically, the MP is responsible for assigning tasks to tugboats and allocating battery-swapping operations to stations (Step 2). It achieves this by relaxing the sequencing constraints, thereby simplifying the problem. Based on the solution from the MP, the SP determines the optimal sequence of battery-swapping operations at each station to minimize the total completion time (Step 3). Once the SP is solved, a UB to the original ETSBS problem is obtained. If the optimal solution has not been obtained or the termination condition is not met, Benders cuts will be generated and added to the MP (Step 4). This process continues iteratively until the termination condition is satisfied. Once the termination condition is met, the final solution is output, and the algorithm concludes (Step 6). If the termination condition is not satisfied, Benders cuts are generated and added into the MP (Step 4). The algorithm continues to iterate and refine the solution until the termination condition is met. Upon meeting the termination condition, the algorithm outputs the final solution (Step 6), marking the end of the process. Note that in the SP, we can always obtain feasible sequences for the battery-swapping stations, indicating that only optimality cuts are generated.

Next, we delve into the LBBD algorithm and describe the MP, SP, and the Benders cuts in depth.

4.1. Master Problem

In the MP, the sequencing of battery-swapping operations at the battery-swapping stations, i.e., $x_{kr{k}^{\prime }{r}^{\prime }}(k,k^{\prime }\in K,k\ne k^{\prime },r,r^{\prime }\in R)$ are relaxed while the other key variables, concerning the battery-swapping decision $w_{kr}$, the task assignment $v_{jkr}$, and the battery-swapping allocation $y_{krm}$ are to be determined. Therefore, the completion time of each battery-swapping $C_{kr}$ cannot be exactly calculated. We define the relaxed makespan as ${\tilde{C}}_{max}$. To compute ${\tilde{C}}_{max}$, we define another auxiliary non-negative variable ${\mathrm{\Pi }}_m$ indicating the completion time of battery-swapping station m. Then, we can obtain ${\tilde{C}}_{max}\ge {\mathrm{\Pi }}_m,\forall m\in H$. Similarly, the relationship between completion time ${\mathrm{\Pi }}_m$ and $C_{kr}$ can be described as ${\mathrm{\Pi }}_m\ge C_{kr}-M(1-y_{krm})$, i.e., if the rth battery-swapping operation of tugboat k is allocated to station m ($y_{krm}=1$), the completion time of the station is at least that of the demand. Based on the description and definition above, the model of the MP (denoted as model $\mathcal{MP}$) can be formulated as follows.

$$\begin{array}{ccc}\hfill & min{\tilde{C}}_{max}\hfill & \hfill \\ \hfill \mathrm{s}.\mathrm{t}.\\ \hfill & \left(6\right)-\left(13\right), \left(16\right)-\left(18\right),\mathrm{and}\mathrm{to}:\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill & {\tilde{C}}_{max}\ge {\mathrm{\Pi }}_m\hfill & \hfill \forall m\in H\end{array}$$

(21)

$$\begin{array}{ccc}\hfill & {\mathrm{\Pi }}_m\ge 0\hfill & \hfill \forall m\in H\end{array}$$

(22)

$$\begin{array}{ccc}\hfill & Benderscuts\hfill & \hfill \end{array}$$

(23)

Objective function (20) minimizes the relaxed makespan. Constraints (21) make sure that the makespan is no less than the completion time of any station m. Constraints (22) state that ${\mathrm{\Pi }}_m$ is non-negative. (23) is the Benders cuts that were generated in the SP and added to MP during the iterations.

The above MP is now weak and loose, making the LB hard to lift. To strengthen it, several valid inequalities are provided as follows.

$$\begin{array}{ccc}\hfill & {\tilde{C}}_{max}\ge C_{kr}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(24)

$$\begin{array}{ccc}\hfill & C_{kr}\ge C_{k,r-1}+t{w}_{kr}+{\displaystyle \sum _{j\in J}}{p}_j{v}_{jk,r-1}\hfill & \hfill \forall k\in K,r\in R\end{array}$$

(25)

$$\begin{array}{ccc}\hfill & {\mathrm{\Pi }}_m\ge {\displaystyle \sum _{k\in K,r\in R}}t{y}_{krm}\hfill & \hfill \forall m\in H\end{array}$$

(26)

Constraints (24) and (25) are consistent with (2) and (3). Constraints (26) means that the completion time of station m is at least the product of the battery swapping time t and the times of the battery-swapping operations performed at this station. After the MP is solved, some key variables, including $w_{kr},v_{jkr}$ and $y_{krm}$ are fixed and they are passed to the SP as parameters.

4.2. Subproblem

Given the solution to MP, the allocation of battery-swapping operations to the stations are determined. Then, the SP needs to solve a series of SMSPs at each station m. For each station m, the jobs are those battery-swapping operations allocated to this station, i.e., $y_{krm}=1$, and the objective function is to minimize the completion time ${\mathrm{\Pi }}_m$. We first formulate the subproblem into a MILP model at each station m, and then propose a dispatching heuristic to solve the SPSM for each m.

4.2.1. MILP Model

To formulate the SMSP at station m, we define a set ${\overline{K}}_m$ containing all the jobs, i.e., ${\overline{K}}_m=\left\{(k,r)\right|y_{krm}=1,\forall k\in K,r\in R\}$. The combination $(k,r)$ indicates the rth battery-swapping operation of tugboat k. We also define two binary variables $x_{kr{k}^{\prime }{r}^{\prime }}$ and $z_{kr}$. $x_{kr{k}^{\prime }{r}^{\prime }}$ takes value one if battery-swapping operations $(k,r),(k^{\prime },r^{\prime })\in {\overline{K}}_m,(k,r)\ne (k^{\prime },r^{\prime })$ and operation $(k,r)$ is performed immediately before $(k^{\prime },r^{\prime })$, and 0 otherwise. $z_{kr}$ equals one if operation $(k,r)$ is the first operation performed at station m, and 0 otherwise. Then, for station m that $|{\overline{K}}_m>0|$, the MILP model (denoted as model ${\mathcal{SP}}_m$) can be formulated as follows.

$$\begin{array}{ccc}\hfill & {\mathcal{SP}}_m:min{\mathrm{\Pi }}_m\hfill & \hfill \\ \hfill \mathrm{s}.\mathrm{t}.\end{array}$$

(27)

$$\begin{array}{ccc}\hfill & {\mathrm{\Pi }}_m\ge C_{kr}\hfill & \hfill \forall (k,r)\in {\overline{K}}_m\end{array}$$

(28)

$$\begin{array}{ccc}\hfill & {\displaystyle \sum _{(k,r)\in {\overline{K}}_m}}{z}_{kr}=1\hfill & \hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill & z_{k^{\prime }{r}^{\prime }}+{\displaystyle \sum _{(k,r)\in {\overline{K}}_m}}{x}_{kr{k}^{\prime }{r}^{\prime }}=1\hfill & \hfill \forall (k^{\prime },r^{\prime })\in {\overline{K}}_m\end{array}$$

(30)

$$\begin{array}{ccc}\hfill & {\displaystyle \sum _{(k^{\prime },r^{\prime })\in {\overline{K}}_m}}{x}_{kr{k}^{\prime }{r}^{\prime }}\le 1\hfill & \hfill \forall (k,r)\in {\overline{K}}_m\end{array}$$

(31)

$$\begin{array}{ccc}\hfill & x_{kr{k}^{\prime }{r}^{\prime }}+x_{k^{\prime }{r}^{\prime }kr}\le 1\hfill & \hfill \forall (k,r),(k^{\prime },r^{\prime })\in {\overline{K}}_m\end{array}$$

(32)

$$\begin{array}{ccc}\hfill & C_{kr}\ge t-M(1-z_{kr})\hfill & \hfill \forall (k,r),(k^{\prime },r^{\prime })\in {\overline{K}}_m\end{array}$$

(33)

$$\begin{array}{ccc}\hfill & C_{k^{\prime }{r}^{\prime }}\ge C_{kr}+t-M(1-x_{kr{k}^{\prime }{r}^{\prime }})\hfill & \hfill \forall (k,r),(k^{\prime },r^{\prime })\in {\overline{K}}_m\end{array}$$

(34)

$$\begin{array}{ccc}\hfill & x_{kr{k}^{\prime }{r}^{\prime }}\in \{0,1\}\hfill & \hfill \forall (k,r),(k^{\prime },r^{\prime })\in {\overline{K}}_m\end{array}$$

(35)

$$\begin{array}{ccc}\hfill & C_{kr}\ge 0\hfill & \hfill \forall (k,r)\in {\overline{K}}_m\end{array}$$

(36)

The objective function (27) minimizes the completion time of battery-swapping station m. Constraints (28) indicate that the completion time of station m is at least the completion of any battery-swapping operation. Constraints (29) limit that each station has only one first-performed operation. Constraints (30) make sure that an operation is either the first-performed one or a predecessor of another. Similarly, constraints (31) enforce that an operation can be the predecessor of at most one other operation. Constraints (32) indicate that an operation cannot simultaneously be the predecessor and a successor of another. Constraints (33) and (34) mean that if an operation is the first-performed one at station m, its completion time is at least the battery-swapping time t; otherwise, it will be no earlier than the completion time of its predecessor plus t.

Typically, the SMSP with makespan minimization can be optimally solved with the earliest release date rule. However, in the ETSBS investigated in this paper, the stations are not independent since each tugboat can perform battery-swapping operations multiple times, and they may be allocated to different stations. Among the operations, constraints (34) must be respected, which make sure that the rth operation cannot start after the completion of the previous ones.

$$\begin{array}{ccc}\hfill & C_{kr}\ge C_{k,r^{\prime }}+(r-r^{\prime })t\hfill & \hfill \forall k\in K,r,r^{\prime }\in R,r^{\prime }<r\end{array}$$

(37)

Due to the inter-dependence of different battery-swapping operations of one tugboat that may be allocated to different stations, the subproblem can hardly use the ERD rule to sequence the battery-swapping operations for each station m. Thus, we propose a dispatching heuristic to efficiently determine the sequence and completion time of each battery-swapping operation and eventually minimize the makespan of all the stations.

4.2.2. A Dispatching Heuristic

The dispatching of the battery-swapping operations are conducted under multifold constraints, including the sequencing constraints for the operations allocated at the same station and those among operations of the same tugboats. To find appropriate positions for each operation, we design a dispatching heuristic as described in Algorithm 1.

Algorithm 1: Dispatching heuristic.

4.3. Benders Cuts

After the MP and SP are solved, respectively, a complete and feasible solution is generated, and the completion time of each battery-swapping station m, i.e., ${\mathrm{\Pi }}_m$ has been figured out. We define ${\widehat{\mathrm{\Pi }}}_m$ as the exact value of the completion time of m obtained in Section 4.2, then we can add the following Benders cuts in the subsequent iterations.

$$\begin{array}{ccc}\hfill & {\mathrm{\Pi }}_m\ge {\widehat{\mathrm{\Pi }}}_m({\displaystyle \sum _{(k,r)\in {\overline{K}}_m}}{y}_{krm}-|{\overline{K}}_m|+1)\hfill & \hfill \forall m\in H\end{array}$$

(38)

In cuts (38), ${\overline{K}}_m$ are operations that allocated to station m in the incumbent iteration and ${\widehat{\mathrm{\Pi }}}_m$ is the completion time of m calculated in Section 4.2. ${\mathrm{\Pi }}_m$ and $y_{krm}$ are variables in model $\mathcal{MP}$. Therefore, cuts (38) make sure that if the same set of operations as in ${\overline{K}}_m$ is allocated again to station m, its completion time ${\mathrm{\Pi }}_m$, correspondingly, is at least ${\widehat{\mathrm{\Pi }}}_m$. The cuts (38) are added to constraints (23) in MP to lift up the LB.

5. Case Study

In this section, we conduct a case study to show how the LBBD obtains the globally optimal solution for the scheduling and battery-swapping of electric tugboats. Due to the lack of practical data from ports that simultaneously concerns the berthing or unberthing tasks, tugboats, and berths with battery-swapping stations, we use some of the data from [17,24] and generate the other necessary ones to provide a complete and clear perspective showing how our solution methods obtains effective, efficient, and quantitative operational plans for port managers. In this case, the port builds two berths with battery-swapping stations, and each battery-swapping operation takes half an hour. Four tugboats are equipped in the port, each with battery capacities $b_k=\{800,800,400,400\}$, respectively. The parameters of the tasks are listed in

Table 2. Parameters of the tasks.

j	${\mathit{e}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}$	$\overline{\mathit{k}}$	j	${\mathit{e}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}$	$\overline{\mathit{k}}$	j	${\mathit{e}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}$	$\overline{\mathit{k}}$	j	${\mathit{e}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}$	$\overline{\mathit{k}}$
1	106	60	1	21	83	160	1	41	77	180	3	61	88	110	3
2	114	140	1	22	116	80	2	42	95	70	3	62	89	180	3
3	93	120	1	23	63	60	2	43	117	170	3	63	103	130	4
4	137	110	1	24	102	160	2	44	95	180	3	64	115	160	4
5	107	120	1	25	128	180	2	45	114	100	3	65	115	90	4
6	72	100	1	26	97	90	2	46	110	120	3	66	73	100	4
7	103	80	1	27	136	70	2	47	75	150	3	67	115	90	4
8	90	80	1	28	85	180	2	48	135	130	3	68	128	170	4
9	85	70	1	29	62	120	2	49	73	130	3	69	102	160	4
10	149	120	1	30	99	100	2	50	79	130	3	70	106	180	4
11	94	130	1	31	117	110	2	51	69	80	3	71	93	180	4
12	96	110	1	32	86	80	2	52	110	180	3	72	63	180	4
13	108	120	1	33	65	70	2	53	71	140	3	73	96	140	4
14	138	130	1	34	75	100	2	54	119	150	3	74	119	110	4
15	62	150	1	35	104	80	2	55	81	60	3	75	101	90	4
16	110	150	1	36	81	110	2	56	73	120	3	76	62	110	4
17	111	60	1	37	68	160	2	57	60	130	3	77	87	160	4
18	89	170	1	38	87	70	2	58	124	90	3	78	66	70	4
19	81	120	1	39	68	170	2	59	87	100	3	79	99	170	4
20	117	80	1	40	106	160	2	60	64	120	3	80	145	180	4

. Note that $\overline{k}$ in the table indicates the tugboat eligible for the task, i.e., ${\alpha }_{\overline{k}j}=1$.

At first, an enumeration-based heuristic (EBH) is applied that includes three main steps.

First, from the first task on, assign as many tasks as possible to each working period of the eligible tugboat and figure out the minimum number of battery-swapping operations each tugboat needs. Second, calculate the earliest ready time of each battery operation. For example, a total 720 and 640 min of tasks are assigned to the first and second working periods of the first tugboat, respectively. Thus, we can obtain the earliest ready time of the first two battery-swapping operations as 720 and 1390 (i.e., 720 + 30 + 640), respectively.

Second, sort the operations in non-decreasing order of the earliest ready time.

Third, allocate the ordered battery operations one by one to the battery-swapping stations. For the rth operation of tugboat k with the earliest ready time ${\tau }_k$, enumerate the battery-swapping stations and find the first station m that is idle at time ${\tau }_k$. Update the completion time $C_{kr}$ and ${\mathrm{\Pi }}_m$. If all the stations are busy at that time, allocate j to the station $m^{\prime }$ with the earliest completion time, i.e., $y_{kr{m}^{\prime }}=1|m^{\prime }=argmin{\mathrm{\Pi }}_m$. Then, update $C_{kr}$ and ${\mathrm{\Pi }}_m$.

Fourth, continue the enumeration until all the battery-swapping operations are allocated, and then output the total completion time.

5.1. Result of the Case Study

Following the above EBH, a feasible solution is obtained with a total completion time of 2560, and the number of battery operations performed on the two battery-swapping stations is 12 and 1, respectively. After that, we apply the LBBD algorithm proposed in this paper to solve the case. An optimal solution is quickly obtained, which is described with Gantt Charts shown in Figure 3. From the figure, we can observe that the total completion time obtained by LBBD is 2460, slightly shorter than that of EBH. Also, we can find that the first station performed six battery-swapping operations, while the second swapped batteries eight times, which is more balanced than EBH.

The case study shows that the LBBD algorithm can obtain optimal solutions for the practical-sized problem with 80 tasks, four tugboats, and two battery-swapping stations. Compared with the enumeration-based method that is often applied in reality, the LBBD obtains the solution with shorter completion time and more balanced allocation for the battery-swapping stations. Next, we conduct more comprehensive numerical experiments to evaluate the performance of LBBD.

5.2. Exploratory Analysis Across Diverse Scenarios

For a thorough examination of our case study, we have designated the initial case data as a baseline scenario. Subsequently, we have constructed a series of alternative scenarios, each characterized by a singular adjustment to a key parameter while holding all others constant. This approach allows us to isolate the impact of each parameter variation. Each of these scenarios has been solved using the CPLEX solver, enabling us to assess the distinct effects on both computational efficiency and solution quality.

Scenario 1: Varying Battery-Swapping Duration.

This scenario explores the repercussions of fluctuating battery-swapping time t, spanning a spectrum from 10 to 120 min. The analysis reveals a direct correlation between extended battery-swapping times and increased completion times. Notably, despite these variations in t, the underlying scheduling patterns for both tugboats and battery-swapping stations align with those illustrated in Figure 3. This observed congruity is likely attributable to the fact that modifications in t predominantly alter the completion time without substantially impacting the allocation of tasks to specific working periods of the tugboats, the allocation of battery-swapping operations to the stations, or the sequencing of operations at each station.

Scenario 2: Varying Battery-Swapping Capacities.

In this scenario, we explore the effects of battery capacity on operational outcomes by comparing two cases: one with high battery capacity, where $b_k=800$ for all tugboats $k\in \{1,2,3,4\}$, and another with low battery capacity, where $b_k=400$. All other parameters mirror those of the baseline scenario. Our findings reveal that the completion times for the high- and low-capacity cases are 1820 min and 2460 min, respectively. The computational times required to solve these cases were 360.47 s and 246.66 s, respectively. The results indicate that tugboats equipped with higher battery capacities can substantially reduce the completion time by accomplishing more tasks within each working period, thereby reducing the frequency of battery swaps. Nonetheless, the scheduling of tugboats with higher battery capacities proves to be relatively more complex.

Scenario 3: Elevated energy consumption.

As detailed in Table 2, the energy consumption $e_j$ originally spans the range $[60,150]$, with an average of 96.04. To assess the impact of heightened energy consumption on our solutions, we have generated a new set of parameters where $e_j$ is adjusted to range from $[80,150]$, raising the average to 109.14. All other parameters align with those listed in Table 2. The computational outcomes reveal that the objective function value for the high energy consumption scenario is 2590 min, marking an increase of 130 min over the baseline scenario. This increase can be attributed to the reduced number of tasks assignable to each tugboat’s working period due to higher average energy consumption per task, resulting in a higher frequency of battery-swapping operations. Also, a total of 17 battery-swapping operations were conducted, which is 3 more than the number observed in the baseline scenario. Additionally, the computational time for the high energy consumption scenario was 193.45 s, which is shorter than that of the baseline scenario.

Another conclusion can be drawn regarding the environmental sustainability of electric tugboats. Based on the study by Ergüven et al. in Izmit Bay, Turkey, fuel-powered tugboats produced 1.26 tons of CO₂, 20.6 kg of NO_x, and 11.3 kg of SO_x per hour during 7834 activities between March and May 2020 [33]. In this case, for 80 tasks taking 163.67 h, electric tugboats could at least reduce emissions by approximately 206.22 tons of CO₂, 3.37 tons of NO_x, and 1.85 tons of SO_x. This highlights the significant environmental benefits of electric tugboats over fuel-powered ones.

6. Numerical Experiments

In this section, we conduct numerical experiments using randomly generated instances to validate the effectiveness of the proposed MILP model and LBBD algorithm. Since there is no readily available benchmark for our ETSBS, we first introduce the data generation method. We then apply both the commercial solver CPLEX and the LBBD algorithm to solve the instances. The results are subsequently reported and analyzed.

6.1. Data Generation

To the best of our knowledge, there is currently no literature that addresses the joint optimization of tugboat task scheduling and battery-swapping station scheduling, resulting in the lack of benchmark instances. In fact, only [17] have studied the scheduling of electric tugboats considering charging operations, assuming a fixed recharging time of 90 or 120 min each time. This is somewhat analogous to our ETSBS problem, as we also assume a constant battery-swapping time of 20 min. Given this context, we have referred to relevant papers and generated instances randomly based on reasonable assumptions.

In this section, 80 instances are randomly generated, each with a varying combination of number of tasks, tugboats, and battery-swapping stations $\left(\right|J|,|K|,|H\left|\right)$. Specifically, $\left|J\right|=\{20,40,\dots ,100\}$, $\left|K\right|=\{4,6,8,10\}$, and $\left|H\right|=\{1,2,4,6\}$. For each task $j\in J$, the processing time $p_j$ is randomly generated from a uniform distribution in the interval $[60,180]$ of minutes, and the energy consumption $e_j$ is randomly generated from a uniform distribution in the interval $[30,100]$. For each tugboat k, the battery capacity $b_k$ is randomly chosen from $\{400,800\}$, and the fixed battery-swapping time $t=20$ of minutes. Both the MILP models and the LBBD algorithm are coded in C++ linking with CPLEX 12.10 and run on a computer with an Intel Core CPU i7-13700H at 2.40 GHz with 16 GB RAM. A time limit of 3600 s is set for each run. Next, we apply CPLEX and LBBD to solve all 80 instances and then analyze their computational results.

6.2. Effectiveness of the Valid Inequalities for LBBD

In Section 4.1, we propose valid inequalities (24)–(26) to strengthen the MP. To validate the effectiveness of the inequalities, we apply LBBD with the inequalities (denoted as LBBD) and the one without them (denoted as LBBD-) to solve all 80 instances, respectively. The results are shown in

Table 3. Effectiveness of the valid inequalities for LBBD.

$\left|\mathit{J}\right|$	$\mathit{Obj}$ (min)			$\mathit{Time}$ (s)
	LBBD	LBBD-	${\mathit{Df}}^{\mathbf{1}}\%$	LBBD	LBBD-	${\mathit{Df}}^{\mathbf{2}}\%$
20	50.00	90.00	44.44	0.09	0.52	82.42
40	492.50	816.88	39.71	1.59	2.79	42.82
60	1087.50	1554.38	30.04	21.02	21.09	0.32
80	1876.25	2195.00	14.52	231.62	173.56	−33.45
100	2394.38	2700.00	11.32	851.35	1232.49	30.92
Average	1180.13	1471.25	19.79	221.14	286.09	22.70

. The first column of the table shows the number of tasks $\left|J\right|$, and the following three columns compare the objective function values (denoted as $Obj$) of the two versions and the percentage of the differences $D{f}^1\%$, which is calculated using Equation (39). The last three columns, similarly, are the comparison between the two in terms of computational time (denoted as $Time$). Their percentage of difference $D{f}^2\%$ is calculated using Equation (40). Note that each entity in Table 3 is the average value of 16 instances with the same number of tasks but different numbers of tugboats and battery-swapping stations.

$$\begin{array}{cc}\hfill & D{f}^1\%={\displaystyle \frac{Ob{j}_{(LBBD-)}-Ob{j}_{\left(LBBD\right)}}{Ob{j}_{(LBBD-)}}}\times 100\%\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & D{f}^2\%={\displaystyle \frac{Tim{e}_{(LBBD-)}-Tim{e}_{\left(LBBD\right)}}{Tim{e}_{(LBBD-)}}}\times 100\%\hfill \end{array}$$

(40)

From Table 3, we can find that both LBBD and LBBD- can provide solutions for all 80 instances with up to 100 tasks. Generally, the average objective function value of LBBD is 1180.13 min, which is 19.79% lower than that of LBBD- (1471.25). This means that the augmentation of valid inequalities can significantly help the algorithm find high-quality solutions and shorten the total makespan of the tugboat recharging system. More specifically, we can find that the $Obj$ of both LBBD and LBBD- increases fast as the instances get larger, but the value of $D{f}^1\%$ decreases in this process. The reason might be that larger instances are of higher complexity, which poses challenges for both algorithms. Also, the average computational time of LBBD is 22.70% shorter than LBBD, indicating that the valid inequalities have made contributions to improve the efficiency of the algorithm. To conclude, the valid inequalities contribute to both improving the solution quality and saving computational time. Therefore, we use LBBD with the valid inequalities in the following analysis.

6.3. Computational Results of CPLEX and LBBD

In this section, we firstly apply CPLEX to solve all 80 instances and compare the results with those of the LBBD. We present the computational results in

Table 4. Computational results of CPLEX and LBBD.

$\left|\mathit{K}\right|$	$\left|\mathit{J}\right|$	$\mathit{Obj}$ (min)		$\mathit{Time}$ (s)
		CPLEX	LBBD	CPLEX	LBBD
4	20	200.00	200.00	1.48	0.10
	40	922.50	922.50	57.79	1.91
	60	2425.00	2425.00	827.92	34.59
	80	$--$	3640.00	$--$	552.48
	100	$--$	4187.50	$--$	2233.19
6	20	0.00	0.00	0.17	0.06
	40	632.50	632.50	27.92	0.99
	60	907.50	907.50	657.85	11.23
	80	$--$	1815.00	$--$	137.41
	100	$--$	2487.50	$--$	480.43
8	20	0.00	0.00	0.29	0.09
	40	232.50	232.50	47.56	1.92
	60	630.00	630.00	982.84	20.48
	80	1355.00	1337.50	2981.79	114.51
	100	$--$	1562.50	$--$	243.76
10	20	0.00	0.00	0.56	0.11
	40	182.50	182.50	17.32	1.55
	60	387.50	387.50	274.12	17.79
	80	$--$	712.50	$--$	122.07
	100	$--$	1340.00	$--$	448.01

. In the table, the first two columns present the number of tugboats $\left|K\right|$ and tasks $\left|J\right|$. The next two columns show the objective function values obtained by the two methods, while the last two are their computational time. The symbol “$--$” indicates that at least one of the four instances with the same $\left(\right|K|,|J\left|\right)$ but different number of stations $\left|H\right|$ is not successfully solved within the time limit of 3600 s.

The table reveals a notable difference in the computational capabilities of CPLEX and LBBD. While CPLEX struggles to solve most groups of instances with $\left|J\right|\ge 80$, LBBD consistently obtains solutions for all instances, regardless of size. This demonstrates that LBBD’s computational capacity is superior to that of CPLEX, making it a more promising candidate for real-world applications. In large ports, where the number of tasks can reach into the hundreds, the ability to efficiently solve large-scale problems is crucial. LBBD’s success in handling such instances highlights its potential to be effectively adapted to practical scenarios, where complex constraints and extensive datasets are common. Furthermore, it is evident that LBBD not only outperforms CPLEX in terms of computational capacity but also demonstrates significantly higher computational efficiency. For instance, with $\left|J\right|\le 60$, the average computational time of CPLEX is 241.32 s, which is nearly 32 times that of LBBD (7.57 s). As $\left|J\right|$ grows from 20 to 60, the computational time of CPLEX rises sharply while LBBD maintains high efficiency. Specifically, When $\left|J\right|=60$, LBBD obtains solutions in approximately 20 s or less, while CPLEX’s computational time ranges from 274.12 s to 982.84 s—several dozen times longer than that of LBBD.

6.4. The Effects of $\left|K\right|$ and $\left|H\right|$

Fundamental facility development is the one of the essential factor affecting the capacity and efficiency of the tugboat operation. In this problem, we proceed to analyze the effects of the number of tugboats $\left|K\right|$ and battery-swapping stations $\left|H\right|$ on the solution quality and the computational efficiency of the ETSBS. The results are presented in

Table 5. The effects of $\left|K\right|$ and $\left|H\right|$.

$\left|\mathit{K}\right|$	$\left|\mathit{H}\right|$	$\mathit{Obj}$ (min)	$\mathit{Time}$ (s)
4	1	2356.00	177.07
	2	2080.00	232.06
	4	2400.00	803.57
	6	2264.00	1045.13
	Average	2275.00	564.45
6	1	1488.00	21.32
	2	868.00	66.34
	4	1196.00	170.32
	6	1122.00	246.13
	Average	1168.50	126.03
8	1	844.00	24.95
	2	790.00	133.08
	4	662.00	72.69
	6	714.00	73.90
	Average	752.50	76.15
10	1	606.00	39.60
	2	458.00	158.96
	4	612.00	81.25
	6	422.00	191.82
	Average	524.50	117.91

and illustrated in Figure. Note that since CPLEX cannot provide nearly all the instances with $\left|J\right|\ge 80$ with all the combinations of ($\left|K\right|,\left|H\right|$), we only analyze the computational results obtained by LBBD.

As shown in Table 5, both $Obj$ and $Time$ decrease steadily as $\left|K\right|$ increases. For example, when $\left|K\right|$ grows from four to ten, $Obj$ decreases from 2275.00 to 524.5, and $Time$ drops from 564.45 s to 117.91 s. This means that equipping more tugboats can significantly improve the efficiency of performing berthing or unberthing assistance. At the same time, the number of tasks assigned to each tugboat decreases, and fewer battery-swapping operations will be needed, making the allocation and sequencing of battery-swapping operations to the shore stations less complicated. Moreover, we can also find that given a fixed $\left|K\right|$, the increase in $\left|H\right|$ might make the problem more challenging. For example, when $\left|K\right|=4$, the average computational time increases from 177.07 s to 1045.13 s as $\left|H\right|$ grows from one to six. The complexity of battery-swapping operations can be elucidated from the standpoint of battery-swapping stations. Essentially, the allocation and sequencing of these operations can be modeled as a parallel machine scheduling problem. The introduction of additional stations significantly complicates the scenario. It not only increases the number of allocation options but also introduces intricate interdependencies between the battery-swapping operations. Moreover, this complexity gives rise to numerous subproblems and necessitates the use of Benders cuts in the LBBD algorithm. Collectively, these factors render the problem more challenging to solve, thereby resulting in longer computational times. The change in $\left|H\right|$, however, does not have a significant effect on the $Obj$. Figure 4 provides a vivid perspective that an increase in $\left|K\right|$ results in a decrease in both $Obj$ and $Time$, while an increase in $\left|H\right|$, when $\left|K\right|$ is fixed, causes longer computational times.

7. Conclusions

This paper investigates the unmanned electric tugboat scheduling considering battery-swapping operations (ETSBS), which integrates task assignment of tugboats, allocation of battery-swapping operations to shore stations, and sequencing of operations at each station to determine the start and completion times of each battery swap. As the electrification of port operations accelerates globally, the limited availability of energy-replenishment facilities makes efficient scheduling of battery-swapping operations increasingly critical. By jointly optimizing these decisions, the ETSBS enables a globally optimal solution that enhances both tugboat operational efficiency and port resource utilization. We formulate the problem as a MILP model aimed at minimizing the makespan of battery-swapping operations, and develop an LBBD algorithm that decomposes the problem into an MP and an SP, connected via Benders cuts in an iterative manner. The MP assigns tasks to tugboat working periods and allocates battery-swapping operations to stations, while the SP addresses a series of Single Machine Scheduling Problems (SMSPs) with makespan minimization, which is solved by a dispatching heuristic to handle inter-dependencies among battery-swapping operations of the same tugboat. A case study involving 80 tasks, four tugboats, and two battery-swapping stations demonstrates the effectiveness of the proposed model and algorithm. Additionally, we have verified the significant environmental benefits of using electric tugboats. Specifically, electric tugboats could reduce emissions by approximately 206.22 tons of CO₂, 3.37 tons of NO_x, and 1.85 tons of SO_x for 80 tasks with a total processing time of 163.67 h. Numerical experiments on 80 randomly generated instances, with up to 100 tasks, 10 tugboats, and six battery-swapping stations, demonstrate that LBBD significantly outperforms CPLEX in both scalability and efficiency, solving all instances within a reasonable computational time and achieving comparable solution quality much faster. Further analysis reveals that increasing the number of tugboats reduces makespan and simplifies the problem, whereas increasing the number of battery-swapping stations increases complexity with longer computational time.

Future research could extend the model to incorporate multi-criteria decision-making, which balances cost, efficiency, and environmental impact to achieve sustainable port operations. Additionally, addressing real-world operational disruptions, such as tugboat failures, sudden weather changes, and vessel delays, is another area worthy of investigation. With the large-scale adoption of electric tugboats and ongoing infrastructure development, future work should also focus on case studies using realistic port operational data to better reflect actual tugboat operations and enhance the credibility of the methods proposed.