Dual-Objective Optimization of Port Tugboat Scheduling with Heterogeneous Service Capabilities

Abstract

As critical hubs in the global supply chain, a port’s competitiveness and sustainability are directly impacted by the efficiency and carbon emissions of its tugboat scheduling. This paper addresses the scheduling optimization of heterogeneous tugboat fleets, aiming to balance these dual objectives. A Mixed-Integer Linear Programming (MILP) model is constructed to minimize vessel waiting time and total carbon emissions, considering key real-world constraints such as tidal windows and channel capacity. Given the model’s complexity, an improved multi-objective evolutionary algorithm is designed, which significantly enhances the performance for solving large-scale instances. A case study based on actual data from Xiamen Port shows that the proposed model and algorithm can effectively generate a series of Pareto-optimal schedules, providing a decision-making basis for port authorities to achieve green and efficient tugboat scheduling.

1. Introduction

As the core support of global trade and international logistics, the shipping industry plays an irreplaceable role in promoting economic globalization and regional economic integration. Ports, as crucial hubs in this system, find their operational efficiency directly determining the performance and service quality of the entire shipping chain [1]. Within port operations, the arrival and departure of vessels are critical links, where tugboats serve as essential resources for ensuring safe berthing and unberthing, thereby enhancing overall efficiency. However, the continuous growth in international shipping demand has led to a sharp increase in vessel numbers, presenting ports with multiple challenges, including limited resources, congested channels, and complex tidal variations [2]. Consequently, achieving an efficient and orderly organization of vessel movements under these constraints to reduce waiting times and improve shipowner satisfaction has become a central issue in port management.

In practice, the tugboat fleet at a port is often “heterogeneous,” meaning it consists of a mix of tugboats with different horsepower, sizes, and functionalities [3]. This heterogeneity arises from several factors: first, the port needs to serve vessels of various sizes and types, where large vessels require high-power tugboats, while smaller ones may only need low-power tugs, thus balancing economy and functionality; second, different operational tasks (such as pushing, towing, and escorting) impose varying performance requirements on the tugboats; finally, the fleet is often developed over many years and includes tugboats procured at different times with diverse technical specifications. While this heterogeneity enhances the operational flexibility of the fleet, it also makes the scheduling and matching of tugboats exceptionally complex [4]. A core challenge in port scheduling is how to select the optimal combination of tugboats from a heterogeneous fleet based on the needs of the vessels and the characteristics of the operations, while maximizing scheduling efficiency and economic benefits under various real-world constraints.

Meanwhile, green development and the low-carbon transition have emerged as key strategic directions for the global port and shipping industry [1]. Tugboat operations, characterized by frequent maneuvering and high energy consumption, are a major source of carbon emissions in ports [5]. Growing international concern over emissions has prompted the International Maritime Organization (IMO) and various governments to introduce multiple reduction targets and policies. These initiatives, such as the 2023 IMO GHG Strategy and amendments to the EU FuelEU Maritime and EU ETS, are driving a shift in port operations from an “efficiency-first” paradigm to one that balances both efficiency and environmental protection [6,7,8]. In this context, tugboat scheduling must evolve to incorporate carbon emission control alongside traditional efficiency objectives, supporting the construction of green ports by ensuring that scientific and rational operational arrangements effectively reduce environmental impact [2].

In recent years, the academic community has conducted extensive research on the optimization of tugboat scheduling, mainly focusing on the following three aspects: firstly, an efficiency-oriented approach, with some studies focusing on how to reduce vessel waiting times and improve berthing and unberthing efficiency by optimizing tugboat allocation and scheduling [2]; secondly, a cost-control-oriented approach, where other studies emphasize minimizing operational costs as the objective, establishing mathematical models to balance tugboat scheduling and economic benefits [9]; and thirdly, a resource-and-environment-constraint-oriented approach, with some research attempting to incorporate real-world factors such as tidal windows, berth resources, and channel capacity into scheduling models to improve the practical feasibility of the scheduling results [10]. However, most existing research focuses on a single objective and rarely considers both carbon emission control and scheduling efficiency simultaneously. There is a significant research gap, especially in concurrently addressing tidal impacts, tugboat matching, and carbon reduction strategies.

To address the aforementioned research gap, this paper focuses on the coordinated optimization of tugboat scheduling efficiency and carbon emission control. At the model level, this paper constructs a comprehensive Mixed-Integer Linear Programming (MILP) model that, for the first time, systematically integrates multiple real-world constraints such as tugboat heterogeneity, tidal windows, multi-state carbon emission accounting, and channel capacity within a unified optimization framework. At the methodological level, considering that the NP-hard nature of the problem makes it difficult for exact algorithms to solve large-scale instances, this paper designs an improved NSGA-II algorithm aimed at efficiently generating a high-quality set of Pareto optimal solutions. Finally, at the application level, this paper designs case studies based on empirical data from a typical port to validate the effectiveness of the proposed model and algorithm, and deeply analyzes the impact of different scheduling strategies on operational efficiency and environmental costs.

The main contributions and innovations of this paper are reflected in the following three aspects:

1.

Model Innovation: A comprehensive Mixed-Integer Linear Programming (MILP) model is constructed, which integrates the dual objectives of efficiency and carbon emissions within a unified framework and comprehensively considers multiple constraints such as tugboat heterogeneity and tidal windows.

2.

Algorithmic Innovation: While existing research has established NSGA-II as an effective algorithm for similar scheduling problems, the unique operational complexities of tugboat scheduling, such as dynamic tidal windows and intricate multi-tug collaboration, pose significant challenges to standard implementations. To address this gap and the NP-hard nature of the model, this paper proposes an improved NSGA-II algorithm. By incorporating problem-specific genetic operators and adaptive mechanisms tailored to these challenges, our approach enhances both the quality of solutions and the efficiency for solving large-scale scheduling problems.

3.

Application and Management Implications: Empirical research is conducted based on real port data to deeply analyze the impact mechanism of tidal influences and carbon emission constraints on tugboat scheduling decisions, providing a scientific basis and management implications for green port operations.

The remainder of this paper is structured as follows: Section 2 reviews the relevant literature. Section 3 details the constructed mixed-integer linear programming model, including the problem description, objective functions, and constraints. Section 4 designs the improved NSGA-II algorithm for solving the model. Section 5 validates the effectiveness of the model and algorithm through a case study based on real port data and provides a deep analysis of the results. Section 6 concludes the paper and discusses future research directions.

2. Related Work

Research in port operations management has established a multi-layered collaborative framework centered on berth allocation, quay crane scheduling, truck scheduling, and vessel traffic sequencing [11,12,13]. However, as a core auxiliary resource for ensuring the safe berthing and unberthing of large vessels, research on autonomous tugboat scheduling remains relatively weak [14,15,16]. Highlighting this importance, studies from a European context, such as Paulauskas et al. [17], have emphasized the critical role of tugboats in improving navigational safety, particularly within the constrained environments of ports. Tugboat scheduling is essentially a decision-making process for service sequences, tugboat allocation, and departure times within a multi-tug, multi-vessel, and multi-task framework. It constitutes a typical multi-to-multi allocation combinatorial optimization problem, which is highly structurally isomorphic to classic NP-hard problems such as the knapsack problem, flow shop scheduling problem, and traveling salesman problem [3,18]. However, tugboat scheduling requires real-time incorporation of complex constraints such as tidal windows, horsepower classification, differences in pilot qualifications, and carbon emission accounting. These factors make it difficult to directly apply existing general models, which must be separated from the broader context of port research and subjected to specialized and refined modeling [4,19].

In recent years, research on tugboat scheduling optimization has gradually shifted from a single consideration to a simultaneous consideration of efficiency and carbon emissions, but the impact of factors such as tides and tugboat heterogeneity has not been adequately considered. For example, Yu [20] aimed to minimize the maximum completion time of tugboats. Jia et al. [3] focused on reducing vessel waiting times. The Mixed-Integer Linear Programming (MILP) model constructed by Yu [20] for the Port of Singapore only considered tugboat horsepower and operation duration constraints, completely ignoring carbon emission constraints. Lashgari et al. [21] further linearly transformed fuel consumption into carbon tax costs. Such single-dimensional methods can easily lead to an imbalance between efficiency and environmental sustainability—compressing operation times often increases carbon emissions due to high-speed empty runs, while overemphasizing low-carbon measures may prolong vessel turnaround time and reduce the overall port throughput. Li et al. [22] introduced fuel consumption into the objective function but still used empirical factors to estimate carbon emissions and did not incorporate the dynamic tidal constraints of channel capacity. Zhong et al. [2] proposed the concept of “tidal windows” but only applied it to joint berth-quay crane scheduling, treating tugboats as an infinitely available auxiliary resource, resulting in a 54 percentage point difference in tugboat utilization between high tide and low tide periods. Ren et al. [23] optimized total fuel consumption. However, existing studies have failed to consider the effects of factors such as tides and tugboat heterogeneity while simultaneously considering efficiency and carbon emissions.

In terms of solution algorithms, considering the large scale of practical scenarios, most existing literature uses heuristic algorithms represented by the multi-objective genetic algorithm NSGA-II and its various improved versions to solve the model [24]. For example, Jiang et al. [25] improved the fast non-dominated sorting to increase the convergence speed by 22%, Stanimirovic et al. [26] used linear weighting to transform a multi-objective problem into a single-objective problem, Jia et al. [3] found that the hierarchical genetic algorithm still suffered from premature convergence in large-scale instances, and Zhang et al. [27] improved NSGA-II by introducing a uniparental crossover operator to enhance the diversity of the solution set.

In summary, existing research has made some progress on the tugboat scheduling problem, but it has failed to simultaneously optimize efficiency and carbon emissions, and comprehensively consider the impacts of tidal constraints and tugboat heterogeneity, making it difficult to effectively solve large-scale problems in practical scenarios. In response, this paper proposes for the first time a “carbon-time” dual-objective framework: while minimizing the waiting time of vessels in port to ensure the level of port service, it also minimizes the total carbon emissions of tugboats to meet the IMO 2030 emission reduction targets. An MILP model is constructed, and an improved NSGA-II algorithm is proposed. The effectiveness of the model is verified through experiments, providing theoretical support and a decision-making basis for achieving green and efficient port operations.

3. Problem Definition and Optimization Model

Port tugboat scheduling is a critical aspect of port operations, and its efficiency directly impacts vessel turnaround times and the overall throughput of the port. This problem involves assigning suitable tugboats to a series of inbound and outbound vessels and planning their service times, making it a complex combinatorial optimization problem. To accurately describe and solve this problem, this paper constructs a Mixed-Integer Linear Programming (MILP) model. The model aims to achieve the dual optimization objectives of minimizing total vessel waiting time (representing operational efficiency) and total tugboat carbon emissions (representing environmental benefits). In the modeling process, this paper fully considers key operational constraints, including channel capacity limits, tidal time windows, transfer times for tugboats between different berths, and the total availability of tugboat resources. By solving this model, we aim to obtain a set of Pareto-optimal scheduling solutions that balance operational efficiency and environmental benefits.

To construct the optimization model, this paper considers the following assumptions:

1.

Tugboats have only one berthing base, and all tugboats are located at the berthing base in their initial state. In addition, the tugboat berthing base does not coincide with the vessels berths.

2.

Tugboats can travel directly from the previous service point to the next service point.

3.

The berth allocation for vessels is known and does not need to be optimized in the model.

4.

The vessel is moored in the shortest possible time.

3.1. Model Description and Symbol Definitions

This study considers the joint scheduling of a group of vessels’ arrivals and departures and the allocation of tugboats within a given scheduling horizon (total duration H) (see Figure 1). The port has tugboats of different types (indexed by $u\in U$) and levels (${\mathit{Level}}_u$), with the quantity of each type u being $M_u$. The set of vessels to be scheduled is $V=\{1,2,\dots ,n\}$, which are divided into different types (indexed by $i\in \{1,2,\dots ,I\}$). Specific individual vessels are distinguished by the index j, denoted as $V_{ij}$.

Each vessel $V_{ij}$ arrives at the anchorage at an estimated time $A_{ij}$, and its loading and unloading operation time at the berth is fixed at $S_{ij}$. One of the model’s objectives is to minimize the vessel’s waiting time, including the waiting time at the anchorage $W_{ij}^A$ and the waiting time after completing loading and unloading at the berth $W_{ij}^B$. Different types of vessels have different sensitivities to time, which are reflected by the weight parameter $w_i$, while the importance of the two waiting times in the objective function is adjusted by the coefficients ${\alpha }_W$ and ${\beta }_W$.

The inbound and outbound operations of vessels are the core of the scheduling. The inbound and outbound operations of vessel $V_{ij}$ require a minimum of $R_{ij}^{\mathrm{in}}$ and $R_{ij}^{\mathrm{out}}$ tugboats, respectively, with base durations of $T_{ij}^{\mathrm{in}}$ and $T_{ij}^{\mathrm{out}}$. Using more efficient tugboats u can effectively reduce the operation time, with the time reductions for inbound and outbound operations being $∆T_{ij}^{\mathrm{in}}\left(u\right)$ and $∆T_{ij}^{\mathrm{out}}\left(u\right)$, respectively. The core task of the model is to decide on the allocation of tugboats and the start times of the operations.

Some vessel types belong to the set $TD$ that requires tidal operations. If vessel $V_{ij}$ needs to enter the port with the tide (indicated by the binary variable ${\delta }_{ij}^1$) or leave the port with the tide (${\delta }_{ij}^2$), its operation start time must fall within the time windows $[T{W}_{ij}^1,T{W}_{ij}^2]$ and $[T{W}_{ij}^3,T{W}_{ij}^4]$, respectively.

Tugboat scheduling is closely coupled with vessel operations. The carbon emission rate per unit of time for a tugboat varies in different states, including the working state $C_{\mathrm{work},u}$, idle state $C_{\mathrm{idle},u}$, and the transfer state between different tasks $C_{\mathrm{trans},u}$. The time required for a tugboat to transfer from serving vessel $(i^{\prime },j^{\prime })$ to serving vessel $(i,j)$ is $T_{\mathrm{trans},u}^{i^{\prime }{j}^{\prime },ij}$. The model needs to calculate the total number of transfers $N{T}_{uk}$ and the total idle time $T{I}_{uk}$ for each tugboat $u_k$ to optimize carbon emissions.

The port’s channel resources are limited. The channel can accommodate a maximum of $Q^{\mathrm{ch}}$ vessels at the same time, which can be seen as a set of capacity resources L (indexed by l). The sailing times for vessel $V_{ij}$ in the channel during inbound and outbound operations are $S{T}_{ij}^{\mathrm{in}}$ and $S{T}_{ij}^{\mathrm{out}}$, respectively. To ensure safety, a minimum safety interval $t_s$ must be maintained between any two vessels passing through the same channel area. The model manages the allocation and temporal relationships of channel resources through a series of auxiliary variables. Variables $a_{ij}^{l,\mathrm{in}}$ and $a_{ij}^{l,\mathrm{out}}$ assign the channel occupancy of a vessel to a certain capacity slot l. Variables $z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{in},\mathrm{in}}$ and $z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{out},\mathrm{out}}$ are used to determine the sequence of operations for any two vessels. Furthermore, variables $b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{in}}$, $b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{out},\mathrm{out}}$, $c_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{out}}$, and $c_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{out},\mathrm{in}}$ finely describe the sequence relationship between different vessels and different sailing directions (inbound or outbound) in the same capacity slot l. In addition, the model uses M as a sufficiently large positive number for linearizing constraints.

To achieve the above scheduling objectives, the model needs to determine the following core decision variables: First, the exact start time for vessel $V_{ij}$ to enter the port, $T{S}_{ij}$, and the start time to leave the port, $D{S}_{ij}$, which are continuous variables. Second, the tugboat allocation plan needs to be specified, using binary variables $y_{ij}^{uk,\mathrm{in}}$ and $y_{ij}^{uk,\mathrm{out}}$ to decide whether tugboat $u_k$ (the k-th tugboat of type u) participates in the inbound and outbound services of vessel $V_{ij}$, respectively. To linearize the model, we further introduce the following auxiliary decision variables: $A{T}_{ij}^{\mathrm{in}}$ and $A{T}_{ij}^{\mathrm{out}}$, continuous variables representing the actual duration of the inbound and outbound operations of vessel $ij$, respectively; and $W{D}_{uk,ij}^{\mathrm{in}}$ and $W{D}_{uk,ij}^{\mathrm{out}}$, continuous variables representing the actual working duration of tugboat $uk$ when serving the inbound/outbound tasks of vessel $ij$, respectively.

3.2. Optimization Model

3.2.1. Objective Functions

Objective 1: Minimize Weighted Total Waiting Time

$$min{Z}_1=\sum _{i\in V}\sum _{j\in V_i}{w}_i·({\alpha }_W·W_{ij}^A+{\beta }_W·W_{ij}^B)$$

(1)

The objective function (1) aims to minimize the weighted total waiting time for all vessels. The vessel time weight parameter $w_i$ reflects the sensitivity of different vessels to waiting time, and the weight coefficients ${\alpha }_W$ and ${\beta }_W$ can adjust the importance of anchorage waiting versus post-berth waiting.

Objective 2: Minimize Total Tugboat Carbon Emissions

$$\begin{array}{cc}\hfill min{Z}_2=& \sum _{u\in U}\sum _{k\in U_u}\sum _{i\in V}\sum _{j\in V_i}(W{D}_{uk,ij}^{\mathrm{in}}+W{D}_{uk,ij}^{\mathrm{out}})·C_{\mathrm{work},u}\hfill \\ \hfill & +\sum _{u\in U}\sum _{k\in U_u}N{T}_{uk}·T_{\mathrm{trans},u}^{\mathrm{avg}}·C_{\mathrm{trans},u}\hfill \\ \hfill & +\sum _{u\in U}\sum _{k\in U_u}T{I}_{uk}·C_{\mathrm{idle},u}\hfill \end{array}$$

(2)

The objective function (2) aims to minimize the total carbon emissions of the tugboats, which includes three parts: emissions during the working state, transfer state, and idle state, comprehensively reflecting the environmental impact of tugboat scheduling.

3.2.2. Constraints

This section outlines the constraints that define the feasible solution space for the tugboat scheduling problem. These mathematical formulations are categorized to address distinct operational aspects.

$$A{T}_{ij}^{\mathrm{in}}=T_{ij}^{\mathrm{in}}-\sum _{u\in U}\sum _{k\in U_u}{y}_{ij}^{uk,\mathrm{in}}·\Delta T_{ij}^{\mathrm{in}}\left(u\right)$$

(3)

$$A{T}_{ij}^{\mathrm{out}}=T_{ij}^{\mathrm{out}}-\sum _{u\in U}\sum _{k\in U_u}{y}_{ij}^{uk,\mathrm{out}}·\Delta T_{ij}^{\mathrm{out}}\left(u\right)$$

(4)

$$W{D}_{uk,ij}^{\mathrm{in}}\le M·y_{ij}^{uk,\mathrm{in}}$$

(5)

$$W{D}_{uk,ij}^{\mathrm{in}}\le A{T}_{ij}^{\mathrm{in}}$$

(6)

$$W{D}_{uk,ij}^{\mathrm{in}}\ge A{T}_{ij}^{\mathrm{in}}-M·(1-y_{ij}^{uk,\mathrm{in}})$$

(7)

$$W{D}_{uk,ij}^{\mathrm{out}}\le M·y_{ij}^{uk,\mathrm{out}}$$

(8)

$$W{D}_{uk,ij}^{\mathrm{out}}\le A{T}_{ij}^{\mathrm{out}}$$

(9)

$$W{D}_{uk,ij}^{\mathrm{out}}\ge A{T}_{ij}^{\mathrm{out}}-M·(1-y_{ij}^{uk,\mathrm{out}})$$

(10)

$$D{S}_{ij}\ge T{S}_{ij}+A{T}_{ij}^{\mathrm{in}}+S_{ij}$$

(11)

$$\sum _{u\in U}\sum _{k\in U_u}{y}_{ij}^{uk,\mathrm{in}}\ge R_{ij}^{\mathrm{in}}$$

(12)

$$\sum _{u\in U}\sum _{k\in U_u}{y}_{ij}^{uk,\mathrm{out}}\ge R_{ij}^{\mathrm{out}}$$

(13)

$$\sum _{i\in V}\sum _{j\in V_i}{y}_{ij}^{uk,\mathrm{in}}\le 1$$

(14)

$$\sum _{i\in V}\sum _{j\in V_i}{y}_{ij}^{uk,\mathrm{out}}\le 1$$

(15)

$$z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{in},\mathrm{in}}+z_{i^{\prime }{j}^{\prime },ij}^{\mathrm{in},\mathrm{in}}=1$$

(16)

$$\begin{array}{c}\hfill T{S}_{i^{\prime }{j}^{\prime }}\ge T{S}_{ij}+A{T}_{ij}^{\mathrm{in}}-M(1-z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{in},\mathrm{in}})\end{array}$$

(17)

$$z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{out},\mathrm{out}}+z_{i^{\prime }{j}^{\prime },ij}^{\mathrm{out},\mathrm{out}}=1$$

(18)

$$\begin{array}{c}\hfill D{S}_{i^{\prime }{j}^{\prime }}\ge D{S}_{ij}+A{T}_{ij}^{\mathrm{out}}-M(1-z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{out},\mathrm{out}})\end{array}$$

(19)

$$\sum _{i\in V}\sum _{j\in V_i}{y}_{ij}^{uk,\mathrm{in}}·(1-z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{in},\mathrm{in}})\le 1$$

(20)

$$\sum _{i\in V}\sum _{j\in V_i}{y}_{ij}^{uk,\mathrm{out}}·(1-z_{ij,i^{\prime }{j}^{\prime }}^{\mathrm{out},\mathrm{out}})\le 1$$

(21)

$$\sum _{l\in L}{a}_{ij}^{l,\mathrm{in}}=1$$

(22)

$$\sum _{l\in L}{a}_{ij}^{l,\mathrm{out}}=1$$

(23)

$$b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{in}}+b_{i^{\prime }{j}^{\prime },ij}^{l,\mathrm{in},\mathrm{in}}\ge a_{ij}^{l,\mathrm{in}}+a_{i^{\prime }{j}^{\prime }}^{l,\mathrm{in}}-1$$

(24)

$$T{S}_{i^{\prime }{j}^{\prime }}\ge T{S}_{ij}+S{T}_{ij}^{\mathrm{in}}+t_s-M(1-b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{in}})$$

(25)

$$b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{out},\mathrm{out}}+b_{i^{\prime }{j}^{\prime },ij}^{l,\mathrm{out},\mathrm{out}}\ge a_{ij}^{l,\mathrm{out}}+a_{i^{\prime }{j}^{\prime }}^{l,\mathrm{out}}-1$$

(26)

$$D{S}_{i^{\prime }{j}^{\prime }}\ge D{S}_{ij}+S{T}_{ij}^{\mathrm{out}}+t_s-M(1-b_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{out},\mathrm{out}})$$

(27)

$$c_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{out}}+c_{i^{\prime }{j}^{\prime },ij}^{l,\mathrm{out},\mathrm{in}}\ge a_{ij}^{l,\mathrm{in}}+a_{i^{\prime }{j}^{\prime }}^{l,\mathrm{out}}-1$$

(28)

$$D{S}_{i^{\prime }{j}^{\prime }}\ge T{S}_{ij}+S{T}_{ij}^{\mathrm{in}}+t_s-M(1-c_{ij,i^{\prime }{j}^{\prime }}^{l,\mathrm{in},\mathrm{out}})$$

(29)

$$T{S}_{ij}\ge D{S}_{i^{\prime }{j}^{\prime }}+S{T}_{i^{\prime }{j}^{\prime }}^{\mathrm{out}}+t_s-M(1-c_{i^{\prime }{j}^{\prime },ij}^{l,\mathrm{out},\mathrm{in}})$$

(30)

$${\delta }_{ij}^1·T{W}_{ij}^1\le T{S}_{ij}\le {\delta }_{ij}^1·T{W}_{ij}^2+(1-{\delta }_{ij}^1)M$$

(31)

$$\begin{array}{c}\hfill {\delta }_{ij}^1·T{W}_{ij}^1\le T{S}_{ij}+A{T}_{ij}^{\mathrm{in}}\le {\delta }_{ij}^1·T{W}_{ij}^2+(1-{\delta }_{ij}^1)M\end{array}$$

(32)

$${\delta }_{ij}^2·T{W}_{ij}^3\le D{S}_{ij}\le {\delta }_{ij}^2·T{W}_{ij}^4+(1-{\delta }_{ij}^2)M$$

(33)

$$\begin{array}{c}\hfill {\delta }_{ij}^2·T{W}_{ij}^3\le D{S}_{ij}+A{T}_{ij}^{\mathrm{out}}\le {\delta }_{ij}^2·T{W}_{ij}^4+(1-{\delta }_{ij}^2)M\end{array}$$

(34)

$$W_{ij}^A=T{S}_{ij}-A_{ij}$$

(35)

$$W_{ij}^B\ge D{S}_{ij}-(T{S}_{ij}+A{T}_{ij}^{\mathrm{in}}+S_{ij})$$

(36)

$$T{S}_{ij}\ge A_{ij}$$

(37)

$$N{T}_{uk}\ge \sum _{i\in V}\sum _{j\in V_i}(y_{ij}^{uk,\mathrm{in}}+y_{ij}^{uk,\mathrm{out}})-1$$

(38)

$$\begin{array}{c}\hfill T{I}_{uk}\ge H-{\displaystyle \sum _{i\in V}}{\displaystyle \sum _{j\in V_i}}(W{D}_{uk,ij}^{\mathrm{in}}+W{D}_{uk,ij}^{\mathrm{out}})-N{T}_{uk}·T_{\mathrm{trans},u}^{\mathrm{avg}}\end{array}$$

(39)

Constraint (11) ensures that a vessel can only begin its outbound operation after completing its inbound operation and loading/unloading. This constraint considers the time-shortening effect of large tugboats. Constraint (12) ensures that the number of tugboats assigned to vessel $V_{ij}$ for its inbound phase meets the minimum required total. Constraint (13) ensures that the number of tugboats assigned to vessel $V_{ij}$ for its outbound phase meets the minimum required total. Constraints (14) and (15) ensure that each tugboat can only participate in the operation of one vessel at a time. Constraints (16)–(19) introduce binary variables to represent the relative order of operations, rather than enforcing a fixed sequence, allowing operations at different berths to proceed in parallel, maintaining the effectiveness of tugboat resource constraints, and providing greater flexibility for scheduling optimization. Constraints (20) and (21) ensure that a single tugboat is not assigned to multiple vessels during the same time period. Constraints (22)–(30) are to ensure that the total number of vessels navigating the channel at any given time does not exceed the preset capacity, in order to maintain safe intervals. Constraints (31) and (32) ensure that vessels requiring tidal assistance for inbound operations complete their inbound journey within the specified tidal window. Constraints (33) and (34) ensure that vessels requiring tidal assistance for outbound operations complete their outbound journey within the specified tidal window. Constraint (35) defines the waiting time of a vessel at the anchorage. Constraint (36) defines the waiting time of a vessel at the berth, considering the time-shortening effect of large tugboats. Constraint (37) ensures that a vessel can only begin its inbound journey after arriving at the anchorage. Constraint (38) calculates the number of transfers for tugboat $u_k$ based on the number of vessels it services. Constraint (39) calculates the idle time for tugboat $u_k$, which is the total time minus the working and transfer times.

4. Design of the Improved NSGA-II Algorithm

The collaborative tugboat scheduling problem is a typical NP-hard problem. Through the linearization reconstruction described above, the model proposed in this paper is a Mixed-Integer Linear Programming (MILP) model. Theoretically, for small to medium-sized instances, commercial solvers (such as CPLEX, Gurobi) can be used to obtain the exact solutions of the Pareto optimal front. However, considering the large scale of port scheduling problems in the real world, exact algorithms often face the curse of dimensionality and cannot find solutions within a reasonable time. Therefore, this paper adopts the Pareto-optimality-based multi-objective genetic algorithm NSGA-II (Non-dominated Sorting Genetic Algorithm II) to solve the problem, and improves and optimizes the NSGA-II framework for the characteristics of the collaborative tugboat scheduling problem to achieve an efficient search for the Pareto optimal solution set.

4.1. Core Algorithm Architecture

The algorithm adopts a phased iterative optimization strategy. Each evolutionary cycle includes the following key steps, and the complete algorithm flowchart is shown in Figure 2. The algorithm is based on the NSGA-II framework, generating initial solutions through intelligent population initialization (integrating four strategies: FCFS, cost-priority, tidal urgency, and a hybrid heuristic), and takes minimizing total vessel waiting time and minimizing total tugboat carbon emissions as dual optimization objectives. The core process includes constructing the Pareto front through non-dominated sorting, maintaining population diversity using crowding distance, an improved binary tournament selection mechanism, genetic operations such as tide-aware crossover and adaptive mutation, and local search improvements for tidal window matching, tugboat resource optimization, and channel conflict resolution. The algorithm detects stagnation and diversity crises through a convergence monitoring mechanism. When the restart conditions are met, a restart mechanism is triggered to preserve elite individuals and reset parameters. Finally, an elitism strategy ensures the preservation of high-quality solutions.

4.2. Chromosome Structure

When applying the mathematical model constructed in this paper to an evolutionary algorithm, several challenges must be addressed. Firstly, the model includes integer, continuous real, and binary variables, resulting in diverse decision variable types that are difficult to handle uniformly with standard encoding methods. Secondly, there are complex logical and temporal coupling relationships among the decision variables, particularly in the three core decision dimensions: vessel service order, tugboat class selection, and service time optimization. To address this, this paper designs a three-segment hybrid encoding chromosome structure that can accurately represent decision variables and adapt to subsequent genetic operations.

4.2.1. Overall Chromosome Structure

To address the characteristics of the collaborative scheduling problem of tugboats and pilots, a three-segment hybrid encoding chromosome structure is designed. The total length of the chromosome is $5n$ genes, where n is the number of vessels to be scheduled. The three-segment chromosome adopts a hybrid encoding strategy, and the specific configuration of each substring is shown in Figure 3.

The first substring of the chromosome is the vessel sequencing substring, with gene positions from 1 to n, using permutation encoding to represent the service priority order of the vessels. Each gene stores a permutation value of a vessel index, and by sorting this substring, a unique vessel service sequence can be determined. For example, for 3 vessels, a possible sequence is $[2,1,3]$, which means the scheduling system will service vessel 2, vessel 1, and vessel 3 in order.

The second substring is the tugboat class selection substring, located at gene positions $n+1$ to $3n$, using integer encoding. The length of this substring is $2n$, where the first n genes encode the preferred tugboat class for each vessel’s inbound service, and the next n genes correspond to the class preference for their outbound service. The integer value of the gene represents the tugboat class, for example, 1 can represent the lowest class, and a larger value indicates a higher class. As an example, in a scenario with two vessels and three tugboat classes, a possible substring $[1,2,3,2]$ means: vessel 1 prefers class 1 and class 2 tugboats for inbound and outbound services, respectively, while vessel 2 prefers class 3 and class 2 tugboats for inbound and outbound services, respectively.

The third substring is the time optimization strategy substring, with gene positions distributed from $3n+1$ to $5n$, using real-number encoding. Each gene in this substring is a continuous value in the range $[0.0,1.0]$, corresponding to the time optimization coefficient for a vessel, used to control the application intensity of the time reduction strategy based on tugboat-vessel matching. The core mechanism of this strategy is to shorten the standard service time by assigning higher-class or better-matched tugboats to vessels, leveraging their performance advantages. During the decoding phase, the system will retrieve the corresponding time reduction value from a time reduction database based on the vessel type and the assigned tugboat type to calculate the actual service time, which is the standard service time minus the maximum time reduction value among all assigned tugboats. The mechanism of this substring gene value is as follows: 0.0 means forcing the use of the lowest-class tugboat without applying any time reduction; 1.0 means prioritizing the selection of the tugboat combination that provides the maximum time reduction; values in between are used to proportionally adjust the tugboat selection preference. In this way, this gene substring can effectively optimize tugboat allocation, significantly reduce vessel waiting time while ensuring operational safety, and thereby improve the overall throughput efficiency of the port.

4.2.2. Encoding and Decoding Mechanism

For a scheduling problem involving n vessels, the chromosome encoding process follows these steps: First, a random permutation of $\{0,1,\dots ,n-1\}$ is generated to complete the vessel sequence encoding. This permutation directly determines the service priority of each vessel through its sort order. Second, in the tugboat class encoding stage, the system randomly assigns a class l from the set of available classes $\{L_{\mathit{min}},\dots ,L_{\mathit{max}}\}$ based on each vessel’s tonnage and service requirements. Finally, to achieve time optimization encoding, the system generates a random optimization coefficient $\alpha$ in the range $[0,1]$ for each service, which will be used to control the reduction ratio of the standard service time.

The chromosome decoding process is completed by a scheduling simulator, which translates the abstract gene encoding into a concrete and feasible schedule, as detailed in

Table 1. Chromosome decoding workflow.

Input: Chromosome C

Output: Feasible schedule S

Initialization

Extract ship permutation P, tug preferences L, and optimization factors O from chromosome C.

Initialize empty schedule S.

For each vessel $v_i$ in permutation P:

For each service $j\in \left\{\mathrm{inbound},\mathrm{outbound}\right\}$:

Assign tugboats based on preferences $L(v_i,j)$ and current availability.

Calculate actual service time $t_{\mathrm{actual}}$ using standard time and optimization factor $O(v_i,j)$.

Determine optimal start time $t_{\mathrm{start}}$ by:

- Identifying valid time windows for Tidal, Berth, and Channel constraints.

- Computing the intersection of these windows to find feasible time slots.

- Selecting the time from feasible slots that minimizes vessel waiting time.

Update schedule S with new service details $(v_i,j,t_{\mathrm{start}},t_{\mathrm{actual}})$.

Finalization

return complete schedule S.

. The decoder first extracts the vessel service sequence, tugboat class preferences, and time optimization coefficients from the chromosome. It then iterates through each vessel service in the sequence and performs a four-stage decoding process. This process involves, first, Tugboat Allocation, which assigns suitable tugboats to the current service based on the preferences defined in the genes and real-time tugboat availability; second, Service Time Calculation, which calculates the actual service duration by combining the standard service time with the time optimization coefficient; third, Optimal Start Time Determination, which identifies valid time windows under key constraints such as tides, berths, and channels, computes the intersection of these windows to form a feasible time domain, and selects the optimal moment from this domain that minimizes vessel waiting time to be the service start time; and finally, Schedule Update, which adds the newly generated service arrangement to the schedule and updates resource statuses. By iteratively processing all services, a complete, conflict-free schedule is finally generated.

4.3. Initial Solution Generation

The final convergence performance of an evolutionary algorithm largely depends on the quality and diversity of the initial population. A completely random initialization strategy can ensure population diversity, but its blindness often leads to slow convergence, especially in a solution space with complex constraints. To solve this problem, this study adopts a hybrid initialization strategy that combines domain knowledge with random exploration, generating high-quality initial solutions through four complementary heuristic rules, laying a good foundation for the algorithm’s evolutionary process.

4.3.1. Heuristic Construction Strategy Framework

The initial population construction is based on two complementary heuristic strategies, each designed for different optimization dimensions of the problem, achieving a comprehensive improvement in population quality through systematic combination. These strategies fully consider the multiple constraints and optimization objectives of the tugboat scheduling problem, exploring high-quality initial solution spaces from different perspectives while ensuring solution feasibility. To enhance the adaptability of the strategies, this study also introduces an adaptive weight adjustment mechanism, allowing the algorithm to automatically adjust optimization strategies based on the characteristics of the specific problem, improving the pertinence and effectiveness of the solution process.

1.

First-Come, First-Served (FCFS) Construction Strategy

The FCFS construction strategy follows the natural order of vessel arrival times, reflecting the principle of fairness in port scheduling. The core idea of this strategy is to build a strict time sequence according to the estimated time of arrival (ETA) of the vessels, ensuring that vessels arriving earlier receive service first. Mathematically, this sorting criterion can be expressed as ${\pi }_{\mathrm{FCFS}}:\mathrm{sort}(V,\mathrm{key}={\eta }_i^{\mathrm{ETA}}),\forall i\in V$, where V is the set of vessels to be scheduled, and ${\eta }_i^{\mathrm{ETA}}$ is the ETA of vessel i.

In the tugboat allocation phase, this strategy uses a nearest-neighbor-first principle, assigning the nearest available tugboat to each vessel to minimize the tugboat’s empty travel distance and improve resource utilization efficiency. Specifically, for each vessel i, its optimal tugboat assignment can be obtained by solving ${\tau }_{ij}^{*}=arg{min}_{j\in T}{d}_{ij}$, where T is the set of available tugboats, $d_{ij}$ is the distance between vessel i and tugboat j, while satisfying the tugboat capability constraint ${\mathrm{capability}}_j\ge {\mathrm{requirement}}_i$. In addition, considering the impact of tidal windows on vessel arrivals and departures, the strategy includes a tidal adjustment mechanism. It calculates the match between the vessel’s service time and various tidal windows and selects the scheme with the minimum time adjustment, i.e., $∆t_i={min}_{w\in W}|t_i-t_w^{\mathrm{optimal}}|$, where W is the set of tidal windows and $t_w^{\mathrm{optimal}}$ is the optimal service time within window w.

2.

Tidal Urgency Construction Strategy

The tidal urgency construction strategy is specifically designed for the strictness of tidal constraints, prioritizing vessels that are most severely restricted by tidal windows to reduce the risk of tidal constraint violations. The core of this strategy is to establish a tidal urgency index $U_i^{\mathrm{tidal}}={\displaystyle \frac{1}{{min}_{w\in W}|t_i-t_w^{\mathrm{start}}|+ϵ}}$, where $t_i$ is the estimated service time for vessel i, $t_w^{\mathrm{start}}$ is the start time of tidal window w, and $ϵ$ is a very small positive number to avoid division by zero. A higher value of this index indicates that the vessel is more severely affected by tidal constraints and should be scheduled with priority.

During the tugboat allocation process, this strategy employs a tide-aware allocation mechanism that considers not only the distance between the tugboat and the vessel but also incorporates a tidal penalty into the decision model. The specific allocation policy is ${\tau }_{ij}^{\mathrm{tidal}}=arg{min}_{j\in T}(d_{ij}+\lambda ·\mathrm{tide}\text{\_}{\mathrm{penalty}}_{ij})$, where $\lambda$ is the tidal penalty coefficient used to balance distance cost and the risk of tidal violation, and $\mathrm{tide}\text{\_}{\mathrm{penalty}}_{ij}$ quantifies the potential tidal delay when tugboat j services vessel i. Furthermore, this strategy has dynamic adjustment capabilities, continuously updating the schedule using a rolling horizon optimization method based on real-time tidal forecast information to improve the system’s adaptability to tidal changes.

To ensure a uniform distribution of the initial population in the solution space and maintain an appropriate level of diversity, this paper designs a population distribution control mechanism that guarantees the quality of the initial population while preserving its diversity, laying a solid foundation for subsequent evolutionary search.

In terms of individual composition, the algorithm uses an empirical proportional allocation method to construct the initial population. For example, individuals generated by the FCFS, tidal urgency, and hybrid heuristic construction strategies can be allocated to the initial population at a ratio of 50%, 30%, and 20%, respectively, to achieve a balanced optimization by integrating multiple optimization criteria. In addition, to further enhance the algorithm’s global exploration capability and avoid getting trapped in local optima, a small number of randomly initialized individuals can be inserted during the iteration process.

Furthermore, during the construction of initial solution individuals, to enhance the algorithm’s exploratory power, this paper introduces a multi-level diversity enhancement mechanism to expand the search scope of the solution space from different dimensions. First, a slight perturbation is applied to all heuristically constructed individuals except the first one. For example, the perturbation process can follow the pattern $\mathrm{perturbation}\left(x\right)=x+ϵ$, where $ϵ\sim \mathcal{N}(0,{\sigma }^2)$ is a random perturbation following a normal distribution with an intensity parameter $\sigma =0.1$. Second, based on the perturbation, the algorithm further generates multiple local variants for each heuristically constructed individual. Priority adjustment variants explore local improvements by swapping adjacent vessels while keeping most of the scheduling structure unchanged, generating $k=3$ such variants for each individual. Local optimization variants perform local reordering within a neighborhood of size $\delta =2$ to find a better vessel service sequence. Third, tugboat reallocation variants keep the vessel service sequence unchanged and only optimize the tugboat allocation scheme, exploring different resource configuration possibilities.

4.3.2. Initial Solution Quality Assessment and Control Mechanism

Although the aforementioned hybrid initialization strategy aims to generate a high-quality and diverse initial population, the combination of heuristic rules and random perturbations may still produce infeasible solutions that do not satisfy complex constraints, or redundant individuals that are too similar in the solution space. To address this, to ensure the effectiveness of the population during the evolutionary process, this paper designs a quality control system that includes feasibility verification, quality assessment, and redundancy elimination. Specifically, this system involves three stages: first, a comprehensive constraint feasibility check is performed on chromosome individuals, including tugboat capability constraints, tidal window constraints, and safe separation intervals between vessels, with a penalty value set for individuals that do not meet the constraints. Second, on this basis, a quality assessment function quantitatively evaluates each individual, comprehensively considering its cost and degree of constraint violation. Third, to avoid having too many similar individuals in the population, a redundancy elimination mechanism is introduced, which screens individuals whose quality similarity exceeds a given threshold, retaining only the one with higher quality, thereby ensuring the quality and diversity of the initial population.

4.4. Main Genetic Steps

4.4.1. Selection Operation

For an individual i in the Pareto front $F_l$, its crowding distance $c_d\left(i\right)$ is defined as:

$$c_d\left(i\right)=\sum _{m=1}^2{\displaystyle \frac{f_m^{i+1}-f_m^{i-1}}{f_m^{max}-f_m^{min}}}$$

(40)

where $f_m^{max}$ and $f_m^{min}$ are the maximum and minimum values of the m-th objective function in the population, respectively, and $f_m^{i+1}$ and $f_m^{i-1}$ are the objective function values of the individuals adjacent to individual i in the sorted objective space.

To further enhance the distribution of the solution set on the Pareto front and avoid solution clustering, the algorithm introduces a niching technique based on a clearing mechanism. The core of this technique is the adaptive adjustment of the niche radius $r_{\mathrm{niche}}$, which can change dynamically based on the distribution density of the current Pareto front. Its calculation formula is

$$r_{\mathrm{niche}}=\alpha ·{\displaystyle \frac{\sqrt{{(Z_1^{max}-Z_1^{min})}^2+{(Z_2^{max}-Z_2^{min})}^2}}{\sqrt{|F_1|}}}$$

(41)

where $\alpha$ is the niche radius factor and $|F_1|$ is the size of the Pareto front.

The selection operator of this algorithm adopts an improved binary tournament mechanism based on Pareto dominance and crowding distance. In each selection round, the algorithm randomly selects two individuals from the current population and compares their non-dominated sorting ranks. The individual with the lower rank (i.e., on a better Pareto front) wins. If the two individuals have the same rank, the one with the larger crowding distance is selected (indicating that the solutions around this individual are sparser), to maintain population diversity. For infeasible individuals, the algorithm uses an adaptive penalty function to adjust their fitness values, putting them at a disadvantage in the competition, thereby ensuring that feasible solutions have a higher competitive advantage during the selection process.

In constrained optimization problems, the traditional static penalty function method is highly sensitive to the setting of the penalty coefficient. A fixed coefficient value is unlikely to maintain optimal constraint handling performance throughout all stages of the algorithm’s evolution. To solve this problem, this paper designs a penalty function mechanism that adaptively adjusts based on the population state to achieve a dynamic balance between constraint handling and objective optimization. The core of this mechanism is the adaptive adjustment of the penalty coefficient. Its penalty coefficient $\lambda \left(t\right)$ is dynamically updated according to the proportion of feasible solutions in the current population, with the formula

$$\lambda \left(t\right)={\lambda }_0·(1+\beta ·{\displaystyle \frac{P_{\mathrm{feasible}}^{\mathrm{target}}-P_{\mathrm{feasible}}^{\mathrm{actual}}\left(t\right)}{P_{\mathrm{feasible}}^{\mathrm{target}}}})$$

(42)

where ${\lambda }_0$ is the initial penalty coefficient, and $P_{\mathrm{feasible}}^{\mathrm{target}}$ and $P_{\mathrm{feasible}}^{\mathrm{actual}}\left(t\right)$ are the target and current generation’s proportion of feasible solutions, respectively. To handle the problem of search space contraction that may be caused by over-constraint, the algorithm also introduces a constraint relaxation mechanism. This mechanism enhances the algorithm’s exploration capability by dynamically adjusting the priority weights of different types of constraints and transforming some non-critical constraints (such as tugboat usage preferences) into soft constraints, while ensuring that critical constraints are met.

4.4.2. Crossover Operation

Through the selection operator, the algorithm screens out parent individuals with higher fitness. The core task of the crossover operation is to explore higher-quality solutions by recombining the chromosomes of the parents while preserving these excellent gene segments. However, traditional crossover operators are very prone to producing infeasible offspring when dealing with the complex constraints of this problem. For this reason, this paper designs a set of problem-specific crossover operators based on the “identify–preserve–repair” model, which preferentially preserves the excellent gene segments from the parents and ensures the feasibility of the offspring through a dedicated repair mechanism.

The crossover operators considered in this paper include a tide-aware crossover operator, a tugboat optimization crossover operator, and a channel conflict resolution crossover operator. The principle of the tide-aware crossover operator is detailed below as an example. This operator is specifically designed for the critical hard constraint of tidal time windows. Its core principle is as follows: first, the operator identifies a set of “critical vessels” $C_{\mathrm{critical}}$ by analyzing the proximity of each vessel’s service time to the tidal window boundaries in the parent individuals, where the service time is less than a preset threshold (e.g., 1 h) from the boundary. Then, during the crossover process, a protective strategy is adopted for the gene positions corresponding to the critical vessels, preferentially inheriting the gene values with better tidal adaptability (i.e., service times that fall more safely within the window) from the parents. For non-critical vessels, standard crossover operations (such as uniform crossover or single-point crossover) are used to enhance exploration.

After the crossover operation is completed, the algorithm immediately performs a tidal constraint check on the newly generated offspring. If any service time is found to deviate from a valid tidal window, a local adjustment repair mechanism is initiated. It repairs the deviation by shifting the service time to the nearest valid tidal window boundary, thereby ensuring the feasibility of the offspring solution (see Figure 4).

Similarly, this paper also designs two other operators that follow the same “identify–preserve–repair” model:

• Tugboat Optimization Crossover Operator: This operator identifies and preferentially preserves efficient tugboat allocation patterns associated with large vessels or complex operations from the parents using a tugboat-vessel matching matrix. Its repair mechanism resolves resource conflicts that may arise after crossover by adjusting time windows or reassigning tugboats.
• Channel Conflict Resolution Crossover Operator: This operator focuses on identifying pairs of vessels in the parents that may have spatio-temporal channel conflicts and preserves conflict-free sequence segments by adjusting the service order substring during crossover. Its repair mechanism operates through an iterative time adjustment procedure: after detecting a conflict, it systematically postpones the service start time of the second vessel in the conflicting pair until the minimum safe separation is met. The procedure continues to check and postpone all subsequent affected services while re-validating them against other constraints such as tidal windows.

To dynamically leverage the advantages of different crossover operators, this paper uses a roulette wheel selection mechanism to integrate these three crossover operators. The selection probability of each operator is dynamically adjusted based on its success rate in producing high-quality offspring during the historical evolution process. Specifically, the selection probability of an operator $P_{\mathrm{operator}}$ is determined by the ratio of its success rate $\mathit{success}\text{\_}{\mathit{rate}}_{\mathrm{operator}}$ to the total success rate of all operators, i.e.,

$$P_{\mathrm{operator}}={\displaystyle \frac{\mathit{success}\text{\_}{\mathit{rate}}_{\mathrm{operator}}}{{\sum }_k\mathit{success}\text{\_}{\mathit{rate}}_k}}$$

(43)

This adaptive mechanism allows the algorithm to more frequently select the better-performing crossover operators based on the current search state.

4.4.3. Mutation Operation

The crossover operation focuses on exploring the solution space by recombining existing excellent genes, while the mutation operation is the key to maintaining population diversity and enabling the algorithm to escape local optima. To strike a balance between exploration and exploitation, this paper adopts an adaptive mutation strategy based on population diversity monitoring and designs problem-oriented mutation operators.

• Tugboat Optimization Mutation Operator
  This operator focuses on the local optimization of tugboat allocation schemes, improving resource utilization efficiency by fine-tuning tugboat class selections. For the tugboat class selection genes in the chromosome, mutation occurs with a certain probability. The new mutated class $l_i^{\prime }$ is randomly selected from the set of available classes but must satisfy the condition that its difference from the original class $l_i$ does not exceed 1, i.e., $|l_i^{\prime }-l_i|\le 1$, to achieve fine-grained search. After mutation, the balance of tugboat resource allocation is re-checked, and resource balance is achieved by fine-tuning the tugboat classes of adjacent vessels, ensuring that the mutated individual can improve overall scheduling efficiency.
• Channel Conflict Resolution Mutation Operator
  This operator actively resolves potential channel conflicts by adjusting the vessel service order and time optimization strategy. It first identifies pairs of vessels at risk of conflict based on their service time windows and channel capacity constraints. Then, it swaps the positions of the conflicting vessels in the vessel sequencing substring, with the probability of swapping being proportional to the severity of the conflict. At the same time, it adjusts the corresponding genes in the time optimization strategy substring by increasing or decreasing the service time offset to further resolve the conflict. After mutation is complete, a comprehensive constraint check is performed on the new individual to ensure it satisfies all hard constraints.

To further enhance the algorithm’s global search capability, this paper designs an adaptive mutation rate mechanism. When it is detected that the algorithm is trapped in a local optimum or the population diversity is severely insufficient (e.g., no improvement in the optimal objective value for 50 consecutive generations, or the diversity index is below a critical threshold of 0.05), an emergency mutation mechanism is triggered. Under this mechanism, the mutation probability is significantly increased to 0.5, and the mutation scope is expanded to 30% of the entire chromosome, thereby achieving a drastic perturbation of the solution space. To protect high-quality solutions from being destroyed, this mechanism is only applied to non-elite individuals; elite individuals on the current Pareto front are preserved. After the emergency mutation is executed, the algorithm’s stagnation counter is reset, and it returns to the standard evolutionary process.

5. Case Study

5.1. Scenario Setting

To validate the effectiveness of the improved NSGA-II algorithm proposed in this paper, a case study of a port tugboat operation scheduling scenario was constructed based on actual data from Xiamen Port, and a series of computational experiments were conducted on this basis. The case study constructed in this research aims to simulate complex scheduling operations in a typical medium-sized port over a 36 h planning period. Spatially, the port is set up as a unified port area containing a single anchorage, a shared channel, and multiple berths, where the channel can accommodate a maximum of four vessels operating simultaneously at any given time.

In terms of task scale, a total of 23 vessels, including bulk carriers, container ships, and tankers, need to be scheduled within the planning period. The inbound and outbound operations of some vessels are strictly limited by tidal times. To handle these tasks, the port is equipped with a total of 31 tugboats, divided into three different classes. The service relationship between vessels and tugboats follows a class-based compatibility and substitution principle, meaning that tugboats of a higher class than the minimum requirement can be used to improve operational efficiency. The minimum class requirements and the number of tugboats needed for various types of vessels are detailed in

Table 2. Vessel tug service requirements.

Vessel Type	Minimum Tugboat Level	Inbound Tug Demand	Outbound Tug Demand
Bulk Carrier	Type 1	1–3 units	1–2 units
Container Ship	Type 1	2–4 units	1–2 units
Tanker	Type 1	2–3 units	1–3 units

.

To more accurately simulate the real operating environment, tugboats of different classes have significant performance differences, mainly in terms of operational efficiency and carbon emissions. The detailed performance parameters of the tugboats, including the quantity of each class and their carbon emission rates in working, shifting, and idle states, are listed in

Table 3. Tugboat performance and emission parameters.

Parameter	Unit	Type 1 Tug	Type 2 Tug	Type 3 Tug
Quantity	units	12	11	8
Carbon Emission Rate
Working State	kg CO2/hour	40.0	60.0	80.0
Shifting State	kg CO2/hour	25.0	35.0	45.0
Idle State	kg CO2/hour	8.0	12.0	16.0

. In terms of operational efficiency, higher-class tugboats can significantly shorten service times, with the specific average service durations shown in

Table 4. Average inbound/outbound service time by tug type (hours).

Vessel Type	Type 1 Tug	Type 2 Tug	Type 3 Tug
Bulk Carrier	1.86/ 1.49	1.55/ 1.24	infeasible
Container Ship	infeasible	1.78/ 1.43	1.43/ 1.14
Tanker	2.13/ 1.70	1.77/ 1.42	1.42/ 1.13

infeasible: This tug-vessel combination is not applicable due to operational constraints.

. It is important to note that these service times have been specifically calibrated to represent a high-congestion scenario, creating a challenging test case to rigorously evaluate the algorithm’s performance under pressure. This case study integrates multiple challenges, including limited resources, diverse tasks, and complex constraints, aiming to comprehensively test the overall performance of the algorithm.

5.2. Algorithm Performance Comparison Experiment

To verify the effectiveness of the improved NSGA-II algorithm proposed in this paper, this section selects the standard NSGA-II algorithm and the exact solver CPLEX as benchmarks for performance comparison. CPLEX is used to solve the MILP model constructed in this paper, aiming to provide an optimal solution reference for small-scale instances. All algorithms were run in the same hardware environment (CPLEX 12.8, IBM Corporation, Armonk, NY, USA and AMD Ryzen 5 4600H processor, Advanced Micro Devices, Santa Clara, CA, USA), with the maximum running time for both CPLEX and the heuristic algorithms set to 3600 s.

Analyzing the results in

Table 5. Performance comparison of different algorithms.

No.	Scenario Parameters			Weighted-Sum Objective (Lower Is Better)				Running Time (s)
	V	T	H	CPLEX	Std	Imp	GAP (%)	CPLEX	Std	Imp
S1	5	8	12	10,250	10,380	10,375	0.05	5.6	748.75	421.37
S2	8	12	24	16,120	16,450	16,380	0.43	25.8	1756.29	576.12
S3	10	15	24	19,850	20,320	20,150	0.84	89.3	2353.53	715.03
M1	15	20	36	29,450	30,180	29,680	1.66	455.2	2315.63	1023.71
M2	20	25	36	38,920	40,250	39,350	2.24	1873.1	2269.28	1194.48
M3	25	30	48	49,200	48,900	47,250	-	3600 *	3547.92	1442.64
L1	35	40	48	72,450	68,350	63,890	-	3600 *	3605.72	2189.82
L2	50	55	72	128,250	96,480	86,320	-	3600 *	3604.66	3533.45
L3	80	70	72	-	174,200	134,850	-	3600 *	3616.19	3606.43

Note: V = Vessels; T = Tugs; H = Horizon (h); Std = Standard NSGA-II; Imp = Improved NSGA-II; * = time limit reached.

, several conclusions can be drawn. First, CPLEX demonstrated its superiority in solving small to medium-sized scenarios (S1-M2), but as the scale of the scenario increased, its performance bottleneck quickly emerged. Starting from instance M3, CPLEX reached the 3600 s computation time limit. In instances L1 and L2, the quality of the solutions it found within the specified time was already far inferior to that of the heuristic algorithms. In the largest instance, L3, CPLEX could not even find a feasible solution within the time limit. This fully exposes the NP-hard nature of the problem and highlights the necessity of developing specialized heuristic algorithms for large-scale real-world scenarios. Second, the standard NSGA-II algorithm performed reasonably well on small-scale problems, but its performance declined significantly as the problem size and complexity increased. Especially in large-scale instances (L1–L3), the quality of the solutions it found deteriorated sharply, indicating that standard genetic operators are difficult to effectively explore the vast and highly constrained solution space, easily getting trapped in low-quality local optima. Third, the improved NSGA-II algorithm proposed in this paper can find near-optimal solutions for small-scale problems. On large-scale problems where both the standard NSGA-II and CPLEX fail, it can still consistently obtain high-quality solutions, and the performance gap with the standard NSGA-II widens rapidly. For example, in instance L1, the solution quality of the improved algorithm is about 6.52% better than the standard algorithm, and this improvement reaches 10.53% in instance L2. These results provide strong evidence for the effectiveness of the improvement strategies designed in this paper (such as problem-specific genetic operators and adaptive mechanisms), ensuring the reliability and superiority of the algorithm in complex real-world scenarios.

To evaluate the convergence performance of the algorithm, the average values of the two objective functions (weighted waiting time, total carbon emissions) were tracked over 200 generations of evolution, and the results are shown in Figure 5. In the early stages of evolution (approximately the first 50 generations), both curves show rapid decline and sharp fluctuations, indicating that the algorithm is conducting a global search over a wide range of the solution space. As the evolution progresses, the curves gradually stabilize, signaling that the algorithm has successfully transitioned from global exploration to local exploitation of high-quality solution regions. Particularly after about 150 generations, the fluctuation of the objective values significantly decreases, eventually stabilizing within an optimal range. This process clearly demonstrates that the proposed algorithm has good convergence properties. More importantly, the results show that the scheduling fault-tolerance mechanism introduced to cope with scheduling failures did not undermine the convergence characteristics of the improved NSGA-II algorithm. This mechanism ensures that if any service cannot be scheduled at its ideal time due to resource conflicts, it is not discarded by the system but is automatically postponed to the earliest future time point that satisfies all hard constraints. This “guarantee scheduling, penalize delay” design provides a solid foundation for the stable convergence of the algorithm by ensuring the integrity of every solution.

Figure 6 shows the final Pareto Front obtained by the algorithm at the 200th generation. Each point on this front represents a non-dominated scheduling solution, and together they form the optimal solution set for the problem. The front clearly reveals the typical conflicting relationship between the two objectives of total waiting time and total carbon emissions, meaning that an improvement in one objective comes at the cost of the other, which is consistent with the fundamental principles of multi-objective optimization. For example, solutions in the lower-left part of the front achieve higher scheduling efficiency (lower waiting time) at the expense of some environmental benefits, while solutions in the upper-right part do the opposite. Although the number of high-quality non-dominated solutions is limited under the current relatively strict constraints (4 in the figure), this set of solutions still provides decision-makers with distinctly different policy options, ranging from “efficiency-first” to “environment-first.” It is worth emphasizing that, thanks to the aforementioned scheduling fault-tolerance mechanism, each point on the front corresponds to a complete scheduling plan covering all 23 vessels, thus ensuring the reliability and practical applicability of the decisions.

To further investigate the micro-structure and feasibility of the solutions, this paper conducted a visual analysis of a typical solution (Solution 1) from Figure 6, and its corresponding vessel service Gantt chart is shown in Figure 7. The Gantt chart intuitively displays the time arrangement of all 23 vessels’ in-service and out-service operations within the 36 h planning period, once again verifying that the algorithm can generate a complete schedule covering all tasks without “discarding” any services. It can be observed from the chart that the execution of services is not uniformly distributed over the timeline; the start times of some services are significantly later than their estimated time of arrival (ETA). This precisely reflects the algorithm’s proactive postponement of schedules to meet hard constraints such as tidal windows, channel capacity, or tugboat availability, and these necessary delays constitute the main cost source of the objective function $Z_1$ (waiting time). Furthermore, the start and end times of vessel services in the chart are closely connected but do not overlap, demonstrating the fine-grained time-division multiplexing management of limited tugboat resources. Overall, this scheduling plan is a reasonable result that effectively balances economic and environmental objectives while satisfying all physical and regulatory constraints.

From the results above, it can be concluded that the improved NSGA-II algorithm proposed in this paper is an effective and robust tool for solving this type of multi-objective tugboat scheduling problem. Its core advantage lies in the scheduling fault-tolerance mechanism, which successfully addresses the problem of scheduling failures caused by complex constraints. By handling resource conflicts through forced postponement rather than direct discarding, this mechanism ensures the completeness and comparability of all solutions, which is a fundamental prerequisite for effective optimization. On this basis, the algorithm can efficiently generate a set of Pareto optimal solutions that clearly reveal the quantitative trade-offs between different operational objectives (e.g., efficiency and environmental protection), providing port decision-makers with a rich set of data-driven policy options. Finally, through the visual analysis of the convergence process, the Pareto front, and specific scheduling plans, the optimization behavior of the algorithm and the quality of the results are highly interpretable.

5.3. Sensitivity Analysis Experiment

To investigate the profound impact of the composition of the port’s tugboat fleet on overall scheduling performance, this section conducts a sensitivity analysis on the proportion of high-grade tugboats (defined as Grade 2 and Grade 3 tugboats) in the total number of tugboats. We established five different configuration scenarios where the proportion of high-grade tugboats was 25%, 40%, 55%, 70%, and 85%, respectively, and conducted a thorough comparative analysis of their Pareto optimal solution sets.

Figure 8 visually demonstrates the trade-off relationship between “total waiting time” and “total carbon emissions” for scheduling plans under different tugboat configurations. Each scatter point in the figure represents a non-dominated solution, and the boundary formed by these points is the Pareto front for that scenario. A better tugboat configuration will have its Pareto front closer to the origin of the coordinate axes (lower left), meaning it can achieve higher operational efficiency at a lower environmental cost.

From the evolutionary trend in Figure 8, two key patterns can be observed:

First, there is a significant improvement effect. As the proportion of high-grade tugboats gradually increases from 25% to 70%, the Pareto front shows a continuous and significant shift towards the lower left. This clearly indicates that increasing the investment in high-efficiency tugboats can effectively break system bottlenecks, leading to an improvement in the potential optimal performance of the entire scheduling system. For example, comparing the 25% and 70% configurations, at the same level of total waiting time, the minimum carbon emissions for the 70% configuration are much lower than for the 25% configuration. Similarly, at the same level of carbon emissions, the 70% configuration can achieve shorter waiting times.

Second, there is a diminishing marginal return. Although increasing the number of high-grade tugboats leads to performance improvement, the magnitude of this improvement is not linear. The movement of the Pareto front is greatest during the increase from 25% to 55%, indicating that the return on investment for high-grade tugboats is highest in the initial stages when tugboat resources are relatively scarce. However, when the configuration proportion increases from 70% to 85%, the Pareto fronts of the two scenarios almost overlap, and the optimization effect nearly stagnates. This reveals a clear law of diminishing marginal returns: once the number of high-grade tugboats exceeds a certain critical point, other factors in the system (such as channel capacity, number of berths) become the new bottlenecks. At this point, continuing to add more high-grade tugboats does not bring significant additional benefits and may even lead to resource waste.

6. Conclusions

This paper focuses on the optimization challenge of port tugboat scheduling under the dual objectives of “efficiency-environment.” The main contributions are twofold: In terms of model construction, this paper proposes a comprehensive Mixed-Integer Linear Programming (MILP) model that, through linearization techniques, integrates for the first time multiple key constraints such as tugboat heterogeneity, tidal windows, multi-state carbon emissions, and channel capacity within a unified mathematical framework. In terms of solution method, to address the NP-hard nature of the model, an improved NSGA-II algorithm is designed, which achieves efficient solutions for large-scale instances by introducing problem-specific genetic operators and adaptive mechanisms.

Through a case study based on real port data, this paper validates the effectiveness of the proposed model and algorithm. The experimental results show that the improved algorithm can converge effectively and generate a set of high-quality Pareto optimal solutions, clearly revealing the quantitative trade-off relationship between economic and environmental benefits. Furthermore, a sensitivity analysis of the tugboat class configuration discovered the law of diminishing marginal returns from investing in higher-class tugboats, providing important data support and management insights for port resource allocation strategies.

However, this study has several limitations that open clear avenues for future research. The primary limitation is the assumption of a deterministic operating environment, which does not account for real-world uncertainties like variable vessel arrival times. To address this, a crucial future direction is to incorporate stochastic or robust optimization techniques, which would produce more resilient schedules capable of absorbing operational disruptions. Furthermore, while the proposed algorithm is effective, tackling even larger-scale or real-time instances could benefit from exploring more advanced solution methods, such as hyper-heuristics or deep reinforcement learning. Finally, the model’s scope could be broadened from a tug-centric view to a holistic port operations perspective by integrating other key resources, such as pilot and berth allocation, into a unified framework for collaborative optimization.