An Emergency Scheduling Model for Oil Containment Boom in Dynamically Changing Marine Oil Spills: Integrating Economic and Ecological Considerations

Abstract

Marine oil spills pose substantial risks to human society and ecosystems, resulting in significant economic and ecological consequences. Timely containment of oil films is a complex and urgent task, in which the efficient scheduling of oil containment booms plays a crucial role in reducing economic and ecological losses caused by oil spills. However, due to dynamically changing marine oil spills, the length of boom required and the losses caused by oil spills are inherently uncertain. This study aims to optimize the containment of oil films, exploring the interrelationships among oil films, spill losses, and scheduling decisions for booms. By incorporating economic and ecological losses into decisions, this study proposes a scheduling model for oil containment booms to minimize spill-related losses while reducing scheduling time. Additionally, an improved Multi-Objective Grey Wolf Optimization algorithm is used to solve the problem. A hypothetical case study is then conducted in the Zhoushan sea area of the East China Sea. The proposed scheduling scheme achieves a containment time of 8.9781 h and reduces total spill losses to CNY 313.68 million. Compared with a scheme that does not consider spill losses, the proposed method achieves a nearly 24% reduction in losses while maintaining comparable efficiency.

1. Introduction

Oil spills have been recognized as originating from maritime transportation, oil rig drilling, and natural incidents [1]. As global terrestrial resources have become increasingly scarce, coastal industrial nations have shifted their development focus toward exploring marine resources, particularly offshore oil resources. This trend has not only advanced offshore oil extraction technologies, but also facilitated the rapid growth of maritime oil transportation logistics. However, the risk of marine oil spills has significantly increased. In an incident on 30 April 2024, in Spain, 25,000 to 30,000 L of fuel leaked in Ceuta due to a crack in the fuel tank. On 15 December 2024, two Russian tankers carrying more than 9000 tons of heavy fuel oil collided near the port of Taman in the Kerch Strait of the Black Sea. The spill released a substantial amount of toxic oil, threatening thousands of birds, dolphins, and other marine species [2]. Given the serious consequences of oil spills, timely containment is essential for effective cleanup operations [3]. Oil containment booms are essential emergency tools used to reduce pollution by restricting the drift and diffusion of oil films. In addition, oil containment booms are able to divert and channel oil films along predetermined paths [4,5]. These facilitate oil collection and improve the efficiency of skimming and other techniques [6].

Oil spills exhibit time-varying characteristics due to environmental factors such as wind, ocean currents, and waves. These factors cause oil films to undergo continuous changes in location, shape, and extent, making emergency containment a complex and dynamic task [7]. As oil films drift and diffuse over time, the length and quantity of containment booms required also vary.

Beyond dynamic marine environments, the consequences of oil spills are also highly uncertain, especially in terms of economic and ecological impacts. We argue that the losses caused by oil spills cannot be ignored. An oil spill incident of the British Petroleum Deepwater Horizon occurred in 2010 is an example. The incident was a major disaster with substantial environmental and economic impacts, totaling estimated damage of USD 36.9 billion [8,9,10]. The severity of these losses depends on factors such as the oil film area, the economic value of affected zones, and the timeliness of containment operations.

Given these challenges, oil spill response decisions must not only be timely but also consequence-sensitive, prioritizing areas with higher potential losses. However, existing studies primarily focused on physical cleanup following an accident, with limited research on reducing the resulting damage. The initial containment is the most important stage in controlling the economic and ecological consequences of oil spills. As boom deployment directly influences the scale of pollution and the severity of the resulting damage. Timely scheduling of booms ensures the effectiveness of oil containment, which prevents further drift and diffusion of oil films and facilitates subsequent recovery operations [11,12]. This study aims to develop a scheduling model that incorporates time-varying spill characteristics and evaluates economic and ecological consequences to support more effective boom deployment decisions.

Additionally, given the high requirements for rescue efficiency in oil spills, decision-makers must consider the rescue time. Many studies have used rescue time throughout oil spills as one of the objective functions [13]. Therefore, a multi-objective scheduling model is proposed to provide decision-makers with a scheduling plan for oil containment booms by balancing scheduling time and spill losses.

The main contributions of this paper lie in the following aspects:

(1) This study highlights the critical role of oil containment booms in limiting the drift and diffusion of oil films. Delayed containment exacerbates both economic and ecological damage. The interrelationships among oil films, spill losses, and scheduling decisions for oil containment booms are considered in this study.

(2) Time-varying parameters are introduced to simulate the dynamic characteristics of oil films affected by environmental forces such as wind, currents, and waves. This approach captures the uncertainty in boom demand resulting from dynamically changing oil spills.

(3) A multi-objective emergency scheduling model for oil containment boom is developed, which incorporates time-varying spill characteristics. Furthermore, economic and ecological losses are quantified and embedded as objective functions. Considering the complexity and variability of oil spill impacts, the model’s scheduling plan aims to minimize both economic and ecological losses while ensuring timely containment.

(4) Oil spills require quick and effective solutions. The improved Multi-Objective Grey Wolf Optimization algorithm is applied to enhance computational efficiency while maintaining solution quality, and enables faster decision-making in responding to dynamic and complex oil spill scenarios.

The rest of this paper is organized as follows. Section 2 reviews the related work with the scope of the study. Section 3 analyzes the problem characteristics. Section 4 develops a scheduling model for oil containment boom. Section 5 presents a case study in the Zhoushan sea area of the East China Sea to illustrate and validate the proposed model. Finally, conclusions are presented in Section 6.

2. Literature Review

Several studies have attempted to incorporate the time-varying nature of oil spills into their response strategies. Zhong and You [14] developed a bi-criterion MILP model for oil spill response planning, which integrates oil weathering dynamics. Ye et al. [15] developed an emergency response system that took weathering conditions and waste residue status as input to obtain a response plan. Souza Júnior et al. [16] analyzed oil weathering and dispersion patterns by SAR sensors. These studies highlight the complexity of oil behavior at sea, which poses challenges in responding to it [17]. However, existing research still falls short in addressing the response operations of oil film dynamic characteristics, particularly in the early stages of containment. These variations influence the length of the required containment booms.

The significant impact of oil spills has posed severe challenges to sustainable development. Several studies have considered the variability of spill impacts. Bi and Si [18] developed a dynamic risk assessment model for oil spills, which integrates numerical simulations to reflect the changing losses. Zhao et al. [19] highlighted the volatility of marine disaster losses caused by environmental instability. They introduced the ENN-GRNN-DI model to predict direct economic losses. These studies confirm that the severity of oil spill impacts depends on oil film area, the value of the affected region, and the time of containment. Delayed containment results in extended pollution duration, which amplifies environmental degradation and increases recovery costs [20]. However, meteorological data, oil spill data, socio-economic data, and so forth related to oil spills are rarely effectively integrated, which hinders the control of accident losses [21,22]. Decision-makers must not only act quickly but also prioritize areas with higher potential damage. Currently, there are limited studies on how to reduce losses caused by oil spills.

In response to marine oil spills, researchers have investigated methods for scheduling emergency resources, among which containment booms are included. Xu et al. [23] proposed a multi-objective emergency resource dispatching model for oil spills to minimize dispatch time and the number of storage locations. While their model incorporates resource priority mechanisms to reflect real-world constraints, it does not account for the time-varying nature of oil spills. Li et al. [24] proposed a method for emergency material dispatch in large-scale oil spills to minimize sailing time and balance distribution. While the model demonstrates practical relevance, it lacks the integration of environmental dynamics and consequence-based decision-making that is essential for effective spill containment in complex marine conditions. Huang et al. [25] proposed a dynamic optimization model to minimize time and cost, considering the drift of multiple oil films. However, it focuses primarily on scheduling time and cost, lacking explicit evaluation of economic and environmental impact. Zhang et al. [26] proposed a dynamic location-routing model with split delivery, incorporating time-varying demands and transportation uncertainties. However, the study fails to prioritize high-impact spill areas, which weakens its practical value in complex real-world scenarios. Li et al. [27] developed a multi-objective optimization model with time window constraints to improve the scheduling of oil spill cleanup materials for complex incidents. Although it incorporates cost and time factors and uses an improved genetic algorithm for efficiency, it primarily focuses on logistics and neglects environmental sensitivity and the dynamic behavior of oil films. Zhang and Lu [28] developed a model that considers multiple resource types, split deliveries, and vehicle–resource matching. A hybrid PSO-PGSA algorithm was used for optimization. While it addresses dynamic demand and vehicle–resource matching, it overlooks scheduling time and may fall short in ensuring timely containment. Dong et al. [29] proposed a multi-objective optimization method for MSAR resource allocation, incorporating accident prediction and hotspot analysis. While effective in reducing response time and cost, it does not account for the dynamic behavior of oil films. These studies have made contributions to improving marine oil spill emergency capabilities. It can also be found that in order to make prompt decisions in oil spills, heuristic algorithms have been widely applied to optimize the scheduling of emergency resources.

Overall, prior studies have made contributions in optimizing emergency resource scheduling, but they largely neglect the integration of dynamically changing oil film characteristics and oil spill consequences. Compared to these previous studies, the model proposed in this paper demonstrates several advantages. The model integrates the dynamic changing behavior of oil films through time-varying parameters, which enhances the model’s adaptability to marine conditions. Additionally, unlike models focusing solely on time or cost, this study explicitly incorporates both economic and ecological loss functions into the objective formulation.

3. Problem Description

Multiple oil films appear on the sea surface when marine oil spills occur. Due to the influence of environmental factors, the spilt oil is likely to be broken into several dispersed oil films [23]. Furthermore, oil spills may extend to multiple sea areas, forming multiple dispersed oil films over a larger region. Consequently, boom deployment vessels must be arranged to deploy booms to numerous affected areas. It is worth noting that the emergency operation of oil spills is often associated with an emergency decision-making organization. The organization assigns tasks to relevant emergency divisions during emergency operations. To demonstrate the process for scheduling oil containment booms, an illustration is shown in Figure 1. The operation framework for scheduling oil containment booms comprises four critical stages: information collection, data analysis, decision-making, and scheduling operations. During the information collection phase, real-time data on oil spills, sea conditions, and weather are integrated with database resources. However, the economic and ecological losses in affected areas may vary throughout the scheduling process due to changing factors. Therefore, it is necessary to predict the dynamic changes in the oil film and the resulting losses during the data analysis phase. Subsequently, the model is used to optimize scheduling strategies. The primary objective of this process is to minimize scheduling time and oil spill losses. Once oil spills are detected, booms must be transported from the emergency equipment depot. Upon arrival at the spill site, boom deployment vessels deploy booms to contain oil films.

3.1. Dynamic Changes of Oil Films

When oil is spilt into the marine environment, weathering processes, including physical and chemical alterations, dynamically change its properties [30]. These changes must be considered to develop effective scheduling plans. Diffusion leads to an expansion of the contaminated area. Moreover, drift changes the location of the oil film and affects the transportation routes of boom deployment vessels. The dynamic nature of oil films implies that the impact on the economic and ecological environment continuously changes. In this study, we use mathematical formulas to quantify the characteristics of the oil film.

3.1.1. Drift of Oil Films

Affected by wind, waves, and currents, the oil film continues to drift, which changes its geographical location. Lagrangian transport models are useful tools extensively used in oil spill modeling. The drift process of oil films is simulated by a Lagrangian model, where the oil films are represented by numerous particles. The transport mechanisms include advection driven by water currents and mixing caused by turbulence [31]. Al-Rabeh et al. [32] use the Lagrangian particle-tracking technique to solve the advection–diffusion equation. They highlighted the critical role of the advective velocity field in pollutant transport. Similarly, Chen et al. [33] thought that oil spill drift primarily consists of advection and turbulent diffusion processes. Surface flow, wind effects, current speed, and current direction are the key factors influencing advection. The following functions are proposed based on these foundational studies.

$$L{L}_t^j=L{L}_0^j+(M_L+\alpha H_{10}\cos \theta )t$$

(1)

$$B{B}_t^j=B{B}_0^j+(M_B+\alpha H_{10}\sin \theta )t$$

(2)

As shown in Formulas (1) and (2), $L{L}_t^j$ and $B{B}_t^j$ are the horizontal and vertical coordinates of the oil film $j$ at time $t$, respectively. $L{L}_0^j$ and $B{B}_0^j$ represent the horizontal and vertical coordinates of the initial oil film $j$, respectively. $M_L$ and $M_B$ are advective velocity components in the horizontal and vertical directions, respectively. $H_{10}$ is the wind speed at the height of 10 m above sea level, $\theta$ is the wind angle, and $\alpha$ is the coefficient of wind-induced currents.

3.1.2. Diffusion of Oil Films

The three-stage extended model proposed by Fay [34] remains one of the most representative research studies on the spreading behavior of marine oil spills. Liu and Leendertse [35] proposed an improved oil spill diffusion model after integrating the three-phase oil film diffusion model, as follows:

$$\epsilon =1-{\displaystyle \frac{{\rho }_o}{{\rho }_w}}$$

(3)

$$f\left(t,V_j\right)=0.61{\left[1.3{\left(\epsilon g{V}_j\right)}^{{\displaystyle \frac{1}{2}}}t+2.1{\left({\displaystyle \frac{\epsilon g{V_j}^2}{\sqrt{v_w}}}\right)}^{{\displaystyle \frac{1}{3}}}\sqrt{t}+5.29{\left({\displaystyle \frac{{\delta }^2{t}^3}{{\rho }_w^2{v}_w}}\right)}^{{\displaystyle \frac{1}{2}}}\right]}^{{\displaystyle \frac{1}{2}}}$$

(4)

As shown in Formulas (3) and (4), $f\left(t,V_j\right)$ is the oil film diameter, ${\rho }_o$ represents oil density, ${\rho }_w$ represents seawater density, $g$ represents acceleration of gravity, $V_j$ represents oil spill volume at the $j-th$ oil film, $v_w$ represents viscosity coefficient, and $\delta$ represents net surface tension coefficient.

3.1.3. Dynamics of Oil Concentration

Mackay et al. [36] studied the temporal dynamics of oil concentration beneath a dispersed slick, which demonstrated an approximately linear relationship with time. The temporal evolution of the plume area and the dispersed oil volume depends on factors such as slick evaporation and undispersed oil remaining on the surface. Both of them can be expressed as time-dependent variables given sufficient data. This approach provides a tractable pattern for understanding the concentration dynamics of oil.

$$C_j^t=\left(V_j{\rho }_o{\displaystyle \frac{\sqrt{\frac{\pi }{4\gamma t}}}{s_j}}\right)\exp \left(-h_j\sqrt{{\displaystyle \frac{\pi }{4\gamma t}}}\right)$$

(5)

As shown in Formula (5), $C_j^t$ is the concentration of the oil film $j$ at time $t$, $V_j$ represents the $j-th$ oil volume, ${\rho }_o$ and $\gamma$ denote the oil density and vertical diffusivity, respectively, $s_j$ is the spill area of the $j-th$ oil film, and $h_j$ indicates the depth of seawater affected by the oil spill.

3.2. Economic Losses of Marine Oil Spills

Economic losses caused by marine oil spills primarily stem from cost, transportation expenses, and damage to related marine industries such as fisheries and tourism [37]. Oil spills damage marine ecosystems and severely impact regional economies by reducing fisheries’ output, impairing aquaculture, and restricting tourism activities. Commercial fisheries and aquaculture suffer losses from direct mortality, habitat destruction, and harvesting restrictions. Simultaneously, tourism is severely affected due to beach and waterfront contamination [38]. The economic losses from oil spills are directly influenced by the properties of oil films. James and Lee [39] proposed the pollution concentration–value loss curve to quantify the reduction in resource value caused by pollution. Modeled using a logistic function, this curve represents the relationship between pollution concentration and value loss [40]. As shown in Formulas (6)–(8), this study applies the approach to oil spills, enabling the estimation of economic damage by linking oil film concentration with industrial value. Combined with the pollution loss rate $U_j$, the economic loss of the industry $ec$ caused by the oil film $j$ can be calculated and recorded as $E{S}_j^{ec}$.

$$c={\displaystyle \frac{C_j}{C_0}}$$

(6)

$$U_j={\displaystyle \frac{1}{1+a_z\exp (-b_{z}c)}}$$

(7)

$$E{S}_j^{ec}=U_j{K}_j^{ec}A{S}_j$$

(8)

$C_j$ represents the concentration of the oil film $j$. $C_0$ represents the target concentration of water quality. $a_z$ and $b_z$ are dimensionless parameters, which are related to the characteristics of oil films. $K_j^{ec}$ represents the unit economic output value of the industry $ec$ at oil film $j$. $A{S}_j$ represents the area affected by oil film $j$.

$$RC={\displaystyle \sum _{j=1}^n{A}_{j}CK}$$

(9)

$$RT={\displaystyle \sum _{i=1}^{rv}\frac{S_{i}C{I}_i}{V{I}_i}}$$

(10)

In addition, as shown in Equations (9) and (10), the boom costs $RC$ and transportation expenses $RT$ in economic losses cannot be ignored. $CK$ represents the unit cost of the oil boom. $A_j$ is the length of booms required at oil film $j$. $C{I}_i$ represents the transportation cost per unit time for the boom deployment vessel $i$. $V{I}_i$ and $S_i$ are the velocity and the total transportation distance of boom deployment vessel $i$, respectively.

3.3. Ecological Losses of Marine Oil Spills

Oil spills disrupt ecosystem functions and services, resulting in long-term environmental degradation. The timely scheduling of booms is essential to contain ecological losses and preserve marine biodiversity. Furthermore, the ecological damage caused by oil spills is closely related to the properties of the oil films, with changes in it directly affecting the extent of ecological loss [41]. According to the Technical Guidelines for Marine Ecological Damage Assessment (GB/T 34546-2017) issued by the State Oceanic Administration of China in 2017, marine oil spills not only impair the ecological service functions, but also consume part of the environmental capacity [42]. Therefore, the ecological damage caused by oil spills includes two components. The ecological losses of marine oil spills are formulated as follows:

$$h{y}_j=h{y}_{ed}A{S}_j{\mu }_j{t}_{j}Ge$$

(11)

$$f{w}_j=h_{j}A{S}_{j}FA$$

(12)

As shown in Formulas (11) and (12), $h{y}_j$ represents the loss of marine ecosystem service functions caused by oil film $j$, $f{w}_j$ represents the environmental capacity loss caused by oil film $j$, $h{y}_{ed}$ represents the unit value of ecological type $ed$, ${\mu }_j$ represents the impact level of oil film $j$ on the marine ecosystem, $t_j$ represents the recovery time required for the marine ecosystem at oil film $j$, $G$ represents the oil spill toxicity coefficient, $e$ represents the sensitivity conversion rate, $FA$ represents the unit cost of wastewater treatment, and $h_j$ represents the depth of seawater affected by oil film $j$.

3.4. Scheduling Decision-Making

Few studies have explored the impact of dynamically changing marine oil spills on both accident losses and boom scheduling. As shown in Figure 2, due to the dynamic characteristics of oil films, there exist interdependent relationships among oil films, spill losses, and scheduling decisions for oil containment booms. Firstly, there is no doubt that changes in oil films affect accident losses and scheduling decisions. They influence the scope and extent of oil spill pollution, which lead to uncertainty in losses. Meanwhile, it is necessary to consider the impact of oil film drift on the transportation route. Accurately predicting the location of the contained oil film is able to reduce scheduling time and related costs. Secondly, to mitigate the consequences of oil spills, scheduling plans that consider both economic and ecological losses are affected by changes in oil spill losses. The value of the affected area plays a crucial role, as regions with higher economic importance suffer greater economic losses. Finally, scheduling decisions can, in turn, influence the changes in oil films and oil spill losses. For instance, the efficient scheduling of booms can control oil films, effectively suppressing their diffusion. In addition, losses are also impacted by the duration of pollution treatment efforts. Prioritizing certain oil films for containment may cause uncontrolled oil films to change continuously, potentially causing more significant economic and ecological losses elsewhere. Therefore, it is vital to analyze the interrelationships among oil films, spill losses, and scheduling decisions for oil containment booms.

As shown in Figure 3, influenced by external factors, the oil film changes continuously. Under surface tension, oil films at sea diffuse, resulting in the expansion of contaminated areas. Additionally, the drift of oil films may change the transportation routes of vessels.

As explained above, it is necessary to analyze changes in oil films and oil spill losses by integrating monitoring data with database information. This ensures the timely scheduling of booms to control oil spills. A scheduling model for oil containment booms in dynamically changing marine oil spills is proposed in this study.

4. Method

Boom scheduling in marine oil spills is urgent and complex. Containment is the primary task during an oil spill emergency operation [43]. A scheduling model for oil containment booms in dynamically changing marine oil spills considering both economic and ecological losses is developed in this section [44,45,46,47,48,49]. Beyond basic scheduling constraints, the model emphasizes the interrelationships among oil films, spill losses, and boom deployment decisions. The objective is to minimize the economic and ecological impacts of oil spills while ensuring containment efficiency. This multi-objective design ensures that the model not only supports efficient response operations but also reflects the uncertainty of oil spill impacts, providing more robust and balanced containment strategies.

4.1. Model Assumptions

Considering the complexity of the problem, the following assumptions are made:

(1) The state of oil films can be identified through technology, including the amount, location, and shape of the oil films.

(2) Each emergency equipment depot is equipped with boom deployment vessels for boom scheduling, and the coordinates of emergency equipment depots are known.

(3) The relevant parameters of the impact of weather and sea conditions on oil spills are known. Following the oil spill accident, the wind and wave environment at sea remains stable. Similarly, boom deployment vessels are assumed to transport at a constant speed.

(4) Once the vessel has deployed booms, the oil films will be controlled, preventing further pollution of the marine environment.

(5) Considering the diffusion of the oil film, the oil film’s longest diagonal distance at the time of treatment is the length of oil booms required.

4.2. Notations and Definitions

The lists of notations and definitions used in the model look like this:

• Sets and indexes

$K$: Set of emergency equipment depots, $K=\left\{k|k=1,2,3,\dots ,m\right\}$;

$J$: Set of oil films, $J=\left\{j|j=1,2,3,\dots ,n\right\}$;

$H$: Set of all nodes in the scheduling networks, $H=K\cup J$;

$I$: Set of boom deployment vessels, $I=\left\{i|i=1,2,3,\dots ,rv\right\}$;

$EC$: Set of economic industries, $EC=\left\{ec|ec=1,2,3,\dots ,\tau \right\}$;

$ED$: Set of ecological types, $ED=\left\{ed|ed=1,2,3,\dots ,\omega \right\}$.

2.

Model parameters

$m$: Number of emergency equipment depots;

$n$: Number of oil films;

$rv$: Number of boom deployment vessels;

$\tau$: Number of economic industries;

$\omega$: Number of ecological types;

$df$: Deployment speed at which vessels deploy booms;

$V{I}_i$: Speed of boom deployment vessel $i$;

$N_i$: Number of oil films assigned to boom deployment vessel $i$;

$r_{ij}$: Sequence of oil film $j$ in the scheduling task of boom deployment vessel $i$;

$Q{C}_i^{\max }$: Maximum capacity of boom deployment vessel $i$;

$V_j$: Oil spill volume of oil film $j$;

$Q_j$: Spilt quantity of oil film $j$;

$\theta$: Wind angle;

$V_o$: Oil film drift speed;

$M_L$: Velocity of water flow in the horizontal direction;

$M_B$: Velocity of water flow in the vertical direction;

${\rho }_o$: Oil density;

${\rho }_w$: Seawater density;

$A_j^t$: Length of booms required for oil film $j$ at time $t$;

$A_j$: Length of booms required for oil film $j$ when it is controlled;

$A{S}_j$: Area affected by oil film $j$ when it is controlled;

$S_i$: Total transportation distance of boom deployment vessel $i$;

$E{P}_i$: Emergency preparation time of boom deployment vessel $i$;

$T_{iq}$: Moment when boom deployment vessel $i$ arrives at node $q$, $q\in J$;

$T_{pi}$: Moment when boom deployment vessel $i$ leaves node $p$, $p\in J$;

$t_{ipq}$: Transportation time required for boom deployment vessel $i$ from node $p$ to node $q$;

$t_{iqk}$: Transportation time required for boom deployment vessel $i$ from node $q$ to port $k$;

$T_i$: Completion time of boom scheduling for boom deployment vessel $i$;

$L_t^i$: Horizontal coordinate of the boom deployment vessel $i$ at time $t$;

$B_t^i$: Vertical coordinate of the boom deployment vessel $i$ at time $t$;

$L{L}_t^j$: Horizontal coordinate of the oil film $j$ at time $t$;

$B{B}_t^j$: Vertical coordinate of the oil film $j$ at time $t$;

$L_j^i$: Horizontal coordinate of the boom deployment vessel $i$ arriving at oil film $j$;

$B_j^i$: Vertical coordinate of the boom deployment vessel $i$ arriving at oil film $j$;

$d_{ij}^t$: Distance between boom deployment vessel $i$ and oil film $j$ at time $t$;

$\overrightarrow{a}$: Vector from the oil film $r_{f+1}^i$ to the oil film $r_f^i$;

$\overrightarrow{b}$: Vector of oil film drift;

${\beta }_t^i$: Angle between the direction from oil film $r_{f+1}^i$ to oil film $r_f^i$ and the direction of oil film drift at time $t$;

$I{S}_j$: Industrial losses caused by oil film $j$;

$I{A}_{ec}$: Range of economic industries $ec$;

$H{Y}_j$: Losses of ecological service functions caused by oil film $j$;

$D{R}_{ed}$: Range of ecological type $ed$.

3.

Decision variables

$X_{ji}$: 1, if boom deployment vessel $i$ is responsible for oil film $j$, and 0 otherwise;

$Y_{pq}^i$: 1, if boom deployment vessel $i$ transports from node $p$ to node $q$, and 0 otherwise.

4.3. Mathematical Model

Both scheduling time and spill losses are considered optimization objectives in this study. The time at which the vessels complete the deployment of booms to all oil films is defined as the final scheduling time. Therefore, it is necessary to calculate the transportation time and boom deployment time for each vessel.

$$T_{pi}=0,\hspace{1em}\forall i\in I,\forall p\in K$$

(13)

$$L_j^i=L{L}_t^j+\left(M_L+\alpha H_{10}\cos \theta \right)t_{ipq},\hspace{1em}\forall i\in I,\forall p,q,j\in J$$

(14)

$$B_j^i=B{B}_t^j+\left(M_B+\alpha H_{10}\sin \theta \right)t_{ipq},\hspace{1em}\forall i\in I,\forall p,q,j\in J$$

(15)

$$d_{ij}^t=\sqrt{{\left(L{L}_t^j-L_t^i\right)}^2+{\left(B{B}_t^j-B_t^i\right)}^2},\hspace{1em}\forall i\in I,\forall j\in J$$

(16)

$$R_i=\left\{\left(j,r_{ij}\right)\mid j\in J,r_{ij}\in \left\{1,2,\dots ,N_i\right\}\right\},\hspace{1em}\forall i\in I$$

(17)

$$r_f^i=\left\{j\mid \left(j,r_{ij}\right)\in R_i,r_{ij}=f\right\},\hspace{1em}\forall i\in I$$

(18)

Formula (13) indicates the departure time of boom deployment vessel $i$. Formulas (14) and (15) represent the coordinates where boom deployment vessel $i$ arrives at oil film $j$. Formula (16) indicates the distance between boom deployment vessel $i$ and oil film $j$ at time $t$. Formula (17) represents the set of oil films controlled by boom deployment vessel $i$ and their scheduling order. Formula (18) represents the index of the $j-th$ oil film assigned to boom deployment vessel $i$.

As shown in Figure 4, due to the drift of oil films, the distance between oil films changes dynamically when vessel $i$ completes boom deployment at the node and proceeds to the next node. Consequently, the transportation time to the next node also varies with the changes in oil films [50].

First, based on the relative locations of oil film $r_f^i$ and oil film $r_{f+1}^i$, as well as the drift direction of the oil film, the angle ${\beta }_t^i$ between the direction of oil film $r_{f+1}^i$ relative to oil film $r_f^i$ at time $t$, and the drift direction of oil film $r_{f+1}^i$ can be calculated. Subsequently, using the law of cosines, the distances among oil film $r_{f+1}^i$ at time $t$, boom deployment vessel $i$ at oil film $r_f^i$ at time $t$, and the location where boom deployment vessel $i$ arrives at oil film $r_{f+1}^i$ after its drift can be determined. To calculate the transportation time between oil film $r_f^i$ to the next node $r_{f+1}^i$, the following formula is proposed:

$$V_o=\sqrt{{\left(M_L+\alpha H_{10}\cos \theta \right)}^2+{\left(M_B+\alpha H_{10}\sin \theta \right)}^2}$$

(19)

$$\overrightarrow{a}=\left(L_t^i-L{L}_t^j,B_t^i-B{B}_t^j\right),\hspace{1em}\forall i\in I,\forall j\in J$$

(20)

$$\overrightarrow{b}=\left(M_L+\alpha H_{10}\cos \theta ,M_B+\alpha H_{10}\sin \theta \right)$$

(21)

$$\begin{array}{ll}{\beta }_t^i\hfill & =\arccos \left({\displaystyle \frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\left|\overrightarrow{a}\right|\cdot \left|\overrightarrow{b}\right|}}\right)\hfill \\ & =\arccos \left({\displaystyle \frac{\left(L_t^i-L{L}_t^j\right)\cdot \left(M_L+\alpha H_{10}\cos \theta \right)+(B_t^i-B{B}_t^j)\cdot \left(M_B+\alpha H_{10}\sin \theta \right)}{\sqrt{{\left(L_t^i-L{L}_t^j\right)}^2+{\left(B_t^i-B{B}_t^j\right)}^2}\cdot \sqrt{{\left(M_L+\alpha H_{10}\cos \theta \right)}^2+{\left(M_B+\alpha H_{10}\sin \theta \right)}^2}}}\right),\hfill \\ & \forall i\in I,\forall j\in J\hfill \end{array}$$

(22)

$$\begin{array}{l}{t}_{ipq}={\displaystyle \frac{\sqrt{{\left(L{L}_t^j-L_t^i\right)}^2+{\left(B{B}_t^j-B_t^i\right)}^2}\cdot \sqrt{V{I}_i^2-{\left(V_o\sin {\beta }_t^i\right)}^2}}{V{I}_i^2-{V_o}^2}}\\ \hspace{1em}-{\displaystyle \frac{\sqrt{{\left(L{L}_t^j-L_t^i\right)}^2+{\left(B{B}_t^j-B_t^i\right)}^2}\cdot V_o\cos {\beta }_t^i}{V{I}_i^2-{V_o}^2}},\hspace{1em}\forall i\in I,\forall p,q,j\in J\end{array}$$

(23)

Formula (19) represents the drift velocity of the oil film. Formula (20) represents the vector from oil film $r_{f+1}^i$ to boom deployment vessel $i$ located at oil film $r_f^i$ at time $t$. Formula (21) represents the vector of the oil film drift. Formula (22) represents the angle between the direction of oil film $r_{f+1}^i$ relative to oil film $r_f^i$ at time $t$, and the drift direction of oil film $r_{f+1}^i$. Formula (23) defines the transportation time of boom deployment vessel $i$ from oil film $r_f^i(r_f^i=p)$ to the next node $r_{f+1}^i(r_{f+1}^i=q)$, considering both the speed of the vessel and the influence of external environment. As shown in Figure 4, the formula is derived based on the cosine law, which is applied to construct the geometric relationship between the navigation direction of the vessel and the vector of oil film drift.

$$V_j={\displaystyle \frac{Q_j}{{\rho }_o}},\hspace{1em}\forall j\in J$$

(24)

$$A_j^t=0.61{\left[1.3{\left(\epsilon g{V}_j\right)}^{{\displaystyle \frac{1}{2}}}t+2.1{\left({\displaystyle \frac{\epsilon g{V}_j^2}{\sqrt{{\nu }_w}}}\right)}^{{\displaystyle \frac{1}{3}}}\sqrt{t}+5.29{\left({\displaystyle \frac{{\delta }^2{t}^3}{{\rho }_w^2{\nu }_w}}\right)}^{{\displaystyle \frac{1}{2}}}\right]}^{{\displaystyle \frac{1}{2}}},\hspace{1em}\forall j\in J$$

(25)

$$T_{iq}={\displaystyle \sum _{p\in J}\left(T_{pi}+t_{ipq}\right)Y_{pq}^i},\hspace{1em}\forall i\in I,\forall q\in J$$

(26)

$$T_{pi}=T_{iq}+{\displaystyle \frac{A_j{X}_{ji}}{df}},\hspace{1em}\forall i\in I,\forall p,q,j\in J$$

(27)

$$T_i=E{P}_i+{\displaystyle \sum _{p\in H}{t}_{ipq}{Y}_{pq}^i}+{\displaystyle \sum _{j\in J}\frac{A_j{X}_{ji}}{df}},\hspace{1em}\forall i\in I,\forall q\in J$$

(28)

$$S_i={\displaystyle \sum _{p\in H}{t}_{ipq}V{I}_i{Y}_{pq}^i}+t_{iqk}V{I}_i,\hspace{1em}\forall i\in I,\forall q\in J,\forall k\in K$$

(29)

Formula (24) represents the volume of the oil film $j$. Formula (25) represents the length of booms required for oil film $j$ at time $t$. As referenced in the oil spill diffusion model, the diameter of the oil film at a moment can be estimated based on its spread characteristics [35]. In actual emergency operations, two boom deployment vessels work as a coordinated pair to drag and deploy the boom simultaneously. Therefore, the required boom length is determined by the maximum diameter of the oil film at time $t$. Formula (26) represents the time when the boom deployment vessel $i$ arrives at the node $q$. Formula (27) represents the moment when boom deployment vessel $i$ leaves node $p$ for the next node. Formula (28) represents the completion time of boom scheduling for vessel $i$. Formula (29) represents the total transportation distance of boom deployment vessel $i$.

The losses caused by oil spills are primarily reflected in the economic and ecological damage [51]. Specifically, economic losses include boom costs $RC$, transportation expenses $RT$, and losses $IS$ related to marine industries. Ecological losses include the loss of ecological service functions $HY$ and the loss of marine environmental capacity $FW$.

$$RC={\displaystyle \sum _{j=1}^n{A}_{j}CK}$$

(30)

$$RT={\displaystyle \sum _{i=1}^{rv}C{I}_i\left({\displaystyle \sum _{p\in H}{t}_{ipq}{Y}_{pq}^i+t_{iqk}}\right)},\hspace{1em}\forall q\in J,\forall k\in K$$

(31)

$$I{S}_j=\left\{\begin{array}{l}{\displaystyle \sum _{ec=1}^{\tau }\frac{K_j^{ec}A{S}_j}{1+a_Z\exp \left(-b_{Z}c\right)},\hspace{1em}\left(L{L}_t^j,B{B}_t^j\right)\in I{A}_{ec},\forall ec\in EC,\forall j\in J}\hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise\hfill \end{array}\right.$$

(32)

$$IS={\displaystyle \sum _{j=1}^{n}I{S}_j}$$

(33)

$$H{Y}_j=\left\{\begin{array}{l}h{y}_{ed}{\mu }_j{t}_{j}GeA{S}_j,\hspace{1em}\left(L{L}_t^j,B{B}_t^j\right)\in D{R}_{ed},\forall ed\in ED,\forall j\in J\hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise\hfill \end{array}\right.$$

(34)

$$HY={\displaystyle \sum _{j=1}^{n}H{Y}_j}$$

(35)

$$FW={\displaystyle \sum _{j=1}^n{h}_{j}A{S}_{j}FA}$$

(36)

Formula (30) represents the cost of booms required for oil spill containment, which is primarily determined by the total length of booms used. Formula (31) represents transportation expenses. This cost depends on the number of vessels involved and their respective transportation expenses, reflecting the direct cost of deploying resources for emergency response. Formula (32) represents the industrial losses caused by oil film $j$. If the oil film is within an industrial area, the economic losses to local industries must be calculated. If the oil film is outside industrial areas, the industrial loss is considered zero. Formula (33) represents the total losses of related industries affected by oil spills. These losses are based on both the extent of pollution and the economic value of the industries exposed to the spill. Formula (34) identifies whether oil film $j$ is located in an ecological protection zone, and if so, determines the corresponding ecological type. It then estimates the ecological service loss by evaluating the polluted area and the value of the impacted ecosystem. Formula (35) represents the total losses of ecological service functions. Formula (36) represents the losses of marine environmental capacity.

$${\mathrm{Min} \mathrm{F}}_1=\underset{i\in I}{\max }\left\{E{P}_i+{\displaystyle \sum _{p\in H}{\displaystyle \sum _{q\in J}{t}_{ipq}{Y}_{pq}^i+{\displaystyle \sum _{j\in J}\frac{A_j{X}_{ji}}{df}}}}\right\}$$

(37)

$$\begin{array}{ll}{\mathrm{Min} \mathrm{F}}_2& =RC+RT+IS+HY+FW\\ & ={\displaystyle \sum _{j=1}^n{A}_{j}CK+{\displaystyle \sum _{i=1}^{rv}C{I}_i\left({\displaystyle \sum _{p\in H}{\displaystyle \sum _{q\in J}{t}_{ipq}{Y}_{pq}^i+{\displaystyle \sum _{q\in J}{\displaystyle \sum _{k\in K}{t}_{iqk}}}}}\right)+{\displaystyle \sum _{j=1}^{n}I{S}_j}}}\\ & +{\displaystyle \sum _{j=1}^{n}H{Y}_j}+{\displaystyle \sum _{j=1}^n{h}_{j}A{S}_{j}FA}\end{array}$$

(38)

$${\displaystyle \sum _{q\in J}{Y}_{kq}^i=1,\hspace{1em}\forall i\in I,\forall k\in K}$$

(39)

$${\displaystyle \sum _{p\in K}{\displaystyle \sum _{q\in K}{Y}_{pq}^i=0}},\hspace{1em}\forall i\in I$$

(40)

$${\displaystyle \sum _{q\in J}{Y}_{kq}^i}-{\displaystyle \sum _{p\in J}{Y}_{pk}^i}=0\hspace{1em}\left(p\ne q\right),\hspace{1em}\forall i\in I,\forall k\in K$$

(41)

$${\displaystyle \sum _{i=1}^{rv}{N}_i}=n$$

(42)

$${\displaystyle \sum _{j\in J}{A}_j{X}_{ji}}\le Q{C}_i^{max},\hspace{1em}\forall i\in I$$

(43)

$${\displaystyle \sum _{p\in H}{Y}_{pq}^i}+{\displaystyle \sum _{p\in H}{Y}_{qp}^i}\le 2,\hspace{1em}\forall i\in I,\forall q\in H$$

(44)

$${\displaystyle \sum _{i\in I}{X}_{ji}}=1,\hspace{1em}\forall j\in J$$

(45)

$$Y_{pq}^i\left(T_{iq}-T_{ip}\right)\ge 0,\hspace{1em}\forall i\in I,\forall p,q\in J$$

(46)

$$X_{ji}\in \left\{0,1\right\},\hspace{1em}\forall i\in I,\forall j\in J$$

(47)

$$Y_{pq}^i\in \left\{0,1\right\},\hspace{1em}\forall i\in I,\forall p,q\in J$$

(48)

The first objective (37) minimizes the total scheduling time, which consists of the emergency preparation time, the transportation time, and the boom deployment time. The second objective (38) aims to minimize the economic and ecological losses caused by oil spills. To achieve the trade-off between these two objectives, the model is formulated as a bi-objective optimization problem, and solutions are evaluated based on their Pareto optimality. Constraint (39) is a constraint on the starting site of the boom deployment vessel. Constraint (40) indicates that the boom deployment vessel cannot travel between emergency equipment depots. Constraint (41) indicates that the boom deployment vessel must return to its starting point at the end. Constraint (42) indicates that each oil film must be controlled during the emergency operation. Constraint (43) indicates that for each boom deployment vessel $i$, the boom amount carried by boom deployment vessel $i$ cannot exceed its maximum carrying capacity. Constraint (44) indicates that each oil film can be served no more than once. Constraint (45) indicates that the booms required for each oil film are sourced from a single boom deployment vessel. Constraint (46) indicates that the nodes along the route of boom deployment vessels occur sequentially. Binary integer constraints for the decision variables are given in Constraints (47) and (48).

4.4. The Improved Multi-Objective Grey Wolf Optimization Algorithm

Multi-objective scheduling models are typically NP-hard problems, especially in dynamically changing marine oil spills. Therefore, determining a unique, non-dominated solution that satisfies all the optimization objectives is challenging. Given the complexity of a multi-objective and multi-constraint problem, it is necessary to use heuristic algorithms to obtain optimal solutions efficiently [52]. Recently, the Multi-Objective Grey Wolf Optimizer (MOGWO) has emerged as a promising swarm intelligence algorithm, offering excellent convergence performance and requiring fewer adjustable parameters [53]. The MOGWO algorithm mimics the social hierarchy of grey wolves, dividing the population into alpha, beta, and delta leaders, followed by the rest of the wolves [54]. Guided by the leaders, the algorithm iteratively updates the positions of wolves to converge toward optimal solutions. However, the traditional MOGWO algorithm converges slowly in the later stage, making it susceptible to local optimization [55]. To efficiently solve the proposed multi-objective and multi-constraint model, this study employs an improved Multi-Objective Grey Wolf Optimization (IMOGWO) algorithm. This approach not only reflects the trade-offs between scheduling time and oil spill losses, but also supports decision-making by offering a solution set that balances both objectives. The solution process of the IMOGWO algorithm is shown in Figure 5. The enhancement integrates crossover and mutation operators from genetic algorithms, improving exploration ability and enhancing computational efficiency. The solution steps are as follows.

Step 1: Initialization phase. At first, it is necessary to set essential parameters, including population size, maximum iterations, archive size, and grid parameters. An initial population of grey wolves is randomly generated within the search space. Each grey wolf represents a candidate solution.

Step 2: Fitness evaluation and archiving of the initial population. After initialization, each wolf’s fitness value is computed based on the objective functions. The non-dominated solutions are stored in an external archive.

Step 3: Iterative optimization process. The best wolves (alpha, beta, and delta) are selected in the archive. The remaining wolves adjust their positions based on the weighted influence of the three leaders. Then, single-point crossover and random mutation are performed with probabilities of 20% and 10%, respectively. All updated wolf positions will be checked against the boundary constraints.

Step 4: Fitness evaluation and archiving. The newly generated solutions are evaluated for fitness. Their dominance is determined relative to the existing solutions in the archive. The archive is updated by storing new non-dominated solutions and removing inferior solutions, resulting in a better Pareto front.

Step 5: Termination and results output. The algorithm continues iterations until reaching the maximum iteration count. It outputs the final Pareto front last.

5. Case Study

In this section, we take the East China Sea as a hypothetical case study area to design a marine oil spill accident. A hypothetical case study is presented to test the effectiveness of the scheduling model for oil containment booms in dynamically changing marine oil spills. The proposed model is solved using MATLAB 2023a to obtain a scientific scheduling scheme for oil containment booms. All experiments are performed on the same computer equipped with a 13th Gen Intel(R) Core (TM) i5-13500H processor, 16 GB of RAM, and a Windows 11 operating system.

5.1. Case Description

An oil spill accident occurred in the Zhoushan sea area, China. It involved a large amount of fuel oil leakage, resulting in marine pollution. With the movement of the damaged vessel, as well as the effects of drift and diffusion, oil spills were divided into 11 oil films. Figure 6 illustrates the emergency rescue scenario for this accident. Boom deployment vessels from three emergency equipment depots in Zhoushan Islands were designated for spill containment. To improve containment efficiency, two boom deployment vessels operated as a group.

In these scenarios, the related parameters of three groups of boom deployment vessels were different, as shown in

Table 1. Data on the groups of boom deployment vessels.

Group No.	Coordinate		Emergency Preparation Time (h)	Velocity (Knots)	Unit Cost of Transportation (CNY/h)
	Longitude	Latitude
a	122.029° E	29.994° N	0.8	15.75	6680
b	121.977° E	30.055° N	0.6	16.98	7020
c	122.227° E	30.24° N	1.15	13.24	6140

. Additionally, the unit cost of the oil boom was 170 CNY/m, and the deployment speed was 30 m/min. To determine the oil boom scheduling plan more accurately, the plan was officially issued when boom deployment vessels were about to reach the accident area (1.5 h after the accident).

At this time, the coordinates and quantities of these oil films were monitored using technology, as shown in

Table 2. Data on the oil films.

Oil Film No.	Oil Quantity (t)	Coordinate
		Longitude	Latitude
1	6.11	122.596° E	29.899° N
2	3.14	122.543° E	29.975° N
3	1.57	122.466° E	30.041° N
4	4.01	122.512° E	29.958° N
5	3.39	122.441° E	30.018° N
6	10.74	122.499° E	30.001° N
7	7.48	122.568° E	29.904° N
8	5.87	122.607° E	29.863° N
9	6.21	122.536° E	29.935° N
10	5.84	122.577° E	29.936° N
11	4.46	122.455° E	29.983° N

.

The impact of external factors on the oil films cannot be ignored during boom deployment.

Table 3. Data on the oil film dynamic characteristics.

Parameter	Value	Unit	Parameter	Value	Unit
Sea water density	1050	kg/m3	Gravity acceleration	9.8	N/kg
Oil density	830	kg/m3	Viscosity coefficient	0.0000017	${\mathrm{m}}^2$/s
Water flow velocity in east-west direction	−0.5	km/h	Net surface tension coefficient	0.03	N/m
Water flow velocity in north-south direction	0.3	km/h	Vertical diffusivity	0.001	${\mathrm{m}}^2$/s
Wind speed at 10 m height	18	km/h	Wind angle	152	°
Wind-induced current coefficient				0.025

provides information on these conditions. To ensure the realism of the simulation scenario, these parameters of the oil spill incident were derived with reference to official oil spill investigation reports published by the Maritime Safety Administration of the People’s Republic of China.

Reducing the economic and ecological losses caused by oil spills through effective containment is one aspect of the strategic efforts in responding to such accidents. Specifically, this study aims to optimize the scheduling of oil containment booms. By promptly restricting the drift and diffusion of oil films, this approach reduces economic and ecological impacts while balancing scheduling time and accident losses. Figure 7 illustrates the spatial distribution of economic industries in the affected sea area.

Table 4. Data on the economic industries in Zhoushan.

Industry Type	Fisheries	Aquaculture	Tourism
Annual unit output value of industry (CNY/hectare/year)	52,110	643,180	435,578

provides the data of economic industries in Zhoushan related to this oil spill accident. The information is based on statistical data and relevant documents issued by the Zhoushan Municipal People’s Government.

Referring to the Technical Guidelines for Marine Ecological Damage Assessment (GB/T 34546-2017) issued by the State Oceanic Administration of China, the affected area is classified as an estuary and bay ecosystem, with an estimated value of CNY 182,950 per hectare per year [42]. To assess the ecological losses caused by the oil spills,

Table 5. Data on the ecological loss evaluation.

Parameter	Value	Unit	Parameter	Value	Unit
Recovery time	5	year	Permissible pollutant concentration	0.01	mg/L
Affected sea depth	0.01	m	Unit cost of wastewater treatment	46	CNY/t
Oil spill toxicity coefficient	1		Impact lever on marine ecosystems	50%
Sensitivity conversion rate	0.03

provides parameters relevant to the evaluation.

5.2. Algorithm Analysis

In order to measure the capability of the proposed IMOGWO algorithm, a comparison is made with the MOGWO algorithm. The parameters of these algorithms are listed in

Table 6. Values for the parameters.

Algorithm	Parameter	Value
IMOGWO	Number of grey wolves	200
	Maximum number of iterations	300
	Archive size	150
	Crossover rate	0.2
	Mutation rate	0.1
MOGWO	Number of grey wolves	200
	Maximum number of iterations	300
	Archive size	150

. Since the optimal solution obtained by the multi-objective algorithm is not unique, this study employs the ideal point method to determine the best scheduling plan. The best solution with the minimum Euclidean distance between each solution in the archive and the theoretical ideal point is selected [56].

Each algorithm is run 10 times, with the results evaluated and compared in

Table 7. Comparison of results in IMOGWO and MOGWO algorithms.

No.	IMOGWO			MOGWO
	Scheduling Time (h)	Spill Losses (CNY)	CPU (s)	Scheduling Time (h)	Spill Losses (CNY)	CPU (s)
1	8.9781	313,676,279.8	111.63	9.0296	321,080,170.9	188.56
2	9.0217	314,019,532.5	156.35	8.9781	325,138,754.2	218.43
3	8.9398	321,812,921.7	132.74	9.0296	318,425,464.1	212.02
4	9.0296	321,429,977	113.68	9.1527	319,144,091.7	173.12
5	9.0652	320,754,878.8	140.81	9.0296	316,776,877	182.72
6	8.9781	315,315,824.8	140.27	8.9781	315,845,136.3	197.00
7	9.0992	315,502,075.5	137.20	9.0652	320,753,957.8	194.63
8	9.1198	320,400,740.5	121.53	9.298	319,177,021.2	200.39
9	8.9398	320,565,434.7	148.76	9.0296	316,776,877	232.79
10	9.1446	318,772,335.3	139.92	9.2431	321,093,914.6	216.89
avg	9.03159	318,225,000.1	134.289	9.08336	319,421,226.5	201.655

and Figure 8. It is observed that the IMOGWO algorithm exhibits a significant advantage in computational time. The ability to generate optimal solutions quickly allows managers to issue scheduling plans without delays. Additionally, the IMOGWO algorithm does not degrade the quality of the solution. As illustrated in Figure 8, boxplots clearly reflect the performance distributions of both algorithms. In terms of scheduling time, the IMOGWO algorithm demonstrates a slightly lower median value and a narrower interquartile range. For oil spill losses, the IMOGWO algorithm not only achieves a lower median loss but also exhibits less variability, confirming its robustness in minimizing economic and ecological impacts. Most notably, in computational efficiency, the IMOGWO algorithm significantly outperforms the MOGWO algorithm with a reduced median CPU time. Due to the probability-based global search mechanism of the MOGWO algorithm, it struggles to find optimal solutions within an acceptable timeframe. In this study, the crossover and mutation operators from the genetic algorithm are incorporated into the MOGWO algorithm. This enhancement balances the search capability and enhances the computational performance. These results verify that the enhancements incorporated into the IMOGWO algorithm are effective in accelerating convergence.

The convergence curves of the best-performing solutions from 10 independent runs are illustrated in Figure 9. From Figure 9a, the two algorithms show comparable convergence speed and ultimately reach similar scheduling time values. Figure 9b reveals a clear advantage of IMOGWO algorithm, which reduces spill-related losses more rapidly and achieves better final values. However, the stepped pattern in MOGWO algorithm reflects premature convergence and stagnation. This is because the IMOGWO algorithm integrates genetic algorithm operators, including single-point crossover and random mutation. These operators enhance the algorithm’s exploration capability, enabling it to escape local optima and better maintain population diversity. This method allows the IMOGWO algorithm to provide a more reliable solution set for emergency scheduling tasks.

5.3. Analysis of Oil Containment Boom Scheduling Results

Ten runs are performed with the IMOGWO algorithm to obtain ten sets of Pareto frontiers. In this study, the optimal solution in 10 runs is selected as the optimal oil boom scheduling plan. In actual emergency response operations, to improve the efficiency of oil spill containment, two boom deployment vessels operate as a pair to simultaneously tow and deploy booms around the oil film. This operation causes the oil to gradually converge toward the center during the deployment process [57]. Therefore, the length of boom required for each oil film is determined by its maximum diameter rather than the full perimeter. This strategy allows effective encirclement without necessitating the entire oil film boundary to be covered. In this scenario, the scheduling time is 8.9781 h, and the spill losses are CNY 313,676,279.8. The results of oil films being controlled by booms are presented in

Table 8. Results of oil films being controlled by oil containment booms.

Oil Film No.	Coordinate		Oil Film Area (km2)	Length of Booms (m)	Boom Deployment Time (h)
	Longitude	Latitude
1	122.576° E	29.909° N	0.50124	800	1.3939
2	122.515° E	29.989° N	0.47467	778	1.3565
3	122.455° E	30.047° N	0.14708	433	0.75509
4	122.490° E	29.969° N	0.40226	716	1.2488
5	122.425° E	30.026° N	0.30354	622	1.0847
6	122.476° E	30.012° N	0.80896	1016	1.7709
7	122.554° E	29.911° N	0.38697	703	1.2248
8	122.580° E	29.876° N	0.67721	929	1.6203
9	122.503° E	29.951° N	0.8536	1043	1.8191
10	122.542° E	29.953° N	0.85326	1043	1.8187
11	122.440° E	29.990° N	0.29804	617	1.0749
Total			5.70683	8700

.

Additionally, under this scheduling scheme, the economic and ecological consequences caused by oil spills are shown in

Table 9. Economic and ecological losses caused by oil films.

Oil Film No.	Area Affected by Oil Film (km2)	Ecological Losses (CNY)	Economic Losses
			Industrial Losses (CNY)	Cost of Booms (CNY)	Transportation Expenses (CNY)
1	2.0077	3,678,286.47	10,461,923.06	136,000	58,033.13
2	2.5156	4,608,803.68	13,108,535.70	132,260
3	0.68138	1,248,371.88	43,824,947.95	73,610
4	1.8	3,297,769.67	9,379,642.64	121,720
5	1.3177	2,414,136.44	57,394,813.33	105,740
6	2.9751	5,450,719.64	15,503,145.28	172,720
7	1.2739	2,333,903.89	6,638,178.72	119,510
8	3.1027	5,684,548.02	16,168,208.91	157,930
9	4.164	7,628,988.62	21,698,661.24	177,310
10	4.2809	7,843,220.03	22,307,986.39	177,310
11	1.1338	2,077,318.15	49,387,136.96	104,890
Total	25.25278	46,266,066.5	265,873,180.2	1,479,000

. It can be seen that deploying oil booms is crucial to contain oil spills. The total oil film area controlled by booms is 5.71 km². However, the total area affected by oil films is much larger at 25.25 km², indicating that oil film is not static. The dynamically changing marine oil spills lead to further economic and environmental damage. If boom deployment takes too long, the oil film may change further.

Economic losses are categorized into industrial losses, boom costs, and transportation expenses. The cost of booms and transportation expenses are the direct costs of emergency efforts. Although boom costs and transportation expenses are included in the model, they constitute only a small fraction of the total loss. In fact, compared to the substantial industrial and ecological damage, these direct emergency costs are negligible. However, to construct a more comprehensive and realistic scheduling model, these costs are still incorporated into the optimization process. Industrial losses are the most substantial, totaling CNY 265.87 million. These losses arise from disruptions to fisheries, tourism, and aquaculture. Coastal industries rely heavily on clean water and healthy ecosystems. However, exposure to oil pollution leads to significant declines in revenue. The industrial losses associated with each oil film vary based on the affected area and industrial value. The highest economic loss was CNY 57.39 million, caused by oil film 5.

Ecological losses arise from the damage to marine and coastal ecosystems. The total ecological losses amount to CNY 46.27 million. These losses depend on the affected area and ecological type. The most ecological loss was CNY 7.84 million, related to oil film 10. Larger affected areas and higher ecological value lead to more serious ecological damage. These ecosystems take years to recover. If emergency efforts fail to contain oil drift and diffusion, the impact on the environment will be more severe.

Figure 10 illustrates the scheduling routes of oil containment booms. The routes of Group a, Group b, and Group c are marked with different colors. Controlled oil film indicates where the oil film is ultimately contained. Each group starts from an emergency equipment depot. It can be observed that it is necessary to schedule as early as possible for those oil films that could result in severe consequences.

The operational details of boom deployment groups are shown in Figure 11 and

Table 10. Scheduling scheme of oil containment booms.

Group No.	Scheduling Planning	Length of Booms (m)	Completion Time of Boom Deployment (h)
a	7→1→8	2432	7.3834
b	11→4→2→10	3154	8.9781
c	3→5→6→9	3114	8.7595

. Each group follows a sequence of emergency preparation, transportation, and boom deployment. The scheme determines the sequence of scheduling and the number of booms required for each group. The timeline shows the tasks assigned to each group and their execution times. Group a is responsible for three oil films. The total boom length used is relatively short compared to the other groups. Group b handles four oil films, requiring the most booms, and takes the longest to complete. The final grouping and scheduling plan comprehensively consider the spatial distribution of oil films, the severity of economic and ecological losses, and the overall completion time. Oil films in closer proximity are grouped to reduce vessel transit time, while those with higher loss values are prioritized to minimize damage. This strategy ensures a balance between operational efficiency and loss control.

In this study, we compare the proposed model with a model that does not account for these losses. Under the same accident scenario, a single-objective scheduling plan is studied that focuses solely on minimizing scheduling time. An Improved Grey Wolf Optimization (IGWO) algorithm is applied, using a population size of 200 and 300 maximum iterations. The algorithm is executed 10 times, with a minimum scheduling time of 8.9188 h. However, under this scheduling plan, the oil spill loss amounted to CNY 412,788,925.6. Without accounting for economic and ecological losses, the scheduling plan fails to prioritize oil films with greater industrial and environmental impact. As a result, the pollution spreads further, increasing overall spill-related damage. In contrast, under the proposed multi-objective scheduling model that considers economic and ecological losses, the scheduling time is 8.9781 h, which is only slightly longer than 8.9188 h. However, the reduction in spill losses by nearly 24% demonstrates the effectiveness of the proposed model. This approach ensures that scheduling prioritizes locations with greater value and achieves rapid containment.

Through numerical experiments, it can be found that the proposed boom deployment model has strong potential for integration into real-world emergency response systems. The model can assist decision-makers in formulating containment strategies. Moreover, the IMOGWO algorithm enhances solution efficiency under complex and dynamic conditions, making it suitable for decision support systems.

6. Conclusions

This study highlights the critical role of efficient boom scheduling in responding to marine oil spills. A scheduling model for oil containment booms in dynamically changing marine oil spills is developed. The model incorporates time-varying oil film parameters, economic and ecological loss quantification, and deployment constraints to better reflect the uncertainty of marine oil spills. The main conclusions are as follows:

(1) Delayed containment leads to severe economic and ecological consequences. Simulations in the Zhoushan sea area show that, compared to a model focused only on scheduling time, the proposed model achieves similar efficiency while cutting spill losses to CNY 313.68 million—a 24% reduction. The results highlight the dynamic nature of spills: while oil films cover 5.71 km², the affected area expands to 25.25 km².

(2) This innovative approach provides decision-making support for effective oil spill response planning. By developing a multi-objective scheduling model, the study demonstrates how integrated consideration of economic and ecological losses can significantly enhance containment effectiveness and reduce spill impact.

(3) Our experiments further demonstrate that the IMOGWO algorithm outperforms the MOGWO algorithm. By integrating the crossover and mutation operators, the IMOGWO algorithm enhances computational efficiency while maintaining high-quality solutions.

The proposed model enables managers to formulate oil containment boom scheduling strategies. However, there are still several limitations that require further research. First, regarding the prediction of oil films, this study accounts for various dynamic changes, such as spill location, area, and concentration. However, the treatment of complex environmental factors remains simplified in the current study. Future research could incorporate more sophisticated models to improve the accuracy of oil spill prediction and boom demand estimation. Second, the simulation scenarios in this study are based on a single hypothetical case in the Zhoushan sea area under fixed environmental conditions. While this helps to demonstrate the feasibility of the proposed model, it limits the generalizability of the findings. Future research could consider dynamic scheduling problems arising from environmental changes. Lastly, the current case study is based on hypothetical scenarios. In future work, applying the proposed model to real oil spill cases with comprehensive field data would enhance the credibility of the approach.