1. Introduction
Due to the recent increase in coastal erosion caused by climate change, the frequency of stranded accidents has been rising in Korea, particularly involving fishing boats and small ships where the ship’s bottom contacts the seabed, making navigation difficult. Although numerical modeling studies to predict coastal erosion are ongoing, such as by Park et al. (2025), accidents involving stranded ships can lead to not only casualties and property damage but also secondary consequences like marine pollution. Therefore, the refloating operation to move a stranded ship to a safe area is essential. This refloating operation is carried out through a refloating plan, which consists of a series of refloating actions such as adding, removing, or transferring cargo to different compartments within the ship (Varsami et al., 2014).
The ultimate goal of a refloating plan is to minimize the refloating force, which is the force required to lift the hull and recover the stranded ship. Unlike normal floating conditions, a stranded ship is acted upon by a ground reaction force at the point of contact with the seabed, in addition to gravity and buoyancy. The magnitude of this ground reaction force is proportional to the refloating force. Therefore, the objective of the refloating plan is to incrementally reduce the magnitude of the ground reaction force by appropriately selecting available refloating actions. If the ground reaction force becomes zero, the ship’s position and attitude can change, creating a risk of it being pushed by small forces and becoming stranded again. For this reason, the ground reaction force must be reduced gradually according to a specified ground reaction force improvement rate. Furthermore, the cost of performing each refloating action must also be considered when developing the refloating plan.
Currently, the HECSALV program from Hebert-ABS is predominantly used for establishing refloating plans. However, HECSALV does not account for domestic conditions and units; instead, plans are formulated by manually inputting refloating actions based on an expert’s experience. In other words, there are limitations in effectively considering various alternatives, and the program also fails to account for the costs associated with the refloating operation. Therefore, the development of an optimization-based program for establishing refloating plans is required to simultaneously consider both the magnitude of the ground reaction force and the associated costs.
Previous related research includes studies that estimate the ground reaction force by considering the contact points between the ship and the ground in actual stranding environments, as well as those that analyze the structural damage to stranded ships. Simonsen et al. (1997) developed an analytical model to calculate the structural damage and the forces acting on the bottom of a ship when it moves while stranded on a reef. By applying this model to actual stranding incidents, they improved the predictive accuracy for bottom damage on stranded ships. Pineau et al. (2023) proposed an analytical model for the damage occurring to stranded ships by analyzing the ship’s structural resistance while simultaneously considering its surge and heave motions. These studies model and analyze the damage that occurs to a ship based on its motion while stranded, enabling its prediction. Im et al. (2025) analyzed the impact of a clump weight (CW) from a mooring system (connecting a floating offshore wind turbine to an anchor point) contacting the seabed, confirming that seabed contact affects the motion stability and structural reliability of the floating structure. Seo et al. (2024) proposed a method to simulate real-world ground reaction force distribution. They first calculated the magnitude of the ground reaction force at a specific point on the ship’s hull using force equilibrium conditions and then distributed this force to two or multiple contact points based on a force distribution model. Additionally, Lee et al. (2022) proposed a method to derive the final equilibrium state by using force and moment equilibrium equations. They iteratively calculated the changes in the magnitude and points of application for gravity, buoyancy, and ground reaction force resulting from refloating actions like cargo shifting, addition, or removal. However, their work did not extend to a three-dimensional analysis. In relation to costs, Hussein et al. (2016) proposed charting a method for the rapid refloating of a stranded double-hull tanker. This method involves transferring weight from cargo tanks to ballast tanks while considering the ship’s structural strength, with the goal of minimizing risks and damages during an actual stranding incident.
Most of the studies reviewed above focus on predicting damage to stranded ships, calculating the magnitude of the ground reaction force acting on them, and simulating the actual ground reaction force by distributing it over points or areas. However, there is a lack of research on establishing plans for the execution of actual refloating operations and on the optimization problem of considering both the refloating force and associated costs. To establish an effective refloating plan, it must be possible to comprehensively and quantitatively evaluate the forces and moments each refloating action has on the stranded ship, the resulting changes in the hull’s attitude, and the feasibility of maintaining equilibrium. Based on this evaluation, there is a need to develop a plan using multi-objective optimization techniques that can simultaneously minimize both the ground reaction force and costs.
Therefore, in this study, we numerically calculate the equilibrium state of a stranded ship using a modified nonlinear hydrostatic analysis method (Kim and Ham, 2025), which adapts the existing nonlinear hydrostatic analysis method of Park et al. (2018) to be applicable to stranded ships. Based on this, we established an optimal refloating plan by applying a genetic algorithm. The type of refloating action and the amount of cargo were set as design variables to create a plan that minimizes both the ground reaction force and costs while maintaining the ship’s equilibrium.
2. Dynamics of Stranded Ship
2.1 Forces Acting on Stranded Ship
A stranded ship is subjected to three forces. Gravity (FG) due to the hull’s weight, buoyancy (FB) generated by the submerged volume of the hull, and the ground reaction force (FR), which is the force supporting the ship from the ground where the ship’s bottom is in contact. In particular, gravity must be calculated carefully, while operational data should be referenced, the calculation must also reflect the current stranding situation, as cargo or other weights may have shifted during the incident.
These forces act as illustrated in Fig. 1. Gravity acts at the ship’s center of gravity, buoyancy at the center of buoyancy, and the ground reaction force at the point of stranding. The ground reaction force is equal to the loss of buoyancy that occurs when a part of the hull emerges from the water during the stranding. That is the difference between gravity and buoyancy (Eq. (1)). Usually, when a ship runs aground, it is in contact with the ground over an area rather than at a single point. In such cases, the calculated ground reaction force must be distributed over the contact area, taking the ship’s attitude into account. Furthermore, the shift in the center of rotation, which depends on the size and shape of the contact area, must also be considered. In this study, however, we assume a single point of support, as shown in Fig. 1, rather than an area, and propose a method for calculating the ship’s equilibrium attitude under this assumption.
2.2 Calculation Method to Find Equilibrium Position of Stranded Ship
For a ship to maintain equilibrium, the forces and moments acting on it must be equilibrium (Beer et al., 2012). For a stranded ship, the ground reaction force must be considered in addition to the gravity and buoyancy that act on a typical ship. Park et al. (2018) proposed a nonlinear hydrostatic analysis method that derives a ship’s equilibrium attitude by iteratively calculating the infinitesimal changes in its attitude in response to changes in the acting forces and moments in a static environment (Eq. (2)).
(2)
F: The sum of forces acting on a ship
MT: Sum of transverse moments due to force
ML: Sum of longitudinal moments due to force
MG,T, MB,T: Transverse moment due to gravity and buoyancy
MG,L, MB,L: Longitudinal moment due to gravity and buoyancy
δz: Infinitesimal changes in immersion of the ship on the O-frame
δφ: Infinitesimal changes in heel of the ship on the O-frame
δθ: Infinitesimal changes in trim of the ship on the O-frame
At this time, the force due to gravity has no change from the infinitesimal changes of immersion, heel, and trim, and the transverse and longitudinal moments due to gravity have no change from the infinitesimal change of immersion. Likewise, as it is an infinitesimal attitude change, the transverse and longitudinal moments due to gravity are unchanged by the infinitesimal change of trim and the infinitesimal change of heel, respectively. The internal differential terms of Eq. (2) are referenced from Park et al. (2018).
Kim and Ham (2025) apply this to a stranded ship. For a stranded ship, unlike a conventional ship, the influence that the ground reaction force (occurring from the contact of the lower hull with the ground) has on the ship’s infinitesimal attitude changes must be considered. Furthermore, the attitude change of a stranded ship becomes dependent on the point of stranding. Therefore, unlike the free-floating condition assumed in conventional analysis methods, a modification of the attitude analysis frame based on the fixed point of stranding is necessary. Additionally, the attitude change of the stranded ship must be defined not in the O-frame (Original frame), which is based on the ship’s center, but in the R-frame (Reaction-based frame), which is based on the point of stranding. The infinitesimal attitude changes (δφ, δθ) due to heel and trim in both coordinate systems are the same as before, but the infinitesimal attitude changes (δz) due to immersion and the forces and moments acting on the stranded ship require transformation into the R-frame.
The velocity of a point on a rotating body is expressed as the cross product of the angular velocity and the position vector from the axis of rotation to that point. Since the stranded ship rotates about the point of application R of the ground reaction force, the velocity of the ship’s center O with respect to point R is as shown in Eq. (3).
ErO/R: The vector from point R to point O
Er̃O/R: ErO/R Skew-symmetric matrix represented as the matrix of the vector’s cross product (Blundell and Harty, 2004)
However, the angular velocity component is expressed through the changes (φ, θ, ψ) in the ship’s motion using the transformation formula proposed by Shabana (2009) (Eq. (5)).
(6)
In Eq. (6), the yaw (ψ) component unnecessary for the stranding equilibrium calculation is set to zero, and the unnecessary changes in x and y are also eliminated to derive the relationship between the z velocity component and the ship’s attitude change (Eq. 7).
By substituting the differential term with a small attitude change, we derive the relationship equation between the center of the R-frame and the center of the stranded ship. Using this, we derive the relationship equation J matrix (Eq. (8)) between the small attitude change (δφR, δθR) of the stranded ship and δz, and incorporate this into Eq. (2).
The force and moment terms of Eq. (2) must be transformed into R-frame values using the J matrix. By multiplying both sides of the equation by the transpose of the J matrix, the internal variables within each term are converted into their R-frame values. When the equation is then rearranged in terms of the stranded ship’s attitude change, Eq. (9) is derived. This equation calculates the infinitesimal change in the stranded ship’s attitude based on the magnitude of the forces and moments acting upon it.
(9)
Once the infinitesimal change in the stranded ship’s attitude is calculated via Eq. (9), this change is used to update the existing attitude. According to the updated attitude, the magnitude and points of application of the forces and moments acting on the stranded ship will also change, which requires the change in attitude to be recalculated.
Fig. 2 is a flowchart of the process for determining the equilibrium attitude of the stranded ship through iterative calculations. First, information such as the stranded ship’s attitude and the point of stranding is input. Based on this information, the forces and moments acting on the stranded ship are calculated. The calculated forces and moments are then compared with a threshold value to determine if the ship is in equilibrium. If the values are smaller than the threshold, the ship is considered to have reached equilibrium. If they are larger, the infinitesimal change in the stranded ship’s attitude is calculated in the R-frame using Eq. (4). This calculated value is then transformed into the infinitesimal attitude change in the O-frame using Eq. (8). The calculated infinitesimal attitude change is used to update the existing attitude. Based on the updated information, this process is repeated until the stranded ship reaches equilibrium, thus determining its final equilibrium attitude.
3. Optimization of Refloating Plan
To perform an optimization, it’s necessary to first define the problem being targeted. This requires clearly establishing the design variables, constraints, and objective functions suitable for the problem’s structure. To achieve this, an analysis of the structural characteristics of the refloating plan must be conducted beforehand.
3.1 Design Variables
A refloating plan is a series of operational plans for managing the cargo inside a stranded ship to safely recover it, and it is composed of different refloating actions. A refloating action is defined by the Action type (Action), such as the addition, removal, or movement of cargo; the compartments of the stranded ship where the work is performed (RFrom, RTo); and the amount of cargo handled in each operation (Load). In this study, the action type and the corresponding cargo amount for each individual refloating action that constitutes the refloating plan were set as the design variables. For example, assume there is a stranded ship with two compartments, A and B, loaded with 500 t and 400 t of cargo, respectively. If 50 t of cargo is moved from compartment A to compartment B for the ship’s refloating, the composition of the refloating action is as follows: The Action is the movement of cargo, the compartment where the cargo originates (RFrom) is A, the cargo’s destination compartment (RTo) is B, and the amount of cargo moved (Load) is 50 t (Eq. (11)).
It’s assumed that only one operation—whether adding, removing, or transferring cargo—is performed at a time. In other words, an operation to add cargo to one compartment while simultaneously removing it from another isn’t performed. Likewise, during a cargo transfer, cargo isn’t moved from a single compartment to two or more other compartments. This is a measure to prevent sudden changes in the ship’s attitude and is a method that’s applied in real-world, on-site operations.
3.2 Constraints
During the execution of a refloating plan, the ship’s equilibrium must be maintained. The equilibrium status of the stranded ship is checked after each refloating action using the equilibrium attitude calculation method from Kim and Ham (2025). If the calculation shows that equilibrium isn’t reached, that specific refloating action is judged as having failed to maintain the ship’s equilibrium. To recover the stranded ship and prevent it from stranding again, the ground reaction force must be minimized. However, if the reduction in ground reaction force is too large, it can cause a loss of equilibrium and lead to structural problems. Accordingly, this study introduces a minimum ground reaction force improvement rate (FR_min_improved) to ensure the ground reaction force is improved beyond a certain level (thus reducing the refloating force), while also limiting any negative impacts from this reduction. Therefore, in this study, the constraints were defined as follows: the equilibrium condition must be met after the refloating plan is applied, and the ground reaction force improvement rate must be greater than or equal to a preset threshold (Eq. (12)).
FR_inial: Magnitude of ground reaction force acting on the initial stranded ship
FR_optimal: Magnitude of ground reaction force acting on the stranded ship calculated according to optimized design variables
FR_min_improved: Minimum ground reaction force improvement required for refloating a stranded ship
3.3 Objective Functions
The main objective of a refloating plan is to minimize the total cost of recovering a stranded ship. This cost is composed of the ground reaction force (which is proportional to the refloating force) and the operational cost associated with each refloating action. The ground reaction force is improved according to the minimum ground reaction force improvement rate condition, and the cost for each refloating action is calculated by multiplying the unit cost coefficient for that action type (CAction_i) by the corresponding amount of cargo (Eq. (13)).
n: The number of refloating actions comprising the refloating plan
Actioni: ith refloating action
CAction_i: Actioni’s cost coefficient
LoadAction_i: Actioni’sLoad
The unit cost coefficient for each action type varies depending on the type of cargo tank and the refloating action, and it was established based on expert opinion. For example, CAction_i for a W.B.T (Water ballast tank) is lower than that for a C.O.T (Crude oil tank). By refloating action type, CAction_i is highest for cargo removal, followed by addition, and is lowest for movement (Table 1).
As the range of design variables—such as the stranded ship’s compartments and cargo amounts—expands and the number of combinations for refloating actions increases, the number of possible solutions grows exponentially. In this study, NSGA-II (Non-dominated sorting genetic algorithm II), a metaheuristic optimization technique that mimics natural selection and genetic principles, was applied (Deb et al., 2002). NSGA-II is well-suited for this problem as it can derive global optimal solutions for various combinations of design variables, making it effective for handling a complex search space.
4. Application
4.1 Program for Optimum Refloating Plan
In this study, we developed a GUI (Graphical user interface) based program to simulate a ship’s stranding situation in 3D and perform optimization. Fig. 3 shows the four detailed modules used to develop the program and the relationships between them. Each module performs the following functions:
User interface module: It visualizes the ship’s refloating plan and information in 3D and makes the program easy for the user to use.
Model structure module: The ship is constructed with a hull form and compartments that are created based on a Geometry framework.
Equilibrium module: It uses the modified nonlinear hydrostatic analysis method to iteratively calculate the infinitesimal change in the ship’s attitude. This calculation is repeated until the forces and moments acting on the ship are below a threshold value, meaning equilibrium has been reached.
Optimization module: It sets up the refloating plan optimization problem by defining the design variables (the Action: cargo movement, addition, or removal; RFrom: the target cargo compartment; RTo,: the destination compartment for movement; and Load: the amount of cargo), the constraints (the equilibrium maintenance condition and the minimum ground reaction force improvement rate condition), and the objective function (the refloating cost). The module then uses the Equilibrium module to calculate the forces acting on the stranded ship and its equilibrium attitude, and proceeds with the optimization based on the NSGA-II algorithm.
The program developed in this study is shown in Fig. 4. It was developed based on C# and WPF (Windows presentation foundation), and it uses the Eyeshot library to visualize the stranded ship and its surrounding environment in 3D. The hull form and compartments can be viewed, and attitude changes are displayed across four split screens.
The program’s execution process is as follows. First, the stranded ship’s specifications and point of stranding are input, and the ship is visualized. Next, its compartments are modeled, and the amount of cargo in each is set. This is used to calculate the ship’s initial attitude. After this, a list of possible refloating actions is automatically generated, and the optimization calculation begins. Through optimization, a refloating plan is established from the list of possible actions that minimizes both the ground reaction force and cost.
Fig. 5 shows a graph of the ship’s ground reaction force and attitude changes according to the refloating plan. This allows for the analysis of the effect that each refloating action has on the ground reaction force and the attitude of the stranded ship.
4.2 Test for Single Action Effect on Stranded Ship
In this section, the attitude and ground reaction force of the stranded ship are calculated as the result of a single refloating action using optimized refloating planning program. The compartments of the stranded ship used for the test are shown in Fig. 6, and its specifications are summarized in Table 2. The weight and center position of the cargo or ballast water stored in each compartment are as shown in Table 3.
The cargo amount (Load) for each refloating action was fixed at 100 tons, and tests were performed by dividing the problem into three Cases based on the type of refloating action. The possible Actions are cargo addition, removal, and movement. The target RFrom and RTo can be any of the nine compartments, and these two design variables cannot be the same compartment. Case 1 is the addition of cargo to RFrom NO.1 C.O.T; Case 2 is the removal of cargo from RFrom NO.3 C.O.T; and Case 3 is the movement of cargo from RFrom NO.2 C.O.T to RTo NO.1 C.O.T (Fig. 7). The resulting reduction in ground reaction force and the cost for each case are shown in Table 4. When comparing the addition, removal, and movement of the same amount of cargo, moving it (as in Case 3) results in the greatest improvement in ground reaction force and is also the least expensive option.
The improvement in ground reaction force and the cost coefficient differ depending on the type of refloating action. When other design variables, such as the cargo amount, are also considered, the number of possible solution combinations increases exponentially. Using Model 1 as an example, for the six W.B.Ts there are 6 addition, 6 removal, and 30 movement actions. For the three C.O.Ts, there are 3 addition, 3 removal, and 6 movement actions. This results in 54 types of refloating actions alone, and when the cargo amount is considered, even more combinations exist. Therefore, to establish an optimal refloating plan that simultaneously minimizes both ground reaction force and cost, effective optimization is necessary.
4.3 Optimization of Refloating Plan
4.3.1 Definition of target ship
In the optimization process for minimizing the ground reaction force and cost, the ground reaction force must be reduced beyond a certain level to ensure the feasibility of refloating. In this study, the ground reaction force improvement rate is defined as the ratio of the reduction in ground reaction force (achieved through a refloating action) to the current ground reaction force. This study targets a very large crude-oil carrier (VLCC). The problem was divided into multiple Cases based on different preset values for the minimum ground reaction force improvement rate, and optimization was performed for each case. The specifications of the model used are shown in Table 5 and Fig. 8, while the initial cargo amount for each compartment and the points of application of forces are presented in Table 6.
Based on the initial ground reaction force acting on the stranded ship, an optimization study was conducted by setting the minimum ground reaction force improvement rate to 5%, 10%, 15%, and 20%, respectively. The minimum ground reaction force improvement rate prevents the ground reaction force from becoming zero or decreasing too rapidly, which could cause significant changes in the stranded ship’s attitude and position and lead to a loss of structural stability. The possible Action are cargo addition, removal, and movement. The target RFrom and RTo can be any of the 25 compartments, and these two design variables cannot be the same compartment. Furthermore, during a cargo movement, cargo can only be transferred between compartments of the same type (i.e., from a W.B.T to another W.B.T, or from a C.O.T to another C.O.T). Even without considering the cargo amount, there are 240 possible refloating actions for the 15 C.O.T compartments and 110 possible refloating actions for the 10 W.B.T compartments. When the cargo amount is included as a variable, the number of possible choices becomes immense.
4.3.2 Case 1: 5% reduction of ground reaction force
Optimization was performed for Case 1, in which the minimum ground reaction force improvement rate was set to 5%. The resulting refloating plan consists of the following three refloating actions (Fig. 9).
(5) Action 1 = (Transfer, NO.1 W.B.T (S), NO.5 W.B.T (P), 423 t)
(6) Action 2 = (Transfer, NO.2 W.B.T (S), NO.4 W.B.T (P), 676 t)
(7) Action 3 = (Transfer, NO.1 W.B.T (S), NO.3 W.B.T (P), 330 t)
As each refloating action was performed, the ground reaction force acting on the stranded ship decreased from an initial 139,124 t to 133,729 t, 133,052 t, and finally 132,459 t, representing a total reduction of 5.1% from the original value. The Trim changed from an initial −3.4 deg to −3.46, −3.52, and −3.49 deg, while the Heel changed from an initial −8.4 deg to −10.08, −10.42, and −10.71 deg (Fig. 10).
4.3.3 Case 2: 10% reduction of ground reaction force
Optimization was performed for Case 2, in which the minimum ground reaction force improvement rate was set to 10%. The resulting refloating plan consists of the following three refloating actions (Fig. 11).
(8) Action 1 = (Transfer, NO.1 C.O.T (S), NO.4 C.O.T (P), 4120 t)
(9) Action 2 = (Transfer, NO.2 W.B.T (S), NO.4 W.B.T (P), 1901 t)
(10) Action 3 = (Transfer, NO.3 C.O.T (S), NO.5 C.O.T (P), 751 t)
As each refloating action was performed, the ground reaction force acting on the stranded ship decreased from an initial 139,124 t to 128,137 t, 125,825 t, and finally 124,596 t, representing a total reduction of 11.5% from the original value. The Trim changed from an initial −3.4 deg to −3.63, −3.77, and −3.81 deg, while the Heel changed from an initial −8.4 deg to −13.49, −14.82, and −15.31 deg (Fig. 12).
4.3.4 Case 3: 15% reduction of ground reaction force
Optimization was performed for Case 3, in which the minimum ground reaction force improvement rate was set to 15%. The resulting refloating plan consists of the following three refloating actions (Fig. 13).
(11) Action 1 = (Transfer, NO.1 C.O.T (S), NO.3 C.O.T (P), 4354 t)
(12) Action 2 = (Transfer, NO.2 C.O.T (S), NO.5 C.O.T (P), 5404 t)
(13) Action 3 = (Transfer, NO.1 C.O.T (S), NO.3 C.O.T (P), 2458 t)
As each refloating action was performed, the ground reaction force acting on the stranded ship decreased from an initial 139,124 t to 129,018 t, 120,527 t, and finally 117,546 t, representing a total reduction of 15.6% from the original value. The Trim changed from an initial −3.4 deg to −3.57, −3.84, and −4.03 deg, while the Heel changed from an initial −8.4 deg to −13.78, −16.42, and −19.90 deg (Fig. 14).
4.3.5 Case 4: 20% reduction of ground reaction force
Optimization was performed for Case 4, in which the minimum ground reaction force improvement rate was set to 20%. The resulting refloating plan consists of the following three refloating actions (Fig. 15).
(14) Action 1 = (Transfer, NO.1 C.O.T (C), NO.3 C.O.T (P), 7324 t)
(15) Action 2 = (Transfer, NO.1 C.O.T (S), NO.2 C.O.T (P), 6947 t)
(16) Action 3 = (Transfer, NO.2 C.O.T (C), NO.4 C.O.T (P), 6034 t)
As each refloating action was performed, the ground reaction force acting on the stranded ship decreased from an initial 139,124 t to 127,373 t, 113,542 t, and finally 109,577 t, representing a total reduction of 21.3% from the original value. The Trim changed from an initial −3.4 deg to −3.67, −4.08, and −4.32 deg, while the Heel changed from an initial −8.4 deg to −17.53, −22.74, and −28.83 deg (Fig. 16).
5. Conclusions and Future Works
In this study, we developed a program that establishes an optimal refloating plan to minimize the ground reaction force acting on a stranded ship and the recovery cost, by utilizing a modified nonlinear hydrostatic equation and the NSGA-II genetic algorithm. The program was designed to quantitatively analyze the changes in the hull’s equilibrium attitude and the corresponding changes in the ground reaction force according to various refloating actions, such as adding, removing, or moving cargo within the stranded ship’s compartments. By setting the refloating actions as design variables and applying the genetic algorithm, an optimization process was performed. The results of applying this to the various ship models presented in Fig. 6 and Fig. 8 confirmed that it is possible to derive an optimal refloating plan that effectively reduces the ground reaction force and cost. This study established a refloating plan based on the external forces and equilibrium state of the stranded ship. It also presents the possibility for future research to expand this into a more sophisticated program suitable for the characteristics of domestic waters by additionally incorporating marine environmental conditions of the stranding site, such as tidal changes, water depth, and currents.
In future research, we intend to comprehensively reflect the dynamic behavior of the stranded ship due to environmental loads such as ocean currents and waves, as well as suction and friction forces from mud. Furthermore, for verification purposes, we plan to proceed with on-site application through collaboration with domestic private maritime rescue companies.
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