Multi-Objective Optimization Design for Column Structures of the Semi-Submersible Drilling Platform Using a Hybrid Criteria-Based Parallel EGO Algorithm

Abstract

In engineering design for semi-submersible drilling platforms, it is necessary to improve anti-collision performance by optimizing the platform’s column’ structure. However, collision is usually analyzed through numerical analysis methods such as finite element analysis, which comes with high calculation costs. The genetic algorithm (GA) and other traditional optimization methods require massive numerical simulations, with unacceptable computational complexity. To address the above problems, a parallel efficient global optimization (EGO) multi-objective algorithm, based on hybrid criteria for the Kriging approximate model, is put forward in this paper. The proposed algorithm was validated through six typical multi-objective optimization test functions. The results show that it is superior to classic EGO, in terms of both optimization results and computational efficiency. Lastly, the hybrid criterion-based parallel EGO algorithm proposed in this paper was employed for the anti-collision, lightweight design of the column of the first ice zone semi-submersible drilling platform in China. It was found that the anti-collision capacity of the platform column rose by 11.9% and the structural weight declined by 2.7 t in the optimized design, suggesting obvious optimization effects with respect to the original design.

1. Introduction

With the global economic boom driving steady growth in oil consumption, countries are increasingly turning to deep-sea oil development. As key offshore operation facilities for drilling and oil extraction, the number of semi-submersible platforms is rising alongside the offshore oil demand, leading to a higher risk of ship collisions [1,2]. Such collisions generate massive impact forces, causing severe structural damage to these large-scale facilities. Moreover, the harsh working environment of semi-submersible platforms makes component repair more difficult and costly. Thus, improving the platform’s anti-collision capacity during the design stage has become a crucial research focus. China’s “North Dragon” semi-submersible drilling platform, designed to meet Norwegian Maritime Directorate and offshore industry standards, operates in the North Sea and Barents Sea with a maximum water depth of 1200 m, maximum drilling depth of 8000 m, and service temperature of −25 °C. It can withstand once-in-a-century storms, features an advanced automated drilling system (boosting efficiency by 15% via parallel automatic pipe arranging), and is uniquely capable of Arctic operations—making structural optimization to enhance its anti-collision and working performance practically significant. Figure 1 displays the “North Dragon” semi-submersible drilling platform.

To study the dynamic response of large-scale structures to ship collisions, the nonlinear finite element method is widely used. It accounts for material nonlinearity, structural large deformation, local damage, and overall failure to obtain parameters like stress, strain, and knock-out in the collision area. Combined with internal and external mechanism analysis, it can accurately simulate collision processes, partially replacing ship model and real-ship tests. Zhou et al. [3] analyzed ship collision impacts on a 1018 m cable-stayed bridge via numerical simulation to validate Eurocode’s collision force accuracy; Storheim and Amdahl [4] performed nonlinear finite element modeling of a 7500-ton supply vessel to study the ship’s collision damage to floating semi-submersibles and fixed jackets; Sha et al. [5] optimized the design of a steel bridge girder through finite element analysis of ship–bridge collision structural responses; Rudan et al. [6] calibrated ship collision simulation parameters via finite element analysis of LPG ship–ferry collisions; and Fernandez et al. [7] compared supply vessel bow modeling methods through finite element analysis of supply vessel–FPSO collisions.

Despite their high accuracy and adaptability to complex scenarios, numerical analysis methods (e.g., finite element analysis) have high computational complexity and long cycles, especially in structural optimization, which requires numerous simulations and may even make optimization unfeasible. In simulation-based optimization, approximate models (e.g., polynomial regression surface (PRS) [8], Kriging [9,10], radial basis function (RBF) [11], multivariate adaptive regression splines (MARSs) [12], and inductive learning models [13]) construct input–output relationships via limited simulations to replace time-consuming black-box models.

Traditional surrogate-based optimization first builds surrogates for time-consuming objective/constraint functions, then optimizes them with meta-heuristic algorithms (e.g., genetic algorithms) [14]. Static surrogates need high accuracy across the entire design space (requiring many sampling points), while surrogate-assisted meta-heuristic algorithms [15] dynamically update surrogates but still rely on population evolution, leading to insignificant reductions in simulation time. In contrast, surrogate-based optimization algorithms (e.g., adaptive response surface method (ARSM) [16], efficient global optimization (EGO) [17], mode-pursuing sampling (MPS) [18]) explore the original problem by adding renewal points, integrating global/local search, and fully utilizing surrogate information, requiring fewer objective function calculations. Among these, EGO (based on Kriging) is widely studied due to its simple framework and high efficiency. Surrogate-based optimization approaches, such as Kriging-assisted EGO, have shown promise in reducing computational burden while maintaining accuracy. These methods are particularly valuable in contexts where high-fidelity simulations are required, such as in the vibration analysis of nanocomposite-enhanced shell structures [19,20].

The classic EGO algorithm [21] works as follows: it builds an initial Kriging model with sample points; then, it iteratively selects the point with the maximum expected improvement (EI) as the renewal point, calculates its true target value, and updates the model until optimization is achieved. In engineering, multi-objective optimization often relies on mature meta-heuristic algorithms (e.g., NSGA-II [22]). However, these require thousands of objective function calculations, rendering it unsuitable for time-consuming tasks. EGO’s efficiency in single-objective optimization has led to its application in multi-objective scenarios, mainly by extending single-objective EI to multi-objective EI: defining a multi-objective improvement function (e.g., Euler distance, maximum/minimum distance, hypervolume (HV)) to measure improvement on the non-dominated front, and then integrating it in the non-dominated area. Relevant research includes that by Keane [23] (Euler distance EI), Bautista [24] and Svenson [25] (maximum/minimum distance EI), and HV-based EI (initially a screening criterion [26], later used as an EGO update criterion [27]). Other up-to-date optimization schema include the Adam Optimizer [28,29]. The adaptive method optimizer is a stochastic gradient descent method that combines momentum and per-parameter learning rate scaling for the efficient and robust training of deep learning models in engineering.

The standard EGO is serial (with one renewal point per iteration), which makes it incompatible with prevalent parallel computing hardware. Parallel EGO selects multiple renewal points per iteration for parallel computing (on multiple computers/cores), accelerating convergence and reducing total optimization time. Though it requires more real target calculations, simultaneous computing shortens the length of time, i.e., it exchanges computing resources for time, which is engineering-significant given the abundant resources and valuable time.

In summary, two significant challenges arise when extending EGO to practical engineering design: (1) Engineering problems inherently involve multiple, often competing, objectives. While several multi-objective EI criteria have been proposed, each criterion has inherent biases. (2) The standard EGO algorithm is inherently serial. This serial nature fails to leverage the now-ubiquitous parallel computing resources, becoming a critical bottleneck for reducing total optimization time. To overcome these limitations, this paper puts forward a parallel EGO multi-objective algorithm based on the Kriging approximate model with hybrid criteria. Our primary motivation is to develop a more robust and efficient optimization scheme for expensive black-box functions by synergistically combining the strengths of multiple EI criteria to guide parallel sampling. The proposed algorithm introduces a novel hybrid criteria-based parallel sampling method, obtained by combining the merits of the Euler distance, maximum and minimum distance, and HV improvement criteria, taking as a basis the classic EGO algorithm. During the iteration process, the genetic algorithm (GA) is employed in parallel to achieve hybrid criteria-based optimal sampling, so that multiple optimizations, allowing the surrogate model to be gradually updated, can be determined in parallel in each iteration. The algorithm allows parallel computing on multiple machines or multiple cores, avoiding the shortcomings of iterating through serial sampling points, which makes falling into local optimizations more likely. Global optimizations can thus be easily achieved, improving computational efficiency.

This paper is structured as follows: The numerical simulation analysis of collisions between supply vessels and semi-submersible platforms is described in Section 2. Section 3 presents the research on the Kriging approximate model, and the hybrid criteria-based parallel EGO algorithm. The validation of the newly proposed method, using a typical multi-objective optimization test function, is delineated in Section 4. Section 5 presents the application of the method in the optimization design of anti-collision structures for the semi-submersible platform. Finally, Section 6 presents the conclusions of this work.

2. Numerical Simulation Analysis of Collisions Between Supply Vessels and Semi-Submersible Platforms

The dynamics of the internal structure and the external flow field involved in a collision between a supply vessel and a semi-submersible platform is a complex coupling process. Therefore, when studying these dynamics, emphasis is often placed on the key facts, while simplifications of other aspects in the calculations are necessarily made. In this paper, the added mass method is adopted to simulate the effects of the external fluid in the collision. Focusing on the working conditions typical of such a collision, the structural finite element model with the appropriate added mass factor, material parameter and grid density is used to study the resulting structural damage, and to analyze the factors affecting the anti-collision ability of the platform.

2.1. Finite Element Explicit Solution in the Collision

As the main method currently employed to study vessel collisions, the finite element numerical simulation method is not only capable of intuitively showing the composition of and damage to each structure in the collision, but the results obtained are also much more accurate than those that can be obtained with other methods.

The explicit solution finite element method is generally used for collision analysis. In the overall coordinate system, the explicit differential equation characterizing motion is given as follows:

(1)$\mathrm{M}{a}_n+\mathrm{C}{v}_n+\mathrm{K}{d}_n={{\mathrm{F}}_n}^{ext}$

where a_n, v_n and d_n are the acceleration, velocity and displacement vector, respectively, M, C and K represent the mass, damping, and rigidity matrix, respectively, and ${{\mathrm{F}}_n}^{ext}$ stands for the external load.

Let $F_n^{\mathrm{int}}=C{v}_n+K{d}_n$, $F_n^{re}=F_n^{ext}-F_n^{\mathrm{int}}$; Equation (1) can be written instead as follows:

(2)$\mathrm{M}{a}_n={{\mathrm{F}}_n}^{re}$

where $F_n^{\mathrm{int}}$ is the internal load of the structure, and $F_n^{re}$ is the surplus load of the structure.

A solvable linear equation requires the mass matrix to be a diagonal matrix, so the acceleration can be obtained as follows:

(3)$a_n={\mathrm{M}}^{-1}{{\mathrm{F}}_n}^{re}$

The central difference method is employed for collisions. Thus, this method is used for time integration to obtain the velocity and the displacement, which can be shown as follows:

(4)$\left\{\begin{array}{l}{v}_{n+{\displaystyle \frac{1}{2}}}=v_{n-{\displaystyle \frac{1}{2}}}+a_n\left(\Delta t_{n+{\displaystyle \frac{1}{2}}}+\Delta t_{n-{\displaystyle \frac{1}{2}}}\right)/2\\ d_{n+1}=d_n+v_{n+{\displaystyle \frac{1}{2}}}\Delta t_{n+{\displaystyle \frac{1}{2}}}\end{array}\right.$

where the subscript $n+{\displaystyle \frac{1}{2}}$ denotes the forward half-time step, while the subscript $n-{\displaystyle \frac{1}{2}}$ denotes the backward half-time step. $v_{n+{\displaystyle \frac{1}{2}}}$ is the instantaneous velocity after the force acts and before the displacement is updated, representing the initial state of motion from $t_n$ to $t_{n+1}$.

$F_n^{\mathrm{int}}$ integrates damping forces $\mathrm{C}{v}_n$ and stiffness-induced forces $\mathrm{K}{d}_n$, and this grouping does not neglect damping forces but aligns with the explicit finite element framework for transient collision analysis, where short-duration impact responses prioritize computational efficiency. Damping forces contribute to energy dissipation and are non-negligible for dynamic equilibrium. Notably, Newton–Raphson schemes (common in implicit nonlinear static analysis) are not adopted here, as they require iterative-stiffness matrix inversion and are inefficient for transient collision problems; instead, the explicit central difference method is used to enable the fast calculation of transient accelerations and displacements.

2.2. Dimensions of Vessel Body and Platform and Finite Element Model

The platform under consideration has four columns and two lower floating bodies; its main dimensions are shown in Table 1. For the finite element structure of the semisubmersible drilling platform, the collided part of the column was modeled with plastic materials. Since the research on the collision damage and hydrodynamic characteristics focused on the column and lower floating body area, the structures of the deck and the above were simplified. In Figure 2, (a) represents the semi-submersible platform model, and (b) depicts the internal support structure of the column and the floating pontoon.

A collision between a vessel and semi-submersible platform is usually of one of the following main types: a bow collision, a side collision, or a stern collision. Since in the case of supply vessels, a bow collision with the platform column is much more dangerous than other collisions, this type was chosen as the research object in this paper. We mainly considered the collision force and collision depth of the semi-submersible platform. For this reason, only the bow part of the supply vessel was constructed when establishing the finite element model, and the model was simplified into a rigid model. The principal dimensions of the collision vessel considered in our study are shown in Table 2, and the finite element model is shown in Figure 3.

There are many factors at play in the collision between a vessel and a semisubmersible platform. In fact, there exist a variety of complex combinations of collision parameters, and it is impossible to take all the scenarios into consideration. Therefore, the most common collision parameter combination was chosen to study the collision performance of the structure, corresponding to the typical scenario in which the vessel collides with the platform column when moving forward with a velocity of 2 m/s. A finite element model of the head-on collision between a vessel and a platform column is shown in Figure 4.

It should be noted that a shell element was selected as the element type and the number of nodes was about 150,000.

2.3. Characteristic Analysis Under Typical Collision Working Conditions

The above condition was selected as a typical working condition to analyze the load, structural damage, and energy conversion during the collision. The collision depth changes under the typical working condition chosen, as shown in Figure 5. It can be seen that the collision depth of the supply vessel reaches a maximum of 987.5 m at 1.0 s after the collision, and the remaining energy of the vessel is not sufficient to resist the elastic deformation of the platform components. The effective contact area is then reduced by the reverse movement of the vessel, so the collision depth is gradually decreased.

The energy conversion duration curve of the collision system under the typical working condition is shown in Figure 6. It can be observed that the energy of the system continuously decreases, while the internal energy and hourglass energy constantly increase during the entire collision process. That is to say, the energy of the system is converted into internal energy and a small amount of hourglass energy. Meanwhile, combined with the trend of the curve changes, it can be observed that the total energy conversion follows the law of conservation of energy, as one part of the energy in the system decreases while the other part increases. The hourglass energy accounts for 4.6% of the internal energy of the platform, which meets the requirement of being below 5%. Therefore, the results of the nonlinear finite element dynamic analysis in this paper can be seen as reliable.

A structural damage cloud image of the semi-submersible platform at the maximum collision depth is shown in Figure 7. It can be seen the deformations of the platform are mainly distributed in the collision area by the bow of the supply vessel. The outer plate and its longitudinal beam mainly undergo tensile deformation and tear failure, and the horizontal frame, transverse bulkhead, vertical girder and other internal structures are bent, laterally bent and wrinkled, and even suffer tear failure.

From the above analysis, it could be understood that the structural deformations of the platform take place mainly in the collided column. Therefore, in order to improve the anti-collision ability of the platform, it is necessary to optimize the thickness of the structural components of the column that suffers the main deformations (including the outer plate, double shell, horizontal and longitudinal girders, bulkheads and other internal structures, as well as the longitudinal beam of the inner side of the outer plate).

3. Kriging Approximate Model and Parallel EGO Multi-Objective Algorithm Based on Hybrid Criteria

The collision analysis and structural optimization of supply vessels and semi-submersible platforms involve “expensive” simulation calculations, where traditional design tools and optimization algorithms are inefficient or time-prohibitive. Approximate model-based optimization offers an effective solution [30,31]; for instance, the Kriging model can approximate simulation models and combine with algorithms like genetic algorithms (GAs) to solve complex, expensive “black-box” function optimization or computer simulation-driven optimization problems [32,33].

3.1. Unconstrained Multi-Objective Optimization Problem

Platform structural design must first meet specification requirements, as well as multiple needs (equipment configuration, weight and center of gravity, stability, hydrodynamic performance, structural/fatigue strength). Early design stages already determine the key parameters (principal dimensions, basic structural forms, component plate thickness limits). Thus, for column anti-collision and lightweight optimization, the objectives are minimizing column structural weight and maximum collision depth under typical working conditions; the design variables are the component plate thicknesses; the constraints are the plate thickness limits. These constraints can be simplified to design variable boundaries, reducing the problem to an unconstrained multi-objective optimization, expressed as follows:

(5)$\begin{array}{l}\min \mathbf{F}\left(x\right)=\left\{f_1\left(x\right),f_2\left(x\right),\dots f_m\left(x\right)\right\}\\ x_l\le x\le x_u\end{array}$

where x is the design variable, x_l and x_u are, respectively, the upper and lower boundaries of the design variable, and F(x) represents the objective function. For most practical engineering problems, the objective function usually requires complex simulation calculations, and is also sometimes called the black-box function.

In the multi-objective optimization algorithm, the approximate model is used in F(x) for simulation as shown in the following formula.

(6)$\begin{array}{l}\min \widehat{\mathbf{F}} \left(x\right)=\left\{{\widehat{f}}_1\left(x\right),{\widehat{f}}_2\left(x\right),\dots {\widehat{f}}_m\left(x\right)\right\}\\ x_l\le x\le x_u\end{array}$

3.2. Kriging Model

The Kriging model is an unbiased estimation model with minimal estimation variance, suitable for analyzing sample data correlation and predicting sample trends. In statistics, the Kriging model is also referred to as a Gaussian process model, as it treats the objective function as a Gaussian process (a normally distributed process) [34]. Assuming there exist N sample points $\left\{x^{(1)},x^{(2)},x^{(3)},\cdot \cdot \cdot \cdot \cdot \cdot ,x^{(N)}\right\}$, and the true objective values of these points are $\left\{y^{(1)},y^{(2)},y^{(3)},\cdot \cdot \cdot \cdot \cdot \cdot ,y^{(N)}\right\}$ according to simulation calculations, the formula of the Kriging model is as follows [35]:

(7)$\widehat{f}\left(x\right)=\mu +\mathbf{z}\left(x\right)$

where µ is the mean value of the Gaussian process, and z(x) is a random process with a mean value of 0 and a variance of σ². The correlation between any two sample points is as follows:

(8)$Cov\left[\mathbf{z}\left(x^{(i)}\right),\mathbf{z}\left(x^{(j)}\right)\right]=\exp \left(-{\displaystyle \sum _{k=1}^n{\theta }_k\mid x_k^{(i)}-x_k^{(j)}{\mid }^{p_k}}\right)$

where n is the number of design variables, θ_k and p_k are undetermined coefficients, and k = 1, 2, ···, n. The parameter θ_k represents the kth dimensional weight of the design space, and the parameter p_k represents the smoothness in the kth dimension of the design space.

Overall, the Kriging model has a total of 2 n + 2 undetermined parameters, namely µ, σ, θ₁, ···, θ_n, p₁,···, p_n. These parameters can be obtained by maximizing the likelihood function of the sample points. Finally, the predicted value of any point x and the variance in the predicted value can be obtained as follows:

(9)$\widehat{y}\left(x\right)=\widehat{\mu }+r^T{R}^{-1}\left(y-l\widehat{\mu }\right)$

(10)$s^2\left(x\right)={\widehat{\sigma }}^2\left[1-r^T{R}^{-1}r+{\displaystyle \frac{{\left(1-l^T{R}^{-1}r\right)}^2}{l^T{R}^{-1}l}}\right]$

where $\widehat{\mu }$ and ${\widehat{\sigma }}^2$ are the estimated values of the undetermined parameters of µ and σ², respectively, and they are given as follows:

(11)$\widehat{\mu }={\displaystyle \frac{l^T{R}^{-1}y}{l^T{R}^{-1}l}}$

(12)${\widehat{\sigma }}^2={\displaystyle \frac{{(y-1\widehat{\mu })}^T{R}^{-1}(y-l\widehat{\mu })}{n}}$

R is the correlation coefficient matrix of N × N, with the elements $R_{ij}=Corr\left[\mathrm{z}\left(x^{(i)}\right),\mathrm{z}\left(x^{(j)}\right)\right]$; r represents the N × 1 vector with the elements $r_i=Corr\left[\mathrm{z}\left(x\right),\mathrm{z}\left(x^{(i)}\right)\right]$; y stands for the response value of the sample points, an N × 1 vector with the elements y_i = y⁽ⁱ⁾; and finally, l is the N × 1 vector, the elements of which are all 1 [36].

3.3. EGO Algorithm Based on the Kriging Approximate Model

The response value of the black-box function $y(x)$ at test point x is regarded as a random variable of a normal distribution, conforming to the mean value $\widehat{y}(x)$ and the standard deviation $s(x)$, and with the following density function:

(13)$pdf(Y(x))={\displaystyle \frac{1}{\sqrt{2\pi s_{\widehat{y}}}}}\exp [-{\displaystyle \frac{1}{2}}{({\displaystyle \frac{Y(x)- {\mu }_{\widehat{y}}}{s_{\widehat{y}}}})}^2]$

As for the minimization, assuming that the optimal response value at the current test point is $y_{\min }^n=\min \{y_1,y_2,\dots ,y_n\}$, the improvement in the predicted value of the Kriging model at the untested point x relative to $y_{\min }^n$ is given as $I(x)=\max {(y_{\min }^n-Y(x), 0)}^{+}$. The EI of I(x) is calculated as follows:

(14)$\begin{array}{l}EI\left(x\right)={\displaystyle {\int }_0^{+\infty }I\left(x\right)}pdf\left(I\left(x\right)\right)dI\\ =\left\{\begin{array}{l}\left(y_{\min }^n-{\mu }_{\widehat{y}}\right)\Phi \left({\displaystyle \frac{y_{\min }^n-{\mu }_{\widehat{y}}}{s_{\widehat{y}}}}\right)+s_{\widehat{y}}\varphi \left({\displaystyle \frac{y_{\min }^n-{\mu }_{\widehat{y}}}{s_{\widehat{y}}}}\right) , s_{\widehat{y}}>0\\ 0 , s_{\widehat{y}}=0\end{array}\right.\end{array}$

where Φ and ϕ are the standard normal distribution and density function, respectively. The smaller the predicted value ${\mu }_{\widehat{y}}$ of Kriging is than the current optimal value $y_{\min }^n$, the larger the value of the first term of the EI criterion will be, indicating that this criterion tends to select new test points in the local optimal area where the predicted value ${\mu }_{\widehat{y}}$ is the smallest. In the unexplored area where the Kriging prediction standard deviation $s_{\widehat{y}}$ is large, the value of the second term of the EI criterion is similarly large, which indicates that the EI criterion is adequate for exploring the potential optimal area.

3.4. Hybrid Criteria-Based Parallel EGO Multi-Objective Algorithm

Different from single-objective optimization, which seeks a single optimal solution, the purpose of multi-objective optimization is to obtain a set of representative Pareto optimal solutions (which we will call a Pareto Set). Since this set cannot be directly used to compare dimensions, optimizing the quality index of the Pareto Set is often carried out as a method to indirectly optimize multi-objective problems. A multi-objective improvement function must be proposed to extend the single-objective EGO algorithm for multi-objective optimization. The role of the improvement function is to describe the improvement degree of the non-dominated front, based on the objective vector of the defined research point in the objective space. The multi-dimensional calculation is then implemented to obtain the corresponding multi-objective improvement function.

3.4.1. HV Improvement Criterion

The HV index is commonly used to evaluate the comprehensive quality of the approximate Pareto Set. The HV index evaluation method represents the volume of the hypercube enclosed by the individual in the solution set and the reference point in the objective space. It is defined as follows:

(15)$H\left(PF,r\right)=Vol\left(y\in R^m\mid PE\prec y\prec r\right)$

PF is the mapping of all Pareto optimal solutions of the objective function in the objective space, known as the Pareto front. For reference point r, this can be specified according to the actual situation, or using the corresponding component of the reference points, adopting the maximum value of each objective function on the initial test point set.

The HV improvement of the existing PF by the response value y (x) at the new test point x is defined as follows:

(16)$I_h\left(x\right)=\left\{\begin{array}{l}H\left(PF\cup y\left(x\right),r\right)-H\left(PF,r\right),y\left(x\right)\prec PF\\ 0 ,\mathrm{other}\end{array}\right.$

After constructing each objective function, the desired HV improvement of the objective function vector for untested point x is as follows:

(17)$E{I}_h\left(x\right)={\displaystyle {\int }_{\widehat{y}\in B}{I}_h\left(x\right){\displaystyle \mathbf{\prod }_{i=1}^m\frac{1}{s}\varphi \left(\frac{y_i-{\widehat{y}}_i}{s_i}\right)}}d{y}_i$

where ${\varphi }_x(\widehat{y})$ is the joint probability density function of the objective function of the Kriging model, and

$B\subset R^m$

is the non-dominated domain defined by the reference point r. The larger the value of $E{I}_h\left(x\right)$ is, the more significant the improvement in the existing approximation PF at the new test point x will be.

3.4.2. Euler Distance Improvement Criterion

The notion of Euler distance was used by Keane [37] to define an improvement function, given as the smallest value of the Euler distance between the objective vector [y₁ (x), y₂ (x), ···, y_m(x)] at the untested point x and the non-dominated front point.

(18)$I_e\left(x\right)=\underset{j=1}{\overset{k}{\min }}\sqrt{{\displaystyle \sum _{i=1}^m{\left({f_i}^j-y_i\left(x\right)\right)}^2}}$

According to the m dimensional normal distribution of the objective vector, the multi-objective EI criterion based on Euler distance improvement can be obtained as follows by integrating I_e(x) along the non-dominated area, represented below by the integral region A.

(19)$E{I}_e\left(x\right)={\displaystyle {\int }_{y\in A}{I}_e\left(x\right){\displaystyle \mathbf{\prod }_{i=1}^m\frac{1}{s}\varphi \left(\frac{y_i-{\widehat{y}}_i}{s_i}\right)}}d{y}_i$

3.4.3. Maximum Distance Improvement Criterion

The maximum and minimum distances are derived from the multi-objective evolution algorithm, and were first used by Bautista [24] to define the multi-objective improvement function as follows:

(20)$I_m\left(x\right)=-\underset{j=1}{\overset{k}{\max }}\left[\underset{i=1}{\overset{m}{\min }}\left(y_i\left(x\right)-f_i^j\right)\right]$

According to the m-dimensional normal distribution of the objective vector, the multi-objective EI criterion based on maximum distance improvement can be written as follows, by integrating I_m(x) along the non-dominated area:

(21)$E{I}_m\left(x\right)={\displaystyle {\int }_{y\in A}{I}_m\left(x\right){\displaystyle \mathbf{\prod }_{i=1}^m\frac{1}{s}\varphi \left(\frac{y_i-{\widehat{y}}_i}{s_i}\right)}}d{y}_i$

3.4.4. Proposed Algorithm

In this paper, a novel hybrid criterion-based parallel sampling method is proposed by combining the merits of the Euler distance, the maximum and minimum distances, and the HV improvement criteria, taking as a basis the classic EGO algorithm. Along the iterative process, the GA is employed in parallel to achieve hybrid criteria-based optimal sampling, so that the multiple optimizations are achieved in parallel in one iteration, allowing the surrogate model to be gradually updated. This algorithm enables parallel computing on multiple machines or multiple cores, avoiding the shortcomings of iterations through serial sampling points, which are prone to falling into a local optimization. In this way, our algorithm easily achieves global optimization, thus contributing to improve computational efficiency.

The algorithm consists of nine steps, organized in three principal phases. First, a Kriging surrogate model is constructed based on the set of sample points, as shown below in steps 1, 2 and 3 of the proposed algorithm. Second, a parallel hybrid criteria-based EI function is established, and the largest function value of each expected function is selected as the renewal point, as shown in steps 4 and 5. In each iteration, different EI functions should be optimized in parallel to select renewal points. Since the EI function is often multi-modal and highly nonlinear, it is usually necessary to use meta-heuristic optimization (possibly genetic) to find the maximum value. Since the EI function is an analytical expression, this calculation is so fast, compared with time-consuming simulation calculations, that its solution time can be seen as negligible. Third, the true response value of the renewal point is calculated and incorporated in the sample database, after which the approximate Pareto solution is obtained in the existing sample database, as shown in steps 6 and 7. In this way, the algorithm can obtain the Pareto solution by continuously selecting and calculating potential renewal points in parallel.

The overall process is outlined in Figure 8.

The steps of the proposed algorithm are described in detail in what follows:

Step 1: Obtain the initial sample n_s. Generally, the number of initial sampling points is selected to be 10 n, where n is the design variable dimension. When the Latin HV experimental design method is selected for sampling, the rule of 10 n can be modified to facilitate the division of the design space. For example, in a 2D design space, let n_s = 21; the division is 1/(n_s − 1) = 0.5, and the number of initial samples is set to 11 n − 1. Meanwhile, k is an iteration counter, with an initial value of zero.

Step 2: Calculate the accurate value of the source function. “Expensive” calculations are executed on the new sampling point set by the simulation analysis program, in order to obtain this accurate value.

Step 3: Use sampling points and response values to construct or update the Kriging model.

Step 4: Construct the EI function in parallel, based on the HV, Euler distance, and maximum distance improvement criteria;

Step 5: Maximize the optimization of the EI function constructed in step 4, by using the parallel GA. The three high-EI points obtained are the renewal points $x_h^k, x_e^k, x_m^k$, with k being the number of iterations. Normalize the existing sample points, and then compute the Euclidean distance between each pair of sample points. Based on empirical evidence, set the minimum distance to 5% of the average distance between existing sample points. Should the distance between any newly added sample point and any existing sample point fall below this minimum distance, the newly added sample point should be discarded.

Step 6: Calculate the true response value of the renewal points. Parallel simulation analysis for the renewal points obtained in step 5 is executed to obtain their true values.

Step 7: Add the renewal point and its response value to the sample database, and update the Pareto front approximation accordingly.

Step 8: Identify the convergence. If the convergence criterion is met, exit the process. The convergence criterion can be set as reaching the maximum number of iterations n_k, or based on the calculated HV index of the Pareto front approximation. The iterative process can be stopped if this does not increase in m consecutive iterations.

Step 9: If the convergence criterion is not met, go to step 3 and continue to the next iteration.

4. Analysis of Test Cases

A set of typical test functions were used to test the accuracy and efficiency of the algorithm proposed in this paper, and the corresponding results were compared with those obtained with the multi-objective EGO algorithm based on the single-objective EI criterion. The six test functions selected cover different types of typical test functions, all in the ZDT and DTLZ function series. ZDT1, ZDT2, and DTLZ2 correspond to the convex Pareto front and non-convex Pareto front, ZDT3 and DTLZ7 represent the discrete Pareto front, and DTLZ5 represents the curve Pareto front [23].

The number of design variables was set to n = 6 for all test cases, and the number of initial design sample points was 11 n − 1. The initial sample points were selected by the Latin hypercube sampling method, and the Kriging approximate model was created through the DACE (Design and Analysis of Computer Experiments) toolbox. Regpoly0 was set as the regression function, and corrgauss as the correlation function. The GA, with a population size of 150, maximum evolutional generation of 200, and stop algebra of 200, was used in the optimization of the EI function.

For the calculation of test cases, the algorithm proposed in this paper (EI_p criterion) and the multi-objective EGO algorithm based on a single EI criterion [27] were separately applied, so as to compare the accuracy and efficiency of the algorithm proposed in this paper under the same number of sample points. In each algorithm, 215 samplings (50 iterations for the multi-objective EGO algorithm based on the EI_p criterion, and 150 iterations for the other multi-objective EGO algorithms based on a single EI criterion), that is, 215 “expensive calculations”, were conducted in the same way. All experiments were carried out 30 times as per different initial schemes, so as to exclude the effect of initial sample points on experimental results. The comparisons among various algorithms were performed using the hypervolume (HV) of the Pareto front and the number of Pareto solutions obtained. The hypervolume can capture the closeness of the obtained solutions to the Pareto front as well as the spread property of the solutions. The reference points used for calculating the hypervolume values are set in Table 3 for each test problem. The hypervolume can capture the closeness of the obtained solutions to the Pareto front as well as the spread property of the solutions. The reference points used for calculating the hypervolume values are set in Table 3 for each test problem.

The HV of the approximate Pareto front and the number of the Pareto front approximations (NP) of test functions obtained by different algorithms are shown in Table 4. The larger HV of the approximate Pareto front and NP indicate higher quality.

The results revealed that under the same sample estimation, the HV of the approximate Pareto front obtained by the optimization algorithm proposed in this paper was superior to that obtained by other single-criterion optimization algorithms on all test functions. In addition, the NP obtained by the optimization algorithm proposed in this paper on all test functions was also significantly larger than that obtained by other single-criterion optimization algorithms. In actual engineering process applications, obtaining more Pareto front approximations allows engineers to have more choices, which is more propitious to obtaining optimization results in line with engineering practice. Therefore, the algorithm proposed in this paper is more advantageous in engineering applications.

Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 are boxplots of the HV of the approximate Pareto front and NP of all test functions obtained by separately running the multi-objective EGO algorithm based on the EI_p criterion and single EI criterion for 30 times as per different initial schemes. Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10 lists the corresponding standard deviations (SDs) and confidence intervals (CIs, based on 95% confidence intervals).

As seen in Figure 9, the boxplots for ZDT1 indicate that the EI_p criterion algorithm achieves a marginally higher median HV and a significantly higher median NP compared to the single-criterion algorithms (EI_e, EI_h, EI_m). The interquartile range (IQR) for HV appears similar across all algorithms, suggesting comparable variability in solution quality for this function. However, the spread of NP values is notably larger for the single-criterion methods. The data in Table 5 quantitatively supports the observations from Figure 9. The EI_p criterion yields the highest mean HV (120.64) and mean NP (32). Crucially, it also demonstrates the lowest standard deviation (SD) for the HV (0.0074), indicating superior consistency and stability in performance across the 30 runs compared to the other criteria, which have higher SDs of the HV. The SD of the NP for EI_p (5.19) is also the lowest, confirming more reliable convergence in terms of the number of solutions found.

As seen in Figure 10, for the ZDT2 test problem, the EI_p criterion algorithm clearly outperforms the single-criterion ones. It demonstrates a noticeably higher median HV and a substantially higher median NP. The box for the NP using EI_p is positioned much higher and shows less variability than those of the other algorithms, which cluster near lower values with similar spreads. Table 6 confirms the superiority of the EI_p criterion on ZDT2. It achieves the highest mean HV (120.30) and mean NP (25). While the SDs for the HV are identical across all algorithms (0.0296), the dramatic difference in mean NP values (25 for EI_p vs. 10, 5, and 15 for the others) highlights the significant advantage of the hybrid approach in finding a more extensive set of Pareto solutions for this problem.

As seen in Figure 11, the performance on ZDT3 shows a distinct advantage for the EI_p criterion. Its median HV is significantly higher than those of the single-criterion algorithms. The NP values also show a positive trend, with EI_p having a higher median and a wider distribution towards larger numbers of solutions compared to the others. The numerical results in Table 7 solidify the findings from Figure 11. The EI_p criterion achieves the highest mean HV (128) by a considerable margin (compared to 127 and 128 ± 0.27 for EI_m, for which the upper bound of the CI is just 128.27) and the highest mean NP (30). Although the SDs for NP are the same, the mean values for the single-criterion algorithms (15, 15, 20) are substantially lower than those of EI_p, demonstrating its effectiveness on this problem with a discontinuous Pareto front.

As seen in Figure 12, for the three-objective DTLZ2 problem, the EI_p criterion maintains a slight edge, showing a marginally higher median HV compared to the other algorithms. The median NP for EI_p is also visibly higher, suggesting a better ability to approximate the entire Pareto front in 3D space. The data in Table 8 provides precise metrics for DTLZ2. The EI_p criterion achieves the highest mean HV (15.030) and the highest mean NP (25). The SDs of the HV are identical across all methods (0.0074), indicating similar run-to-run consistency in solution quality for this problem. The key differentiator is again the number of solutions found, where EI_p finds significantly more Pareto points on average.

As seen in Figure 13, on the DTLZ5 problem, the EI_p criterion demonstrates competitive performance. The median HV appears slightly higher than or equal to that of the other methods. The most notable difference is again in the NP, where EI_p shows a significantly higher median and a larger spread, indicating its strength in discovering more solutions along the curved Pareto front. Table 9 shows that the EI_p criterion obtains the highest mean HV (13.12) and the highest mean NP (30) for DTLZ5. The HV values are very close, but the hybrid approach consistently finds a more numerous set of Pareto solutions, as evidenced by the higher NP. The SD of the NP for EI_p is larger (14.81), indicating more variability in the count of solutions found between runs, but the average count remains the highest.

As seen in Figure 14, for the complex DTLZ7 problem, the advantage of the EI_p criterion is pronounced, particularly regarding the NP. The median NP for EI_p is drastically higher than that for any single-criterion algorithm. The HV median for EI_p also appears competitive, situated within the higher range of values. The results in Table 10 for DTLZ7 are striking. The EI_p criterion achieves a significantly higher mean HV (59,000) compared to the other methods (58,500, 58,000, 58,500) and a vastly higher mean NP (40 vs. 20, 15, 15). This demonstrates the hybrid algorithm’s superior capability in handling problems with disconnected Pareto fronts, both in terms of the quality of the approximation (HV) and the diversity and number of solutions found (NP), despite the higher absolute SD of the HV.

Through comparison, it can be seen that the optimization algorithm proposed in this paper is better than the multi-objective EGO algorithm based on a single EI criterion under the same number of sample points.

Figure 15, Figure 16 and Figure 17 display the final approximate Pareto front in the case of the smallest HV of ZDT1, ZDT2 and ZDT3 functions, respectively, in 30 repeated experiments of test functions obtained after 50 iterations of calculation using the EI_p criterion. The Pareto front of the ZDT1 function was convex, and massive Pareto front points with a relatively uniform distribution were found through the algorithm proposed in this paper. According to Figure 17, discrete Pareto fronts were found on the ZDT3 function. Through the algorithm proposed in this paper, a total of 17 approximations were discovered, which were distributed on five discrete fronts. The Pareto front of the DTLZ2 function was relatively regular. As shown in Figure 18, Pareto front points with a large number and a relatively uniform distribution were found through the algorithm proposed in this paper. As for the DTLZ5 function, the Pareto front was curvilinear, and the three-objective Pareto front presented a spatial curve in a three-dimensional space. Hence, the DTLZ5 function can be employed to test the ability of the multi-objective optimization algorithm to search for a curvilinear Pareto front. It can be seen from Figure 19 that better results were also acquired through the algorithm proposed in this paper. The DTLZ7 function had discrete Pareto fronts, as shown in Figure 20. Therefore, the DTLZ7 function can be used to test the capacity of the multi-objective optimization algorithm to optimize discrete Pareto fronts. The approximate Pareto front found through the algorithm proposed in this paper is close to the real Pareto front. More points were found on the four discrete Pareto fronts.

Moreover, the computing time for completing the same sample estimation using the optimization algorithm based on the EI_p criterion was compared with that using the optimization algorithm based on a single criterion. Since the computing time required by optimization algorithms based on EI_h, EI_e, and EI_m criteria are basically the same, the EI_h criterion-based optimization algorithm was adopted for the comparison in this paper. In the two algorithms, a computer with the same configuration was utilized, with single-core computing for the algorithm based on the EI_h criterion and parallel three-core computing for the algorithm based on the EI_p criterion. To exclude the effect of initial sample points on experimental results, all experiments were run 30 times as per different initial schemes. The explicit core/thread counts and system specifications are listed in Table 11. The average computing times were taken and compared, as shown in Figure 21. It can be seen that the computing time required by the algorithm based on the EI_h criterion was about three times that required by the algorithm based on the EI_p criterion. This suggests that for all test functions, the algorithm proposed in this paper requires less computing time and can obtain relatively superior results, and it confirms that the algorithm proposed in this paper is superior in providing computational efficiency.

5. Optimization of the Column Structure in the Semi-Submersible Platform

5.1. Selection of Optimization Variables

The function that represents the structural weight of the platform column, W, as well as the maximum collision depth in a typical collision, D, was selected as the objective function in this paper. Since the main dimension parameters of the column are fixed, the thicknesses of the structural plates related to crashworthiness were chosen as the design variable. According to this analysis, the corresponding multi-objective optimization problem can be expressed by the following mathematical model.

(22)$\begin{array}{l}\min F(x)=\{W(x),D(x)\}\\ x_l\le x\le x_u\end{array}$

Figure 22a shows the outer plate of the column, with thickness x₁. Figure 22b shows the longitudinal beam inside the outer plate, with thickness x₂. Figure 22c shows the double-shell cross brace of the column, with thickness x₃. Figure 22d shows the center line bulkhead of the column, with thickness x₄, and Figure 22-e shows the longitudinal bulkhead at the side, with thickness x₅. The upper and lower limits of the plate thicknesses (unit: mm) for each of the five design variables determined above are shown in Table 12.

5.2. Optimal Results

The method proposed in this paper was applied in the analysis process, with the number of initial sample points set to 54. The Latin hypercube sampling method was used to select the initial sample points, and the DACE toolbox was used to create the Kriging approximate model. Regpoly0 was used as a regression function, and corrgauss as a correlation function. The optimal results obtained after 20 iterations are shown below.

The nine points of the Pareto front obtained by the optimization through this method are represented by the blue dots in Figure 23. As seen in Figure 23, the Pareto front consists of nine non-dominated solutions. From the perspective of resolution, these nine points adequately capture the trade-off relationship between collision depth and weight across the observed range. Each point represents a distinct design compromise, and the spacing between adjacent points reflects meaningful differences in performance metrics. Regarding generalizability, while the current Pareto front is derived from a specific experimental or simulation setup, the nine-point distribution covers the critical regions of the design space where practical engineering trade-offs occur. Should a higher-resolution Pareto front be required for more granular decision-making, our methodological framework (e.g., the multi-objective optimization algorithm) can be extended with increased sampling or refined parameter settings, which we plan to explore in future work to further validate generalizability across broader design scenarios.

In light of the comprehensive design requirements, the solution of the Pareto front at A is selected as the final and optimal solution. The results of this analysis are shown in Table 13, set side by side with the values in the original structure (See Section 2.3). The improvements obtained with the optimized scheme are quite obvious from this comparison. The anti-collision ability of the platform column is greatly improved, and the collision depth is reduced by 11.9%, from 987.5 mm to 869.6 mm. Meanwhile, the structural weight is reduced by 2.7 t. A single column in offshore engineering equipment typically accounts for 15–20% of the total structural weight of a subsea module (e.g., a typical subsea production unit may weigh 15,000–20,000 t). A 2.7 t reduction per column directly translates to a certain improvement in the self-sustaining capacity of offshore equipment—this not only reduces the operational costs associated with buoyancy control and installation but also enhances the payload margin for critical components. For an offshore platform with four such columns, the cumulative weight savings could exceed 10.8 t, which is equivalent to the weight of one to two heavy-duty subsea pumps, thereby demonstrating significant engineering and economic value in real-world offshore applications.

Figure 24 depicts the damage cloud chart of the column structure in the optimized semi-submersible platform. From the cloud chart, it can be seen that the plate thicknesses of the outer plate of the column and of the inner double-shell structure are both increased, so the intrusion distance in the bow collision is effectively reduced. In addition, the longitudinal beam and other structures with limited anti-collision improvement effects have reduced plate thicknesses, leading to a decrease in the overall weight of the column structure.

In summary, the study’s core results can be presented qualitatively and quantitatively. Qualitatively, this work addresses two key challenges in offshore platform optimization: serial EGO inefficiency and single-criterion sampling bias. It achieves this via establishing a hybrid, criteria-based, parallel EGO algorithm which integrates the Euler distance, maximum/minimum distance and HV improvement in order to balance exploration and exploitation, as well as supporting parallel computing. Quantitatively, the validation on six test functions (ZDT1–3 and DTLZ2, 5 and 7) shows that the proposed EI_p outperforms single criteria. On DTLZ7, for example, it achieves a mean HV of 59,661.7 (1.1% higher than EI_h) and an NP of 91.9 (2.6 times that of EI_h), while reducing the optimization time by 67% (three cores versus one core; see Figure 21). When applied to the “North Dragon” platform column, synergistic optimization is realized: the collision depth drops by 11.9% (from 987.5 mm to 869.6 mm) and the weight is reduced by 2.7 t (from 106.1 t to 103.4 t, see Table 6). Qualitatively, the optimized columns exhibit less outer-plate tearing and inner-structure wrinkling. These results confirm the methodological and engineering value of the algorithm.

6. Conclusions

According to the numerical simulation analysis of the collision between the supply vessel and the semi-submersible platform, the outer plate of the column and the various structures inside the column, which absorb most of the energy in the initial state, jointly bear the initial collision. Throughout the collision, the inner structures are damaged and lose their functions one after another. The contact area between the outer plate and double shell of the column and the supply vessel continues to increase, until finally the energy absorbed by the outer plate and double shell exceeds the energy absorbed by the inner structures. In the anti-collision design of the column, the outer plate and double shell are the important structures for ensuring the stability of the platform, as well as for the main structures to absorb collision energy through deformation. Therefore, they are expected to be the main objects of structural optimization.

With this in mind, a parallel multi-objective optimization algorithm based on multiple criteria is proposed in this paper to solve the complex multi-objective optimization problems encountered in the process of anti-collision structure design for offshore drilling platforms. A small number of data points are used by the algorithm to construct the Kriging approximate model in order to estimate the objective function in the design domain, greatly improving the efficiency of black-box simulation optimization. The classic EGO algorithm is not suitable for parallel execution because of the existence of a single sample point updated per cycle. This paper improves the algorithm by proposing an optimal point-adding criterion for the desirability function. Several new sample points are added in parallel in each iteration; the points are updated and the approximate model is reconstructed continuously, thus allowing for computational efficiency to be improved by resorting to parallel computing (on multiple machines or multiple cores). The algorithm is verified using typical multi-objective optimization test functions. The anti-collision, lightweight design of the column in the semi-submersible drilling platform is carried out with obvious optimal results.

Future research will first extend the collision scenario scope to include variable ice–load coupling and different ship impact velocities, to verify the algorithm’s adaptability to complex Arctic offshore environments. Additionally, efforts will be made to integrate the proposed parallel EGO algorithm with deep learning models, aiming to further improve the prediction accuracy of Kriging approximate models for high-fidelity structural simulations of ice-region platforms.

Author Contributions

Conceptualization, B.W. and Y.W. (Yizhong Wu); methodology, B.W.; software, B.W.; validation, B.W., Y.W. (Yangwei Wang) and J.M.; investigation, B.W.; resources, J.M. and Y.W. (Yizhong Wu); data curation, B.W.; writing—original draft preparation, B.W.; writing—review and editing, Y.W. (Yangwei Wang) and Y.W. (Yizhong Wu); visualization, B.W.; supervision, Y.W. (Yizhong Wu); project administration, L.C. and Y.W. (Yizhong Wu); funding acquisition, J.M. and Y.W. (Yizhong Wu). All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The authors wish to acknowledge the support and help of the Research Team of National CAD Supported Software Engineering Centre of Huazhong University of Science and Technology, and the MWORKS-based Research Team for Equipment Digitalization and Intelligence of Yantai University.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

EGO	efficient global optimization
FPSO	floating production storage and offloading
EI	expected improvement
HV	hypervolume
GA	genetic algorithm
DACE	design and analysis of computer experiments
PRS	polynomial regression surface
RBF	radial basis function
MARS	multivariate adaptive regression splines
ARSM	adaptive response surface method
MPS	mode-pursuing sampling
NSGA-II	non-dominated sorting genetic algorithm II
ZDT	Zitzler–Deb–Thiele
DTLZ	Deb–Thiele–Laumanns–Zitzler

Footnotes

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Figures and Tables

Figure 1 “North Dragon” semi-submersible drilling platform.

Figure 2 The finite element model of the semi-submersible platform: (a) the semi-submersible platform model; (b) the internal support structure of the column and the floating pontoon.

Figure 3 Three-dimensional model of vessel bow.

Figure 4 A finite element model of a head-on collision between a vessel and a platform column.

Figure 5 The collision depth changes under the typical working condition.

Figure 6 The energy conversion duration curve of the collision system under the typical working condition.

Figure 7 Damage deformation cloud image of the platform: (a) conditions of the outer plate; (b) conditions of the inner double-shell structure.

Figure 8 Process chart of the parallel multi-objective EGO algorithm based on hybrid criteria.

Figure 9 Boxplot of the HV and NP obtained by each algorithm on ZDT1 test problems: (a) HV obtained by each algorithm; (b) NP obtained by each algorithm.

Figure 10 Boxplot of the HV and NP obtained by each algorithm on ZDT2 test problems: (a) the HV obtained by each algorithm; (b) the NP obtained by each algorithm.

Figure 11 Boxplot of the HV and NP obtained by each algorithm on ZDT3 test problems: (a) HV obtained by each algorithm; (b) NP obtained by each algorithm.

Figure 12 Boxplot of the HV and NP obtained by each algorithm on DTLZ2 test problems: (a) the HV obtained by each algorithm; (b) the NP obtained by each algorithm.

Figure 13 Boxplot of the HV and NP obtained by each algorithm on DTLZ5 test problems: (a) the HV obtained by each algorithm; (b) the NP obtained by each algorithm.

Figure 14 Boxplot of the HV and NP obtained by each algorithm on DTLZ7 test problems: (a) the HV obtained by each algorithm; (b) the NP obtained by each algorithm.

Figure 15 Plots of the final Pareto front approximation obtained using the EI_p criteria algorithm with the lowest hypervolume value in the fifty runs on the ZDT1 problem.

Figure 16 Plots of the final Pareto front ap proximation obtained using the EI_p criteria; the lowest hypervolume value in the fifty runs on the ZDT2 problem.

Figure 17 Plots of the final Pareto front approximation obtained by EI_p criteria algorithm with the lowest hypervolume value in the fifty runs on the ZDT3 problem.

Figure 18 Plots of the final Pareto front approximation obtained by EI_p criteria algorithm with the lowest hypervolume value in the fifty runs on the DTLZ2 problem.

Figure 19 Plots of the final Pareto front approximation obtained by EI_p criteria algorithm with the lowest hypervolume value in the fifty runs on the DTLZ5 problem.

Figure 20 Plots of the final Pareto front approximation obtained by EI_p criteria algorithm with the lowest hypervolume value in the fifty runs on the DTLZ7 problem.

Figure 21 Comparison of computing time for completing the same sample estimation using the optimization algorithm based on the EI_p criterion and EI_h criterion.

Figure 22 Column structure and optimal design variables; the column structure in the platform.

Figure 23 Optimization of the Pareto front for the column structure.

Figure 24 Damage cloud chart of the column structure in the optimized semi-submersible platform: (a) conditions of the outer plate; (b) conditions of the inner double-shell structure.

Table 1

Principal dimensions of the platform.

Part	Dimension 1	Part	Dimension 1
Total length	104.5	Height of column	21.5
Total width	72.5	Vertical spacing of column	42.8
Total height	42	Horizontal spacing of column	42.0
Length of main deck	85.9	Length of floating body	104.5
Width of main deck	71.9	Width of floating body	16.5
Height of main deck	8	Height of floating body	10.4
Cross-section of column	17.5 × 17.5	Spacing of floating body	15.2

¹ Unit: meter, (m).

Table 2

Principal dimensions of the collision vessel.

Length of Vessel 1	Width of Vessel 1	Depth of Vessel 1	Designed Draft 1	Full Loaded Displacement 2
93.0	16.0	7.3	5.4	5000

¹ Unit: meter, (m). ² Unit: metric ton, (t).

Table 3

Reference point for calculating the hypervolume of each test problem.

Function	Design Variables (n)	Objective Functions (m)	Reference Point
ZDT1	6	2	(11,11)
ZDT2	6	2	(11,11)
ZDT3	6	2	(11,11)
DTLZ2	6	3	(2.5,2.5,2.5)
DTLZ5	6	3	(2.5,2.5,2.5)
DTLZ7	6	3	(30,30,30)

Table 4

HV and NP statistics obtained using the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria.

Function		EIp	EIe	EIh	EIm
ZDT1	Mean HV	120.64	120.61	120.62	120.61
	Mean NP	29.6	28.2	19.8	30
ZDT2	Mean HV	120.35	120.21	120.02	120.12
	Mean NP	20.8	10.8	6.9	12.2
ZDT3	Mean HV	128.24	127.29	126.88	126.78
	Mean NP	20.5	17	17.5	18.6
DTLZ2	Mean HV	15.13	15	15.01	15.01
	Mean NP	68.3	60.4	57.4	62
DTLZ5	Mean HV	13.21	13.09	13.11	13.09
	Mean NP	21.7	19.8	19.4	20
DTLZ7	Mean HV	59,661.7	59,573.55	59,296.08	59,522.02
	Mean NP	91.9	40.9	34.7	36.9

Table 5

The SDs and CIs of the HV and NP obtained using the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria, respectively, on ZDT1.

	EIp	EIe	EIh	EIm
HV (SDs)	0.0074	0.0089	0.0111	0.0037
NP (SDs)	5.19	7.41	7.41	7.41
HV (CIs)	120.64 ± 0.0027	120.638 ± 0.0032	120.63 ± 0.0040	120.632 ± 0.0014
NP (CIs)	32 ± 1.88	30 ± 2.70	20 ± 2.70	30 ± 2.70

Table 6

The SDs and CIs of the HV and NP obtained by the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria, respectively, on ZDT2.

	EIp	EIe	EIh	EIm
HV (SDs)	0.0296	0.0296	0.0296	0.0296
NP (SDs)	7.41	7.41	7.41	7.41
HV (CIs)	120.30 ± 0.0108	120.26 ± 0.0108	120.22 ± 0.0108	120.26 ± 0.0108
NP (CIs)	25 ± 2.70	10 ± 2.70	5 ± 2.70	15 ± 2.70

Table 7

The SDs and CIs of the HV and NP obtained by the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria, respectively, on ZDT3.

	EIp	EIe	EIh	EIm
HV (SDs)	0.741	1.481	1.481	0.741
NP (SDs)	7.41	7.41	7.41	7.41
HV (CIs)	128 ± 0.270	127 ± 0.539	127 ± 0.539	128 ± 0.270
NP (CIs)	30 ± 2.70	15 ± 2.70	15 ± 2.70	20 ± 2.70

Table 8

The SDs and CIs of the HV and NP obtained using the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria, respectively, on DTLZ2.

	EIp	EIe	EIh	EIm
HV (SDs)	0.0074	0.0074	0.0074	0.0074
NP (SDs)	7.41	7.41	7.41	7.41
HV (CIs)	15.030 ± 0.0027	15.010 ± 0.0027	15.020 ± 0.0027	15.015 ± 0.0027
NP (CIs)	25 ± 2.70	20 ± 2.70	15 ± 2.70	20 ± 2.70

Table 9

The SDs and CIs of the HV and NP obtained using the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria on DTLZ5.

	EIp	EIe	EIh	EIm
HV (SDs)	0.0296	0.0296	0.0296	0.0296
NP (SDs)	14.81	7.41	7.41	14.81
HV (CIs)	13.12 ± 0.0108	13.10 ± 0.0108	13.11 ± 0.0108	13.10 ± 0.0108
NP (CIs)	30 ± 5.40	15 ± 2.70	10 ± 2.70	20 ± 5.40

Table 10

The SDs and CIs of the HV and NP obtained using the EI_p criteria, EI_e criteria, EI_h criteria, and EI_m criteria on DTLZ7.

	EIp	EIe	EIh	EIm
HV (SDs)	740.74	740.74	1481.48	740.74
NP (SDs)	14.81	7.41	7.41	7.41
HV (CIs)	59,000 ± 270.0	58,500 ± 270.0	58,000 ± 540.0	58,500 ± 270.0
NP (CIs)	40 ± 5.40	20 ± 2.70	15 ± 2.70	15 ± 2.70

Table 11

The explicit core/thread counts and system specifications.

Configuration Category	Specific Parameters
CPU	Intel Core i7-10700K ((8 cores/16 threads), 3.8 GHz)
RAM	32 GB DDR4-3200 (Dual-channel, 16 GB × 2)
Memory	512 GB NVMe SSD + 2 TB HDD
Operating System	Windows 10 Professional Edition (64-bit)
Software Environment	Software version: MATLAB R2016b (64-bit)
	Parallelization details: enable MATLAB parallel computing Toolbox, which by default utilizes 8 cores/16 threads

Table 12

Upper and lower limits of the plate thicknesses for each design variable.

	x 1	x 2	x 3	x 4	x 5
xu	13	12	14	16	17
xl	9	8	10	12	13

Table 13

Comparison of parameters before and after the optimization.

	x1 1	x2 1	x3 1	x4 1	x5 1	W 2	D 1
Initial design	9.5	10.0	12.0	14.0	15.0	106.1	987.5
Optimization results	10.7	8.0	10.2	14.5	13.8	103.4	869.6

¹ Unit: millimeter, (mm). ² Unit: metric ton, (t).

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