A Bayesian Optimization Algorithm for the Optimization of Mooring System Design Using Time-Domain Analysis

Abstract

Dynamic analysis can consider the complex behavior of mooring systems. However, the relatively long analysis time of the dynamic analysis makes it difficult to use in the design of mooring systems. To tackle this, we present a Bayesian optimization algorithm (BOA) which is well known as fast convergence using a small number of data points. The BOA evaluates design candidates using a probability-based objective function which is updated during the optimization process as more data points are achieved. In a case study, we applied the BOA to improve an initial mooring system that had been designed by human experts. The BOA was also compared with a genetic algorithm (GA) that used a pre-trained surrogate model for fast evaluation. The optimal designs that were determined by both the BOA and GA have a 50% lower maximum tension than the initial design. However, the computation time of the GA needed 20 times more than that of the BOA because of the training time of the surrogate model.

1. Introduction

Mooring systems provide station keeping performance at a designated location. The design of mooring systems needs to consider numerous combinations of multiple design variables such as the pattern, weight, diameter, and length of the mooring lines, and this makes the optimization of mooring system design difficult in a manual manner. There are research papers related to the optimization of mooring system design for an improved station-keeping performance. Monteiro et al. [1] studied optimization using particle swarm optimization (PSO) and Mirzaei et al. [2] used a genetic algorithm (GA), in which the evaluation used the offset result of static analysis. Shafieefar and Rezvani [3] presented a GA for mooring systems optimization, in which the evaluation model of the GA used the offset results in a frequency domain. Pillai et al. [4] also presented a GA for mooring systems optimization with a random forest based surrogate model for assessment. Surrogate models in general indicate black box models which are substitutes for high fidelity models and used to decrease computational cost. Li et al. [5] used a gradient-based search algorithm for the design optimization of mooring systems. This also used a surrogate model for fast assessment. More optimization research for mooring systems is described in Qiao et al. [6].

In previous studies, optimization has often been performed for minimizing the offset that was calculated by statics analysis in a frequency domain. However, West et al. [7] addressed frequency–domain analysis, which simplifies the analysis model to estimate the response of the mooring system and does not consider the hydrodynamic forces such as the damping and restoring forces on mooring lines. Moreover, Hall and Goupee [8] mentioned that if quasi-static mooring analysis is performed, the hydrodynamic forces are neglected. Thus, the measured mooring tension and platform response are significantly underpredicted. Therefore, the optimization results that use static analysis need to be investigated further using time-domain analysis in the later stage. On the other hand, dynamic analysis can consider the dynamic forces and has higher accuracy than static analysis. However, the computation of the dynamic analysis is more complex and increases the computing time. This could matter in the optimization process that often investigates numerous design alternatives. To tackle this, using a surrogate model as an evaluation function is a quite promising approach but there are also challenges because the development of the model needs numerous data points for accurate prediction.

To take only the benefits of the dynamic analysis, this paper applies a Bayesian optimization algorithm (BOA) to the design of mooring systems. The BOA needs a relatively smaller number of data points for fast convergence in design optimization using a probability-based evaluation function.

To the best of our knowledge, there is no research that uses a BOA for the optimization of mooing systems design. Instead of mooring systems, Elsas et al. [9] applied a BOA to the optimization of steel risers’ initial configuration, in which the BOA was compared to other optimization methods such as a GA and the results showed the superior performance of the BOA. The remaining sections of this paper describe the mooring system design problem and present the optimization process of the BOA in a more detailed manner. This paper then applies the BOA to a mooring system design problem in a case study in which the design result of the BOA was compared with that of a GA and a human expert.

2. Mooring System Design Problem

The performance of mooring systems is dependent on design factors such as the pattern, weight, diameter, and length of the mooring lines as well as environmental conditions. There has been research that investigated the design factors. For instance, Sabziyan et al. [10] investigated the motion of semi-submersibles according to the number and angle of mooring lines. Park et al. [11] investigated the angle of the mooring line and maximum tension for a multi-point moored floating production storage and offloading (FPSO).

Table 1. Factors for the mooring system design.

Parameter	Unit
Water depth	M
Footprint radius	×WD (Water Depth)
Line diameter	M
Pre-tension	% of MBL
Top chain length	M
All line length	M
Outer bundle angle	Degree
Inner bundle angle	Degree
Middle wire length	M
Bottom chain length	M

presents factors that affect the performance of mooring systems significantly. One is the water depth of the installation site. The value of footprint radius (FPR) is a three times value of the water depth, which is a typical value for a catenary-type mooring system. Mooring lines have different minimum breaking loads (MBL) according to the line diameter. Therefore, the diameter should be selected carefully to meet the MBLs. The pretension is determined using the reserve buoyancy in the maximum storage condition, in which all the facilities of the offshore structure are installed. This pretension is about 5%, 10%, and 15% of the MBL of the line for general offshore mooring systems. The pretension of the catenary type mooring system is caused mainly by the weight of the mooring line. A catenary mooring system used in deep water uses a chain–wire–chain type to prevent excessive load. At this time, most of the loads that cause the pretension of the mooring system are determined by the suspended line. In the mooring system of the chain–wire–chain, the pretension generated by the suspended line is mostly caused by the weight of the top chain. Therefore, the top chain length is selected considering the stability of the system. The length of the entire line of the mooring system is selected by the footprint radius. However, there are cases where it is necessary to prevent uplift force depending on the type of anchor or extreme environmental conditions. In this case, set the footprint radius to three times or more to prevent it by increasing the length of the line under it. The outer bundle angle (OBA) and the inner bundle angle (IBA) indicate the angle of the entire bundle and the angle between lines in the bundle, respectively. Park et al. [11] evaluated the effect of OBA and IBA on line tension. As in these studies, the combination of various OBA and IBA shows various results. The middle wire length and the lower chain length determine the suspended line weight other than the upper chain weight depending on the length. In addition, the dynamic effect according to the change in the length of the wire also appears in various ways. Figure 1 illustrates the parameters and design variables described in this research, in which the blue circles indicate the parameters, and the purple triangles indicate the design variables.

3. Dynamic Analysis of Mooring Systems in the Time Domain

In mooring system design, the analysis often includes both static and dynamic analyses, and the static analysis precedes the dynamic analysis. The static analysis finds the equilibrium position and direction of each element of the model. Thus, all forces and moments such as structures, mooring lines, and environmental external forces are in equilibrium in static analysis. If static analysis is performed, the initial values in static equilibrium can be confirmed. Such values include the draft of the floating body to which the mooring line is connected, the pre-tension of the mooring line, and the suspended line length of the mooring line. Static analysis has an advantage in that its computation time is not usually long unless the mooring system is abnormal. Therefore, the static analysis is used to determine the direction of the initial design of the mooring system. However, the static analysis cannot consider the non-linear effects that occur over time and cannot provide the statistical results of the mooring system. In order to design a mooring system, it is necessary to check whether the results of the mooring system, such as the fairlead tension or the offset of the floating body, satisfy the design criteria. Dynamic analysis in the time-domainis performed to confirm this in more detailed manners. In Orcaflex, one of the tools used for mooring line analysis, dynamic analysis in the time-domainis performed as follows. The time-domainanalysis in Orcaflex is fully nonlinear. Mass, damping, stiffness, and loading are analyzed at every time step. The dynamic analysis then solves Equation (1) in each time step in the time domain.

$$M\left(p,a\right)+C\left(p,v\right)+K\left(p\right)=F_W\left(p,v,t\right)+F_M\left(p,v,t\right)$$

(1)

In Equation (1), $M\left(p,a\right)$ is the system inertia load, $C\left(p,v\right)$ is the system damping load, $K\left(p\right)$ is the system stiffness load, $F_W\left(p,v,t\right)$ is the external load by waves, and $F_M\left(p,v,t\right)$ is the external load by mooring lines. The input values $p, v, t$ and a are the floating body’s position, velocity, simulation time, and acceleration vectors, respectively. Equation (1) is solved for floating bodies and mooring lines. There are several methods for calculating the load by the mooring lines, such as the lumped mass method and the finite element method. Orcaflex uses a lumped mass method.

Calculation is performed by dividing the entire line into nodes and segments, respectively. At this time, the node is assigned the properties (mass, weight, drag, buoyancy) of half of the segments on both sides of the actual pipe. Another model element segment is a straight, massless element that models just the axial and torsional properties of the line. Time–domain dynamic analysis takes more time than calculating static analysis to find static equilibrium because analysis has to be done at every time step. In addition, in order to obtain the statistical results of the mooring system mentioned above, the time–domain dynamic analysis was performed for 3 h according to DNV-GL [12].

Establishing the reliability of results obtained from numerical simulations, such as Orcaflex, requires a verification procedure to ensure their validity. To ensure the reliability of results, cross-comparisons are made among outcomes obtained from multiple numerical simulations under consistent conditions, or the results are validated through model testing. Conducting a model test for a deep-water mooring system may pose challenges in achieving an accurate simulation due to limitations in the experimental facilities. In such cases, truncated mooring is used to overcome the limitations of experimental facilities, as demonstrated by Fan et al. [13] and Ferreira et al. [14]. Since this paper is a study that performs the optimization design of the mooring system rather than requiring accurate simulation results, a separate procedure to verify the reliability of the simulation results was not conducted.

4. Bayesian Optimization Algorithm

BOAs allow for non-linear objective functions, and they are well known as suitable for optimization problems that need to use a relatively long computation time of objective function. The optimization process of the BOAs is carried out over iterations. At each iteration, the BOAs use a function referred to as an acquisition function for the determination of a new data point for further investigation. The BOAs are probability-based methods, in which the BOAs evaluate the new data point and update the probabilities of the objective function. Probabilities of the objective function are called surrogate model or proxy model for BOAs. These models probabilistically estimate the objective value of an arbitrary solution. There are multiple alternatives to build the surrogate models, such as Gaussian process (GP), random forest, and tree-structured Parzen estimator (TPE). In particular, the GP is often used as a surrogate model for BOA, which is also used in this paper. In GPs, an arbitrary solution is defined using finite design variables X, which follow a multivariate Gaussian distribution in which the mean function is denoted $m\left(X\right)$ for any finite variables x, and the covariance function is denoted $k\left(x,x^{*}\right)$. Thus, GPs can be denoted as:

$$f\left(x\right)~ GP\left(m\left(X\right),k\left(x,x^{*}\right)\right).$$

(2)

In this paper, we assume that the prior distribution of $f$ is a multivariate normal distribution with a mean function of 0 and a covariance function K. The covariance function is defined as the exponential square function described in Equation (3). ${\sigma }^2{}_f$, ${\sigma }^2{}_n$, l are hyperparameters of the covariance function, and depending on the values, the posterior of the GP is determined in various ways. However, in this paper, because an arbitrary solution is defined at equal intervals, we believe that the effect of the hyperparameters would be small. Thus, the hyperparameters’ value is assumed to be 1.

$$k\left(x_i, x_j\right)={\sigma }^2{}_{f}exp(-\frac{1}{2l^2}\left(x_i-x_j {)}^2\right)+{\sigma }^2{}_n$$

(3)

In Equation (4), the covariance function represents the correlation between the i-th and j-th sample data. The covariance function K for sample data t composed of design variables X can be expressed as:

$$\mathrm{K}=\left[\begin{array}{cccc}k\left(x_1,x_1\right)& k\left(x_1,x_2\right)& \cdots & k\left(x_1,x_t\right)\\ k\left(x_2,x_1\right)& k\left(x_2,x_2\right)& \cdots & k\left(x_2,x_t\right)\\ ⋮& ⋮& \ddots & ⋮\\ k\left(x_t,x_1\right)& k\left(x_t,x_2\right)& \cdots & k\left(x_t,x_t\right)\end{array}\right].$$

(4)

Using Equation (3), K can be calculated, and $f_{1:t}$ can be defined based on sample data t. When we have a new sample data $x_{t+1}$, $f_{t+1}$ is represented by a normal distribution with dimension t + 1 as follows:

$$\left[\begin{array}{c}{f}_{1:t}\\ f_{t+1}\end{array}\right] ~ N(0,\left[\begin{array}{cc}\mathrm{K}& \mathrm{k}\\ {\mathrm{k}}^{\mathrm{T}}& k\left(x_{t+1},x_{t+1}\right)\end{array}\right]).$$

(5)

where $f_{1:t}$ and k are defined in Equations (6) and (7), respectively.

$$f_{1:t}={\left[f_1,f_2,\cdots ,f_t\right]}^{\mathrm{T}}$$

(6)

$$\mathrm{k}=\left[k\left(x_{t+1},x_1\right),k\left(x_{t+1},x_2\right),\cdots ,k\left(x_{t+1},x_t\right)\right]$$

(7)

As defined by Williams et al. [15], the posterior distribution of the GP is defined by Equations (8)–(10) for data t + 1 through the prior distribution of the GP defined above.

$$f_{t+1} ~ N\left({\mu }_{t+1},{\sigma }_{t+1}^2\right)$$

(8)

$${\mu }_{t+1}\left(x_{t+1}\right)={\mathrm{k}}^{\mathrm{T}}{\mathrm{K}}^{-1}{f}_{1:t}$$

(9)

$${\sigma }_{t+1}^2\left(x_{t+1}\right)=-{\mathrm{k}}^{\mathrm{T}}{\mathrm{K}}^{-1}\mathrm{k}+k\left(x_{t+1},x_{t+1}\right)$$

(10)

Finally, the objective function can be determined as a mean function and uncertainty region for an arbitrary finite design variable X using the posterior distribution of the GP.

The level of uncertainty of the objective function lower as more data are investigated. In BOA, the acquisition function is used to investigate effective data for tracking the objective function. Probability of improvement (PI) and expected improvement (EI) are often used as the acquisition function. The PI and EI are described in Equations (11) and (12), respectively. $N\left(\mu \left(x^{*}\right),\sigma \left(x^{*}\right)\right)$ in Equation (11) defines the gaussian distribution at any $x^{*}$ in surrogate model GP. $f\left(\widehat{x}\right)$ is the largest value among the currently investigated data in GP. Therefore, the meaning of PI is to determine the acquisition cost by checking only at the probability that the $f\left(x^{*}\right)$ at an any $x^{*}$ value is higher than the largest current investigated data $f\left(\widehat{x}\right)$. However, in Equation (12), which means EI, the term $\left(f\left(x^{*}\right)-f\left(\widehat{x}\right)\right)$ is added to Equation (11). The additional term introduced into the equation takes into account the impact of standard deviation. For example, there are two data points x₁ and x₂ that have a same PI value but x₁ has a higher standard deviation than x₂. In this case, the x₁ has a higher EI value because of the additional term in Equation (12). In BOAs, EI is mainly used because EI rather than PI applies the acquisition cost considering the effect of standard deviation. Thus, in this paper, we use EI as an acquisition function.

$$PI\left[x^{*}\right]={{\displaystyle \int }}_{f\left(\widehat{x}\right)}^{\infty } N\left(\mu \left(x^{*}\right),\sigma \left(x^{*}\right)\right)df\left(x^{*}\right)$$

(11)

$$EI\left[x^{*}\right]={{\displaystyle \int }}_{f\left(\widehat{x}\right)}^{\infty } \left(f\left(x^{*}\right)-f\left(\widehat{x}\right)\right)N\left(\mu \left(x^{*}\right),\sigma \left(x^{*}\right)\right)df\left(x^{*}\right)$$

(12)

5. Case Study

There is an existing mooring system operating for oil and gas platforms in polar regions. The system is point-symmetric, and it was designed by human experts based on mooring systems in similar environmental conditions. This case study used the BOA to investigate further for better design candidates than the existing system. Furthermore, this case study compared the BOA with another optimization method, GA, that used an artificial neural network (ANN) for fast evaluation in optimization.

Figure 2 illustrates the 3D model of the vessel used in this case study and

Table 2. Principle dimensions of the vessel.

Description	Value	Unit
LBP	244	M
Breath	50	M
Draft	18.6	M
Volume	169.641	M3
GMX	1.99	M
GMY	239.24	M
XCG	117.7	M
YCG	0	M
VCG	19.5	M
$k_{xx}$	17.4	M
$k_{yy}$	59.3	M
$k_{zz}$	60.0	M

presents the vessel’s information, in which the LBP, GM, XCG, YCG, VCG, $k_{xx}$, $k_{yy}$, and $k_{zz}$ indicate length between perpendiculars, metacentric height, surge-directional center of gravity, sway-directional center of gravity, heave-directional center of gravity, and radius of gyrations along the surge-directional axis, the sway-directional axis, and the heave-directional axis, respectively.

Figure 3 and Figure 4 present the mooring configuration. Single-point mooring is used, and the mooring system comprises 16 lines with 4 bundles of 4 lines. Each of the 16 lines comprises 3 segments that are top chain, middle wire, and bottom chain.

Table 3. Properties of each segment.

	Top Chain	Middle Wire	Bottom Chain
Type	R4 studless chain	Spiral strand wire	R4 studless chain
Diameter [mm]	171	153	171
Weight in water [kg/m]	556	81	556
Weight in air [kg/m]	640	93	640
Minimum Breaking Load [kN]	26,952	14,830	26,952
Axial stiffness	2.95 × 1012	935.75 × 103	2.95 × 1012
Inertial coefficient [Normal/Axial]	1.0/0.5	1.0/0	1.0/0.5
Drag coefficient [Normal/Axial]	2.4/1.15	1.2/0	2.4/1.15

presents the mooring line information for each segment.

The environmental conditions were defined as a non-collinear condition according to the DNVGL rule. Figure 3 presents environmental directions applied to the mooring system.

Table 4. Environmental conditions.

Wave [JONSWAP spectrum]	HS [m]	14.54
	TP [S]	16.18
	Peak enhancement factor [γ]	2.39
	Direction [degree]	180
Wind [API spectrum]	Speed at 10 m [m/s]	31.86
	Direction [degree]	150
Current	Speed at surface [m/s]	0.66
	Direction [degree]	135

presents the details of the environmental conditions including the wave, wind, and current.

For defining optimization models, parameters and design variables need to be defined. This case study used the water depth, pretension, the diameter and overall length of the lines, and the length of the top chain as parameters.

Table 5. Parameters for the mooring system design.

Parameter	Value	Unit
Water depth	200	M
Footprint radius	3	×WD (Water Depth)
Pre-tension	10	% of MBL
Line diameter	0.171	M
All line length	660	M
Top chain length	60	M

gives detailed information on the parameters.

As design variables, this case study used the OBA, IBA, middle wire length, and bottom chain length of the system.

Table 6. Design variables for the mooring system design.

Design Variables	Range	Interval	Unit
Outer bundle angle	10–60	1	Degree
Inner bundle angle	1–9	1	Degree
Middle wire length	0–350	5	m
Bottom chain length	250–600	5	m

presents the design variables. As mentioned in Park et al. [11], the OBA and IBA are important factors that affect the performance of mooring systems. The mooring system is a point symmetric because it is a single point mooring using a turret system. The value ranges of OBA and IBA were determined to avoid the point-symmetric mooring lines not overlapping. The middle wire and bottom chain lengths affect the stability of the mooring system. Because this case study assumes that the total length of the mooring line is constant, the middle wire and bottom chain lengths are dependent on each other. The maximum value of middle wire length was determined to prevent the wire rope from touching the seafloor and the minimum value of the middle wire was determined considering the weight of the bottom chain, in which the line tension that is caused by the weight of the bottom chain should not exceed the previously selected pretension. Xu and Soares [16] mentioned that the type of anchor or the connection with the bottom of an area affects the interpretation. However, the mooring system optimization in this paper was applied to the mooring system in the conceptual design stage, so the case study for the type of anchor was not considered. In addition, constraints are set to prevent contact of the middle wire rope with the seabed, so the effect on the chains laid on the seabed is similar.

The objective function is to minimize the maximum tension of the mooring system. There are different ways to represent the maximum mooring tension in a mooring system. For instance, there are ways that conservatively evaluate the maximum mooring tension using statistical methods. However, for simplification, this case study used the maximum tension that was observed on the 16 mooring lines in the time–domain analysis for 3 h. Figure 5 is a schematic diagram of the process of obtaining the maximum tension, an example of which is described in Figure 6.

Figure 7 presents the procedure the BOA used in this case study. In the initial point searching step, a probability-based evaluation model was created using 50 initial data points that were selected in a random manner. In the optimal point searching step, the BOA investigated a further 100 data points in which the evaluation model became more accurate and reliable gradually as more data points were investigated.

GA are one of the meta-heuristics and have been used for a wide variety of optimization problems. GAs determine the optimal solution over iterations creating and evaluating new solutions referred to as populations. The creation of a new population uses reproduction operators which are crossover, mutation, and elite operators. The hyperparameters of GAs comprise the population size, the fraction of the reproduction operators, and the termination criterion.

Table 7. The hyperparameters of the genetic algorithm used in the experiment.

Items	Value
Final iteration	500
Population size	100
Number of new chromosomes using crossover	60
Number of mutations	20
Number of elites	20

presents the hyperparameters used in this experiment. The optimization process of the GA stopped when the iteration reached the final iteration. Figure 8 illustrates the representation scheme of chromosomes and reproduction operators used in this experiment. The genes of the chromosomes indicate the design variables which are OBA, IBA, middle wire length, and bottom chain length presented in Table 6.

The GA used an ANN as an evaluation function, the training data of which were created using the time–domain dynamic analyses on the mooring systems. The ANN is a multilayer perceptron (MLP) and the hyperparameters of the MLP are presented in

Table 8. The hyperparameters of the multilayer perceptron.

Hyperparameters	Value
Number of hidden layers	9
Number of neurons	4
Epochs	15,000
Batch size	211
Type of optimizer	Adam
Type of activation function	Sigmoid
Batch Normalization	Use
Type of weight initialization	Grout uniform
Weight regularization	Use

. The hyperparameter values were determined after test trials in which different values were investigated to improve the performance of the MLP. The training of the ANN used 3120 data, which was collected using uniform data sampling and is 10% of total cases which can be defined using the ranges and steps of the design variables in Table 5. Figure 9 illustrates the uniform data sampling.

The performance of the ANN was measured using $R^2$ which is defined in Equation (13) in which $y_i$ is the test data output,$\widehat{y_{ı}}$ is the prediction, and $\overline{y_{ı}}$ is the average of the test data output.

$$R^2=1-\frac{\frac{1}{n}{{\displaystyle \sum }}_{i-1}^n {\left(y_i-\widehat{y_{ı}}\right)}^2}{\frac{1}{n}{{\displaystyle \sum }}_{i-1}^n {\left(y_i-\overline{y_{ı}}\right)}^2}$$

(13)

The $R^2$ of the ANN is 0.87, which is a generally accepted value. The $R^2$ of the ANN used in Liang et al. [17] for instance was also 0.87.

Table 9. Maximum tension and design variables of each mooring system.

	OBA [Degree]	IBA [Degree]	Wire Length [m]	Maximum Tension [kN]
Human experts	42	2	200	15,208
BOA	11	6	120	7837
ANN based GA	15	4	130	8133

presents the maximum tension and the values of the design variables determined by the human experts, the BOA, and the ANN based GA, respectively. This shows that both optimization methods, the BOA and the GA, determined the better maximum tension of the designs that were 50% lower than that of the design of the human experts. Compared with the ANN based GA, the BOA determined the slightly better design. Figure 10 shows the configuration of the mooring systems determined by the human experts, the BOA, and the ANN based GA.

However, when it comes to the computation time, it seems that the BOA has an explicit advantage over the ANN based GA because of the long training time of the ANN. For training the ANN, 3120 dynamic analyses were performed, in which a single analysis took 40 min approximately. The total computation time of the ANN based GA, which comprises training the ANN and the process of the GA, was 2080 h while the computation time of the BOA was 100 h. Because most of the computation time of the ANN based GA is determined by the training time of the ANN, this huge computation time gap can be reduced significantly if the ANN is reusable. However, the reusability of the ANN may be low because MLPs often have overfitting problems. This indicates that the ANN may have low prediction accuracy if there are changes in the environmental conditions. There are some ways to increase the reusability of the ANN. One is to use a huge data set that covers all possible environmental conditions. When it comes to the cost for the data collection, this may be extremely difficult. Another is to use an ANN that needs a small number of training data points but could have good performance. Meta-learning for few-shot learning, for instance, is a promising candidate method which is one type of the ANNs that creates common knowledge across multiple similar problems which enables training of an ANN for a new problem to be possible using a small data set. In fact, few-shot learning can be implemented in different ways, such as transfer learning. More detailed information on few-shot learning is described in Wang et al. [18].

Another advantage of the BOA is the accuracy of the evaluation function. Although the prediction performance of the ANN is relatively high, the prediction errors could arise. In the comparison, for instance, the optimal mooring system was expected to have 8343 kN maximum tension when the system was evaluated using the ANN. However, the actual maximum tension that was determined using the dynamic analysis was 8133 kN. One could consider this prediction error positive but this could affect the optimization results in a negative manner because the better solutions that appeared during the optimization process may not be able to survive because of the inaccurate evaluation; thus we missed the solutions.

6. Conclusions

This paper presented a BOA that uses time-domain analysis in evaluation for better accuracy considering dynamic forces which cannot be considered in static analysis and frequency domain analysis. In a case study, the mooring system designed by the BOA was compared with that by human experts and an ANN based GA. In the aspect of maximum tension, both optimization methods, the BOA and the GA determined better designs than the human experts. However, the optimization time of the GA needed 20 times more than that of the BOA because of the training time of the ANN of the GA. The BOA used only 150 data points in the whole optimization process, but the training of the GA used 3120 data points. This disadvantage of the ANN based GA can be reduced significantly if the training process of the ANN could be decreased with increased reusability. However, due to the overfitting problems of the MLP, it is difficult to reuse the ANN in different environmental conditions. There are some ways to increase the reusability of the ANN. One is to use a huge data set that can cover all possible environmental conditions and another is to use an ANN that is more reusable such as meta-learning for few-shot learning.

In this paper, the prime goal of the mooring system design problem was on the minimization of the maximum tension. However, there could be other performance indicators that need to be considered, such as the lifecycle cost and offset of mooring systems. Therefore, further research is needed for the multi-objective design problems that use the time–domain analysis.

Moreover, the dynamic analysis in the case study was carried out using a single sea state. There are multiple reasons why we did not use multiple sea states, but the prime reason is the high computational cost. In further research, the BOA needs to be investigated more using multiple sea states, in which an important concern would be the selection of representative sea states that greatly affect the performance of the mooring system.

Further research should also be carried out comparing the two methods, the BOA and ANN based GA, using an improved ANN that would be developed using a more advanced model than a MLP such as meta-learning for more efficient learning.