## 2 Methods
### Risk Assessment Model
#### 2.1.3 Multi-Layer Fuzzy Comprehensive Evaluation Model  
Fuzzy comprehensive evaluation is a method based on the principles of fuzzy mathematics, quantifying some fuzzy or difficult-to-quantify factors of problems by applying the membership theory of fuzzy mathematics and distinguishing the membership levels of evaluated objects from multiple factors [26]. This study uses the multi-layer fuzzy comprehensive evaluation method to evaluate the risks that affect the safety of sailing ships. The main steps in a fuzzy comprehensive evaluation include establishing a set of evaluation factors and evaluation comments, determining the weights of evaluation indicators and membership functions, and selecting comprehensive operators. Firstly, this study determines the set of evaluation factors that we need to evaluate the risk of maritime navigation environment, namely establishing hazard assessment indicators and vulnerability assessment indicators. Then, the evaluation level is determined for each indicator, and the fuzzy evaluation matrix and the weight of the evaluation indicators are determined based on relevant data. Finally, the weight of the indicators and the fuzzy evaluation matrix are fuzzy-synthesized, obtaining the final risk assessment result. Multi-layer fuzzy comprehensive evaluation is an improved evaluation which adds a criterion layer based on the single-layer fuzzy comprehensive evaluation. The steps are as follows:  
1. Establish a set of evaluation factors and evaluation comments  
We use the risk assessment indicators mentioned in Section 2.1.1 as the set of evaluation factors for the evaluation model, namely \(U=U_{1},U_{2}\), where \(U_{1}=\text{Wind speed}(u_{1})\), Wave height \((u_{2})\), Ocean currents \((u_{3})\), Visibility \((u_{4})\) and \(U_{2}=\text{Vessel size}\ (u_{5})\), Vessel age \((u_{6})\), Vessel flag\((u_{7})\). Based on Section 2.1.2, we divide the indicator data into five levels, which we mark as the set of evaluation comments for the set of evaluation factors, namely, \(V=\{\text{High risk}\), Higher risk, Medium risk, Lower risk, Low risk\(\}\). 1. Constructing membership functions  
Common membership functions include triangular distribution, trapezoidal distribution, parabolic or semi-parabolic distribution, Gaussian distribution, and sigmoidal distribution [27]. Considering the characteristics of the value range of the indicators selected in this study and the adaptability of the sigmoidal membership function, the sigmoidal membership function is the best choice. The basic formula is as follows. \[f(x,a,c)=\frac{1}{1+e^{-a(x-c)}} \tag{3}\]  
where the magnitude of \(a\) controls the width of the transition area at the relevant level, \(c\) presents the center of the transition area, and return value \(f\) is the degree of subordination of the indicators to risk levels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Risk Value** & **Wind Speed** & **Wave Height** & **Ocean Currents** & **Visibility** \\ \hline High risk & [20.8, \(\omega\)) & [6, \(\omega\)) & [2, \(\infty\)) & [0,3.1) \\ Higher risk & [15.6, 20.8) & [4.5, 6) & [1.5, 2) & [3.1, 6.2) \\ Medium risk & [10.4, 15.6) & [3, 4.5) & [1, 1.5) & [6.2, 9.3) \\ Lower risk & [5.2, 10.4) & [1.5, 3) & [0.5, 1) & [9.3, 12.4) \\ Low risk & [0, 5.2) & [0, 1.5) & [0, 0.5) & [12.4, \(\infty\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Risk grading standards. Step 3 Determine the single-factor evaluation matrix of the sub-criteria layer  
The single-factor evaluation matrices \(E_{k}\) are constructed according to Equation (3), and the results are shown in Equations (4) and (5). The \(\varepsilon_{\rm ij}\) in matrices \(E_{k}\) represents the membership degree of the \(i\)th indicator belonging to the \(j\)th evaluation level. \[E_{1}=