## 3 Fuzzy Set Theory-Based Risk Assessment System for Navigational Obstacles
### Collision Risk Assessment Model
#### 3.3.3 Monte Carlo Simulation Results  
Monte Carlo simulations were performed to estimate the probability of collision by accounting for uncertain variables in situations where statistical data were insufficient because of the small number of collisions between navigational obstacles and ships. This method allows more realistic collision probabilities to be calculated by repeatedly simulating various scenarios. Additionally, it helps to derive the risk management criteria by effectively providing the data required for the regression model [35; 36]. The initial simulation environment for measuring the influence of each variable was as follows: \(N=1,000,000\), \(L=1000\), \(D=1\), \(S=5\), \(V=0.1\), and \(M=1\). Based on this setup, the probability of collision according to the variable range was calculated using (1)-(7), as shown in Figure 6. The variable ranges were determined according to Table 3.  
From the Monte Carlo simulation, \(S\) (size of drifting obstacle) and \(M\) (number of moving ships) were found to be variables that significantly affect the probability of collision, whereas \(D\) (number of drifting obstacles) and \(V\) (speed of drifting obstacles) were found to have no significant effect. Additionally, the R-squared value of the model was very high at  
Figure 6: Collision Probability based on variable range. (**a**) Collision probability relative to the number of drifting objects, (**b**) Collision probability relative to the drift object size, (**c**) Collision probability relative to the drift velocity, and (**d**) Collision probability relative to the number of moving objects. 0.888, indicating that the model explains a large portion of the variance in the probability of collision. The regression model was derived from the Monte Carlo simulation, as shown below. The regression equation for the variables is:  
\[collision\ probability=\beta_{0}+\beta_{1}\cdot D+\beta_{2}\cdot S+\beta_{3}\cdot V +\beta_{4}\cdot M. \tag{8}\]  
Here, the estimated coefficients of the model are \(\beta_{0}=-0.040041\), \(\beta_{1}=-7.1868\times 10^{-7}\), \(\beta_{2}=0.0025018\), \(\beta_{3}=3.7425\times 10^{-6}\), \(\beta_{4}=0.013346\). Therefore, the regression equation for predicting the collision probability by applying the estimated coefficients is:  
\[collision\ probability=-0.040041-7.1868\times 10^{-7}\cdot D+0.0025018\cdot S+3.7425 \times 10^{-6}\cdot V+0.013346\cdot M. \tag{9}\]  
For instance, if the size of the drifting obstacle is 10, the number of drifting obstacles is 5, the speed of the drifting obstacles is 2, and the number of moving ships is 8, the collision probability can be determined by applying the regression model. Based on these values, the calculated collision probability is approximately 0.0917. The collision probability values were utilized to establish five risk levels, with the range for each level defined based on the distribution of collision probabilities. Risk management criteria were subsequently derived from these levels. The range of collision probabilities was determined by identifying the minimum and maximum values from the data collected through simulation results and prediction range analysis. The boundaries for each risk level were calculated as percentiles, using the minimum and maximum collision probability values as the basis. Figure 7 visually depicts the distribution and frequency of the collision probabilities. Specifically, percentiles were calculated by dividing the range between the minimum and maximum collision probability values into equal intervals. These intervals were selected to effectively categorize the collision probabilities into five risk levels, reflecting their distribution and frequency. Table 4 outlines the ranges for each risk level based on collision probabilities, while Figure 7 illustrates the distribution to aid in understanding the process of deriving these risk levels. Additionally, Table 5 provides detailed sections for each risk level, calculated using the minimum and maximum values.