# NSGA-III-XGBoost-Based Stochastic Reliability Analysis of Deep Soft Rock Tunnel
## 3 NSGA-III Algorithm
For Monitoring point A and Monitoring lines BC, AB, AC, AD and AE, the \(R2\_\)score indices of the surrogate models based on NSGA-III-XGBoost, SVR and BP are shown in Figure 4. The corresponding reliability of Monitoring point A and Monitoring lines BC, AB, AC, AD and AE are calculated according to Equations (7) and (8), shown in Figure 5.

From Figure 4, we can note that the \(R2\_\)score indices of NSGA-III-XGBoost-based surrogate models of Monitoring point A and Monitoring lines BC, AB, AC, AD and AE are all greater than 0.945; the \(R2\_\)score indices of SVR-based surrogate models are all less than 0.89, and those of the BP-based surrogate models fluctuate between 0.863 and 0.966. Furthermore, the \(R2\_\)score index of each measuring point/line based on NSGA-III-XGBoost is not less than that based on SVR or BP, and among them, the NSGA-III-XGBoost-based \(R2\_\)score index values have the least fluctuation. From Figure 5, we can note that the reliability index results of NSGA-XGBoost are relatively close to those of BP, and differ significantly from those of SVR. The NSGA-XGBoost-based reliability index results of monitoring lines AB and AC or AD and AE show very little

Figure 4: Comparison of \(R2\_\)score index based on NSGA–XGBoost, SVR and BP. Figure 5: Comparison of tunnel reliability indexes calculated using NSGA–XGBoost, SVR and BP. change. Due to the symmetry of monitoring lines AB and AC or AD and AE, the reliability index of monitoring lines AB and AC or AD and AE should be the same, but there is a big difference in their BP-based results. #### 5.3.2 Comprehensive Reliability Index of Tunnel Initial Support Structure System

The system reliability theory is often used to estimate the failure probability range of a structural system with fewer failure elements. If it is considered that the failure of each part of tunnel initial support structure is completely related, the displacement failure probability of the section can be expressed as Equation (10). \[P_{\text{f\_lower}}=\max(P_{\text{f1}},P_{\text{f2}},\cdots P_{\text{fn}}) \tag{10}\]

From Equation (10), we obtain the failure probability of the tunnel initial support structure as \(P_{\text{f\_lower}}=0.0467\%\). If the failure of each element is considered to be completely independent, the corresponding failure probability can be denoted as Equation (11). \[P_{\text{f\_upper}}=1-\prod_{i=1}^{n}\left(1-P_{\text{fi}}\right) \tag{11}\]

From Equation (11), we obtain the failure probability of the tunnel structure as \(P_{\text{f\_upper}}=0.233\%\). Therefore, the final failure probability of the tunnel structure system in this paper is between \(P_{\text{f\_lower}}\) and \(P_{\text{f\_upper}}\), and can be denoted as Equation (12). \[0.0467\%\leq P_{\text{f}}\leq 0.233\% \tag{12}\]

As the range of \(P_{\text{f\_lower}}\) and \(P_{\text{f\_upper}}\) is narrow, 0.14%, the average value of \(P_{\text{f\_lower}}\) and \(P_{\text{f\_upper}}\) of Equation (12) is taken as the point estimate value, and the corresponding comprehensive reliability index is 2.989. ## 6 Conclusions

A practical stochastic reliability method based on XGBoost and NSGA-III was proposed in this paper. LHS was used to prepare the sampling points for determining the geo-mechanical parameters dataset of XGBoost. The following conclusions were obtained in this paper. 1. For the tunnel initial support structure system with fewer failure elements, using the proposed NSGA-III-XGBoost-MCS method and the system reliability theory can estimate its failure probability range and determine its comprehensive reliability index. 2.