## 2 Reliability Analysis Method of 3D Slopes
### Particle Swarm Optimization Algorithm  
The PSO algorithm begins by setting up a swarm of \(M_{p}\) particles and the search dimension for a single particle is \(N_{p}\). Then, the optimal solutions of the issue are obtained by iteratively updating the positions of the particles in the particle swarm. The equationsfor adjusting the velocity and position of an individual particle, drawing from its best historical position and the best position attained by the swarm, are presented as follows:  
\[v_{ij}(t+1)=\omega v_{ij}(t)+c_{1}r_{1}\big{(}Pb_{ij}(t)-x_{ij}(t)\big{)}+c_{2}r _{2}\big{(}Gb_{j}(t)-x_{ij}(t)\big{)} \tag{6}\]  
\[x_{ij}(t+1)=x_{ij}(t)+v_{ij}(t+1) \tag{7}\]  
In Equations (6) and (7), \(v_{ij}(t+1)\) represents the velocity of particle \(i\) in the \(j\)-th dimension at the \((t+1)\) iteration; \(x_{ij}(t+1)\) represents the position of particle \(i\) in dimension \(j\) at iteration \((t+1)\); \(t\) signifies the iteration count; \(c_{1}\) and \(c_{2}\) are learning factors; \(\omega\) is the inertia weight; \(r_{1}\) and \(r_{2}\) are random numbers between 0 and 1; \(Pb_{j}(t)\) denotes the historical best position of particle \(X_{i}\) at iteration \(t\); and \(Gb(t)\) is the global best position of the swarm at iteration \(t\). The update process for the individual particle's best position \(Pb_{j}(t+1)\) and the global best position \(Gb(t+1)\) during the search for the optimal solution is as follows:  
\[Pb_{ij}(t+1)=\left\{ \begin{array}{lcl}x_{ij}(t+1)&if&F(x_{i}(t+1))<F(Pb_{i}( t))\\ Pb_{ij}(t)&if&F(x_{i}(t+1))\geq F(Pb_{i}(t))\end{array} \right. \tag{8}\]  
\[Gb_{j}(t+1)=\left\{ \begin{array}{lcl}x_{ij}(t+1)&if&F(x_{i}(t+1))<F(Pb_{i}( t))\\ Gb_{j}(t)&if&F(x_{i}(t+1))\geq F(Pb_{i}(t))\end{array} \right. \tag{9}\]  
where \(F(\ )\) represents the fitness function. Building upon the empirical analysis of standardized parameter configurations for the PSO algorithms, this study adopts the widely validated parameter set proposed by [PERSON] and [PERSON] (2007) [32], where the inertia weight coefficient \(\omega\) = 0.7 (\(\omega\) = 0.4\(\sim\)0.9) and acceleration coefficients \(c_{1}\) = \(c_{2}\) = 2.0. These values have been systematically demonstrated to balance global exploration and local exploitation across diverse optimization problems, ensuring robust convergence behavior while maintaining algorithmic stability. The workflow for identifying the critical slip surface and determining the minimum safety factor using the PSO algorithm is summarized as follows:  
Step I: Particle Swarm Initialization. Establish a particle swarm with a given size and a given dimensional search space, and then randomly generate the initial velocities \(v_{i}(0)\) and positions \(x_{i}(0)\) for each particle, and set parameters such as inertia weight \(\omega\), learning factors \(c_{1}\), \(c_{2}\), and maximum number of iterations \(T_{max}\). In this study, the inertia weight coefficient \(\omega\) is 0.7, acceleration coefficients \(c_{1}\) and \(c_{2}\) are 2.0 and the maximum number of iterations \(T_{max}\) is 30. Step II: Fitness Evaluation. Compute the fitness value \(F_{s}\) (safety factor) via the 3D Bishop method (Equation (5)). Update the best position \(Pb(t)\) (i.e., the historically best slope failure surface parameters found by each particle) for each particle and the global best position \(Gb(t)\) (i.e., the best slope failure surface parameters found among the entire particle swarm) across the swarm. Step III: Velocity and Position Update. Update the velocity \(v_{ij}\) and position \(x_{ij}\) of each particle according to the PSO governing equations. Then, define the maximum \(v_{max}\) and minimum \(v_{min}\) velocities. Adjust velocities to these bounds: if \(v_{ij}<v_{min}\), set \(v_{ij}=v_{min}\); if \(v_{ij}>v_{max}\), set \(v_{ij}=v_{max}\). Step IV: Termination Check. Verify whether the iteration count reaches \(T_{max}\). If satisfied, output the optimal solution; otherwise, repeat Steps II-III.