## 3 Results  
In the formula, x is the current solution; x\({}^{\prime}\) is the new solution; f(x) is the objective function value of the solution; and T is the temperature. Based on the above principles, Equations (6) to (8) were optimized by applying the SA algorithm using MATLAB software to obtain a Pareto solution. The initial temperature of the model is set to 100, the termination temperature is 1, the cooling rate is 0.9, and the maximum number of iterations is 5000. The Pareto optimal solution set of the multi-objective optimization problem of the quarry parameters is obtained using this method. In order to meet the actual engineering requirements and improve the economic benefits of the mine, it is recommended to appropriately reduce the width of the inter-columns and the thickness of the top slab left in place. Therefore, the Pareto optimal solution set is sorted in descending order according to the recovery rate, and the top 15 solutions are obtained, as shown in Table 7. Figure 5 shows the Pareto frontier points of the target optimal solution set.  
Figure 4: Flow chart of Pareto simulated annealing algorithm.  
According to Table 7, Optimal Solution 1 (with a stope span of 15 m, pillar thickness of 10 m, and top pillar thickness of 11.9 m) represents the optimized structural parameters, and both the stope stress and displacement are within the controllable range of the limit mining dimensions, meeting engineering design requirements. However, despite these parameters being able to satisfy engineering design needs after optimization, some technical limitations may still arise during actual construction.  
Firstly, the machinery required for construction must have sufficient capacity to handle specific rock types, mining depths, and the precision of stope structural parameters, as the efficiency and reliability of the equipment directly impact construction progress and safety. Secondly, different construction methods (such as explosive blasting or mechanical mining) have varying suitability and economic viability for different rock types, necessitating the selection of appropriate techniques based on specific strata and conditions.  
Additionally, real-time monitoring technologies for stress, displacement, and groundwater levels need to be sufficiently advanced and accurate to provide timely feedback on the stability of the stope, allowing for the formulation of corresponding adjustment plans. Furthermore, the mechanical properties of the rock must be thoroughly studied to address  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Serial Number** & **X1** & **X2** & **X3** & **Y1** & **Y2** & **Y3** \\ \hline
1 & 15.003 & 10.000 & 11.901 & 0.251 & 3.916 & 4.132 \\
2 & 15.000 & 10.005 & 12.129 & 0.247 & 3.916 & 4.114 \\
3 & 15.000 & 10.020 & 12.112 & 0.247 & 3.919 & 4.111 \\
4 & 15.000 & 10.049 & 12.078 & 0.247 & 3.923 & 4.105 \\
5 & 15.002 & 10.053 & 12.078 & 0.247 & 3.925 & 4.104 \\
6 & 15.003 & 10.043 & 12.100 & 0.247 & 3.923 & 4.105 \\
7 & 15.000 & 10.001 & 12.178 & 0.247 & 3.917 & 4.112 \\
8 & 15.000 & 10.000 & 12.180 & 0.247 & 3.917 & 4.112 \\
9 & 15.000 & 10.000 & 12.191 & 0.247 & 3.917 & 4.112 \\
10 & 15.000 & 10.073 & 12.093 & 0.246 & 3.928 & 4.097 \\
11 & 15.000 & 10.070 & 12.100 & 0.246 & 3.927 & 4.097 \\
12 & 15.007 & 10.031 & 12.220 & 0.245 & 3.925 & 4.101 \\
13 & 15.000 & 10.058 & 12.233 & 0.244 & 3.928 & 4.091 \\
14 & 15.000 & 10.087 & 12.195 & 0.244 & 3.932 & 4.086 \\
15 & 15.002 & 10.102 & 12.184 & 0.244 & 3.935 & 4.082 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Optimization results of Pareto simulated annealing algorithm.  
Figure 5: Pareto front points of the Pareto optimal solution set.  
complex geological conditions, especially when encountering brittle or plastic deformation in the strata, which may require more complex reinforcement and support techniques.