## 6 Case Validation
### Optimization of Equipment Safety Investment Programs in Coal Mine S Enterprises  
Since the digital twin coal mine equipment safety investment optimization model relies on complex technology, it may be difficult to implement and maintain, especially in enterprises with limited technical resources. Based on this, the historical investment in January 2024 and the equipment safety investment in February in the future of enterprise S are taken as an example of equipment safety input optimization based on the traditional optimization method, in order to maximize the benefits of the enterprise's equipment safety investment. The specific program optimization process is as follows:  
1. Enterprise S January 2024 (historical)  
Select the equipment safety investment data of enterprise S from April 2023 to December 2023 as the initial data, and carry out the optimization of the enterprise's equipment safety investment in January 2024, according to the equipment safety investment optimization model constructed above, and take the logarithm of both sides of Equation (4), yielding  
\[lnY=InA+\alpha_{1}lnX_{1}+\alpha_{2}lnX_{2}+\alpha_{3}lnX_{3}+\alpha_{4}lnX_{4 }+\alpha_{5}lnX_{5} \tag{10}\]  
Substituting the data processed by the buffer operator (Table 3) into Equation (10) yields  
\[ \begin{cases}4.39=lnA+\alpha_{1}5.51+\alpha_{2}4.99+\alpha_{3}5.33+\alpha_{4 }5.44+\alpha_{5}5.02\\ 4.37=lnA+\alpha_{1}5.53+\alpha_{2}5.02+\alpha_{3}5.35+\alpha_{4}5.48+\alpha_{5 }5.05\\ 4.28=lnA+\alpha_{1}5.61+\alpha_{2}5.18+\alpha_{3}5.38+\alpha_{4}5.50+\alpha_{5 }5.32\\ 4.19=lnA+\alpha_{1}5.71+\alpha_{2}5.32+\alpha_{3}5.49+\alpha_{4}5.63+\alpha_{5 }5.54\\ 4.08=lnA+\alpha_{1}5.74+\alpha_{2}5.40+\alpha_{3}5.56+\alpha_{4}5.69+\alpha_{5 }5.75\\ 4.05=lnA+\alpha_{1}5.81+\alpha_{2}5.35+\alpha_{3}5.62+\alpha_{4}5.80+\alpha_{5 }5.80\\ 3.88=lnA+\alpha_{1}5.86+\alpha_{2}5.45+\alpha_{3}5.69+\alpha_{4}5.87+\alpha_{5 }5.84\\ \end{cases}  \tag{11}\]  
Solving Equation (11) yields  
\[Y=32623.886X_{1}^{-0.23477}X_{2}^{-0.16387}X_{3}^{-0.39346}X_{4}^{-0.30159}X_{ 5}^{-0.030302} \tag{12}\]  
The normalized ratios of the elasticity coefficients \(\alpha\) for various equipment safety investment factors are as follows:  
\[\alpha_{1}:\alpha_{2}:\alpha_{3}:\alpha_{4}:\alpha_{5}=0.21:0.14:0.35:0.27:0.03\]  
Therefore, with the total equipment safety investment amount fixed, the optimal proportions for various equipment safety investments are as follows: coal mine safety transportation facility investment, coal mine emergency facility investment, coal mine \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Set** & **MAE** & **MAPE** & **RMSE** & **R2** & **Running Time** \\ \hline BP & 2.0509 & 0.026533 & 3.1903 & 0.93591 & 1.80 \\ PSO-BP & 1.3068 & 0.015444 & 2.5035 & 0.96053 & 3.61 \\ IPSO-BP & 0.55274 & 0.0072388 & 0.91689 & 0.99471 & 2.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Calculation results of relevant metrics for the test set. safety excavation facility investment, coal mine safety ventilation facility investment, safety education investment = 0.21:0.14:0.35:0.27:0.03. To compare the coal mine equipment safety investment plan of the enterprise for January 2024, the specific data are shown in Table 9:  
Compared with the actual equipment safety investment plan, the optimized plan reduced accident losses by 19.4%, which verifies the effectiveness of the optimization model. Taking S enterprise in February 2024 as an example (future scenario)  
Given that coal mine S enterprise plans to invest CNY 17.23 million in equipment safety in February 2024, using the same initial data from May 2023 to January 2024, the established investment optimization model can calculate the optimized equipment safety investment plan for coal mine S enterprise in February 2024.