## 4 Optimization Model of Equipment Safety Investment in Coal Mines
### Safety Investment and Accident Loss Behavior Model  
Coal mine equipment safety investment optimization is a multi-input optimization problem, that is, it is necessary to reasonably allocate various safety inputs to maximize the safety economic benefits under a certain amount of inputs. Coal mine equipment safety investment includes investment in safe transportation facilities, safe ventilation facilities and safety education. By reasonably adjusting the proportion of these investments, the level of safety production can be improved, the possibility of accidents can be reduced,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Classification Name** & **Connotation** & **Elements** \\ \hline Coal mine safety & Aimed at ensuring the safe and efficient operation of various transportation equipment (such as hoists, conveyor belts, etc.) within the coal mine, to ensure smooth and safe transportation of coal and related materials. & This investment covers expenses for routine maintenance, regular updates, and necessary safety performance inspections of transportation equipment. \\ Coal mine emergency & Aimed at enhancing the coal miner’s rapid response and rescue capabilities during emergencies, to reduce injuries and property damage caused by accidents. & Includes the purchase of emergency rescue equipment (such as lifeboats, first aid kits, etc.), regular maintenance, updates, and emergency drills. \\ Coal mine safety excavation & excavation operations, involving investments in key & Includes regular maintenance and upkeep, technical upgrades, and safety performance assessments of excavation equipment. \\ Coal mine safety ventilation & Aimed at ensuring the effectiveness of the mine & This investment includes the installation of ventilation equipment, routine maintenance, updates, and regular performance inspections. \\ facility \(X_{4}\) & Aimed at enhancing the safety awareness, operational & This investment includes regular safety training courses, emergency drills, the production and distribution of safety promotional materials, and safety education activities. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Safety investment indicators for coal mine equipment.  
and the sustainability and stability of production can be ensured. The Cobb-Douglas production function (C-D) [43], a model in economics, describes the relationship between production factors (such as labor and capital) and output. In optimizing safety investment in coal mine equipment, different safety investments are treated as production factors, and safety-economic benefits as outputs. Analyzing their behavioral relationship is achieved through constructing the Cobb-Douglas production function (\(B_{V}\)).  
The general formulation of the C-D production function is shown below:  
\[Q=AL^{s}K^{\beta} \tag{3}\]  
In the above formula, \(Q\) represents the total output; \(A\) is the determining constant of production technical conditions in a certain period; \(K\) is the capital element input; \(L\) is the input of labor factor; \(\beta\) is the labor elasticity coefficient; \(\alpha\) is the coefficient of capital elasticity.  
\[Y=AX_{1}^{s_{1}}X_{2}^{s_{2}}X_{3}^{s_{3}}X_{4}^{s_{4}}X_{5}^{s_{5}} \tag{4}\]  
In the above equation \(Y\) is the accident loss; \(X_{1}\) denotes the investment in coal mine safety transportation facilities; \(X_{2}\) denotes the investment in coal mine emergency facilities; \(X_{3}\) denotes the investment in coal mine safety excavation facility; \(X_{4}\) denotes the investment in coal mine safety ventilation facility; and \(X_{5}\) denotes the investment in safety education. \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\), and \(\alpha_{5}\) correspond to the capital elasticity coefficients of the individual investments, respectively.