## 2 Optimization Model
### Solution Algorithm
#### 2.4.2 A TOPSIS-Based Method for Optimal Selection  
The TOPSIS method, also known as the approximate ideal solution ranking method, is a typical Multi-Criteria Decision-Making (MCDM) approach [48; 49; 50]. By applying the multi-objective optimization algorithm, the Pareto solution set of the problem can be obtained, and then TOPSIS is used to filter out the optimal solution. The multi-objective optimization algorithm fully explores the solution space to ensure the diversity and global nature of the solution set, while TOPSIS introduces a clear sorting logic in the solution set to make the screening process of optimal solutions clearer and more efficient, which not only avoids the nuisance of manual subjective selection but also improves the decision-making utility of the solution. The steps are as follows:  
Step 1: The total number of alternative schemes is \(n\). Moreover, there are \(m\) elements in the scheme. The alternative schemes are represented in matrix format. \(x_{ij}\) denotes the attribute of the \(i\)-th scheme under the criterion \(\text{C}_{j}\), with incomparable units.  
\[A_{i}=\begin{pmatrix}x_{11},\ldots\ldots&x_{1j},\ldots\ldots&x_{1m}\\ x_{i1},\ldots\ldots&x_{ij},\ldots\ldots&x_{im}\\ x_{n1},\ldots\ldots&x_{nj},\ldots\ldots&x_{nm}\end{pmatrix}i=1,2,\ldots \ldots,n;j=1,2,\ldots\ldots,m \tag{51}\]  
Step 2: Normalize \(x_{ij}\) to ensure consistency. The normalization rules are as follows:  
For benefit criteria (the larger the value, the better):  
\[u_{ij}=\frac{x_{ij}-\min\left\{x_{ij}\right\}}{max\left\{x_{ij}\right\}-min \left\{x_{ij}\right\}} \tag{52}\]where \(u_{ij}\) is the normalized value of \(x_{ij}\); \(max\{x_{ij}\}\) is the maximum value of \(x_{ij}\) across all alternatives for the criterion \(C_{j}\); and \(min\{x_{ij}\}\) is the minimum value of \(x_{ij}\) across all alternatives for the criterion \(C_{j}\).  
For cost criteria (the smaller the value, the better) the following expression is derived:  
\[u_{ij}=\frac{max\{x_{ij}\}-x_{ij}}{max\{x_{ij}\}-min\{x_{ij}\}} \tag{53}\]  
Step 3: Normalize the matrix by processing each element as a vector. The normalization is performed as follows:  
\[r_{ij}=\frac{u_{ij}}{\sqrt{\sum_{i=1}^{n}{u_{ij}}^{2}}} \tag{54}\]  
Step 4: To calculate the weighted normalized decision matrix, apply the weights \(w_{j}\) to each element of the normalized matrix. The elements of the weighted normalized decision matrix are calculated as follows:  
\[z_{ij}=w_{j}u_{ij} \tag{55}\]  
Step 5: Identify the ideal solution and anti-ideal solution as follows:  
\[A^{+}=\left\{z_{1}^{+},z_{2}^{+},\ldots,z_{j}^{+},\ldots,z_{m}^{+}\right\}= \left\{max\;z_{i1},max\;z_{i2},\ldots,max\;z_{im}\right\} \tag{56}\]  
\[A^{-}=\left\{z_{1}^{-},z_{2}^{-},\ldots,z_{j}^{-},\ldots,z_{m}^{-}\right\}= \left\{min\;z_{i1},min\;z_{i2},\ldots,min\;z_{im}\right\} \tag{57}\]  
Step 6: Calculate Euclidean distances between each alternative and ideal and anti-ideal solution.  
\[D_{i}^{+}=\sqrt{\sum_{j=1}^{n}{\left(z_{ij}-z_{j}^{+}\right)}^{2}} \tag{58}\]  
\[D_{i}^{-}=\sqrt{\sum_{j=1}^{n}{\left(z_{ij}-z_{j}^{-}\right)}^{2}} \tag{59}\]  
Step 7: Calculate the distance to the ideal solution for each alternative.  
\[D_{i}^{+}=\frac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}} \tag{60}\]  
The scheme with the largest ideal proximity value represents the optimal scheme, specifically referring to the optimal operational equipment scheme for the climate control system in ice sports venues.