## 3 Case Study
### Results  
#### 3.3.1 Pareto Frontiers  
In order to comprehensively evaluate the performance of different multi-objective optimization algorithms, three commonly used evaluation metrics, namely, computation time, Hypervolume (HV), and Inverted Generational Distance (IGD), are selected for comparative analysis in this work. Among them, the computation time reflects the operation efficiency of the algorithm; HV measures the breadth and diversity of the distribution of the solution set, and the larger value represents the better quality of the solution set; and IGD measures the proximity between the solution set and the theoretical optimal solution, and the smaller value indicates the better convergence of the solution set. The comparison results of the three multi-objective optimization algorithms NSGA-II, MOEA/D, and MOBBO in terms of computation time, Hypervolume (HV), and Inverted Generational Distance (IGD) are shown in Table 3. In terms of computation time, MOEA/D has the highest running efficiency of 86.21 s, which is better than NSGA-II (295.85 s) and MOBBO (237,860.78 s). However, in terms of solution set diversity and convergence, NSGA-II performs the most outstandingly, with an HV value of 6946.8083, which is significantly better than the other algorithms, showing excellent solution set expansion capability, while the lowest IGD value (10.4241) indicates its excellent convergence characteristics in terms of approaching the optimal frontier. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time Period** & **Time** & **Electricity Tariffs (CNY/kWh)** \\ \hline Peak & 10:00–13:00, 17:00–22:00 & 1.71 \\ Flat & 7:00–10:00, 13:00–17:00, and 22:00–23:00 & 1.00 \\ Valley & 23:00–24:00, 0:00–7:00 & 0.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Electricity tariffs of the grid. Figure 5: Outdoor temperature and relative humidity of typical days: (**a**) Outdoor temperature; (**b**) Outdoor relative humidity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metrics** & **NSGA-II** & **MOEA/D** & **MOBBO** \\ \hline Time & 295.85 s & 86.21 s & 237,860.78 s \\ HV & 6946.8083 & 506.7154 & 4567.56 \\ IGD & 10.4241 & 161.1774 & 62.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of performance metrics (time, HV, IGD) among NSGA-II, MOEA/D, and MOBBO algorithms. Considering computational efficiency, solution set diversity, and convergence, NSGA-II is finally selected as the optimal solution algorithm in this work. Although its computation time is slightly higher than that of MOEA/D, NSGA-II has obvious advantages in the quality of the solution set and convergence, which are the core indexes of multi-objective optimization, and it is more suitable for the application of optimal scheduling of the ice risk climate control system in this study. In this study, the initial population size was set to 200 and the number of iterations to 1000. These parameters were carefully selected based on multiple rounds of testing and tuning. The results showed that this configuration strikes an appropriate balance between solution quality and computational efficiency. It ensures sufficient convergence of the optimization process while effectively controlling the computational time, thus avoiding both premature convergence and excessive computational cost. Figure 6 shows the Pareto front obtained under typical temperature and humidity conditions by applying the NSGA-II algorithm [52]. Figure 6 illustrates the best Pareto fronts for multi-objective optimization of the operation of the ventilation, dehumidification, and radiant heating equipment for the spectator stand of an ice sports arena under typical days in summer, transitional seasons, and winter.