## 1 Introduction
Among them, the Non-dominated Sorting Genetic Algorithm (NSGA-II), as a classical benchmark algorithm, combines the genetic algorithm with the non-dominated sorting mechanism and is able to search the global Pareto frontier efficiently, especially in large-scale problems, and has been proven to have a significant computational advantage over commercial optimization tools. In addition, the standard MOEA/D algorithm decomposes the multi-objective problem into multiple sub-problems through weight vectors and utilizes the neighborhood structure and Tchebycheff aggregation method to achieve a rigorous optimization process based on decomposition and neighborhood, which is suitable for scenarios with obvious objective conflicts and clear problem structure [24]. In recent years, MOBBO (Multi-Objective Brown Bear Optimization), as an emerging group intelligence algorithm, integrates the mechanisms of non-dominated sorting, external solution archiving, and brown bear foraging behavior and takes into account the ability of local exploitation and global exploration, which has shown strong performance advantages in high-dimensional and complex multi-objective engineering problems [25]. Based on the above algorithms, three representative multi-objective optimization algorithms, NSGA-II, MOEA/D, and MOBBO, are selected for comparative analysis to systematically evaluate their applicability and differences in multi-objective scheduling problems. These three algorithms have their own characteristics in terms of optimization mechanism, search strategy, and solution set characteristics, which can comprehensively cover the mainstream technical paths in multi-objective optimization, have strong representativeness and complementarity, and provide multi-dimensional technical support for the solution of the multi-objective scheduling model in this study. However, multi-objective optimization algorithms usually generate a large number of candidate solutions while providing high-quality solution sets, which may lead to decision-making difficulties. For this reason, this study further introduces the TOPSIS method [26], which assists the decision-making in selecting the optimal solution by calculating the distances between each solution in the solution set and the ideal solution. The multi-objective optimization algorithm fully ensures the diversity and global nature of the solution set, while the TOPSIS method introduces an objective sorting logic in the solution set. The two-stage optimization method improves the efficiency and practicability of optimal solution selection. The synergistic mechanism of \"diversified search + rational decision-making\" not only reduces the subjective intervention of manpower but also improves the operability and relevance of the solution, which demonstrates the outstanding comprehensive advantages in complex engineering optimization problems. The innovations of this work are as follows:  
\(\bullet\) This study systematically integrates electricity, equipment start/stop, maintenance, and environmental costs for the first time, moving beyond the traditional focus on energy costs to achieve a more comprehensive optimization that considers equipment lifespan and environmental impact. \(\bullet\) An innovative approach combining multi-objective optimization with TOPSIS decision analysis is proposed, following a \"search first, decide later\" strategy. This greatly enhances the model's practicality and decision-making efficiency while incorporating demand-side analysis. \(\bullet\) This study systematically compares three algorithms--NSGA-II, MOEA/D, and MOBBO--assessing their suitability for ice risk climate control. It identifies the optimal algorithm and introduces a demand-side response strategy under time-of-use tariffs to reduce costs through load shifting and improve energy efficiency. The remainder of this paper is organized as follows: Section 2 develops an approximate physical model of the ice sports venue for the multi-objective optimization of its climate control system.