## 2 Optimization Model
### Solution Algorithm
#### 2.4.1 Solution Procedure for Multi-Objective Optimization  
In this work, three algorithms, NSGA-II, standard MOEA/D, and Brown Bear Optimization, are selected to solve the Climate Control Systems Scheduling for Indoor Ice Sports Venues' Spectator Zones problem. \(\bullet\) NSGA-II algorithm  
The NSGA-II (Non-dominated Sorting Genetic Algorithm II), a widely used intelligent optimization algorithm, is adopted to solve the Pareto frontier of the proposed model [42; 43; 44; 45]. The NSGA-II algorithm effectively identifies the optimal solution set, with its core mechanism based on the selection and population update processes guided by each individual's non-dominated rank and crowding distance within the population. The NSGA-II algorithm is characterized by its high efficiency and the use of an elitism strategy to ensure the quality of solutions. The main decision factors for selecting specific parameter values in the NSGA-II operation include population size, selection process, crossover probability, mutation probability, and the number of iterations. Step 1: Initialization. Generate initial populations. Each individual in the population is a feasible operating scheme. The content of the scheme is the start and stop state of the equipment on a time-by-time basis. Code each individual. Calculate the objective function for each individual. Step 2: Fast Undominated Sorting. Sort all individuals as non-dominated and then divide them into a number of Pareto ranks. Select the high rank as the current optimal solution set. Calculate the crowding distance for each individual, which is used to maintain population diversity. Step 3: Select. The binary tournament selection (BTS) process is based on two key factors: rank and crowding distance. The subsequent selection of parental individuals is made for the purpose of crossover and mutation operations. Step 4: Crossover and variation. The utilization of simulated binary crossover (SBX) with polynomial variation facilitates the generation of a population of offspring. It is imperative that all newly generated individuals satisfy the imposed constraints. Figure 4: Flowchart of the Non-dominated Sort Genetic Algorithm with elitist strategy (NSGA-II). Step 5: Elite retention and population renewal. The subsequent step in the process is the amalgamation of the parent and child populations. Subsequent to this, non-dominated sorting should be performed on the newly formed population. It is recommended that individuals with high Pareto rank and high crowding are retained in order to ensure convergence and diversity. Stage 6: Termination of the proceedings. Upon attaining the maximum permitted number of iterations, the process is terminated. The final output is the set of non-dominated solutions, i.e., the Pareto front of the optimal scheme solution. Specifically, (1) the population size should not be set too large or too small. When it is set too small, it is difficult to find the actual shortest path; when it is set too large, the computation time is longer and the convergence tendency is accelerated, but it is easy to fall into the local optimal solution. (2) The crossover probability determines the average number of individuals involved in mating during the evolutionary process. If it is set too high, the algorithm becomes overly random and may lose its search direction. If it is set too low, the convergence speed becomes too slow. (3) The smaller the mutation probability is, the slower the convergence speed becomes. The larger the mutation probability, the more genetic variation occurs. Mutation has a destructive effect on the optimal solution. If the mutation probability is too large, it may cause better search conditions to degrade into worse ones. (4) If the number of iterations is too small, it is difficult to converge to the optimal solution. Increasing the number of iterations can speed up the convergence, but if the number of iterations is too high, it will be difficult to obtain the global optimal solution.