## 3 Many-Objective Joint Parallel Simulation Method  
### Multi-Objective Optimization Algorithm  
The essence of the multi-objective optimization problem is to find the optimal solution to multiple objectives under the constraints of independent variables, as described by Equation (10) as follows:  
\[min[f_{1}(x),f_{2}(x),f_{3}(x),\ldots,f_{n}(x)] \tag{10}\]  
\[s.t.\left\{ \begin{array}{l}lb\leq x\leq ub\\ Aeq\times x=beq\\ A\times x\leq b\end{array} \right. \tag{11}\]  
where \(f_{1}(x)\) is the objective function to be optimized, \(x\) represents the independent variable to be optimized, and Equation (11) is the constraint on \(x\), including the range constraint, linear constraint, and inequality. Figure 5: (**a**) Models in the literature, Reprinted with permission from Ref. [30] 2019, [PERSON]; (**b**) The simulation model in the original manuscript. Figure 6: Comparison of the simulation results of this paper with the literature results. Multi-objective optimization differs fundamentally from single-objective optimization in that the optimization of one objective may degrade the performance of the other objectives. It is nearly impossible to simultaneously optimize all of the objectives; the trade-offs, weighing, and judging must be used to make all objectives as optimal as possible. In addition, the optimal solution to a multi-objective optimization problem is often not a single solution, but a set of optimal solutions. Currently, there are many methods that are widely used to solve multi-objective optimization problems, mainly genetic algorithms, particle swarm algorithms, simulated annealing algorithms, ant colony algorithms, etc. Based on the single-objective genetic algorithm Srinivas [31], the non-dominated genetic algorithm (NSGA) was proposed in 1995. NSGA, which is based on the concept of Pareto optimality, can be used to solve multi-objective problems, despite its high computational complexity and lack of an elite strategy. In 2002, in response to this motivation, [PERSON] developed the most popularly used fast non-dominated ranking genetic algorithm with an elite strategy based on NSGA [32] (NSGA-II). NSGA-II is unquestionably one of the most popular and effective algorithms for solving two- or three-objective multi-objective optimization problems. However, the number of optimization targets in acoustic optimization problems is often huge. In the process of optimizing an anechoic coating, for instance, it is not the characteristics of a specific frequency that must be satisfied, but rather the optimization of the entire frequency band to the greatest extent. This indicates that the number of optimization targets could reach tens or even dozens. The NSGA-II is slightly insufficient for such high-dimensional objective problems. The reasons for this are as follows: First, the proportion of non-dominated solutions in the population rises dramatically as the optimization objective is increased. In addition, the search process is slow and the calculation of the index for maintaining the population diversity is overly complex for the high-dimensional objective. Moreover, the recombination operator's search capability is inefficient. Therefore, using NSGA-II to solve an optimization problem with a large number of objectives remains extremely difficult. Hence, based on NSGA-II, [PERSON] [19] proposed the non-dominated genetic algorithm with reference points (NSGA-III) in 2014. NSGA-III inherited most of the NSGA-II framework but made a significant change in the selection mechanism, abandoning the ranking method by crowding and adopting the extensive introduction of distribution reference points to preserve the population diversity in the high-dimensional space. The primary procedure is depicted in Figure 7.  
Figure 7: NSGA-III algorithm main flowchart. The NSGA-III algorithm resembles the NSGA-II algorithm, as depicted in Figure 4. However, the NSGA-III algorithm modifies the selection process in the following three main ways:  
1.