## 4 Optimization Design Algorithm
### Results  
#### 4.5.1 Comparison of Different Optimization Results  
The different optimization results are illustrated in two aspects. Firstly, to compare the optimization results under different initialization methods, the progress of iteration is displayed in Figures 11 and 12, respectively. Secondly, the genetic algorithm (GA) is often used to solve multi-objective optimization problems. Therefore, to prove the effectiveness of particle swarm optimization, the performance of both algorithms in solving the same multi-objective optimization problem is compared. Identically, the population size of GA is 30 and is allowed to evolve to 60 generations. In the subsequent figures, the bearing capacity and gas consumption of the bearing are normalized. The ordinate corresponds to the normalized gas consumption, and the abscissa corresponds to the normalized bearing capacity. It can be seen from Figure 11a that the selectivity  
Figure 11: Pareto solutions obtained using **(a)** Latin hypercube and **(b)** random initialization. Figure 12: Pareto solutions obtained using **(a)** Latin hypercube and **(b)** random initialization after the 60 th generation. of the Pareto solutions gradually increases from the 1 st to the 30 th generation. The particle swarm gradually approaches the Pareto front in the iterative process, and the Pareto solutions obtained after the 30 th generation fit well with the final Pareto front. Compared with Figure 11b, it is found that the Pareto solutions obtained using random initialization after the 30 th generation are still poorly fitted with the final Pareto front. As the particle distribution obtained by the Latin hypercube is more reasonable, it effectively reduces the amount of computation and improves the speed of finding the optimal solutions. Figure 12 shows the Pareto solutions obtained using different initialization methods after the 60 th generation. It can be seen from Figure 12b that the distribution of the Pareto solutions in the region identified by the red circle is sparse. Compared with Figure 12a, it is found that the Pareto solutions obtained using Latin hypercube initialization show a much wider distribution and more selections. It also proves that using Latin hypercube as a particle initialization method can effectively improve the global search ability. It can be seen from Figure 13 that the Pareto solutions obtained using the two initialization methods after the 60 th generation are in good agreement. The Pareto solutions obtained using GA are surrounded by the Pareto front obtained using particle swarm optimization in Latin hypercube initialization, which proves that particle swarm optimization shows a stronger global search ability and can find better solutions. The performance of different algorithms in solving the multi-objective optimization problem can be assessed using performance metrics, such as the spacing and diversity metric (DM) [31]. The performance metrics can be obtained using Equations (17) and (18), respectively. Spacing assesses the evenness of the solution set, and DM measures the diversity of the Pareto solutions. A more uniform and diverse solution set results in smaller values of spacing and DM. It can be seen from Figure 14 that particle swarm optimization using Latin hypercube initialization shows a better performance in solving the multi-objective optimization problem:  
\[Spacing(A)=\sqrt{\frac{1}{A-1}{\sum\limits_{i=1}^{A}{(\overline{d}-d_{i})}^{2}}} \tag{17}\]  
\[DM=\frac{d_{f}+d_{l}+\sum\limits_{i=1}^{A-1}{\left|{d_{i}-\overline{d}}\right| }}{d_{f}+d_{l}+(A-1)\overline{d}} \tag{18}\]  
Figure 13: Pareto solutions obtained after the 60 th generation. where \(A\) is the number of Pareto solutions, \(d_{i}\) is the Euclidean distance between consecutive solutions, \(\overline{d}\) is the average of all distances, \(d_{i}\), \(d_{f}\), and \(d_{l}\) are the Euclidean distances between the extreme solutions and the boundary solutions.