## 3 Methods
### Optimization Algorithm
#### 3.3.3 Performance Comparison  
The performance of SDPSO and several PSO variants have been compared through five unconstrained benchmark functions by the author [27]. Considering that the stress optimization for vent holes has a constraint on the total area, this paper selects two new benchmark optimization problems with constraints to evaluate the performance of SDPSO and two other classic PSO variants, namely PSO-CK and SPSO [31]. The two benchmark optimization problems (BOP) are taken from [PERSON] et al. [32] and [PERSON] et al. [33] and defined as follows: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**State** & **Mode** & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) \\ \hline Convergence & \(\xi(t)=1\) & 2 & 2 \\ Exploitation & \(\xi(t)=2\) & 2.1 & 1.9 \\ Exploration & \(\xi(t)=3\) & 2.2 & 1.8 \\ Jumping-out & \(\xi(t)=4\) & 1.8 & 2.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Strategies for selecting \(c_{1}\) and \(c_{2}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**State** & **Mode** & \(\mathbf{pbest}\) & \(\mathbf{gbest}\) & \(\tau_{1}(\xi(t))\) & \(\tau_{2}(\xi(t))\) \\ \hline Convergence & \(\xi(t)=1\) & \(p_{1}(t)\) & \(p_{\text{g}}(t)\) & 0 & 0 \\ Exploitation & \(\xi(t)=2\) & \(p_{\text{i}}(t-\tau_{1}(\xi(t)))\) & \(p_{\text{g}}(t)\) & \(\lfloor t\text{-}rand\rfloor\) & 0 \\ Exploration & \(\xi(t)=3\) & \(p_{\text{i}}(t)\) & \(p_{\text{g}}(t-\tau_{2}(\xi(t)))\) & 0 & \(\lfloor t\text{-}rand\rfloor\) \\ Jumping-out & \(\xi(t)=4\) & \(p_{\text{i}}(t-\tau_{1}(\xi(t)))\) & \(p_{\text{g}}(t-\tau_{2}(\xi(t)))\) & \(\lfloor t\text{-}rand\rfloor\) & \(\lfloor t\text{-}rand\rfloor\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Strategies for selecting delayed information and parameters. 1. BOP1: 10-dimensional optimization problem \[ \begin{split}\min:& f(x)=-\left|\frac{\sum\limits_{i=1}^{10} \cos^{4}(z_{i})-\frac{10}{2}\cos^{2}(z_{i})}{\sqrt{\sum\limits_{i=1}^{D}{z_{i} }^{2}}}\right|\\ \text{w.r.t.}:& x_{i},\quad i=1,2,\ldots,10\\ \text{s.t.}:&\left\{\begin{array}{l}g_{1}=0.75- \frac{10}{i1}\sum\limits_{i}z_{i}\leq 0\\ g_{2}=\sum\limits_{i=1}^{10}z_{i}-75\leq 0\\ \mathbf{x}\in[0,10]^{10}\end{array}\right.\end{split} \] (24) where \(\mathbf{z}=\mathbf{x}-\mathbf{o}\), \(\mathbf{o}=[0.03085872,-0.07863229,0.04865115,-0.06908983,-0.08791854,\)\(0.08898264,0.07414324,-0.08652759,-0.02061653,0.05558611]\);
2. BOP2: 13-dimensional optimization problem \[ \begin{split}\min:& f(x)=5\sum\limits_{i=1}^{4}x_{i}-5 \sum\limits_{i=1}^{4}x_{i}^{2}-\sum\limits_{i=5}^{13}x_{i}\\ \text{w.r.t.}:& x_{i},\quad i=1,2,\ldots,13\\ \text{s.t.}:&\left\{\begin{array}{l}g_{1}=2x_{1}+2x_{2}+x_{10}+x_{ 11}-10\leq 0\\ g_{2}=2x_{1}+2x_{3}+x_{10}+x_{12}-10\leq 0\\ g_{3}=2x_{2}+2x_{3}+x_{11}+x_{12}-10\leq 0\\ g_{4}=-8x_{1}+x_{10}\leq 0\\ g_{5}=-8x_{2}+x_{11}\leq 0\\ g_{6}=-8x_{3}+x_{12}\leq 0\\ g_{7}=-2x_{4}-x_{5}+x_{10}\leq 0\\ g_{8}=-2x_{6}-x_{7}+x_{11}\leq 0\\ g_{9}=-2x_{8}-x_{9}+x_{12}\leq 0\\ 0\leq x_{i}\leq 1(i=1,\ldots,9)\\ 0\leq x_{i}\leq 100(i=10,11,12)\\ 0\leq x_{13}\leq 1\end{array}\right.\end{split} \] (25)  
The parameters of the two benchmark optimization problems are set as follows: the particle number \(s=50\) and the maximum generation number \(t_{\max}=200\). The three optimization algorithms (PSO-CK, SPSO, and SDPSO) are repeated 25 times for each benchmark optimization problem to reduce the randomness, respectively. Figure 6 shows the boxplots of the optimal values of the 25 repetitions obtained by PSO-CK, SPSO, and SDPSO for BOP1 and BOP2, respectively. The boxplots provide a graphical depiction of how the optimal values vary over the 25 repetitions. We can see that: (1) for the two benchmark optimization problems, the optimal values obtained by SDPSO are significantly lower than those obtained by PSO-CK and SPSO. (2) The variation of the optimal values of the 25 repetitions obtained by SDPSO is also smaller than that obtained by PSO-CK and SPSO.