## 3 Algorithm Description
### Strengthened Dominance Relation Sort  
Considering the multiple objectives in SEC, the traditional Pareto optimality treats all nondominated solutions equally, making it challenging to evaluate each nondominated individual [53]. Additionally, striking a balance for nondominated solutions between convergence and diversity also poses a challenge [54]. To address these issues, a new measure called strengthened dominance relation [55] is introduced. We improved it by proposing a strengthened dominance relation sort (SDRS) and embedded it in our MOWPS. The fundamental principles of this approach are described below. Assume a problem with \(K\) objective functions \(\mathbf{f}(\mathbf{x})\), denoted as \(f_{c}(\bullet)\) for \(c=\mathbb{N}_{K}\). We define the solution space as \(\Phi\subset\mathbb{R}^{b}\), which contains all solution vectors \(\mathbf{x}\), with dimensionality \(b\). _Convergence degree_: We can obtain a metric for the convergence degree of a solution vector \(\mathbf{x}\in\Phi\) using (24):  
\[Con(\mathbf{x})=\sum_{c=1}^{K}e^{(\phi-1)}f_{c}(\mathbf{x}) \tag{24}\]  
where \(\phi\) is a convergence degree factor. As \(\phi\) increases, the convergent pressure is strengthened, while weakening diversity. _Niche size:_ Given a set of solutions \(\mathbf{Q}\), we can calculate the angle between any pair of solutions \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbf{Q}\) as \(\theta_{\mathbf{xy}}=\arccos(f(\mathbf{x}),f(\mathbf{y}))\). Then, the niche size \(\overline{\theta}\) is defined as the \(\lceil\Upsilon\rceil\mathbf{Q}\rceil\)-th smallest element of  
\[\left\{\min_{\mathbf{y}\in\mathbf{Q}\lfloor\mathbf{x}\rfloor}\theta_{\mathbf{xy}}|\mathbf{x}\in \mathbf{Q}\right\} \tag{25}\]  
where \(\Upsilon\) is the niche size factor, \(\Upsilon\in(0,1)\), and \(\lceil\bullet\rceil\) is the rounding down operation. As \(\Upsilon\) increases, the dominated area expands, and the convergent pressure intensifies. To calculate Con(x) and \(\overline{\theta}\), we need to first normalize each \(\mathbf{x}\in\mathbf{Q}\) with respect to the ideal point and nadir point of Q. The ideal point is a vector containing the optimal value of each objective function, while the nadir point is a vector consisting of the worst value. _Strengthened dominance value:_ The strengthened dominance value \(D(\mathbf{x},\mathbf{y})\) represents how much solution \(\mathbf{x}\) dominates \(\mathbf{y}\), as calculated by (26). Suppose that it is a multi-objective minimization problem. \[D(\mathbf{x},\mathbf{y})=\left\{ \begin{matrix}\max\{0,Con(\mathbf{y})-Con(\mathbf{x})\},& \theta_{xy}\leq\overline{\theta}\lor x\text{ Pareto dominates }\mathbf{y}\\ \max\{0,Con(\mathbf{y})-\frac{\theta_{xy}}{\overline{\theta}}Con(\mathbf{x})\},& \theta_{xy}>\overline{\theta}\end{matrix} \right. \tag{26}\]  
where \(D(\mathbf{x},\mathbf{y})\) is nonnegative (i.e., \(D(\mathbf{x},\mathbf{y})\) is necessarily zero if \(\mathbf{x}\) is inferior to \(\mathbf{y}\)). **Example 4**. : _For a normalized solution vector of \(\mathbf{x}=\) [0.5, 0.5], the dominated regions are depicted in green in Figure 8 under different values of \(\phi\) and \(\gamma\). The results in Figure 8,b indicate that as the value of \(\phi\) increases, the convergent pressure becomes stronger. Conversely, Figure 8,c show that a smaller value of \(\gamma\) results in better diversity. It is worth noting that the upper right corner is always dominated due to the incorporation of Pareto dominance in Equation (26)._  
_SDRS value:_ Each solution \(\mathbf{x}\) is assigned an SDRS value \(\vartheta(\mathbf{x})\), which reflects its degree of domination over others in the entire solution set \(\mathbf{Q}\). Thus, we have:  
\[\vartheta(\mathbf{x})=\frac{\sum\limits_{\mathbf{y}\in\mathbf{Q}\lceil\mathbf{x}\rfloor}D(\bm {x},\mathbf{y})}{|\mathbf{Q}|-1} \tag{27}\]  
A higher value of \(\vartheta(\mathbf{x})\) indicates better solution quality, which helps to generate diverse solutions near the Pareto frontier.