## 3 Optimization Based on ESS-NSGA-III Algorithm
### Optimization of Mating Selection Probability Using Shift-Based Density Estimation  
In the traditional NSGA-III algorithm, the mating process employs a uniform random selection method that chooses individuals from the parent population. This method lacks a mechanism for evaluating the quality of individuals, preventing it from prioritizing individuals with superior characteristics. To address this limitation, this study introduces a novel method for optimizing the mating selection probability based on SDE.  
SDE utilizes a parameter-free metric for assessing inter-individual densities, effectively balancing both convergence and diversity considerations [22]. The SDE metric is mathematically defined by Equations (11) and (12), which quantify the density distribution within the population and account for the relative positions of the individuals in the objective space. By leveraging this approach, the selection process prioritizes individuals that enhance both population diversity and convergence toward the Pareto front; this is expressed as follows:  
\[\text{SDE}(p_{i})=\min_{p_{j}\in P,\,j\
eq i}\sqrt{\sum_{n=1}^{m}\text{sde}(f_ {n}^{\prime}(p_{i}),f_{n}^{\prime}(p_{j}))^{2}}, \tag{11}\]  
\[\text{sde}(f_{n}^{\prime}(p_{i}),f_{n}^{\prime}(p_{j}))=\max\{0,f_{n}^{\prime} (p_{j})-f_{n}^{\prime}(p_{i})\}, \tag{12}\]  
where \(f_{n}^{\prime}(p_{i})\) represents the \(k\)-th normalization objective value for individual \(p_{i}\), which ensures that all objective values are scaled to a common range, facilitating accurate density estimations and comparisons between individuals.  
The specific process for optimizing the mating selection probability based on SDE is as follows:  
1. [label=0., ref=1]
2. Conduct non-dominated sorting on the population \(P_{t}\) to derive the non-dominated layers \(F_{1}\) to \(F_{L}\).
3. For each solution in the first non-dominated layer \(F_{1}\), calculate its SDE value and sort the solutions in non-increasing order. Assign sequence numbers such that the solution with the highest SDE value receives a sequence number of 1, the next receives 2, and so on, with the last solution in \(F_{1}\) receiving a sequence number of \(N_{1}\) (where \(N_{1}\) is the size of \(F_{1}\)). For subsequent non-dominated layers \(F_{2}\), \(F_{3}\), \(\ldots\), \(F_{L}\), continue the sequential numbering based on the previous layer's numbering. This process yields a vector of sequence values, which is expressed as follows: \[R(x)=(r(x_{1}),r(x_{2}),\cdots,r(x_{i}),\cdots,r(x_{N})),\,r(x_{i})\in\{1,2,3, \cdots,N\},\] (13) where \(r(x_{i})\) represents the sequence number of the \(i\)-th solution.
4. Apply a scaling transformation to the sequence values \(R(x)\), mapping them to the range \([N+1,2N]\). Calculate the selection probability \(p(x_{i})\) for each solution \(x_{i}\) using \[p(x_{i})=\frac{2\times(r(x_{i})+N)}{3N^{2}+N},\] (14)  
According to Equation (14), the ratio of the maximum probability value to the minimum probability value is \(2N/(N+1)\in[1,2]\). Compared to random uniform selection, this method significantly increases the probability of generating high-quality offspring during the evolutionary process.