## 3 Algorithm Improvement and Scheme Evaluation
### Improvement Strategy for the NSGA-III Algorithm
#### 3.1.1 Reference Point Improvement Strategy  
Before applying the NSGA-III algorithm, a set of reference points must be provided [31], which serves to guide the population towards the Pareto-optimal front (PF). Traditional algorithms, such as those proposed by [PERSON] and [PERSON] [32], generate a set of fixed reference points at predefined positions on a normalized hyperplane. However, the distribution of the PF in real-world problems is often unknown, and fixed reference points may fail to adapt to dynamic changes during the optimization process. As a result, they may not accurately reflect the true evolution of the PF. To address this issue, we introduce an adaptive reference point generation method based on population information. This method produces a set of reference points that perform well in terms of both convergence and distribution. However, the introduction of new reference points increases the total number of reference points, which can elevate the computational complexity of the algorithm. To mitigate this, we introduce a reference point elimination mechanism to maintain the algorithm's convergence speed.  
(1) Generating New Reference Points:  
In the traditional NSGA-III algorithm, after niche operations, the population \(P_{t+1}\) is generated, and the niche counts \(\alpha\) for different reference points are updated, with the initial niche count for all reference points set to \(\alpha=1\). If the \(h\)-th reference point is not associated with any population members, its niche count \(\alpha_{h}\) is set to zero. These unassociated reference points are considered ineffective for exploring the current PF and are termed \"ineffective reference points\". In such cases, it is optimal to replace the \(h\)-th reference point with a new one to better reflect the direction of the PF. Under these conditions, Formulas (8) and (9) are used to introduce a set of new reference points, with the number of new reference points being equal to the number of ineffective (\(\alpha=0\)) reference points.  
\[h_{new}^{g}=\underset{x\in P_{t}}{min}f_{g}(x)+\varepsilon_{g} \tag{8}\]  
\[\varepsilon_{g}=\delta\Big{(}f_{g}^{max}-f_{g}^{min}\Big{)} \tag{9}\]  
In the formula, \(P_{t}\) is the parent population of the \(t\)-th generation (size N), \(x\) is the individual in the population \(P_{t}\), \(\delta\) is a random number uniformly distributed in (0, 1), \(h_{new}^{g}=\big{(}h_{new}^{1},h_{new}^{2},\cdots,h_{new}^{G}\big{)}\) represents the new reference point of the target \(g\), \(f_{g}^{max}\) and \(f_{g}^{min}\) are the maximum and minimum values of the target \(g\), respectively.  
(2) Elimination of Invalid Reference Points:  
As new reference points are dynamically added, the total number of reference points increases significantly, thereby raising the computational burden of the algorithm. To address this issue, after each iteration, the niche count \(\alpha\) for all reference points is recalculated and recorded. The total niche count \(\alpha\) is set equal to the population size \(N\) (i.e., \(\sum_{h=1}^{H}\alpha_{h}=N\)). Ideally, each reference point should be associated with exactly one solution in the population \(P_{t+1}\). Subsequently, all reference points with \(\alpha=0\) are removed from the set of reference points \(H\). To maintain a uniform distribution of reference points, the original reference points generated by the method of [PERSON] and [PERSON] [32] are always retained. This ensures that the reference points remain systematic and consistent. Consequently, the set of existing reference points consists of two parts: the original reference points (even those with \(\alpha=0\)) and all newly added reference points with \(\alpha=1\).