## 3 Algorithm Improvement and Scheme Evaluation
### Improvement Strategy for the NSGA-III Algorithm
#### 3.1.4 Algorithm Implementation Process:
Optimization of Selection Strategy)
16: Perform non-dominated sorting on \(P_{t}\);
17: Use Improvement 2 to dynamically adjust selection Number of candidate individuals \(K\);
18:\(\zeta\)= adaptive tournament selection (\(P_{t}\), \(g\));
19:
20:\(Q_{t}\) = Recombination & Mutation(\(\zeta\));
21:\(C_{t}=P_{t}\cup Q_{t}\);
22: Perform non-dominated sorting on \(C_{t}\);
23:\(P_{t+1}\) = SelectSolutions (\(C_{t}\), \(g\));
24:
25: if \(g\) > 0 then
26: Add reserved elite solution q to \(P_{t+1}\);
27: end if
28:
29: Use Reference Point Improvement Strategy (Section 3.1.1) to generate new reference points based on current front;
30: Update reference point set \(H\);
31:
32:\(t=t+1\);
33:
34: end while
35:
36: return Non-dominated solutions in \(P_{t}\) as Pareto-optimal front

#### 3.1.5 Algorithm Testing

To evaluate the overall performance of the I-NSGA-III algorithm, a systematic validation was conducted using the DTLZ standard test suite (DTLZ1-DTLZ4) proposed in reference [33]. The population size for all algorithms was set to 70, the number of objectives was set to 3, and the maximum number of iterations is referenced in Table 1. A comparison of the Pareto front obtained by I-NSGA-III for the DTLZ test problems with the true Pareto front of the test problems is shown in Figure 1. The results indicate that the solutions obtained by I-NSGA-III for the DTLZ test problems generally converge to the true Pareto front, demonstrating the feasibility of the I-NSGA-III algorithm. To further validate the performance of the I-NSGA-III algorithm, two test metrics were used to evaluate I-NSGA-III, and comparisons were made with the NSGA-III, NSGA-II, MOEA/D and RVEA multi-objective algorithms. The test metrics are as follows: the Inverted Generational Distance (IGD) metric, where a smaller value indicates better overall algorithm performance [34]; and Hypervolume (HV), where a larger value indicates better convergence and diversity [35]. The main characteristics of all benchmark problems are shown in Table 1. To mitigate the effects of randomness, each algorithm was independently run 20 times on each test problem. The mean and standard deviation of each metric were calculated for comparison. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Function** & **Characteristics** & **Maximum Iteration** \\ \hline DTLZ1 & Linear, multimodal & 200 \\ DTLZ2 & Concave & 500 \\ DTLZ3 & Concave, multimodal & 700 \\ DTLZ4 & Concave, biased & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main features of test problems. Figure 1: Comparison of Pareto Fronts for DTLZ1-4. Table 2 shows the IGD and HV values obtained by applying different algorithms to the three-objective test function. Please be aware that the bold figures in the table indicate the most favorable results that have been attained. The results indicate that the I-NSGA-III algorithm outperforms NSGA-III in terms of solution set convergence, optimization accuracy, and overall performance within the DTLZ test suite. For the IGD metric, I-NSGA-III achieves the smallest values across all test problems, reducing by 5.17% to 50.22% compared to NSGA-III. The only exception is in DTLZ4, where the standard deviation is slightly larger (the NSGA-II score is too low, so its standard deviation is not considered). Regarding the HV metric, I-NSGA-III performs the best across all test problems, with values increasing by 2.71% to 25.51% compared to NSGA-III, and its standard deviations are smaller than those of NSGA-III, enhancing stability. The RVEA algorithm shows the smallest standard deviation in DTLZ4, demonstrating its relative stability in handling complex multimodal problems. To further verify the performance of I-NSGA-III, statistical tests were conducted to compare it with the original NSGA-III algorithm using the Wilcoxon signed-rank test. The detailed experimental design is as follows: (1) For IGD (Inverted Generational Distance), where a smaller value indicates better performance, a one-tailed left-side test was employed.