## 3 Algorithm Improvement and Scheme Evaluation
### Improvement Strategy for the NSGA-III Algorithm
#### 3.1.4 Algorithm Implementation Process:
The null hypothesis (H\({}_{0}\)) was that there is no significant difference between the IGD distributions of I-NSGA-III and NSGA-III, while the alternative hypothesis (H\({}_{1}\)) was that the IGD values of I-NSGA-III are significantly smaller than those of NSGA-III. (2) For HV (Hypervolume), where a larger value indicates better performance, a one-tailed right-side test was utilized. The null hypothesis (H\({}_{0}\)) stated no significant difference between the HV distributions of I-NSGA-III and NSGA-III, and the alternative hypothesis (H\({}_{1}\)) stated that the HV values of I-NSGA-III are significantly greater than those of NSGA-III. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**IGD**} & \multicolumn{2}{c}{**HV**} \\ \cline{3-6}  & & **Median** & **Std.** & **Median** & **Std.** \\ \hline \multirow{6}{*}{DTLZ1} & I-NSGA-III & **6.26 \(\times\) 10\({}^{-4}\)** & **1.01 \(\times\) 10\({}^{-3}\)** & **8.31 \(\times\) 10\({}^{-1}\)** & **1.47 \(\times\) 10\({}^{-1}\)** \\  & NSGA-III & 1.26 \(\times\) 10\({}^{-3}\) & 1.15 \(\times\) 10\({}^{-3}\) & 8.09 \(\times\) 10\({}^{-1}\) & 2.86 \(\times\) 10\({}^{-1}\) \\  & NSGA-II & 3.17 \(\times\) 10\({}^{-2}\) & 1.07 \(\times\) 10\({}^{-3}\) & 7.82 \(\times\) 10\({}^{-1}\) & 3.25 \(\times\) 10\({}^{-1}\) \\  & MOEA/D & 2.20 \(\times\) 10\({}^{-2}\) & 8.56 \(\times\) 10\({}^{-2}\) & 7.93 \(\times\) 10\({}^{-1}\) & 2.73 \(\times\) 10\({}^{-1}\) \\  & RVEA & 2.78 \(\times\) 10\({}^{-2}\) & 1.57 \(\times\) 10\({}^{-1}\) & 8.02 \(\times\) 10\({}^{-1}\) & 1.76 \(\times\) 10\({}^{-1}\) \\ \hline \multirow{6}{*}{DTLZ2} & I-NSGA-III & **2.72 \(\times\) 10\({}^{-4}\)** & **3.99 \(\times\) 10\({}^{-5}\)** & **5.71 \(\times\) 10\({}^{-1}\)** & **1.06 \(\times\) 10\({}^{-5}\)** \\  & NSGA-III & 2.87 \(\times\) 10\({}^{-4}\) & 7.27 \(\times\) 10\({}^{-5}\) & 5.41 \(\times\) 10\({}^{-1}\) & 1.88 \(\times\) 10\({}^{-5}\) \\  & NSGA-II & 8.62 \(\times\) 10\({}^{-2}\) & 4.41 \(\times\) 10\({}^{-3}\) & 5.18 \(\times\) 10\({}^{-1}\) & 3.98 \(\times\) 10\({}^{-1}\) \\  & MOEA/D & 2.75 \(\times\) 10\({}^{-4}\) & 2.76 \(\times\) 10\({}^{-4}\) & 5.51 \(\times\) 10\({}^{-1}\) & 1.21 \(\times\) 10\({}^{-5}\) \\  & RVEA & 2.45 \(\times\) 10\({}^{-4}\) & 4.26 \(\times\) 10\({}^{-5}\) & 5.51 \(\times\) 10\({}^{-1}\) & 7.64 \(\times\) 10\({}^{-5}\) \\ \hline \multirow{6}{*}{DTLZ3} & I-NSGA-III & **1.36 \(\times\) 10\({}^{-2}\)** & 1.42 \(\times\) 10\({}^{-2}\) & **5.54 \(\times\) 10\({}^{-1}\)** & **8.12 \(\times\) 10\({}^{-3}\)** \\  & NSGA-III & 1.71 \(\times\) 10\({}^{-2}\) & 1.78 \(\times\) 10\({}^{-2}\) & 5.15 \(\times\) 10\({}^{-1}\) & 1.45 \(\times\) 10\({}^{-2}\) \\  & NSGA-II & 9.02 \(\times\) 10\({}^{-2}\) & **4.18 \(\times\) 10\({}^{-3}\)** & 4.96 \(\times\) 10\({}^{-1}\) & 1.44 \(\times\) 10\({}^{-1}\) \\  & MOEA/D & 1.91 \(\times\) 10\({}^{-2}\) & 1.52 \(\times\) 10\({}^{-2}\) & 5.22 \(\times\) 10\({}^{-1}\) & 2.41 \(\times\) 10\({}^{-2}\) \\  & RVEA & 2.22 \(\times\) 10\({}^{-2}\) & 3.17 \(\times\) 10\({}^{-1}\) & 5.32 \(\times\) 10\({}^{-1}\) & 1.26 \(\times\) 10\({}^{-2}\) \\ \hline \multirow{6}{*}{DTLZ4} & I-NSGA-III & **3.66 \(\times\) 10\({}^{-4}\)** & 3.40 \(\times\) 10\({}^{-1}\) & **5.51 \(\times\) 10\({}^{-1}\)** & 4.93 \(\times\) 10\({}^{-2}\) \\  & NSGA-III & 3.85 \(\times\) 10\({}^{-4}\) & 3.37 \(\times\) 10\({}^{-1}\) & 4.39 \(\times\) 10\({}^{-1}\) & 1.34 \(\times\) 10\({}^{-1}\) \\ \cline{1-1}  & NSGA-II & 8.39 \(\times\) 10\({}^{-2}\) & 2.74 \(\times\) 10\({}^{-1}\) & 5.23 \(\times\) 10\({}^{-1}\) & 1.33 \(\times\) 10\({}^{-1}\) \\ \cline{1-1}  & MOEA/D & 2.64 \(\times\) 10\({}^{-1}\) & 3.47 \(\times\) 10\({}^{-1}\) & 4.44 \(\times\) 10\({}^{-1}\) & 1.12 \(\times\) 10\({}^{-1}\) \\ \cline{1-1}  & RVEA & 3.82 \(\times\) 10\({}^{-4}\) & **2.67 \(\times\) 10\({}^{-1}\)** & 5.29 \(\times\) 10\({}^{-1}\) & **6.65 \(\times\) 10\({}^{-2}\)** \\ \hline \hline \end{tabular} (Note: the bold numbers in the table represent the best achieved results).

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