# A Two-Archive Many-Objective Optimization Algorithm Based on D-Domination and Decomposition
## 2 Materials and Methods
### Discussion  
The high-dimensional many-objective optimization problem is closely related to our life and production and has been widely used in various fields, such as power scheduling, network optimization, chemical production, structure design and path planning, so the research on the high-dimensional many-objective optimization algorithm has important practical significance. For example, the algorithm proposed can be applied to the operation and scheduling of high-speed railway trains. Due to the continuous growth of passenger traffic volume and the increasing demand for high-speed railway service quality, it is necessary to optimize the operation adjustment method of high-speed railway trains. Taking the punctuality rate of train operation, passenger comfort and delay time as optimization objectives, a multi-objective optimization model is established to minimize the total number of train delays, and the proposed algorithm is applied to provide a fast and reasonable train operation adjustment scheme for dispatchers. Figure 4: Change of ICD value with the number of assessments (10 objective). Figure 5: Change of ICD value with the number of assessments (15 objective). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Problems** & \(\beta\)**= 0.6** & \(\beta\)**= 0.7** & \(\beta\)**= 0.8** & \(\beta\)**= 0.9** & \(\beta\)**= 1.0** \\ \hline MaF1-5 & 9.4293 \(\times\)\(10^{-1}\) & 1.0189 \(\times\)\(10^{0}\) & 9.2574 \(\times\)\(10^{-1}\) & 9.9358 \(\times\)\(10^{-1}\) & 1.0574 \(\times\)\(10^{0}\) \\ \hline MaF2-5 & 9.7363 \(\times\)\(10^{-1}\) & 1.0439 \(\times\)\(10^{0}\) & 8.3417 \(\times\)\(10^{-1}\) & 1.0067 \(\times\)\(10^{0}\) & 8.8881 \(\times\)\(10^{-1}\) \\ \hline MaF3-5 & 9.1630 \(\times\)\(10^{-1}\) & 1.0596 \(\times\)\(10^{0}\) & 1.2256 \(\times\)\(10^{0}\) & 8.6536 \(\times\)\(10^{-1}\) & 8.7178 \(\times\)\(10^{-1}\) \\ \hline MaF4-5 & 1.1585 \(\times\)\(10^{0}\) & 9.8553 \(\times\)\(10^{-1}\) & 9.1146 \(\times\)\(10^{-1}\) & 1.0135 \(\times\)\(10^{0}\) & 1.1021 \(\times\)\(10^{0}\) \\ \hline MaF5-5 & 1.0614 \(\times\)\(10^{0}\) & 8.9829 \(\times\)\(10^{-1}\) & 1.1919 \(\times\)\(10^{0}\) & 8.4124 \(\times\)\(10^{-1}\) & 1.0229 \(\times\)\(10^{0}\) \\ \hline MaF6-5 & 9.0160 \(\times\)\(10^{-1}\) & 1.0544 \(\times\)\(10_{0}\) & 1.1214 \(\times\)\(10^{0}\) & 7.8145 \(\times\)\(10^{-1}\) & 8.8379 \(\times\)\(10^{-1}\) \\ \hline MaF7-5 & 1.0889 \(\times\)\(10^{0}\) & 9.5638 \(\times\)\(10^{-1}\) & 1.0512 \(\times\)\(10^{0}\) & 9.6275 \(\times\)\(10^{-1}\) & 8.8720 \(\times\)\(10^{-1}\) \\ \hline MaF8-5 & 1.1479 \(\times\)\(10^{0}\) & 1.2521 \(\times\)\(10^{0}\) & 1.1092 \(\times\)\(10^{0}\) & 1.1545 \(\times\)\(10^{0}\) & 1.0437 \(\times\)\(10_{0}\) \\ \hline MaF9-5 & 1.3618 \(\times\)\(10^{-1}\) & 1.6021 \(\times\)\(10^{0}\) & 3.1440 \(\times\)\(10^{-1}\) & 1.3218 \(\times\)\(10^{-1}\) & 1.3046 \(\times\)\(10^{0}\) \\ \hline MaF10-5 & 1.0188 \(\times\)\(10^{0}\) & 1.0159 \(\times\)\(10^{0}\) & 9.8462 \(\times\)\(10^{-1}\) & 1.0523 \(\times\)\(10^{0}\) & 9.8052 \(\times\)\(10^{-1}\) \\ \hline MaF11-5 & 9.3733 \(\times\)\(10^{-1}\) & 9.0826 \(\times\)\(10^{-1}\) & 1.0970 \(\times\)\(10^{0}\) & 8.8298 \(\times\)\(10^{-1}\) & 9.7087 \(\times\)\(10^{-1}\) \\ \hline MaF12-5 & 1.2302 \(\times\)\(10^{0}\) & 9.8316 \(\times\)\(10^{-1}\) & 1.1134 \(\times\)\(10^{0}\) & 9.8456 \(\times\)\(10^{-1}\) & 8.0840 \(\times\)\(10^{-1}\) \\ \hline MaF13-5 & 9.1040 \(\times\)\(10^{-1}\) & 8.9130 \(\times\)\(10^{-1}\) & 9.9249 \(\times\)\(10^{-1}\) & 1.0142 \(\times\)\(10^{0}\) & 1.0193 \(\times\)\(10^{0}\) \\ \hline MaF14-5 & 1.1236 \(\times\)\(10^{0}\) & 1.1630 \(\times\)\(10^{-1}\) & 1.3638 \(\times\)\(10^{-1}\) & 3.3340 \(\times\)\(10^{-1}\) & 1.1313 \(\times\)\(10^{0}\) \\ \hline MaF15-5 & 1.1382 \(\times\)\(10^{0}\) & 1.1540 \(\times\)\(10^{0}\) & 1.0589 \(\times\)\(10^{0}\) & 1.1714 \(\times\)\(10^{0}\) & 1.3945 \(\times\)\(10^{0}\) \\ \hline \end{tabular}
\end{table} 
Table 4: IGD mean obtained by Two Arch-D with different \(\beta\).