## 5 Application to the Satellite Module Layout Problem
### Results and Analysis
#### 5.2.2 Analysis of the Assignment  
This section aims at analyzing the upper stage performance. Table 3 reports the total masses of each container for the fixed assignment list [20], the best assignment list obtained using the Heuristic+NSGA-II algorithm proposed in [19] and the best assignment list obtained using the Genetic Algorithm as detailed in Section 4.2 of this paper. Table 4 reports the coordinates of the center of gravity along the \(z\) axis considering only the components or the whole module, i.e., the components in addition to the empty module, and for the three aforementioned assignment lists.  
For the three assignment schemes, it can be noticed that surfaces 2 and 3 are always the heaviest. This contributes to minimize the global inertia along the \(z\) axis. Then, the bottom plate, i.e., surfaces \(S_{3}\) and \(S_{4}\), is more loaded than the top plate, i.e., surfaces \(S_{1}\) and \(S_{2}\). Indeed, the geometrical center between the two plates is situated at the coordinate of 565 along the \(z\) axis, but all the assignment schemes have centers of gravity located lower. The assignment scheme proposed in [19] has the most balanced two plates. This is due to the fact that the heuristic assignment rules described in this paper impose that the two plates are both balanced (with a delta) in terms of mass and component area. This results in positioning the center of gravity of the components close to the geometrical center of the two plates, i.e., close to coordinate 565. However, as the two plates are not centered at the geometrical center of the empty module in the INTELSAT-III model, this results in pulling away the center of gravity of the components from the center of gravity of the empty module. Then, the assignment task resolution proposed in this paper results in the positioning of the components on the two plates such that both the center of gravity of the components and of the empty module coincide. This should provide a better global inertia in the end as well as a more balanced satellite module.  
Figure 11 shows the median and inter quartile range (IQR) of 50 independently initialized convergence curves of the upper level. The IQR is used to evaluate the dispersion of the results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Surface 1** & **Surface 2** & **Surface 3** & **Surface 4** \\ \hline Fixed [20] & 150.63 & 231.18 & 235.22 & 198.42 \\ \hline Heuristic+NSGA-II [19] & 88.33 & 314.20 & 317.29 & 95.63 \\ \hline GA (This paper) & 85.95 & 289.02 & 326.67 & 113.78 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Masses distributions on surfaces.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(z_{CG}\) & **Fixed [20]** & **Heuristic + NSGA-II** & **Proposed Method** \\ \hline Components & 543.02 & 562.9 & 550.92 \\ \hline Whole module & 547.38 & 559.03 & 552.01 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Center of gravity along \(z\) axis without and with the empty module. Coordinate along \(z\) axis of the center of gravity of the empty module: 553.56.  
It is notably observed that the IQR remains low during all the optimization process, which characterizes a good robustness of the upper level with respect to the final interquartile range.