## 4 Results--A Case Study
### Obtained Structures of the Metro Network  
Figure 5 presents the adopted map of population density [37]. Cities are marked according to Table 1.  
Due to avoiding the uncertainty resulting from non-deterministic nature of genetic algorithm, a series of 10 runs of optimization procedure was carried out. The obtained results were similar but not the same. Figure 6 shows as examples two of the obtained stations layouts. The shaded circles mark the degree of satisfying the transportation demands, and bold black circles mark the additional generators of the transportation demands. These generators and relevant number of people demanding the transport have been added after analyzing the list of such facilities at the examined area. As it can be seen, most of the stations are located mainly on highly populated areas, which was expected. However, the detailed locations of stations differ significantly. Some of the transportation demands generators are not served as well (however, it concerns those of lesser significance). The main reason is probably that the number of stations was underestimated, and it was too low to sate all the highly populated areas. Additionally, as was stated, the objective function did not take into account the details of local topography. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Series**} & \multicolumn{3}{c}{**Parameters**} & \multicolumn{3}{c}{**Scaled objective Function**} & \multirow{2}{*}{**S/N**} \\ \cline{2-2} \cline{6-9}  & **PM** & **PX** & **NE** & **1** & **2** & **3** & **4** & \\ \hline
1 & 0.1 & 0.5 & 2 & 0.9190 & 0.9868 & 0.9037 & 0.9506 & \(-\)0.5521 \\
2 & 0.1 & 0.8 & 8 & 0.9733 & 0.9868 & 0.9162 & 0.9162 & \(-\)0.4780 \\
3 & 0.2 & 0.5 & 8 & 0.9394 & 0.9715 & 1.0000 & 0.9506 & \(-\)0.3134 \\
4 & 0.2 & 0.8 & 2 & 0.9318 & 0.9411 & 0.9162 & 0.9557 & \(-\)0.5757 \\ \hline \end{tabular}
\end{table} 
Table 4: Scaled outputs for the second stage according to L4 orthogonal table. Figure 4: Relative importance of control parameters: (**a**) for stations locations optimization and(**b**) for line layout optimization. In the light of such variability of the obtained results, a conclusion arose that the found stations layouts should be regarded just as the preliminary. The determination of the exact location of each station would require further work and far more detailed data. Anyway, such a matter could be expected due to simplicity and rawness of the model. On the other side, the values of fit function were quite close to each other. This is shown in Figure 7, which presents the progress of the optimization process for five runs. The value of fit function was normalized to unity. As it can be noted, all the solutions were very close to the final value of fit function about 2000 th generation. All of them converged in the same manner: the optimization progress was very quick during the first thousand generations; next, it slowed down; and; finally; it came up to the same value. In the next step, the obtained stations layout with the highest value of fit function was the input data for lines topology optimization. The cost of new-build metro line per unit length depends on local conditions, and this amount can vary even by a factor of 12 [38]. This variety comes mainly from difficult urban infrastructure, geology land acquisition, and type of construction. Table 5 shows the indicative costs of the construction of a metro line per one kilometer. Since the considered region includes also some less urbanized areas, a big part of the metro network could be constructed at the ground level. The underground or elevated facilities should be built in the centers of the cities. Figure 5: Map of population density (white—below 250 pers./km\({}^{2}\), black—above 4000 pers./km\({}^{2}\)). Figure 6: Examples of obtained stations layouts. Because the presented idea of building of a metro system is just in the initial phase, no cost analyses, even superficial ones, have been carried out on this issue.