## 4 Results--A Case Study
### Obtained Structures of the Metro Network
In such a situation, no explicit costs were entered into the model and the costs were treated relatively. Therefore, just the simplified map of the building cost coefficient (\(k\)) was introduced (Figure 8). Light gray areas mark the normal cost for less urbanized areas (ratio = 1); dark gray areas mark the increased cost due to dense built-up terrains (ratio = 2), and black areas mark the costs for the city centers (ratio = 4). The optimization of lines topology was carried out for different assumptions concerning the number of lines and the formulation of the objective function according to the below scheme (Table 6). Disabling Equation (8) means the coherency of the metro network is assured, but the average travel time is not minimized. Figure 8: Map of the construction cost coefficient. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Construction Type** & **Costs [US \$ million]** & **Ratio** \\ \hline Ground level & 15–30 & 1 \\ Elevated & 30–75 & 2–2.5 \\ Underground & 60–180 & 4–6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Indicative costs of a metro line per one kilometer [38]. Figure 7: Progress of the process of the station layout optimization. The results of optimization of lines topology are shown in Figure 9. As previously, a series of 10 calculations was carried out for each case, and the best solutions are shown. The same regularity was observed too: despite almost the same values of fit function, the obtained topologies of metro network differed significantly. Because the fit function was determined strictly here, a different explanation is suggested. The possible reason for such an effect is the equifinality of the problem. This means the same desired result can be achieved in different ways. There is a conspicuous difference between pairs of cases A, C and B, D: the latter pair assures the coherency of the metro network, but sometimes it is needed for a long journey between stations despite their physical proximity. In contradiction, very dense network for cases A and C allows for fast travelling between any pair of stations, although sometimes with more than one transfer. This difference of the radiation accessibility is the obvious effect of the different formulations of the objective function. The short summary of the obtained results is shown in Table 7. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Objective Function**} \\  & \multicolumn{3}{c}{**Full (Equation (9))**} & **Equation (8) Disabled** \\ \hline No of & 3 & A & B \\ lines & 4 & C & D \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scheme of the optimization of lines topology. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Relative Cost**} & \multirow{2}{*}{**Longest Trip [km]**} & \multirow{2}{*}{**Average Trip [km]**} & **Relative Total Distance** \\  & & & & **Travelled (Equation (6))** \\ \hline A & 0.98 & 59.6 & 21.7 & 0.61 \\ B & 0.76 & 98.1 & 35.2 & 1.00 \\ C & 1.00 & 60.3 & 22.0 & 0.62 \\ D & 0.76 & 91.3 & 33.6 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Lines topology—a summary of the obtained results. Figure 9: The best solutions of lines topology for all examined cases (according to Table 6). Two quantities (building cost and total distance travelled) are expressed relatively (referring to the maximal values) due to avoid large numbers without meaningful units. Having in mind the non-deterministic nature of applied optimization method, one can easily conclude that the metro network performance does not depend strongly on the number of the lines. In the light of data in Table 6, cases A and C seem to be almost the same; this concerns cases B and D as well. Therefore, a conclusion has appeared that the total construction cost does not depend really on the number of lines. The proposed model does not provide the premises to make a decision on the number of lines: one could expect the lower number of transfers for three lines, but a system with four lines seems to be less prone to disturbances.