## 4 Results--A Case Study
### Selection of the Relevant Values of Optimization Parameters  
An optimization process is controlled by a number of parameters. In the case of genetic algorithm, the most important are the population size, the number of generations, the probability of mutation, and the probability of cross-over. Some additional parameters are possible depending on the details of the problem. The increase of the population size leads to better convergence and allows for avoiding local optima, but causes the calculation time to grow up. The size of population was preliminary determined by trial and error; finally, it was equal to 250 individuals at the first stage (stations locations) and 1000 at the second stage (lines layout). To select the relevant number of generations, the optimization process was monitored, and then it was stopped when the progress of increasing the objective function value significantly slowed down. The most important of the other parameters for the first stage are the probability of mutation (PM), the probability of cross-over (PX), the elite size (ES), and the standard deviation for the station shift (ND, expressed as a fraction of the region diameter). The parameters taken into account in second stage are PM, PX, and ES. A genetic algorithm is a non-deterministic procedure, so there is a need to perform a series of algorithm runs for each combination of parameters. If each parameter took only two values (from a typical range), the full analysis of all possible parameters combinations would require 16 series at the first stage and 8 series at the second stage. Therefore, the values of the parameters were determined by the use of Taguchi method [36]. The L8 and L4 orthogonal tables were used to design these preliminary tests. [PERSON] proposed a procedure of determining the relative importance of the examined parameters. This idea involves an application of a quantity called signal to noise ratio (S/N). This quantity is taken from signal theory, and it expresses the relation between the signal--a desired feature and the noise--undesired disturbing factors. In the considered case, as high value of the objective function as possible is desired, so according to [PERSON] the S/N value should be expressed as follows:  
\[SN_{i}=-10\text{log}_{10}\frac{1}{n}\sum_{j=1}^{n}\frac{1}{X_{ij}^{2}} \tag{21}\]  
Tables 3 and 4 show the selected parameters values and the optimization process outputs (scaled values of the objective function and S/N) for both stages. The input data described a simplified metro system with 30 stations and three lines to make these preliminary calculations quick. Four procedure runs were carried out in each series. All the values of the objective function were referred to the highest one (normalized to unity). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Series**} & \multicolumn{4}{c}{**Parameters**} & \multicolumn{4}{c}{**Scaled Objective Function**} & \multirow{2}{*}{**S/N**} \\ \cline{2-2} \cline{4-9}  & **PM** & **PX** & **NE** & **NS** & **1** & **2** & **3** & **4** \\ \hline
1 & 0.1 & 0.5 & 1 & 5 & 0.9900 & 0.9991 & 0.9923 & 0.9702 & \(-\)0.1075 \\
2 & 0.1 & 0.5 & 1 & 8 & 0.9646 & 0.9920 & 0.9560 & 0.9905 & \(-\)0.2164 \\
3 & 0.1 & 0.8 & 5 & 5 & 0.9672 & 0.9555 & 0.9527 & 0.9559 & \(-\)0.3746 \\
4 & 0.1 & 0.8 & 5 & 8 & 0.9825 & 0.9702 & 0.9899 & 0.9668 & \(-\)0.2001 \\
5 & 0.2 & 0.5 & 5 & 5 & 0.9308 & 0.9590 & 0.9705 & 0.9598 & \(-\)0.4028 \\
6 & 0.2 & 0.5 & 5 & 8 & 0.9879 & 0.9372 & 0.9753 & 0.9619 & \(-\)0.3092 \\
7 & 0.2 & 0.8 & 1 & 5 & 0.9744 & 0.9586 & 0.9410 & 0.9635 & \(-\)0.3624 \\
8 & 0.2 & 0.8 & 1 & 8 & 0.9333 & 0.9613 & 1.0000 & 0.9667 & \(-\)0.3144 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Scaled outputs for the first stage according to L8 orthogonal table. In the next step, the average values of S/N for each parameter and for each value level must be calculated, and then charts showing the relative importance of each parameter can be constructed.