## 6 Experimental Evaluation
### Numerical Results
#### 6.2.3 Quality of Service Results
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & \multicolumn{4}{c}{_Problem Variant \#1_} \\ \(\boldsymbol{\lambda}\) & \(\boldsymbol{EA}\) & \(\boldsymbol{Exact}\) & \(\boldsymbol{Real}\) & \(\boldsymbol{EA/Real}\) & \(\boldsymbol{Exact/Real}\) & \(\boldsymbol{EA-Real}\) & \(\boldsymbol{Exact-Real}\) \\ \hline
30 & 4.67 & 4.68 & 2.97 & 1.57 & 1.58 & 1.70 & 1.72 \\
50 & 5.68 & 5.69 & 4.43 & 1.28 & 1.28 & 1.25 & 1.26 \\
70 & 7.04 & 7.04 & 6.03 & 1.17 & 1.17 & 1.01 & 1.01 \\
90 & 7.47 & 7.47 & 6.81 & 1.10 & 1.10 & 0.66 & 0.66 \\
100 & 7.36 & 7.36 & 6.61 & 1.11 & 1.11 & 0.75 & 0.75 \\ \hline  & & & & \multicolumn{4}{c}{_Problem Variant \#2_} \\ \(\boldsymbol{\lambda}\) & \(\boldsymbol{EA}\) & \(\boldsymbol{Exact}\) & \(\boldsymbol{Real}\) & \(\boldsymbol{EA/Real}\) & \(\boldsymbol{Exact/Real}\) & \(\boldsymbol{EA-Real}\) & \(\boldsymbol{Exact-Real}\) \\ \hline
30 & 7.42 & 7.66 & 2.97 & 2.50 & 2.58 & 4.45 & 4.70 \\
50 & 7.88 & 8.06 & 4.43 & 1.78 & 1.82 & 3.45 & 3.63 \\
70 & 8.38 & 8.43 & 6.03 & 1.39 & 1.40 & 2.35 & 2.40 \\
90 & 8.51 & 8.52 & 6.81 & 1.25 & 1.25 & 1.70 & 1.72 \\
100 & 8.57 & 8.59 & 6.61 & 1.30 & 1.30 & 1.96 & 1.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average objective values (normalized by the number of transfer zones), grouped by tolerance, for both problem variants. Figure 7: Comparison of normalized objective function values, grouped by tolerance, for BSP variant #1. Results in Table 10 indicate that both studied optimization methods significantly improved the quality of service when compared with the real solution. Both the exact method and the EA computed lower values of the waiting time metric for all instance classes. The best improvements of the studied methods over the baseline real solution occurred for problem instances with NP = 70/110 and \(\lambda\) = 30, where both exact and EA computed solutions with a higher number of synchronized lines. In turn, the largest number of synchronized lines were computed by the exact method in problem instances with NP = 110 and \(\lambda\) = 30. Furthermore, better waiting time values were obtained by both exact and EA in problem instances with lower waiting time tolerance from users, with respect to the baseline real solution. A comparison of waiting time values for transfers of all lines (normalized by \(W_{b}\)) is presented in the histograms in Figure 8. Results correspond to scenario 70.63.30.2, which is representative of the results computed for other studied BSP instances. Normalized average waiting time results are reported for the baseline real solution, the EA solution for BSP variant #1 (\(\alpha\) = 0.0), and the exact solution for BSP variant #2 (\(\alpha\) = 0.3). Values of the normalized average waiting time over 1.0 mean that for a number of lines in the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{_Problem Variant \#1_} \\ \(m\) & \(\lambda\) & _Real_ & & _EA_ & & _Exact_ \\ \hline  & \(r\) & \(l_{s}/l_{n}\) & \(r\) & \(l_{s}/l_{n}\) & \(r\) & \(l_{s}/l_{n}\) & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 10: Average synchronizations for the studied methods. reported scenario, the optimization methods were not able to find a combination of offset and headway values that allows lowering the waiting time for transfers under \(W_{b}\). This occurs as a direct consequence of the inter-related design of bus lines and the behavior of passengers in each scenario, since when adjusting offsets and headways to achieve successful transfers between line \(i\) and line \(j\) in transfer zone \(b\) (i.e., for those transfers avg(wait) \(<W_{b}\)), those lines are not synchronized (at least for some trips) with other lines on other transfer zones \(b*\) (i.e., for those transfers, avg(wait) \(>W_{bs}\)). The histograms in Figure 8 report both successful synchronizations (avg(wait) \(<W_{b}\), on the left side of each histogram) and non-successful synchronizations (avg(wait) \(>W_{b}\), on the right side of each histogram).