## 6 Experimental Evaluation
### Numerical Results
#### 6.2.1 Problem Variant #1: Offset Optimization  
Table 4 presents the objective function values achieved by the exact and the EA for the studied BSP instances for variant #1 (offset optimization). The real timetable is reported as baseline for the comparison. Column 'EA vs. exact' compares the results of both proposed approaches (a negative percentage value means a smaller function value for the EA solution). Relative improvements over the real timetable (in percentage values) are also reported. Column 'EA vs. real' reports the relative improvement over the real timetable for the EA solution, and column 'exact vs. real' reports the relative improvement over the real timetable for the exact solution. Results in Table 4 demonstrate that both exact and EA significantly outperform the baseline real solution in all problem instances. EA improved the real solution up to 66,3% in instance 40.37.30.1, and the exact method improved over the real solution up to 66.6% in instance 70.63.30.2. When comparing the EA and the exact solutions, the evolutionary approach demonstrated to be highly efficient at solving the problem: it computed the optimal solution in 54 out of the 75 problem instances studied. In all other instances, the distance to the optimum was below 1%. Overall, improvements over the real timetable were 24.5% (on average) for EA and 24.6% (on average) for the exact solution. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline _id_ & \(p_{R}\) & \(p_{M}\) & 30.37.30.1 & 30.37.50.1 & 30.37.70.1 & 30.37.90.1 & 30.37.100.1 & _avg._ \\ \hline C1 & 0.50 & 0.01 & 7 & 9 & 8 & 9 & 7 & 8.0 \\ C2 & 0.50 & 0.005 & 1 & 3 & 1 & 1 & 1 & 1.4 \\ C3 & 0.50 & 0.001 & 6 & 8 & 6 & 5 & 5 & 6.0 \\ C4 & 0.75 & 0.01 & 8 & 7 & 7 & 7 & 9 & 7.6 \\ C5 & 0.75 & 0.005 & 2 & 1 & 3 & 4 & 3 & 2.6 \\ C6 & 0.75 & 0.001 & 9 & 4 & 9 & 8 & 8 & 7.6 \\ C7 & 0.90 & 0.01 & 5 & 6 & 4 & 6 & 6 & 5.4 \\ C8 & 0.90 & 0.005 & 3 & 2 & 2 & 2 & 2 & 2.2 \\ C9 & 0.90 & 0.001 & 4 & 5 & 5 & 3 & 4 & 4.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._  
Figure 6: Boxplot analysis of parameter configurations for a representative BSP instance. The average improvements of both exact and EA over the baseline solutions, grouped by tolerance and scenario size, are reported in Table 5. Regarding tolerance levels, improvements of EA and the exact algorithm over the real timetable were up to 57.56% and 57.96%, respectively, in scenarios with \(\lambda=30\), which pose the tighter bounds for user waiting times. In less restrictive scenarios, both studied methods improved over 9.74% in regard to the real timetable. Regarding the scenario dimension, both methods computed robust solutions that improve between 23.88% (EA, for scenarios with 30 transfer zones) and 25.72% (exact, for scenarios with 110 transfer zones). The reported improvements on the objective function values grouped by tolerance and problem dimension indicate that for all sizes, the improvements over the real timetable increase as user tolerance decreases. This is a relevant result, which demonstrates that \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _Scenario_ & _EA_ & _Exact_ & _Real_ & _EA vs. Exact_ & _EA vs. Real_ & _Exact vs. Real_ \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: \(Cont.\)