## 4 Results
### Numerical Results
#### 4.3.2 Comparison with the Current Timetables  
To establish a meaningful baseline for comparison, the timetables that are currently used in Montevideo, hereafter current timetables (CT), are used. The enhancement of the best trade-off MILP solutions in comparison to the CT is analyzed via metrics: (i) the count of transfers, as stipulated in objective function (1); (ii) the cost of the bus schedule, calculated as per the objective function (2); and (iii) the number of buses used in the solution (\(Q_{r}^{i}\)). Despite not being an optimization criteria of the MILP model, the number of buses was reported because it is a relevant result for the operational management of the system. The results stemming from the MILP model and the comparison with the Montevideo CT are presented in Table 3. The table reports the values of the CT and the percentage of improvement achieved via both the compromising solution and the extreme solution of the MILP approach for each objective. Notably, the extreme solution that is compared is different when evaluating each objective. The extreme solution is the solution with the largest number of transfers when assessing the number of transfers, whereas it is the solution with the lowest cost when evaluating the cost of the system. Regarding the number of buses, the CT is compared with the compromising solution of the MILP model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**Successful Transfers**} & \multicolumn{3}{c}{**Cost**} & \multirow{2}{*}{\(\Delta\)} \\ \cline{2-2} \cline{4-7}  & **Ideal** & **Trade-Off** & & \(\delta_{T}\) & **Ideal** & **Trade-Off** & \(\delta_{T}\) \\ \hline
40.90.4 & 273.52 & 268.57 & 1.81\% & 6103.68 & 6177.02 & 1.20\% & 2.17\% \\
41.100.2 & 289.12 & 287.72 & 0.48\% & 7052.32 & 7199.76 & 2.09\% & 2.15\% \\
41.30.2 & 284.95 & 273.22 & 4.12\% & 7052.32 & 7212.24 & 2.27\% & 4.70\% \\
41.50.2 & 286.95 & 277.29 & 3.37\% & 7052.32 & 7170.65 & 1.68\% & 3.76\% \\
41.70.2 & 290.47 & 285.84 & 1.59\% & 7052.32 & 7219.65 & 2.37\% & 2.86\% \\
41.90.2 & 290.82 & 285.14 & 1.95\% & 7052.32 & 7137.91 & 1.21\% & 2.30\% \\
42.100.3 & 277.42 & 272.06 & 1.93\% & 7642.51 & 7642.51 & 0.00\% & 1.93\% \\
42.30.3 & 270.57 & 264.93 & 2.08\% & 7644.06 & 7773.12 & 1.69\% & 2.68\% \\
42.50.3 & 275.31 & 268.87 & 2.34\% & 7643.71 & 7788.25 & 1.89\% & 3.01\% \\
42.70.3 & 276.38 & 273.92 & 0.89\% & 7643.40 & 7794.86 & 1.98\% & 2.17\% \\
42.90.3 & 277.76 & 271.58 & 2.22\% & 7633.18 & 7633.18 & 0.00\% & 2.22\% \\ \hline Average & & & 2.11\% & & & 1.52\% & 2.75\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 6: Distance of the best trade-off solution and the ideal vector per \(\lambda\). Overall, the solutions obtained via the MILP approach are significantly superior than the CT, particularly in terms of successful transfers. The average improvement of the compromising solution over the CT in terms of successful transfers is 61.94% for all the instances, achieving the greatest improvement in instance 37.30.1 (172%). Similarly, the extreme solution excels the CT in terms of successful transfers of 65.77% on average, obtaining the best improvement also in instance 37.30.1 (180.29%). Regarding the costs of the system, the improvements of the best trade-off solution over the CT are on average 27.24%, with the greatest improvement on instance 40.100.4 (29.34%). The extreme solution excels the CT on average by 28.33% in terms of costs of the system, obtaining the best improvement simultaneously in three instances with forty bus lines, i.e., instances 40.30.4, 40.70.4, and 40.90.4 (29.98%). In the case of the number of buses, the compromising solutions use on average 11.71% less buses than the CT, achieving the maximum reduction in instance 41.50.2 (20.38%). Only in instance 37.70.1 does the compromising solution of the MILP model use a slightly larger number of buses (1.82% larger).