## 7 Concluding Remarks  
This article studied the bus timetabling synchronization problem. The problem model extends existing ones by defining extended transfer zones for every pair of bus stops. In turn, an improved model was included to account for the transfer demands of passengers, and two variants of the problem were addressed: optimizing the offset for each line, and optimizing the offset and headways for each line. For the second problem variant, the optimization domain was restricted to a certain deviation on the real headways defined for the case study to provide a proper quality of service to users.  
An exact MILP approach and an EA were proposed to solve the problem. Both methods were evaluated for a real case study for the public transportation system in Montevideo, Uruguay. A total number of 75 scenarios were defined using real data about lines, headways, and transfer demands, gathered from the intelligent transportation system of Montevideo. Results were compared with the real timetable currently implemented by the city administration.  
The empirical evaluation indicated that both proposed methods are able to compute solutions that are significantly improved over the real current timetable in Montevideo. The exact method computed the optimal solution for all instances in problem variant #1, improving successful synchronizations up to 66.6% (24.6% on average) over the real timetable. The EA was highly efficient for this problem variant, too, improving the synchronizations up to 66.3% (24.5% on average) over the current timetable. Problem variant #2 poses a significantly harder challenge for optimization, as the number of variables significantly increases for each considered scenario. Despite this, the exact method was able to compute an accurate solution, which provides significant improvements on the number of successfully synchronized trips over the real timetable. Improvements up to 179.9% (66.2% on average) for the exact methods and up to 174.8% (66.2% on average) for the EA. The proposed optimization methods also improved relevant quality of service metrics, such as the average waiting times for transfers, especially in tight problem instances, which reduced up to 57.8% for BSP variant #1 and up to 158.3% for BSP variant #2.  
Both methods were efficient regarding computing times. The exact method provided accurate objective function values within less than a second for problem variant #1. In turn, the proposed EA is useful for solving larger problem instances of problem variant #2 in reduced execution times.  
Future lines of research are related to addressing other BSP variants, e.g., explicitly focusing on the perceived waiting times for users, and modeling the real demand for direct trips from real data on ticket sales provided by the transportation system. Multi-objective versions of the problem can be devised, too, by including the explicit optimization of other relevant functions, such as the operation cost of the bus fleet and other quality of service metrics.  
This research received no external funding.  
Not applicable.  
**Acknowledgments:** The work of [PERSON] and [PERSON] is partly supported by ANII and PEDECIBA, Uruguay. Authors would like to thank [PERSON] and [PERSON] for assistance in data processing for scenarios generation.  
**Conflicts of Interest:** The authors declare no conflict of interest.