## 2 The Bus Synchronization Problem
### Related Work
The first variant considered buses of the same line departing at regular intervals, requiring determination of only the offset, i.e., the departure time of the first bus. The second variant allowed for variable headways between consecutive bus departures of the same bus line within given predefined limits. To tackle both variants, we developed an exact MILP formulation. We applied this formulation to real instances retrieved from the bus network of the city of Montevideo. The results significantly outperformed the current schedules used in the city in terms of the number of successful transfers achieved and average waiting times for users. Our proposed schedules demonstrated clear improvements over the existing schedules, highlighting the efficacy of our approach. Among the authors that addressed this problem with heuristic methods is the work of [PERSON] et al. [41], who proposed a multi-objective optimization model for the bus synchronization problem to obtain solutions which simultaneously balance the objectives of maximizing the number of successful transfers and minimize the deviation from the departure times of the previous timetable. This is performed in order to acknowledge the impact that the redesign of the timetable of the existing schedule can have on the usual trip plans of passengers. They addressed the problem using a non-dominated sorting genetic algorithm (NSGA-II) as a solution approach and performed a computational experimentation in instances based on the city of Shenyang, China. [PERSON] et al. [45] addressed a synchronization problem to maximize the number of synchronized buses and minimize the total waiting time of users. As a resolution strategy, they combined an evolutionary algorithm and a multi-agent system to alternate between the intensification and diversification phases of the metaheuristic. No computational experimentation is performed. Another relevant aspect discussed in the related literature is the behavior of different stakeholders when bus transport parameters are varied, primarily focusing on which parameters can affect the demand of users for the service [46]. For instance, [PERSON] et al. [47] proposed a unified model as a bi-level optimization problem. In the upper level, the fare of the bus operator is optimized, constrained with bus frequency. In the lower level, the travel costs of commuters are minimized as a sub-assignment traffic model. Simulations are conducted for studying how demand varies according to different bus ticket fares to determine tradeoffs that attain benefits for both users and the public transport company. Similar to the model proposed in this work, [PERSON] et al. [48] studied the captured demand depending on the frequency of the transport unit (in this case, the train transport network). Additionally, they studied the behavior of demand given different levels of government subsidies to the ticket fare. The more frequent the transport units depart, the more demand is captured, but the larger the cost of the system. Additionally, [PERSON] [49] studied a bus network feeding an urban train line, analyzing how demand varies depending on the distance of users to the bus stop, which serves as a proxy for the accessibility of the system. [PERSON] and [PERSON] [50] presented a similar work in which not only the departure times of buses are optimized but also the bus capacity in terms of the number of users to be transported. This article makes a valuable contribution by presenting a formulation based on mathematical programming for the timetable synchronization problem that, unlike previous approaches, considers a multi-objective perspective that aims to maximize the number of successful transfers for passengers undertaking multi-leg trips, while simultaneously minimizing the cost of the system based on a specific cost model. This objective, incorporating a cost model, has not been explored in previous multi-objective versions of the problem.