## 2 The Bus Synchronization Problem
### Overall Description of the Problem Model  
The BSP considers two of the most important purposes of a mass transportation system: offering an efficient means for the movement of citizens, while maintaining low costs and fares. The considered model focuses on citizens, offering them an efficient travel experience and short waiting times for passengers that use two or more buses to perform consecutive trips. The concept of a _synchronization_ event is defined as the action of providing passengers transfers whose waiting times are lower than the maximum time passengers are willing to wait. The research proposes addressing the BSP on real scenarios, built using real data about lines, bus stops, traveling times, and passengers performing transfers between lines. The problem model divides a day into several planning periods, considering the regularity of travel demands, travel times, and citizens' behavior. On each planning period, a data analysis approach can be applied to extract common characteristics and steady information to build BSP instances. In the considered problem model, the bus network is represented by a set of bus lines and a set of relevant locations (the _synchronization nodes_ or _transfer zones_), where passengers can transfer from one line to another. Unlike previous formulations of the problem [9; 10], in the proposed model, nodes are not just bus stops but wider transfer zones, formed by separated bus stops for lines \(i\) and \(j\). This way, the model can include several bus stops for several lines (see a graphical description for two generic lines \(i\) and \(j\) in Figure 1). The proposed model explicitly includes the distance between the bus stops for lines \(i\) and \(j\)\(d_{b}^{i,j}\), drawn in red in Figure 1. For the instances solved in this article, it is assumed that the walking speed of pedestrians is constant \(ws=6\) km/h, and the walking time for a distance \(d\) between any pair of bus stops is given by \(wt=d/ws\). The proposed problem model is useful for capturing the reality of contemporary intelligent transportation systems, which do not impose a limitation on the number of transfers for passengers and in which transfers are not limited to specific locations. Instead, transfers can be performed at every pair of bus stops, providing flexibility to passengers when planning a trip. This is the case of the intelligent transportation system in Montevideo, Uruguay, which is the case study in this article [7]. In this scenario, the corresponding synchronization problem is more complex, as many realistic transfers options are available to passengers, which in turn may require them to walk between bus stops. Unlike traditional formulations, the proposed problem model is not focused on maximizing the number of passengers headed from origin to destination, mainly because the main focus is on the user experience when performing transfers, by considering the maximum time passengers are willing to wait for a transfer. This way, the objective function of the optimization model considers the demand of transfers in each transfer zone (pair of bus stops) for all synchronized trips of two bus lines. Previous articles addressing different variants of the timetable synchronization problem have worked under stronger assumptions, i.e., only considering trips for lines to synchronize. Two cases are distinguished in the proposed model. The case with _uniform departing times_ and the case with _non-uniform departing times_. These two cases are described next. In the case with uniform departing times, there are no differences between departing times in the planning period; thus, \(r\) and \(s\) (trips of line \(i\) and line \(j\), respectively) are not relevant for defining the time between consecutive departures (which in fact is constant, i.e., \(F_{i}^{1}\)). This is a realistic assumption when planning for short and medium periods (e.g., in the morning, in the afternoon, in rush hours, etc.).