## 2 The Bus Synchronization Problem
### Related Work  
The topic of bus timetabling has been explored in various studies with different criteria [26]. The most commonly considered optimization objectives include minimizing user waiting time as well as the required number of buses for providing the service or the total travel time of buses, and maximizing the occupation of buses. Only a few works have addressed the optimization of bus frequencies to enhance synchronization between buses or with other modes of transportation [26]. Improving synchronization involves designing trip schedules that ensure buses arrive at transfer zones at convenient times that facilitate users to take other buses in the zone. A convenient transfer should achieve a balance by ensuring that waiting times are brief to enhance passenger Quality of Service (QoS), while also being sufficiently long to enable passengers to transfer smoothly from one line to another [25]. Consequently, the Synchronization Bus Problem aims to determine the departing time for each trip of each bus line in a network in order to maximize the number of successful transfers in synchronization nodes. Additionally, due to its computational complexity, there is a scarcity of research that employs exact methods to solve the Bus Synchronization Problem (BSP). The computational complexity was proved for both single-objective [26] and multi-objective versions of the problem [41]. As far as we are concerned, there is no work that addresses the multi-objective bus synchronization problem with a multi-objective exact resolution algorithm. However, there are a few works that solved, with exact methods, a single-objective version of the problem. In their work, [PERSON] and [PERSON] [42] introduced an exact mathematical formulation of the problem to address small instances. However, due to the computational complexity of the problem, this exact formulation becomes intractable for large instances. Another group of authors that used mathematical formulations for this problem were [PERSON] et al. [25]. In their formulation, the number of trips per bus line was considered fixed. To strengthen the formulation, they proposed some valid inequalities to strengthen the formulation. Theydemonstrated that these valid inequalities offer a stronger formulation of the problem, enabling the solution of instances of practical size that would otherwise be unsolvable with the plain mathematical formulation. In a more comprehensive model, [PERSON] et al. [43] focused on reducing passenger travel time from the start of their journey to their destination as the primary objective, with the synchronization of buses being a subordinate result of this objective. The authors presented two different mathematical formulations: a plain formulation similar to other related works, and a formulation based on set-partitioning. The formulation based on set-partitioning proved to be more competitive to solve larger instances in smaller computing times than the plain formulation. [PERSON] et al. [44] presented a bus synchronization planning problem that, as a novel feature, considers dwelling times. In operating models, the dwelling time refers to the duration that a bus remains stopped at a bus stop. Despite being an important factor to consider for day-to-day operations management, when it comes to models used in the tactical planning stage of the system, the dwelling time is often not taken into account. The authors proposed a MILP model to obtain the specific timetables and the duration of the dwelling periods of the buses that maximize the number of successful transfers. The model is enhanced via the introduction of some valid inequalities. The computational experimentation is performed in real instances of the transport system of Santiago de Chile, showing that solutions obtained with the model have around 70% more transfers than the current base-case operation. In our previous article [18], we focused on addressing two distinct variants of the Bus Synchronization Problem (BSP) specifically during peak hours.