## 3 Related Work  
Timetable synchronization was stated as an important problem for public transportation systems in pioneering research articles by [PERSON] [3]. [PERSON] and [PERSON] [14] presented one of the first proposals for synchronizing schedules on public transportation. The authors studied several objective functions, including weighted sum approaches with transfers and maximum waiting time, and proposed metaheuristics for simple problem versions with uniform frequencies. The approach was evaluated over a case study on Berlin and several cities in Germany. In the experiments, tabu search outperformed simulated annealing in randomized instances, and the authors highlighted the trade-off between operation costs and efficiency. The transit network timetabling problem was studied by [PERSON] et al. [15] to optimize synchronization events at stops shared by several bus lines, by maximizing the number of simultaneous arrivals. A greedy approach was introduced to define custom timetables, properly selecting nodes from the bus network. The article focused on maximizing simultaneous bus arrivals over small problem instances involving few nodes and lines. A subjective metric was proposed by [PERSON] et al. [16] to evaluate synchronizations, including expert knowledge. The proposed synchronization metric was applied to design a heuristic to minimize vehicle operation costs in a case study consisting of small problem instances from Montreal, Canada. Different timetables were found by weighting the components of the cost function. [PERSON] and [PERSON] [17] presented a mixed integer programming model for optimizing transfers in a public transportation network with fixed transfer stations. The problem considered the offset optimization, i.e., setting the departure times of buses, which was solved by an exact method using CPLEX. A second problem variant was formulated, accounting for the extra stopping time of buses at transfer stations. A genetic algorithm was proposed to solve this problem variant over small problem instances involving 14 lines and just three transfer stations. A real case study was presented: the public transportation network in the city of Mashhad, Iran. The proposed methods were able to improve up to 14.5% with a business-as-usual (i.e., no explicit optimization) strategy. A variant of the the synchronization problem including time-windows between travel times was addressed by [PERSON] and [PERSON] [10]. A multi-start iterated local search (MILS) was applied to solve eight scenarios from the public transportation of Monterrey, Mexico, considering from 4 to 40 nodes. In less than 60 seconds, MILS computed accurate timetables for medium-size instances, regarding both upper bounds and an exact Branch & Bound algorithm. MILS was also applied by [PERSON] et al. [11] to solve the multiperiod BSP, to optimize multiple trips of a given set of lines. MILS was able to compute results similar to a variable neighborhood search and a simple population-based algorithm on synthetic instances with few transfer zones. Results for a sample case study using data for a single line of Monterrey demonstrated that maximizing synchronizations for a specific node usually reduces the number of synchronizations for other nodes. A recent article by [PERSON] et al. [18] proposed a model for timetable synchronization assuming a fixed headway for each line. The authors identified specific cases of the problem that are solvable in polynomial time. An approximation algorithm was proposed, based on the maximum directed cut problem. The proposed method was evaluated in a simple network and a large case study in Mashhad, Iran. In turn, a recursive algorithm that executes in quasi-linear time was proposed to minimize the total transfer waiting time in a problem variant that relaxes the fixed headway assumption. An EA was proposed in a previous article [19] for a specific variant of the BSP, which outperformed real timetables and heuristics.