## 2 The Bus Synchronization Problem
### The Cost Model  
As aforementioned, the proposed model considers two different objectives: the number of successful transfers and the cost of the system. For estimating the cost of the system, several aspects of the case study are taken into account. The local authorities in Montevideo conduct regular analyses of the expenses related to the bus system to calculate the kilometer-based fare. This information is utilized to determine the level of state subsidies allocated to the system. Various types of subsidies are applied for public transportation in Montevideo, with two being particularly noteworthy [29]. The first type involves a discounted fuel price that benefits the bus system, whereas the second type entails reduced fares for specific user groups such as students, those retired, and commuters who travel to and from their destinations on a consistent basis. These subsidies have a substantial impact on the cost structure of the system. For instance, the discounted fuel price significantly reduces the cost burden in comparison to other bus systems. In Montevideo, it accounts for only 5% of the total estimated cost [30], as opposed to 13% in comparable public transportation systems [31]. Nevertheless, the discounted fuel price poses a significant cost for the Uruguayan government. Recently, Montevideo has initiated a shift from diesel to electric buses as part of an environmentally conscious approach [5]. The primary objective of this shift is to decrease fossil fuel consumption and mitigate greenhouse gas emissions. The two primary expenses in the system consist of the driver's salary and the cost of fuel or electricity, depending on the type of bus (diesel or electric). These costs correspond to the operating hours (\(U_{VH}\)) for the driver's salary and the distance traveled (\(U_{VKM}\)) for fuel or electricity expenses.The driver's salary totals 9.2 USD per hour, which encompasses an 8 USD basic wage along with 1.2 USD in social charges. In term of fuel, the average consumption for a standard diesel bus operating in Montevideo is 0.396 L per kilometer, with the fuel price in the city being approximately 1.6 USD per liter. The costs models considered for public transportation varies according to their distinct objectives [31]. In this article, the focus is on the problem of determining bus schedule headways without affecting the layout of the bus lines. In cases where the bus line layout remains fixed, the cost functions generally are composed of linear combinations of several unit costs for relevant parameters, such as the bus operating time and the distance traveled [32; 33; 34]. Although more complex models can be devised, practitioners also commonly employ these linear functions since they are simple to understand and use in the decision-making process [35]. Hence, this article employs a cost function that is associated with vehicle operating time and distance-traveled variables. Regarding vehicle operating time, certain operating costs (driver salaries and administrative supervision hours) are directly associated with the number of vehicle operating hours. Consequently, these expenses are appropriately distributed based on vehicle hours. Moreover, vehicle hours are frequently utilized as a proxy for the working hours of employees in cost models, due to their simplicity of calculation [36]. Regarding vehicle distance, several operating costs, including fuel, oil, tires, and vehicle maintenance, are directly related to the distance traveled by buses. Operating costs are determined based on statistical information about operating time and distance traveled [37]. Some linear cost models incorporate the number of buses on the system [33] or consider the number of vehicles needed to meet peak-hour demand, which differs from the demand in off-peak periods [36]. The proposed model focuses on studying time windows during peak hours, assuming a fixed fleet size.