###### Abstract
By assuming that current headways are dimensioned to manage the load of the system along a time-window with regularity and that relatively slow deviations from those values linearly affect surveyed figures, a formulation is devised to allow tuning the system to increase the number of successful transfers. Timetable synchronization is a very relevant problem in this case study, since the bus system in Montevideo is extremely complex. The bus network consists of 145 main bus lines, and each one of them has multiple variants (e.g., outward and return trips, different origins/destinations, and shorter versions), so the total number of bus lines considering all variants is 1383, a remarkably large number when compared to bus networks in other similar cities [7]. In any case, the proposed models and algorithms are fully portable to other systems, as long as transfers are allowed and the main goal is optimizing the number of synchronized trips. The formulation is independent of data particulars, and it can be easily ported to another system. For instance, transfer zones can be arbitrarily set. They are not bound to infrastructure deployment, so an instance can, in fact, be determined from existing transfer stations or a mix of closed and on-street stations. In addition, there is no particular reason to prevent these models from being used with other means of transportation, such as light rail transit, metro networks, or arrangements thereof. This article extends our previous conference article \"Exact and metaheuristic approach for bus timetable synchronization to maximize transfers\" [8]. New contents include: (i) an extended transfer zones model; (ii) the formulation of two problem variants, optimizing offset and headways (variable within a range); (iii) a new and more realistic objective function, properly modeling the transfer demands considering variable headways; (iv) the proposed exact and evolutionary approaches, adapted to solve both addressed problem variants; (v) an exhaustive experimental evaluation of the proposed optimization methods over 45 problem instances for each problem variant. Regarding previous studies in the literature, this article contributes by considering an extended transfer zones proposal to model transfers between different (geographically separated) bus stops of different lines. This formulation is more useful than standard models to capture the reality of modern intelligent transportation systems that do not limit transfers to specific locations, instead allowing them to be performed at every pair of bus stops. The new problem model provides flexibility for passengers who have many realistic transfer options available and also for the bus system administration to design proper timetables. The new formulation also considers the explicit demand of transfer trips for each pair of bus stops that defines a transfer zone, unlike previous formulations that only considered the number of bus trips synchronized [9; 10; 11] or proposals focused on minimizing waiting times between bus trips [12]. Thus, the proposed formulation allows for focusing on the quality of service offered to passengers better than existing approaches. The proposed evolutionary approach is also a contribution, as no previous application of EAs to solve the problem was found in the review of related works. The EA is useful for solving large-dimension problem instances, e.g., by considering a large number of transfer zones in a city scenario. Besides the formulations proposed to model both problem variants, a relevant contribution of the reported research is related to the obtained results for the case study in Montevideo: both the exact and EA approaches are able to compute timetables that are significantly improved over the real timetable applied by the city administration, considered as a reference baseline for the comparison. The article is organized as follows. Section 2 introduces the problem, the proposed model, and the two variants addressed. Section 3 presents a review of related articles.