Synchronization of different transit lines is an important activity to increase the level of service in transportation systems. In particular, for passengers, transferring from one line to another, there may be low-frequency periods and transfer zones where walking is needed, or passengers are exposed to adverse weather conditions and uncomfortable infrastructure. In this study, we define the Bus Lines Synchronization Problem that determines the frequency for each line (regarding the even headway), the timetable (including holding times for buses at transfer stops), and passenger-route assignments to minimize the sum of passenger and operational costs. We propose a nonlinear mixed integer formulation with time-indexed variables which allow representing the route choice for passengers and different types of costs. We implement an iterative heuristic algorithm based on fixing variables and solving a simplified formulation with a commercial solver. We implement our proposed heuristic on the transit network in Santiago, Chile. Numerical results indicate that our approach is capable of reducing operating costs and increasing the level of service for large scenarios.
In transport systems in big cities, the passengers demand varies in space and time. As the provision of direct lines between every origin and destination would be costly, passengers frequently need to transfer from one bus to another while making their trips. Public transport systems are increasingly multimodal and shifting towards fare integration which increases the need of transfer coordination for passengers since they look for the most convenient way to reach their destination. Passengers tend to accept these transfers; however, they require them to be as quick and comfortable as possible. Thus, better transfer conditions should increase demand and influence passenger route choices within the system.
Indeed, it is desirable to schedule and synchronize buses to optimize transfers, particularly under three conditions: low frequencies, low travel time variability, and undesirable waiting circumstances. These three conditions are commonly present during night services, so we will have this period in mind when defining our optimization problem. In these circumstances, vehicles will travel with few passengers so that we can assume no capacity issues during the whole planning period. Finally, we consider even headways for all transit lines since we focus on a frequency-based operation; that is, passengers do not have access to the timetables. Moreover, the case study considers a night period where the frequency is determined based on a minimal level of service instead of the passengers’ demand. The latter characteristic will not only reduce the complexity of our model but will also be easier for passengers to remember the schedule. In urban contexts, transfers often require some walking between the stop where the passenger alights and the stop where he/she will board the next vehicle. This walking time should be considered in the scheduling process so that passengers can reach the next vehicle they would board before it departs from the stop.
Usually, line frequencies (trips per hour) are determined before determining the timetable (departure time for each planned trip). Once these frequencies are obtained, the level of synchronization that can be achieved among different lines strongly depends on how much flexibility is available to determine the departure time of each bus from each stop. For example, if services must operate under an even headway for each line synchronization is much more limited than if depot departure times or headways can be accommodated within certain intervals as it can be addressed in [
Synchronization can not only be improved by adjusting line frequency and timetable. We can also plan some holding time for certain buses at specific stops. Holding has been proposed as a real-time decision to improve successful transfers. For example, the studies of Delgado
Finally, once timetables of a bus network have been changed to improve transferring experience, affecting service frequencies, and holding times have been added, passengers may change the routes they use to reach their destinations. This problem is known as passenger assignment problem (see the review of Desaulniers
In this paper, we deal with the Bus Lines Synchronization Problem which determines the frequency of each line, the timetable (considering holding times) to foster synchronization at transfer stops, and it assigns each passenger to his/her more convenient route to reach his/her destination. In this process, the goal is to minimize the weighted sum of user and operator costs. The rest of this paper proceeds as follows. First, we review related studies present in the literature. Next, we introduce our optimization problem and proposed mathematical formulation. Then, we present details of our solution approach. Finally, we show the numerical results of implementing our approach in a case study based on the transit network of Santiago in Chile.
Synchronizing buses in large networks can be extremely difficult. Passengers often transfer in all directions and between every pair of intersecting lines. Thus, synchronizing transfers at each stop would become an endless process since improving the performance on one synchronization point necessarily will affect the conditions on another. Additionally, improved transfer conditions should affect passengers route choice so, under networks where more than one route is available for some origin-destination pairs, passenger assignment should not be considered as given. Thus, improving transfer conditions will not only benefit current users of this transfer but also attract passengers from alternative routes lacking this coordination. Bookbinder and Désilets [
The synchronization of different lines at a single stop has been addressed by Chakroborty
More recently, Ibarra-Rojas and Rios-Solis [
Integration of timetabling and frequency setting has been addressed in the works of Bookbinder and Désilets [
The synchronization is also of interest in railway-based transit systems to optimize the transfer conditions for passengers. For example, Gou
As it can be seen, the synchronization of different lines in transport systems has been addressed through mathematical programming, heuristics, and meta-heuristics for bus-based and railway-based transit systems (see a literature review in [
Our methodology consists of the following three stages: (i) defining an optimization problem based on the context of the decision-making process, (ii) designing a mathematical formulation for the proposed problem in order to test commercial solvers, and, finally, (iii) designing and implementing a solution algorithm which is validated in an experimental stage using different scenarios for the problem. The mentioned steps of our approach are detailed in the following sections.
To define our problem, we assume a given fixed demand on the transit network for each origin-destination pair in a matrix OD. Moreover, we assume known passengers’ arrival rates and that passengers choose their routes based on the total cost in terms of the walking distance, in-vehicle waiting time, transfer waiting time, and the number of transfers.
For the transit network operation, we consider the following assumptions. There is a specific planning period in which trips with even headways must be provided for each line; we consider single-directed lines that start and end in different (but near) points; travel times between stops and boarding and alighting times at stops are assumed to be deterministic and known during the planning period, and there is a fleet of homogeneous vehicles where each vehicle can be assigned to only one line. Since we consider homogeneous fleet and no effects of drivers on travel times, the turnaround times for all trips of the same line are equal. These characteristics lead to evenly spaced departures and arrivals for each line at all stops. Thus, the fleet size for each line is estimated regarding its turnaround time and headway. Finally, no capacity constraints on the buses are considered; that is, all the passengers can board a bus at each stop. Notice that these assumptions can also be made in multimodal transit networks, but we refer to bus systems since we develop our approach based on the bus transit network in Santiago, Chile.
Based on the above considerations, the Bus Lines Synchronization Problem, called BLSP,
Next, we introduce the necessary elements to define our mathematical formulation which is based on time-indexed events to represent arrivals/departures of buses/passengers and to compute the costs based on travel time, waiting time at first stops, waiting time at transfer stops, walking time, and in-vehicle waiting time.
We highlight that the main elements of our formulation are the routes that can be computed with a preprocessing stage, but their cost will depend on the timetable and the frequencies, and those costs affect the route choice for passengers. As we mentioned before, a route consists of a sequence of stops (boarding/alighting) and a set of transit lines covering those stops. Figure
Route
Now, we present notation for parameters based on demand, different kind of costs, and travel times (without loss of generality, times are in minutes).
where
As we mentioned before, the main elements of our formulation are the routes and the waiting times can be computed if arrival times of passengers and buses at the beginning of trip-legs can be computed. Then, we introduce the following time-indexed auxiliary variables in order to represent arrival/departures, operational costs, and passenger costs considering the time discretization.
In the following, we detail the modeling of constraints, objective function, and the entire proposed mathematical formulation.
Our mathematical formulation is a mixed-integer nonlinear program that considers among other things discrete instants of time for arrivals of passengers to the system. Through this assumption, it was possible to formulate a model that includes neither differential (or integral) calculus nor complex transfer variables with several indexes representing ascending and descending buses and stops. This characteristic will allow us to define a simple sequential algorithm which may be impossible to define if the timetabling formulation was intractable (e.g., [
We define a nonlinear formulation for the BLSP. Thus, it is not possible to guarantee global optimality when implementing a commercial solver. Moreover, we use integer variables which makes the problem more intractable. However, if frequency setting and passenger assignment decisions are fixed, our formulation becomes a mixed-integer linear problem considering only timetabling decisions for each line, that is, departure time of the first trip and holding time at each stop. Moreover, we focus on low-frequency periods where there are few passengers. Thus, frequencies are determined based mainly on the level of service rather than passengers’ affluence; let say, the frequencies should be within a reference set.
On the basis of the above, we propose an iterative heuristic that first implements a frequency rule to determine the even headway subject to a feasible frequency set. Then, it fixes passenger-route assignments and solves the resulting (reduced) formulation for the timetabling problem with a commercial solver. This algorithm is illustrated in Figure
Solution methodology for the Bus Lines Synchronization Problem.
At the beginning of the algorithm frequencies are fixed as the ones used during system operation. If Step
The first trip departs at the beginning of the planning period and holding times are set to zero. Therefore, the departure time of the
For each origin-destination pair
Once previous steps fix the passenger assignments and frequencies, the mathematical formulation is solved implementing the solver of CPLEX to obtain new holding times and departure time for the first trip for all lines.
A new timetable is obtained by defining departure, arrival, and holding times at the optimization phase. Thus, it is possible to compute a new optimal passenger assignment. If the new optimal passenger assignment does not change compared with the previous one, the algorithm continues; otherwise, take the new passenger assignment solution and return to Step
In this final step, the frequency rule defined in Step
The previous algorithm was implemented in a transit network during the night period since relatively low frequencies define an ideal case wherein passenger transfers are needed. Moreover, assumptions of deterministic travel times and a feasible set of frequencies are also acceptable in that night period.
Step
Transantiago operated in 2012 on an area of 680 km2 with 6,298 buses. It provides service with 374 lines covering 2,766 kilometers and 11,165 stops (with an average separation of 0.4 kilometers for consecutive stop). 1,685 million transactions were performed in 2012, wherein 596 million were trips with more than one leg; that is, transfers were performed. The average number of trip-legs per business day was 3,184,289. Regarding operational infrastructure, there are many exclusive lanes.
During the night period, the system operates a subset of 60 lines with low/medium frequencies, covering 4987 stops. There is significantly lower demand, and fare evasion is common at this period leading to capturing only a part of the real OD pairs by using monitoring tools. We study night period to optimize passenger transfers through lines synchronization. Since feeder lines usually share only one or few stops with trunk lines, feeder lines may be synchronized in the system once the operation of the trunk lines is given. Then, we consider the 26 trunk lines of Transantiago covering 2855 stops (see Figure
Trunk lines of the transit network Transantiago in Santiago, Chile. White circles represent the transfer stops considered in our problem definition.
The parameter values used to compute the total generalized cost of routes and the operational costs were defined as follows:
Indeed, the number of routes increases exponentially regarding the size of the transit network. However, it is possible to reduce the set of feasible routes for a specific origin-destination pair by making reasonable assumptions such as limited length (or time) for each route, a limited number of transfers evens, avoiding particular zones of the network, among others. In this study, we consider the following assumptions to define the set of feasible routes. Origin-destination points consist of groups of stops near each other. Routes are limited to three trip-legs (maximum number of transfer events with no monetary cost due to the integrated payment system). Passengers do not transfer to lines previously used in the route. There is a reasonable walking distance for transfer events (which is obtained from data of transfer events collected by the Automatic Fare Collection (AFC)).
To perform our experimental stage, we define three instances sizes based on the number of passengers and the number of origin-destination pairs (see Table
Instances types to analyze the impact of decisions considered in the BLSP.
Instance type | Passengers | OD pairs | Total (min) |
---|---|---|---|
A | 210,000 | 83 | 150 |
B | 7,939 | 414 | 90 |
C | 237,900 | 414 | 90 |
For each one of these instance types in Table
All instances were solved in minutes of computational time by implementing our proposed methodology and it was not possible to obtain information about the global optimality due to limitations with the original formulation and the commercial solver. However, we mainly focus on the analysis of the different costs included in the objective function of BLSP compared with base scenarios in order to present the potential improvements of implementing our proposed approach. The notation is as follows: “Operation” represents the costs related to the fleet size. “Trips” indicate the in-vehicle costs for passengers during their trips. “Walk” exhibits the cost of walking in transfer events. “Holding” represents the costs for passengers caused by holding vehicles. “First” indicates the cost of waiting times to board the first bus. Finally, “Transfer wait” exhibits the costs of waiting at transfer stops.
Table
Numerical results for scenarios A.
Costs ($) | | | | | |
---|---|---|---|---|---|
Operation | 3,407,611 | 2,773,100 | 2,773,100 | | |
Trips | 68,234 | | 68,499 | | 68,308 |
Walk | 1,120 | | 1,097 | 1,120 | 1,471 |
Holding | 0 | 0 | 0 | 389 | |
First wait | 46,464 | | 19,238 | 37,422 | 26,462 |
Transfer wait | 6,125 | 3,531 | | 6,293 | 5,224 |
| |||||
Total Cost | 3,529,554 | 2,864,409 | 2,864,152 | 2,225,225 | |
Savings for scenarios of type A.
Table
Numerical results for scenarios B.
Costs ($) | | | | | |
---|---|---|---|---|---|
Operation | 1,703,806 | 1,818,961 | | 1,809,906 | |
Trips | 2,182,696 | | | 2,191,530 | 2,197,770 |
Walk | 19,636 | | | 32,852 | 42,129 |
Holding | 0 | 189,410 | 0 | | 0 |
First wait | 2,221,525 | 710,454 | 1,364,490 | | 1,181,050 |
Transfer wait | 272,466 | | 119,573 | 113,576 | 156,266 |
| |||||
Total Cost | 6,400,129 | 5,032,701 | 5,390,218 | | 5,281,034 |
Savings for scenarios of type B.
Finally, Table
Numerical results for scenarios C.
Costs ($) | | | | | | |
---|---|---|---|---|---|---|
Operation | 1,703,806 | 1,627,713 | 1,734,388 | 1,603,968 | | 1,214,534 |
Trips | 2,182,696 | | | 2,188,440 | | 2,188,990 |
Walk | 19,636 | | | 23,906 | | 26,601 |
Holding | 0 | | 240,540 | 142,510 | 367,714 | 375,586 |
First wait | 2,221,525 | | 618,411 | 412,922 | 903,477 | 602,113 |
Transfer wait | 272,466 | 71,275 | 59,561 | 52,453 | | 81,863 |
| ||||||
Total Cost | 6,400,129 | 4,441,431 | 4,855,236 | | 4,695,533 | 4,489,687 |
Savings for scenarios of type C.
We highlight that parameter values are critical in the optimization process. For example, in scenarios A using holding times does not provide benefits since there are few trips with transfers. While in scenarios B, the percentage of trips doing transfers is bigger which exhibits that holding buses could be advantageous. To study the trade-off between operating cost and the level of service, multicriteria approaches could be implemented, or parameter tuning strategies should be used in order to represent the subjective value of different measures in different travel conditions.
We have introduced the Bus Lines Synchronization Problem (BLSP), which integrates the frequency setting, demand assignment, and timetabling problems considering synchronization at transfer stops to reduce the weighted sum of passenger and operational costs. We formulated this problem as a mixed-integer nonlinear deterministic program. The proposed problem is suitable to study scenarios of low-frequency services and low travel time variability, for example, the night time where also there is often a reduction of the network size. However, our approach could be implemented to synchronize day time services if we prioritize the transfer zones based on the affluence of passengers (so we could choose only a subset of those zones) and if we only consider zones or planning periods with low variability in travel times or services operating in corridors without congestions.
Since the BLSP is challenging to solve, we develop an iterative solution algorithm that fixes frequencies and passenger routes at each iteration and then the simplified formulation is solved using a commercial solver. We implement our proposed approach on the transit network in Santiago, Chile, and numerical results show significant reductions in operating costs and improvement in the level of service of night services. Besides, our results suggest that using frequencies which are multiples of a common denominator is a good strategy to synchronize services at transfer stops to reduce waiting times.
Step
Previously reported data of costs and origin-destination matrix were used to support this study and are available at [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors are grateful for funding provided under the FONDECYT project (1110720 and 3140358). This research also benefited from the support of the Bus Rapid Transit Centre of Excellence, funded by the Volvo Research and Educational Foundations (VREF) and Centro de Desarrollo Urbano Sustentable (CEDEUS), Conicyt/Fondap/15110020.