Journal of Advanced Transportation

Volume 2019, Article ID 9408595, 13 pages

https://doi.org/10.1155/2019/9408595

## Integrating Frequency Setting, Timetabling, and Route Assignment to Synchronize Transit Lines

^{1}Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, N.L. 66455, Mexico^{2}Departamento de Ingeniería de Transporte y Logística, Pontificia Universidad Católica de Chile, Santiago, 4860, Chile

Correspondence should be addressed to Omar J. Ibarra-Rojas; xm.ude.lnau@jrarrabi.ramo

Received 3 November 2018; Accepted 19 March 2019; Published 7 April 2019

Academic Editor: Rocío de Oña

Copyright © 2019 Omar J. Ibarra-Rojas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. As it can be seen in the study of Sivakumaran et al. [2], the set of frequencies of the network lines affect the level of synchronization that can be achieved. Thus, it seems reasonable to optimize timetables and frequencies simultaneously.

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* et al.* [3] and Hall* et al.* [4] define optimization problems to determine holding time of buses at different stops with the objective of minimizing waiting time at transfer stops among other criteria. Those problems must be solved in a real-time framework using monitoring tools such as Geographical Positioning Systems (GPS) and Automatic Passengers Counting (APC) systems to define the input. Moreover, holdings could be implemented at the planning stage to guarantee that all transferring passengers can board their next bus or reduce waiting times of boarding the first bus. The drawback of adding these holding times is the increment of in-vehicle waiting times, longer cycle times, and therefore a larger fleet size needed to provide the service. Thus, there is a clear trade-off between transferring passengers experience, in-vehicle passengers experience, and operational costs. The waiting time subjective value at stops is much higher compared to the in-vehicle waiting time. Under poor waiting time conditions (e.g., during the night, poor weather conditions, or a threatening urban context), this difference increases significantly, while under low-frequency operation, the expected waiting after a missed bus is high as it can be seen in Boardman et al. [5].

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* et al.* [6]) and should be considered in the bus scheduling process.

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.

#### 2. Related Literature

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 [7] present two approaches for the problem of synchronizing different transit lines: (i) trying to minimize passenger waiting times at transfer zones and (ii) trying to maximize the number of synchronization events at transfer zones. After that, several investigations have been conducted following both directions mainly by solving frequency setting and timetabling problems, as it can be seen in literature reviews of Desaulniers* et al.* [6], Guihaire and Hao [8], and Ibarra-Rojas,* et al.* [9].

The synchronization of different lines at a single stop has been addressed by Chakroborty* et al.* [10], wherein the authors define a mixed integer nonlinear program to schedule trips minimizing the total travel time and solve the proposed optimization problem through genetic algorithms. In the latter work, the authors assume deterministic travel times, random passenger arrivals, and a trunk and feeder network to determine a periodic timetable; that is, there are evenly spaced arrivals for each line at the transfer stop (holding buses is allowed) leading to cyclic timetables with respect to a period of* T *minutes. To maximize the number of pairwise simultaneous arrivals at multiple synchronization stops, Ceder* et al.* [11] define an aperiodic timetabling problem (i.e., there are heterogeneous headways) considering a given number of trips per period (frequency) that must be scheduled and bounds for headways of each line. The authors define an integer linear formulation that becomes intractable by commercial solvers. Then, the authors design a heuristic procedure to generate timetables and this method is tested on several example networks. The latter study is enhanced by Liu* et al.* [12] using a synchronization measure that comprises the ratio of the number of lines where vehicles are arriving simultaneously at a connection stop to the number of all lines passing through the same stop. Then, a nesting tabu search that implements Ceder’s heuristic is designed to obtain feasible solutions for small instances (up to eight lines and three synchronization nodes). Eranki [13] also redefines the model in [11] by redefining synchronization as the event of two trips belonging to different lines that are arriving at a common stop with a separation time within a specific time interval. Then, the algorithm proposed in [11] is adapted to solve the proposed model.

More recently, Ibarra-Rojas and Rios-Solis [1] enhance the aperiodic timetabling problem proposed in [13], considering oriented transfers; that is, passengers may transfer from one line* i* to a line* j* but not necessarily vice versa. Synchronization events are also used to reduce the congestion of buses belonging to different lines at common route segments. The authors indicate that the problem is intractable by commercial solvers and prove that their synchronization timetabling problem for more than three lines at common stops is NP-hard (by defining a polynomial reduction to the Not All Equal 3-Satisfiability problem, NAE-3SAT). Moreover, they develop a preprocessing stage and an Iterated Local Search to generate feasible solutions for large instances. Later, Ibarra-Rojas* et al.* [14] extend the previous study considering multiple planning periods (which may be different for each line) to define synchronization events between trips belonging to different periods and be able to determine a timetable for the entire day. The size of the problem increases considerably when an entire day is optimized. Thus, the authors implement metaheuristics, such as Multi-start Iterated Local Searches and Variable Neighborhood Searches. Indeed, the synchronization events between trips belonging to different planning periods are relevant as it is stated by the study of Ning* et al. *[15] which addresses an optimization problem to reduce transfer waiting time and transfer availability with respect to the first or last train. Finally, Wu et al. [16] enhance the problem of [1] to a biobjective version that optimizes the number of passengers benefited with synchronization and the deviation from an initial timetable. The authors implement a BRKGA II to approximate the Pareto frontier. Although previous studies are flexible enough to consider walking times and the proposed approaches are capable of solving large instances, the authors consider neither the passenger assignment decisions nor holding times.

Integration of timetabling and frequency setting has been addressed in the works of Bookbinder and Désilets [7] and Klemt and Stemme [17]. These studies focus on the minimization of transfer costs optimizing the first departure time and the even headway for each line. Based on these two decisions, the remaining departure times are computed. Bookbinder and Désilets [7] consider variability in bus travel times; thus, there are evenly spaced departures at the origin point but heterogeneous headways at stops (due to travel time variability). The synchronization impact of each timetable is evaluated through simulation. Numerical results indicate that the higher the headway times and the lower the travel time variability, the higher the synchronization benefits. Ting and Schonfeld [18] consider a transit network with multiple lines and transfer stops. Although walking times are ignored, holding times at certain stops are considered. The authors determine an optimal even headway for each line in which headways must be multiple of a given reference. The authors use a stochastic model in which travel times and passenger arrivals are recognized as random inputs.

The synchronization is also of interest in railway-based transit systems to optimize the transfer conditions for passengers. For example, Gou* et al.* [19] address a synchronization timetabling problem in metro systems with particular emphasis in transitional periods (from peak to off-peak hours or vice versa) during which train headway changes and passenger travel demand varies significantly. In the case of intermodal transport systems, Li* et al.* [20] study the transit scheduling problem to optimize the interaction of different services at an intermodal transport network considering demand variability and therefore variable headways. More recently, Guo* et al.* [21] focus on synchronization between train lines and bus services to minimize the total connection time, in particular, for passengers transferring from the first train service to bus lines.

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 [9]). However, to the best of our knowledge, none of these approaches simultaneously consider a time-dependent origin-destination matrix, a transit network composed of several lines, passengers choosing the fastest route to reach their destination, and walking/waiting time needed to transfer from one line to another at transfer points. In this paper, we present an optimization problem that considers all these characteristics. Our significant contributions are the following: (i) an integrated optimization problem for frequency setting, timetabling, and passenger assignment problem to minimize sum of passenger and operation costs; (ii) a mathematical formulation for the proposed optimization problem based on time-indexed variables which avoid differential/integral calculus; and (iii) a heuristic approach to obtain solutions for real-size instances defining a tool for the decision-making process in transit network planning.

#### 3. Materials and Methods

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.

##### 3.1. Problem Definition

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,* determines for a single planning period the frequency for each line (in terms of the even headway), the passenger assignment, and the timetable (in terms of departure time of the first trip of each line and holding times at each stop). The objective is to minimize the total cost based on the walking distance, waiting times to take the first bus and to perform a transfer, the in-vehicle waiting time, a penalization for the number of trip-legs, and operational costs*.

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.

##### 3.2. Mathematical Formulation

*Sets and Parameters*. To design our mathematical formulation for the BLSP, we consider the following sets regarding the routes in the transit network and a discretization of the planning period.(i)*N*: the set of stops in the transit network.(ii)*L*: the set of lines where each line* l* consists of a sequence of stops that should be covered by each trip of the line.(iii): the set of stops covered by line* l*.(iv): the set of arcs of line* l,* that is, the route segments defined by consecutive stops and covered by line* l*.(v): the set of arcs of line* l*, from its first stop until stop* i* in .(vi): the set of all possible routes for all origin-destination pairs.(vii): set of routes that passenger may choose to travel from origin* i* to destination* j*. We highlight that a route* p* may consist of several trip-legs (route segments covered by different lines) which should be connected with a transfer event.(viii): the set of lines covered by route* p*.(ix): the set of boarding stops covered by route* p*. Each boarding stop represents the first stop of a specific trip-leg of route* p*.(x): the set of arcs covered by route* p* starting from boarding stop* i* to the next alighting stop.(xi): the set of instants of times where passenger arrivals may occur (discretization of the planning period).

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 1 shows an example of a route* p *in to travel from stop to stop . The sets of stops and lines covered by route* p* are and , respectively. Notice that route* p* consists of two trip-legs: a trip-leg from* i* to* c* (using line* l*) and a trip-leg from* d* to* j* (using line ). Hence, route* p* leads to passengers walking from alighting stop* c* to boarding stop* d* in order to transfer from* l* to (we represent all transfers events by using two stops).