Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7609572, 13 pages

http://dx.doi.org/10.1155/2016/7609572

## Least Expected Time Paths in Stochastic Schedule-Based Transit Networks

Faculty of Computer Science & Engineering, Ho Chi Minh City University of Technology, VNU-HCM, 268 Ly Thuong Kiet Street, Ho Chi Minh City 740500, Vietnam

Received 14 December 2015; Revised 9 February 2016; Accepted 18 February 2016

Academic Editor: Wuhong Wang

Copyright © 2016 Dang Khoa Vo 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

We consider the problem of determining a least expected time (LET) path that minimizes the number of transfers and the expected total travel time in a stochastic schedule-based transit network. A time-dependent model is proposed to represent the stochastic transit network where vehicle arrival times are fully stochastically correlated. An exact label-correcting algorithm is developed, based on a proposed dominance condition by which Bellman’s principle of optimality is valid. Experimental results, which are conducted on the Ho Chi Minh City bus network, show that the running time of the proposed algorithm is suitable for real-time operation, and the resulting LET paths are robust against uncertainty, such as unknown traffic scenarios.

#### 1. Introduction

The routing problem in a schedule-based transit network involves scheduling decisions made by a traveler, for example, accessing to a stop (station), walking between stops, waiting to board, traveling in-vehicle, alighting, and egressing. These decisions guide the traveler from an origin to a destination with minimum travel costs, such as number of transfers, total travel time, walking time, and waiting time. The decisions of the traveler are not only constrained by the network configuration, that is, transit routes (lines), but also constrained by the schedules of transit vehicles. However, due to the stochastic and time-varying nature of vehicle travel time, as well as the effects of the arrival of a transit vehicle at upstream stop on its arrivals at downstream stops, the arrival times of transit vehicles usually do not follow their schedules. Therefore, the determination of robust routing decisions can greatly affect the quality of the routing service provided under uncertain conditions.

Along with the stochasticity of vehicle travel times and the relationship between vehicle arrival times on the same transit route, there might also exist overlaps between transit routes in the network. Therefore, the arrival times of transit vehicles would be not only stochastic but also fully stochastically correlated. The routing problem with stochastically correlated link travel times has been investigated intensively in highway networks [1–6]. However, its counterpart in transit networks, where vehicle arrival times are considered as stochastically correlated, has not been addressed, while existing works in literature assumed vehicle arrival times to be deterministic [7–13] or statistically independent [14]. The main issue when designing a routing algorithm in a schedule-based transit network with correlated vehicle arrival times is to model the stochastic correlation of vehicle arrival times. This issue is related to the question of how to incorporate the correlation of vehicle arrival times into the routing process, in which not only constraints on transit routes but also constraints on vehicle arrival times are taken into account:(i)A time-dependent model is proposed for stochastic schedule-based transit networks where the correlation of all vehicle arrival times is presented as a scenario. The graph model captures travelers’ decisions, namely, boarding, traveling in-vehicle, alighting, walking, and time constraints of these decisions in each scenario.(ii)A new dominance condition for paths is established with respect to number of transfers and travel times over a set of possible scenarios. Then a formal proof that Bellman’s principle of optimality is valid with nondominated paths is presented. This theoretical establishment enables the use of pretrip online information to reduce uncertainties for more robust LET paths.(iii)An exact link-based routing algorithm is proposed for efficiently determining LET paths, based on Bellman’s principle of optimality. The results from experiments, which are conducted using data from a real-size bus network in Ho Chi Minh City, show that the running time of the proposed algorithm is feasible for online applications. Also, LET paths are shown to be robust in the presence of unknown scenarios.

The remaining of the paper is organized as follows. We present related researches on the routing problem in transit networks in Section 2. In Section 3, we define components used to develop the algorithm for our routing problem. Then we propose the solution algorithm for determining the LET path in Section 4. Various experiments are conducted, and their results are discussed in Section 5. Finally, the conclusion is given in Section 6.

#### 2. Related Work

A transit routing algorithm in literature has been built on the notion of* path* [7, 8, 15] or* hyperpath* [16–18]. A path consists of fixed decisions made by a traveler at stops, which are determined before he/she leaves the origin. In contrast, a hyperpath represents routing strategies in which the traveler is allowed to change his/her decision at each intermediate stop, depending on the previous decisions and what are likely to happen in the future. Routing based on hyperpath was shown to make better travel costs under uncertainty but requires the incorporation of online information and high computational complexity [19].

Treatment for the routing problem in a transit network can be different, depending on the type of transit services, that is, either* headway-based* [15–17] or* schedule-based* [7, 8, 11, 12]. In the former, transit services are represented by transit routes, and arrival/departure times of transit vehicles are not explicitly considered. This results in an approximation in calculating boarding times and in-vehicle travel times. In the latter, transit services are explicitly specified in terms of trips (runs), in which arrival/departure times of transit vehicles at stops are considered. Meanwhile the routing algorithm in a headway-based transit network can employ shortest path algorithms, for example, Dijkstra’s algorithm [20], which are the same as those used for highway networks. A schedule-based transit network requires a time-dependent network presentation where routing processes of travelers are not only constrained by the network topology but also constrained by scheduled arrival/departure times of transit vehicles. Therefore, modeling transit services is the first and important task in solving the routing problem in a schedule-based transit network. As classified by Nuzzolo and Crisalli [21], the representation of a schedule-based transit network can be one of the three forms: the diachronic (time-expanded) network [9, 10, 13], the dual network [22], and the mixed line-based/database supply model (time-dependent model) [11, 23, 24].

In the context where transit services are insufficiently reliable, headways and arrival/departure times of transit vehicles are commonly modeled as random variables with well-known forms of probability distribution, for example, exponentially distributed headways [25, 26], Gaussian distributed headways [27, 28], and Gaussian distributed scheduled times [14]. Along with the stochasticity of transit services, the uncertainty in travelers’ perceptions on different types of travel costs can be also regarded as a source of stochasticity in a transit routing problem [8, 27, 28]. In these works, random weights for different travel cost components, such as transfer penalty, walking time, and waiting time, were incorporated into the routing process.

The routing problem in transit networks has been investigated with various assumptions on many aspects, such as capacity limitation, congestion and overcrowding issues, vehicle capacity, and boarding failures [29]. Nuzzolo and Crisalli [21] investigated various routing models for* low-* and* high-frequency* schedule-based transit networks. In the former, for example, in regional bus or railway networks, routing processes are based on arrival/departure times of transit vehicles [30, 31]. In the latter, typically in urban areas, travelers usually have a large number of options at stops to reach their destination. In this case, arrivals of travelers at stops do not rely heavily on vehicle arrival/departure times but are significantly affected by vehicle congestion, which are defined in literature as situations in which a traveler cannot board the first arriving vehicle and has to wait for next vehicles. Vehicle congestion can be modeled implicitly as increasing discomfort functions [11, 12, 32] or explicitly with vehicle capacity or set availability constraints [33–35].

#### 3. Network Modeling

In this section, we define components used to develop the algorithm for determining LET paths in stochastic schedule-based transit networks.

##### 3.1. Stochastic Schedule-Based Transit Network

We consider transit network , where is the set of stops and is the set of routes. A route, , is a fixed sequence of stops through which transit vehicles run periodically with fixed trips and defined by a set where is the th stop and is the number of stops on route . Let be the number of trips of route over a set of time intervals , where is unit of time and is the last time interval. The universal stochastic scenario set is a set of all known possible scenarios in the network such that where is the occurrence probability of scenario . Each scenario can be defined by a set of stop timeswhere denotes the stop time (scheduled arrival time of a transit vehicle) at the th stop of the th trip on route in scenario and is the universal set of all stop times in all possible scenarios such that

In the context of transit networks, there might exist overlaps among routes. A scenario presents a stochastic correlation of not only stop times on the same route but also stop times on routes sharing the same physical links. The probability of a scenario happening is the full joint probability of all stop times taking place, and stop times are known for each scenario a priori. This allows us explicitly to take into account delays resulting from transfer failures due to late arrivals and their effects on the total travel time in each scenario.

For example, consider the transit network shown in Figure 1 with and . In this network, there are three routes in which routes and provide services from stop to stop with and , and route provides services from stop to stop with and . The stop times of transit vehicles in the network are shown in Table 1 with , , and in which each of the routes has two trips and each scenario has an occurrence probability of .