Reliable shortest path finding in stochastic time-dependent road network with spatial-temporal link correlations: A case study from Beijing
Introduction
Path navigation is one of important auxiliary services for transportation. In the last few years, benefited from the rapid development of communication as well as the cloud computing technology, much attention has been paid to the evolution of vehicle navigation systems. It is well known that vehicle navigation systems can assist people in making smart routing choices as well as help improve the traffic status of the whole road network, such as for routing recommendation and parking message propaganda (Li, Huang, & Lam, 2012). So far, plenty of researches have been devoted to ameliorating route guidance algorithms and the optimization of vehicle navigation systems (Huang, Wu, & Zhan, 2007; Nie, Wu, Dillenburg, & Nelson, 2012; Yang & Zhou, 2014; Li, Chen, Wang, & Lam, 2015).
Most previous studies investigated path-finding algorithms in the deterministic (Li et al., 2012; Schultes, 2007) or static (Chen, Lam, Sumalee, & Li, 2012; Chen, Tong, Lu, & Wang, 2018b) network. Given the origin–destination (OD) pair, the optimal path is confirmed by minimizing the distance or average travel time. However, travel times in subsistent road networks are usually unsteady due to both supply disruptions (e.g., traffic control and accidents) and demand fluctuations (e.g., traffic jam). If the random and time-dependent characteristic of travel times were not involved in routing guidance, the unreliable paths may be recommended to travelers, which result in late arrival. Thus, stochastic and time-dependent networks may be a more appropriate expression form of the practical road networks contrasted with a deterministic or static one. Moreover, in route guidance problems, travelers concern not only economizing travel times but also travel time reliability (Brownstone, Ghosh, Golob, Kazimi, & Amelsfort, 2003; Wu & Nie, 2011; Srinivasan, Prakash, & Seshadri, 2014). Given different trip purposes, the TTR requirements are completely different for travelers (i.e., risk-averse, risk-neutral, and risk-taking). For instance, if a traveler wants to take an interview on-time, he or she may become risk-averse. Thus, travelers’ TTR requirements should be incorporated in stochastic reliable shortest path problems. To date, several kinds of criteria have been generally used in empirical studies to measure TTR, such as on-time arrival feasibility (Nie et al., 2012; Srinivasan et al., 2014; Frank, 1969; Nie & Wu, 2009; Yang & Zhou, 2017), and α-reliable travel time (Chen & Ji, 2005; Ji, Yong, & Chen, 2011; Chen et al., 2013; Zeng, Miwa, Wakita, & Morikawa, 2015; Chen, Li, & Lam, 2016b). Shahabi, Unnikrishnan and Boyles (2013) developed an outer approximation algorithm for the stochastic shortest path problem, in which path costs are determined as a weighted sum of expected cost and cost standard deviation. It is noteworthy that the cumulative distribution function (CDF) of travel times is the foundation of majority of these criteria. Thus, a basic problem for travel time reliability assessment as well as reliable shortest path-finding problems is calculating path TTD.
The spatio-temporal link correlations can be embodied by the phenomenon that the traffic congestions on a link may result in congestion in upstream links and the link may also be in a congested state for a while. Hence, link correlations have a significant influence on TTR (Zeng et al., 2015; Zhang, Shen, & Song, 2017) and should be taken into account in path TTD calculation (Chen, Yu, Chen, & Wang, 2017). The quantitative methods for link correlations usually can be classified into two groups: (1) Markov chains (Ma, Koutsopoulos, Ferreira, & Mesbah, 2017; Ramezani & Geroliminis, 2012) and (2) correlation coefficients (Nie et al., 2012; Chen et al., 2012; Lei, Wang, Lu, & Sun, 2014; Chen, Lam, & Li, 2016a). Ma et al. (2017) and Ramezani and Geroliminis (2012) drew two-dimensional Markov chain diagrams revealing TTDs of adjacent links and applied cluster analysis approaches to determine the travel time distributions of successive links before establishing path travel time distribution. A Markov chain is more suitable to describe the correlation between neighboring links and obtain discrete TTDs; however, continuous probability density function (PDF) or CDF of path travel times failed to be expediently embodied. The Monte-Carlo simulation (MCS)-based approaches have been developed to deal with two reliability-based optimal path finding problems, i.e., the shortest path problem with on-time arrival probability and the minimum travel time budget path problem by accounting for network wide link travel time correlations (Zockaie, Nie, Wu, & Mahmassani, 2013; Zockaie, Nie, & Mahmassani, 2014; Zockaie, Mahmassani, & Nie, 2016). As stated in Fakhrmoosavi, Zockaie, Abdelghany and Hashemi (2019), the computation burden may be the main challenges of most MCS-based approaches for solving path finding problems. Zeng et al. (2015) used coefficients to quantify the link correlations, and they utilized the Lagrangian relaxation method to solve α-reliable path searching problem in stochastic and static networks. In another study, Lagrangian relaxation approach was also utilized to settle RSP problems with link correlations (Zhang et al., 2017). Nevertheless, the assumption that the travel times obey a static normal distribution is generally a foundation of these researches. Srinivasan et al. (2014) indicated that although normality assumption simplifies complicated computational and analytical, the normal distribution imposes impractical restrictions. The lognormal distribution seems to be a more suitable option to represent the natures of travel times (Chen, Shi, Zhang, Lam, Li, & Xiang). However, the computational complexity will be greatly increased when evaluating the path TTR due to taking the stochastic property of travel times and link correlations into account, which often results in non-additive and non-linear conundrums. Thus, an efficient approach of quantifying path travel time reliability with spatio-temporal link correlations and a homologous RSP searching algorithm is supposed to be developed. However, limited research on path finding problems applied actual traffic data due to the challenge in acquiring the travel time information in large actual road networks, which is probably defective in RSP finding problems.
Stochastic path finding approaches afford either prior path navigation (Nie et al., 2012; Chen et al., 2018b; Zeng et al., 2015) or adaptive guidance (Fan, Kalaba, & Moore, 2005; Huang & Gao, 2012; Prakash, 2018) to travelers. According to evaluation indices of the optimal reliable shortest path, previous stochastic path finding approaches can be roughly divided into three classifications: (1) the least expected travel time path finding approach (Hall, 1986; Chen, Yin, & Sun, 2014), (2) the most reliable shortest path finding approach (Nie et al., 2012; Frank, 1969; Nie & Wu, 2009; Nie & Fan, 2006; Xing & Zhou, 2011), and (3) the α-reliable path finding approach (Chen & Ji, 2005; Chen et al., 2013). Hall (1986) explored the least expected travel path searching problem on STD networks for the first time. Nevertheless, the travel time reliability requirement according to different trip purposes failed to satisfy. The target of the most reliable shortest path approach is to maximize the feasibility of arrival in a predefined time limit (Nie et al., 2012; Frank, 1969; Nie & Wu, 2009). Frank (1969) raised the construct of the most reliable path that maximizes the feasibility of arriving at the bourn inside a pre-specified time limit. Xing and Zhou (2011) developed two models for the most reliable shortest path problem with and without link travel time correlations, respectively, where path travel time variability is indicated by its standard deviation. Nie et al. (2012) solved RSP problem in a static and stochastic network, searching the path with the maximum potential of arriving the destination punctually. However, travelers may prefer to choose a requirement for TTR rather than setting a reasonable time limit. The target of the α-reliable path searching approach is to find the path with minimum travel time under a given on-time arrival feasibility (Chen & Ji, 2005; Chen et al., 2013). Chen and Ji (2005) found the α-reliable path whose travel time budget was least guaranteeing the specific probability of punctual arrival. Nevertheless, majority of these studies neglected the spatio-temporal correlations among the links.
Due to the non-additive and non-linear objective functions, a handful of efficient path finding algorithms are presented to figure out RSP problem in stochastic network, containing (1) the non-dominance-based approaches (Chen, Lam, & Li, 2016; Chen, Li, & Lam, 2016), (2) the genetic algorithm (Ji et al., 2011; Liao & Zheng, 2018; Rajabi-Bahaabadi, Shariat-Mohaymany, Babaei, & Chang, 2015), and (3) the Lagrangian relaxation methods (Yang & Zhou, 2017; Zeng et al., 2015; Zhang et al., 2017; Li et al., 2017). Chen et al. (2016a) presented a multi-criteria A* algorithm to search the α-reliable path, which was based on three kinds of effective dominance methods. Ji et al. (2011) came up with the simulation-based multi-objective genetic algorithm to search reliable paths in the stochastic road networks in which the links are spatial correlated. Nevertheless, the simulation procedure is time-consuming especially when required high accuracy. Rajabi-Bahaabadi et al. (2015) proposed a multi-objective path-finding problem to search the best path in stochastic and time-dependent road networks with the help of a non-dominated sorting genetic algorithm. However, the correlation among the link travel times were not considered. Liao and Zheng (2018) proposed a hybrid heuristic algorithm to recommend the personalized routing guidance in a stochastic and time-dependent environment. However, the nature of practical travel times may not conform to static normal distributions. Thus, a more suitable distribution model should be applied into RSP searching problems.
In this study, we concentrate on determining the α-reliable shortest path in a stochastic and time-dependent network with spatial-temporal correlations under a reliability preference pre-specified by travelers. The reliable shortest path finding problem has become a hot issue of research. Nevertheless, when calculating path travel time reliability, existing studies either base on several strict restrictions, or cost a large amount of calculation times. In this study, we try to pad these vacancies by presenting a dynamic moment-matching-based variant A* algorithm STCRSP-DMA* as well as carrying out performance analyses of the algorithm in an actual large road network. The major contributions of this study can be summed up as follows.
- (1)
Practical travel time data collected by probe vehicles are used to calculate link and path travel time reliability. In this study, link travel times’ probability density functions are expressed in form of lognormal distributions, which are better than normal distributions in the aspect of fitting goodness. Then, spatial-temporal correlations between links are quantified by correlation coefficient. The influence of spatial distance, temporal distance and road type on link correlation is calculated and scrutinized.
- (2)
A dynamic moment-matching method (DMM) is employed to compute approximate path TTD parameters, which guarantees the calculation accuracy and efficiency at the same time. And the estimated time of arrival (ETA) for travelers can be calculated efficiently.
- (3)
A DMM-based variant A* algorithm STCRSP-DMA* is proposed to search the RSP in a stochastic and time-dependent network with spatial-temporal correlations. The spatio-temporal correlations between travel times of each link pair included in the path will be calculated when determining the path TTD. The Manhattan distance between one node and the destination is selected as a part of the heuristic function of STCRSP-DMA*. An extensive case study using travel time data collected from a realistic road network in Beijing is executed. The numerical results reveal that STCRSP-DMA* has desirable computation efficiency and search ability.
The remainder of this study is organized as follows. In Section 2, the reliable shortest path in a stochastic and time-dependent network is formulated. Section 3 presents the STCRSP-DMA*, and Section 4 carries out numerical experiments. Finally, conclusions of this study are made in Section 5.
Section snippets
Problem statement
Concentrate on a stochastic and time-dependent network G = (N, A, ϑ) where N (|N| = n) is a group of points, A (|A| = m) is a group of links and ϑ is the period considered. According to pervious work (Prakash, 2018; Rajabi-Bahaabadi et al., 2015; Miller-Hooks & Mahmassani, 2000), the period ϑ is discretized into time intervals represented by ϑ = {0, Δ1,Δ2,…, Δk = k × Δ, …}, where k is a natural number, and Δ is the breadth of an interval. Each point i has a group of successor points ϕ(i),
Solution algorithm
In this study, STCRSP-DMA* is designed to search the reliable shortest path in a stochastic and time-dependent network considering spatial-temporal correlations (STC) of links. Similar to traditional A∗ algorithm (Zeng & Church, 2009), the STCRSP-DMA* algorithm uses a heuristic evaluation function F(i) = G(i) + h(i) as a tag for point i ∈ N. In this function, the G(i) is a travel time budget estimated value from origin r to point i ∈ N, and G(i) = 0 at the origin, h(i) is a travel time
Overview of tested network
To check out the proposed STCRSP-DMA* path searching algorithm, a subsistent road network in Beijing, China, is chosen for case studies in this section. The network consists of 218 nodes and 744 links. In this study, since the data source we can obtain only includes floating car data, we assume that the floating car data can better characterize the running status of the road network. Based on the floating car data of two months, i.e., June and July in 2015, the link travel times can be
Conclusions
This study aims to solve reliable shortest path (RSP) finding problem in stochastic and time-dependent networks considering the spatial-temporal correlations (STC) of links for different travelers. A dynamic-moment-matching-based A* algorithm (STCRSP-DMA*) was put forward to solve the reliable shortest path finding conundrum. STCRSP-DMA* provides a practical routing guidance approach because travelers are able to acquire the α-reliable optimal path RSP easily by setting their travel time
CRediT authorship contribution statement
Peng Chen: Conceptualization, Data curation, Methodology, Validation, Writing - original draft. Rui Tong: Conceptualization, Data curation, Methodology, Validation, Writing - original draft. Bin Yu: Conceptualization, Methodology, Validation, Writing - original draft. Yunpeng Wang: Conceptualization, Methodology, Validation, Writing - original draft.
Declaration of Competing Interest
None.
Acknowledgments
The authors appreciate the National Key R&D Program of China (2018YFB1600500) for support of this research.
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