A Multimodal Hierarchical-based Assignment Model for Integrated Transportation Networks

JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 9, Issue 6, December 2009 Online English edition of the Chinese language journal RESEARCH PAPER

Cite this article as: J Transpn Sys Eng & IT, 2009, 9(6), 130í135.

A Multimodal Hierarchical-based Assignment Model for Integrated Transportation Networks CHEN Shaokuan*, PENG Hongqin, LIU Shuang, YANG Yuanzhou Integrated Transport Research Center of China, School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

Abstract:

A hierarchical network and traffic assignment model is developed for multimodal transportation networks to meet the

prediction of large-scale transportation demand in this paper. In the proposed model, the city center and transportation terminal of different modes are defined as central nodes and transfer nodes respectively to characterize travel behavior of passengers. The generalized cost function and route choice algorithm are also correspondingly improved to suit the proposed model. A computer-aided demand model is also developed to efficiently attain the analysis results for large-scale transportation networks. The applicability and availability of the demand model is illustrated by the case studies over Jing-Hu high speed railway in China. Key Words: multimodal transportation; assignment model; route choice algorithm

1

Introduction

Transportation networks in China have been developed during recent decades with the rapid growth of economy. The overload of different transportation modes (railway, highway, civil aviation etc) has occurred more frequently especially on traditional holidays, such as Spring Festival, National Day and Labor Day. Thus, it has played an important role to provide more available supports for planning and operating multimodal transportation networks by computer-aided decision systems. Relevant researches focused on the methodologies for the optimization and assessment of multimodal transportation system and their application. Pallottino proposed a hyper-path shortest path model and a searching algorithm for multimodal transportation networks[1]. Lozano and Storchi also provided a shortest viable hyper-path algorithm for multimodal transportation networks[2,3]. Except for the shortest path searching algorithms, travelers’ transfer behavior and route choice models should be considered together in multimodal transportation networks. Lo et al. developed a formulation considering transfer characteristics and non-linear fare structure in multimodal transportation networks and transformed a multimodal network to a state-augmented multimodal network for simplifying traffic assignment and

analysis procedures[4]. In addition, much work was recently carried out in the application of exiting studies for multimodal transportation networks, including scheduling transportation systems[5], modeling interregional freight transport in European[6], pricing and rationing policies[7], assessing economic impacts from unexpected events[8], validating a travel simulator with travel information provision[9], accessing to public transport with a hierarchical multimodal choice model[10] and optimizing multimodal transportation of urban freeway corridors[11]. This paper presents a computer-aided demand analysis model for multimodal transportation networks. A hierarchical topology is developed to characterize multimodal transportation networks and then generalized cost functions and route choice algorithms are improved for matching to the proposed topology in this paper. The new demand model for multimodal transportation networks is crucial and necessary for operators and engineers to analyze the travel demand of integrated transportation in large-scale networks.

2

Hierarchical multimodal networks

2.1 Modeling network A transportation network is usually proposed through a traditional graph theory. For the convenience of description, a transportation network is defined as a graph G=(V, A), where

Received date: Jul 13, 2009; Revised date: Oct 21, 2009; Accepted date: Oct 29, 2009 *Corresponding author. E-mail: [email protected] Copyright © 2009, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(08)60093-8

CHEN Shaokuan et al. / J Transpn Sys Eng & IT, 2009, 9(6), 130í135 vi1

v21

v11

v31

v20

v10

vi 0

v30

v22 vi 2

v12

v32

v2 k vik

v1k

only in transportation mode layers; otherwise, they will happen in both the central modal layer and transportation mode layers. 2.2 Generalized travel cost Generalized travel cost is employed to represent the total travel consumption of passengers which is nonlinearly comprised by travel cost, time, comfort and safety, etc. For convenience, linear functions are used to calculate generalized travel cost in practice. One of them is often expressed as follows. Cijm b0m tijm ( xijm )  ¦ blm d ijlm (1) l

m

v3 k

Fig. 1 An example of the topology of HMMTN

V={v1, v2, v3,···, vn} denotes a set of nodes, A={a1, a2, a3,···, am} denotes a set of arcs, vi is a node which represents a transportation hub or transfer, al denotes an arc of two adjacent nodes (vi, vj), V n is the node number of the graph G, and A m denotes the arc number of the graph G. Actually, the topology of multimodal transportation networks (MMTN) is more complex than that of single-modal transportation networks (SMTN). A hierarchical network model will be proposed in this paper referring to the Ref. [3]. An MMTN G=(V, E) is developed by k modal sub-network Gb=(Vb, Eb), where V, E, respectively are the sets of physical nodes and links, b  B represents a transportation mode, B is the set of transportation modes, k B is the number of transportation modes, and Vb V , Eb  E respectively are the sets of nodes and links in the sub-network Gb. A sub-network G0 is defined as the central modal network which represents the center of cities connecting to the hubs of different transport modals. In the proposed model, the node vi is usually divided into different nodes vi0 which denotes the node of central modal and vi1, vi2,···, vik which denote the nodes of traffic modals. Thus, the links between central nodes and traffic nodes represent transfer, whilst the links between traffic nodes figure different traffic modes, e.g. highway, railway and aviation. The topology of the proposed multimodal transportation network in this paper is shown in Fig. 1. An MMTN G=(V, E) is be transformed into an HMMTN GH=(VH, EH) as shown in Fig. 1. In an HMMTN, a transportation network is usually separated into k+1 layers. The 0th layer describes the central modal layer and the 1st, 2nd, 3rd,···, kth layers describe the transportation mode layers. Different transportation mode layers are connected with a transportation center layer. Transportation demand occurs or disappears only in the central modal layer. If the transportation mode is not changed, route choices will happen

where Cij represents the generalized cost on the mth transportation mode between the adjacent nodes i and j, tijm ( x mij ) is the travel time function on the mth transportation m mode between the adjacent nodes i and j, xij is the transportation volume on the mth transportation mode between m adjacent nodes i and j, d ijl is the lth contribution to generalized cost on the mth transportation mode between adjacent nodes i and j, b0m is the value of time on the mth transportation mode, blm is the unit cost by the distance of the lth contribution to generalized cost on the mth transportation mode. The generalized travel cost function in the proposed method is simply defined as follows. (2) Cijm b0m tijm ( xijm )  Pijm  Fijm m

where Pij denotes the transportation cost on the mth m transportation mode between adjacent nodes i and j, Fij is the penalty of other influences except for travel time and cost. m For railway or aviation systems, travel time tij ( xij ) is basically a constant value with transportation volume because the travels of trains or airplanes usually refer to predefined timetables. For highway systems, let tij ( xij ) Lij / Vij , where Vij is the vehicle speed and usually calculated through equations given by China Transportation Planning and Research Institute, for example, 2 ª § xij · º ¨ ¸ « » Vij 80.14 u exp  0.173 ¨C ¸ » « ij ¹ © ¬ ¼ is developed for freeway when x ij / C ij  0.8 , where Cij is the capacity between adjacent nodes i and j. The total generalized travel cost is proposed to depict the comprehensive impact of the total costs from different transportation modes for integrated transportation demand. Two methods are employed to calculate the total generalized travel cost in this paper. One is a weighted function Cij ¦ Om Cijm m

where Ȝm is the weight of the generalized cost on the mth transportation mode and ¦ O 1The other is given by 1 Cij  ¦ exp(TCijm ) T m m

m

CHEN Shaokuan et al. / J Transpn Sys Eng & IT, 2009, 9(6), 130í135 5 2

6

1

7 3

8

assumptions about utilities: First, individuals always select the choice option with maximal utility, and second, each choice option of individuals depends on the characteristics of individuals and the choice options. Logit model is usually used to calculate the probability of the two choice options at least and it is formulated as Pj

exp(DU j ) J

¦ exp(DU ) i

4

9

10

Fig. 2 Topology of an example network

from behavior choice theory, where ș is a disperse parameter associated with travel distance which is estimated through the least square estimation. The former method is widely used but it is difficult to confirm the weight for different transportation modes. The latter is more suitable for practical engineering but a great of filed data is necessary to calculate the disperse parameter ș.

3

Route choice algorithm

i

where Pj denotes the probability of the jth choice option, J denotes a set of choice options, Į is a parameter. If the utility is represented by the negative value of the generalized costs, the model is expressed as Pj

exp(DC j ) J

¦ exp(DC ) i

i

where Ci represents the generalized cost of the ith path in HMMTN. Wang et al. proposed an improved Logit model and redefined the valid paths[12]. The improved Logit model is described as P (i, j , k )

exp[  T ˜ w ( k ) / w ] m

¦

exp[  T ˜ w ( l ) / w ]

l

Route choice algorithms are crucial algorithms for transportation assignment, which include the shortest path, stochastic path choice and valid path searching algorithms. 3.1 Shortest path algorithm A route choice algorithm is usually developed from shortest-path algorithms. Dijkstra and Floyd-Warshall algorithms are the two of the most well-known algorithms for their prevalent applications. Dijkstra algorithm is used to find the shortest paths between single origin and multi destinations. Floyd-Warshall algorithm is employed for the shortest paths between multi origins and multi destinations. The time complexities of the two algorithms are both equal to O(n3). The k-shortest path algorithm is used in the proposed methods. The k-shortest path algorithm is employed to compute the shortest path set involving from the first to the kth shortest path between nodes in transportation networks. The time complexity of the k-shortest path algorithm is also k·O(n3). 3.2 Stochastic path choice algorithm The stochastic path choice algorithms are developed to calculate the probability of path choices from the stochastic utility theory. The utility, the inverse of generalized cost mentioned in value, is considered as the degree of satisfaction to the choice option depending on the characteristics of transportation supply or service, e.g. social and economic factors. The utility, Ui=Vi+İi, is usually figured as stochastic variables, where Ui represents the utility of the ith choice option, Vi represents the certain term of utilities, İi represents the stochastic term of utilities. There are two essential

where P(i, j, k) denotes the probability of transportation demand assigned on the kth valid path, w(k) denotes the generalized cost of the kth valid path, w denotes the average generalized cost of all valid paths, ș is a parameter, m is the number of valid paths. The parameter ș indicates the sensitivity of the difference of generalized costs. When T o 0 , the selected probability of all valid paths is equal to P=1/k. This means that individuals are equally sensitive to the difference of generalized costs. When șĺ’, the selected probability of the shortest path accesses to 1. This shows that individuals are extremely sensitive to the difference of generalized costs and all individuals will choose the shortest path. 3.3 Breadth-first path search algorithm The breadth-first path searching algorithm is a searching algorithm of graphs based on dynamic planning methods. In this paper, we search the valid paths through the breadth-first path searching algorithm and calculate the assignment ratio of transportation demand through the improved Logit model. For more detailed descriptions, the topology of an example network is provided in Fig. 2, where node 1 is defined as the original node. The current nodes and the visiting nodes are respectively stored in the set CurrentQueue and NextQueue. The breadth-first path searching algorithm is developed as follows. Step 1: initializing the set CurrentQueue={1}, NextQueue I . Step 2: if a node adjacent to the nodes in CurrentQueue are visited on the valid path, enter it into the set NextQueue.

CHEN Shaokuan et al. / J Transpn Sys Eng & IT, 2009, 9(6), 130í135

1

2 3 4

1

5 6 3 7 8 4 9 10

2 3 4

5 6 3 7 8 4 9 10

7 8 9 10

7 8 9 10

Fig. 3 Procedure of breadth-first path search algorithm Table 1 Mean transportation price and value of time for case studies 2005 2015 Existing Existing JHHS Highways Highways railways railways railway

Items Mean price ratio (RMB/person km) Value of <200 km time 200–300 km (RMB/h) 300–500 km

0.151

0.187

0.253

0.207

22.34

46.35

15.62

32.26

14.72

29.45

500–700 km

14.05

28.63

700–1 100 km

13.18

26.47

>1 100 km

18.25

38.35

0.450

4

Table 2 Comparing passenger load from the proposed and other models (unit: thousand persons) Different models

BeijingTianjin

TianjinJinan

JinanXuzhou

XuzhouNanjing

NanjingShanghai

Proposed model

32,743

33,240

29,524

33,890

42,076

Four-step model

34,627

39,453

42,860

46,537

48,032

Trend model

29,125

27,802

25,010

30,404

39,358

Step 3: considering the set CurrentQueue=NextQueue, NextQueue I .

Step

4:

repeating

Step

2

and

3

until

of the valid paths from node vj to node vs which are consistent with the transfer constraint. For example, the number of transfer is less than k. CurrentQueue and NextQueue are respectively defined as the queues of the current nodes and the visiting nodes. The details about the proposed algorithm are the following as shown in Fig. 3. Step 1: Initializing, input the origin of OD into the CurrentQueue and let NextQueue I . Step 2: Selecting the node i in the CurrentQueue in turn. Step 3: Selecting the forward node j adjacent to the node i. If the node j in the set VPi, jĺNextQueue and computing the weight of the node j through the improved Logit model. Step 4: If the forward adjacent node is selected in turn, CurrentQueue=NextQueue and NextQueue I . Otherwise, returning to 3. Step 5: If CurrentQueue I , returning to 2. Step 6: Saving the set of valid paths.

the

set

CurrentQueue I .

3.4 Extension study The route choice model proposed before considers hardly some transfer constraints, such as the number and the status of transfers. However, travel behaviors usually depend on the generalized costs, but sometimes the generalized costs could not fully describe the behavior of the travelers. Angelica Lozano and Giovanni Storchi proposed the shortest viable path algorithm considering the number of transfer[2]. In our paper, the route choice algorithm considering the transfer constraints between OD pairs is proposed based on an improved Logit model[13]. The proposed route choice algorithm redefines a set of valid paths VPi VPi '  VPi t , where VPi denotes the set of valid paths considering transfer constraints between vi and destination node, VPi ' denotes the set of valid paths redefined by Leurent[14], VPi t denotes the set

Case studies

The proposed model is employed to forecast the passenger transportation demand over the area along the high speed railway between Beijing and Shanghai (Jing Hu High Speed railway, JHHS railway) in China for case studies in this paper. JHHS railway with 1,318 kilometers long and 21 stations will go through seven large cities or provinces in the eastern of China, including Beijing, Tianjin, Hebei, Shandong, Anhui, Jiangsu and Shanghai. For the convenience of comparison and analysis, the base data including transportation price ratio and value of time in 2005 and 2015 by different transportation modes are listed in Table 1. From Table 1, mean price ratio and value of time in 2015 are obviously higher than those in 2005 because of the development of economy. The value of time in case studies varies with the change of travel distance. From the estimation of the value of time, when it is less than 1,100 kilometers, the value of time is decreasing with the increasing of travel distance; when it is more than 1,100 kilometers, the value of time is increasing with the travel distance. The results of forecasting passenger demand over JHHS railway from the proposed model, four-step model and trend model are respectively shown in Table 2. From Table 2, the results from the proposed model are less than those from the four-step model due to more descriptions of integrated topology for multimodal transportation. The results in trend model depending on the characteristics of social economy are usually less than other models because the transfer among different transport modes and the inducing increment from new JHHS railway are not sufficiently considered. The passenger shares in multimodal transportation network in 2015 are calculated through the proposed model and listed in Table 3.

CHEN Shaokuan et al. / J Transpn Sys Eng & IT, 2009, 9(6), 130í135

Table 3 Passenger Share from the proposed and four-step models (2015) (in %) Proposed model

Case scenarios Highway

Existing railway

JHHS railway

Highway

Existing railway

JHHS railway

Without JHHS railway

43.34

56.66

-

45.32

54.68

-

With JHSS railway

14.96

20.19

64.85

16.24

19.21

64.55

From Table 3, the results from the proposed are the same as those from the four-step model whatever with JHHS railway or not. The results also shows that more passengers will transfer to JHHS railway after JHHS railway comes into operation mainly because the trip time will decrease from about 12 hours in 2005 to 5 hours in 2015 and the transportation service will be obviously improved. The passenger volume on JHHS railway is mainly composed of three parts: transferring passengers from highway, existing railway and inducing passengers. Proportion of passenger volume is 28.79%, 45.75%, and 25.46% respectively.

5

Four-step model

Conclusions

A hierarchical multimodal assignment model is proposed to evaluate the transport demand for integrated transportation networks. The studies focus on the topology of hierarchical networks and the improvement of path searching algorithms. The conclusions are given below through case studies. The proposed method represents more precisely travel costs and transfer behaviors than the four-step model and trend method. For example, the value of time estimated by different travel distance is more practical in transport demand prediction. The predicted demand from the proposed method is lower than those from the four-step method in case studies. The improvements about path choice and transfer behavior in the former method lead to the homogeneous distribution of transport demand over the multimodal transportation network. The studies from the prediction about transport demand over the area along JHHS railway in 2005 show that the prediction results from the proposed method are very close to those from the four-step model. The passenger shares are both about 65% and the total transport demand is composed of the transfer demand from the existing highway and railway, as well as the inducing demand due to the operation of JHHS railway. Their shares are 28.79%, 45.75% and 25.46%, respectively.

Acknowledgements This research was funded by the National Natural Science Key Foundation (70631001), the National Natural Science Foundation of China (70571005), and The Energy Foundation (G-0811-10565).

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