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Comprehensive evaluation of passenger train service plan based on complex network theory
Measurement 58 (2014) 221–229
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Measurement journal homepage: www.elsevier.com/locate/measurement
Comprehensive evaluation of passenger train service plan based on complex network theory Xuelei Meng a,b,1, Yong Qin b,c,2, Limin Jia b,c,⇑ a
School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China c School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China b
a r t i c l e
i n f o
Article history: Received 20 January 2014 Received in revised form 7 May 2014 Accepted 20 August 2014 Available online 7 September 2014 Keywords: Comprehensive evaluation Passenger train Service plan Complex network theory
a b s t r a c t Train service plans evaluation remains a longstanding challenge in railway operation. The train service plan evaluation problem based on complex network theory is investigated in this paper. The train service plan is transferred into a train service network, which is a complex network with the typical characteristics of small-world and degree free. Three key indices of a complex network are listed and we relate them with the train service plan. Based on the three indices, we define the convenience degree for traveling, the transferring times, the travel time, and station clustering coefficients to evaluate the train service plan. A computing case is performed to illustrate the applications of the evaluation model and the algorithm. The results show that the evaluating results can be attained via the method proposed effectively, which can offer supporting information for decision makers of the railway operation bureaus. It proposes a novel, practical and valuable tool to evaluate the train service plan. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Complexity phenomena exist in the transportation system and attract studies of the experts from all over the world. LWR (Lighthill and Whitham and Richards) model is to a traffic simulation model based on the fluid kinematics theories, which first describes the shockwaves and complex phenomenon in the traffic system [1,2]. Daganzo pointed out that the high order modifications lead to a fundamentally flawed structure of LWR model based on ⇑ Corresponding author. Tel./fax: +86 10 51683846. E-mail addresses: [email protected] (X. Meng), [email protected] (Y. Qin), [email protected], [email protected]. com (L. Jia). 1 Address: Post Box 405, School of Traffic and Transportation, Lanzhou Jiaotong University, No.88, Anning West Street, Anning District, Lanzhou 730070, Gansu Province, China. Tel.: +86 15117071015; fax: +86 0931 7678976. 2 Tel./fax: +86 10 51683846. http://dx.doi.org/10.1016/j.measurement.2014.08.038 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
the analysis of the logical flaws [3]. And unreasonable predictions were made by all existing models formulated as a quasilinear system of partial differential equations in speed, density, and (sometimes) other variables but not by the LWR model. Bianca dealt with the kinetic theory framework for the modeling of the complex dynamics of crowds constituted by a large number of individuals (pedestrians) that interacted in a nonlinear fashion in a domain with and without obstacles [4], based on the multi-scale mathematical model which reproduces the predominant features of crowd dynamics by taking into account the distance among pedestrians [5]. Bianca and Coscia dealt with the derivation and the analysis of a new mathematical model for vehicular traffic along a one-way road obtained by the coupling of a uniform and an adaptive discretization of the velocity variable in the framework of the kinetic theory [6]. These publications showed that there were complex dynamics in the crowds and traffic flow.
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There are many publications today on the pedestrian, vehicle and the relation of them, which can bring highlights for us to study the train service plan evaluation problem. The essence of the problem is to measure the fit degree of train service plan and the passenger requirements. Berthelin et al. presented a realistic model namely the Second Order Model with Constraints (in short SOMC). An existence result of weak solutions for this model by means of cluster dynamics was proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely [7]. Helbing considered the empirical data and then reviewed the main approaches to modeling pedestrian and vehicle traffic. He studied the vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well [8]. Kerner analyzed complex spatiotemporal behavior of traffic and devoted to the explanation of freeway traffic congestion, a fact of life for many car drivers. Results of empirical observations of freeway congestion, which exhibit diverse complex spatiotemporal patterns including moving traffic jams, are analyzed [9]. Dogbe showed that the monotonicity properties were propagated under certain assumptions. He gained the results by the method of characteristics and a priori estimates under a monotonicity criterion [10]. Kottenhoff presented practical methods in these areas, costs, valuation methods and handling of package effects and finally how to trade off passengers’ valuations for costs [11]. Many complex systems can be described and studied with a complex network [12]. There are two noteworthy achievements, which are the small-world from model Watts and Strogatz [13] and scale-free network model from Barabasi and Albert [14]. A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is L / log n [13]. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes ~c where c is a parameter goes for large values of k as PðkÞk whose value is typically in the range 2 < c < 3, although occasionally it may lie outside these bounds [14]. Numerous researchers have found that there many complex networks in the real world, such as biology network, Internet, research cooperating network, electricity system, a wireless communication system, traffic network and etc. They all have the small-world network or scale-free network characteristics. With the development of the complexity theories of complex networks, researches begin to pay more attention to the implementation of complex network theories in the railway operation. It began with the complex characteristics analysis of the complex networks in the field of railway study, such as the fractal dimension, average distance and average clustering coefficients. The destination is to point out the properties of the virtual
network on railway operation. Benguigui and Daoud studied the possibility that the suburban railway system is fractal [15] Sen et al. studied the train service network in India, concluded that the network is a typical small-world network with the average path length 2.16 and the average clustering coefficient 0.69 [16]. Latora and Marchiori proposed a more universal complex network analysis model, considering the passenger transportation efficiency and used the model to analyze the railway network of Boston (American) [17]. Katherine and Lisa studied the properties of the bipartite graph, constructed with the railway lines of Vienna (American) and Boston (American) and proved that it is a small-world network [18]. Zhao et al. studied the railway passenger service network based on the service plan of Chinese railway of 2005, and they found that the network was a small-world network with the average path length 3.27 and average clustering coefficient 0.83 [19]. Meng et al. already found that the railway service network in China had power law degree distribution character and analyzed the dynamic properties [20]. And Meng et al. analyzed the efficiency of the train service network under the condition of intentional attack and random breakdown respectively to study the vulnerability of the network and concluded that the network was vulnerable under intentional attack while it had a strong resistance to the random breakdown [21]. Kunimatsu et al. developed a micro-simulation system to simulate both train operation and passengers’ train choice behavior. Then the timetable is evaluated, which is a detailed operation file of line plan [22]. In this paper, we build a model for the train service plan evaluation problem based on the complex theory in order to generate valuable supporting information for the railway operators. We list three indices of the complex network, degree distribution, length of the network path and clustering coefficient to map the evaluation indices of train service plan. We apply the evaluation model in the train service plan evaluating of the Beijing–Shanghai and Shanghai–Hangzhou high speed railway. This is a novel method to evaluate the train service plan and it can afford useful information for the train service plan designers. This paper is structured as follows. Section 1 is the introduction of the train service plan evaluation problem. Section 2 presents the method to convert a train service plan to a train service network. In Section 3, to analyze the key indices of a train service plan, the indices of the train service network are defined based on the introduction of three indices of a complex network. Section 4 gives the computing case, based on the data of Beijing–Shanghai and Shanghai– Hangzhou high speed railway. Section 5 draws a conclusion.
2. Construction method of train service network The train service plan determines the origin stations and destination stations of the trains, the paths of the trains, the stops along the railway, the vehicle types and, etc. The paths of the trains and the stops are the key factors of the service plan, which decides the service quality provided to the public. So it is essential to evaluate the quality of the service plan. The service plan can be transferred into as a complex network [19–21].
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We take all the stations as nodes of the network. Then the edges will be added on the network one by one according to the rules in Ref. [19–21]. If a train passes two stations and stops at the two stations, an edge will be added to the network. If the number of edges between two nodes is greater than 1, then we only draw one edge between the two stations and mark the number of the edges as the weight. Thus, a complex network will be constructed when all the trains are processed. The network is a mapping of the service plan, which can represent the service plan. Here we present a sample to construct the train service network. We take train G1, G15 (Beijingnan–Hongqiao) and G7321 (Hongqiao–Hangzhoudong) as an example to show the method to form the service network. The information is listed in Table 1. There are 7 stations in the network, named Beijingnan, Jinanxi, Nanjingnan, Hongqiao, Jiaxingnan, Hainingxi and Hanzhoudong. As showed in Table 1, G1 has 3 stops, which are marked with dark spots. According to the definition of the train service network, three edges should be added to the network, which are Beijingnan–Nanjingnan, Nanjingnan–Hongqiao and Beijingnan–Hongqiao, shown in Fig. 1. Since G15 has the four stops, Beijingnan, Jinanxi, Nanjingnan, Hongqiao. Then we should add 6 edges on the network. They are Beijingnan–Jinanxi, Beijingnan– Nanjingnan, Beijingnan–Hongqiao, Jinanxi–Nanjingnan, Jinanxi–Hongqiao, and Nanjingnan–Hongqiao. If there is already an edge from a node to another node, we add 1 to the weight of the edges. The network is changed into a different service network, as showed in Fig. 2. In the like manner, we add the edges generated from train G7321 on the network, the network is turned into the one shown in Fig. 3. 3. Definition of the indices of the train service network
2
Beijingnan
Hongqiao
Nanjingnan
2 1 1
2
1
Jinanxi
Hangzhoudong
Haining
Jiaxingnan
Fig. 2. Train service network formed with train G1 and G15.
2
Beijingnan
Nanjingnan
2
Hongqiao 2
1
1 Jinanxi
1
1 Hangzhoudong
1 1
1
1
Haining Jiaxingnan
1
Fig. 3. Train service network formed with train G1, G15 and G7321.
of the service plan. So we should map the indices of the service plan to the service plan indices of the service plan. We will introduce the indices of a complex work and define the indices for evaluating the train service plan in this section. 3.1. Key statistics indices of the complex network The fundamental indices of complex networks are the shortest path, clustering coefficients, betweenness, node degree, average distance and density. The definitions of them are as follows: (1) Degrees and degree distribution
Since the train service network is constructed, we can evaluate the service plan with the service network. The indices of the service network stand for some characteristics
The degree ki of a specified node i is defined as the number of edges connecting with node i. The network average degree is defined as the average degree of all the nodes in the network.
Table 1 Information of the train stops.
hki ¼
No.
Station
G1
G15
1 2 3 4 5 6 7
Beijingnan Jinanxi Nanjingnan Hongqiao Jiaxingnan Hainingxi Hanzhoudong
G7321
Nanjingnan
1
ð1Þ
where N is the number of the nodes in the network. (2) Length of the network path
The length of shortest path from node i to node j is defined as the minimal number of edges which can connect node i and node j. The average path length of the whole network is defined as the average length of the path length between all the nodes.
1 Beijingnan
n 1X ki N i¼1
Hongqiao 1
L¼
X 2 dij NðN 1Þ i6j
ð2Þ
where N is the number of the nodes in the network.
Jinanxi
(3) Clustering coefficient Hangzhoudong
Haining
Jiaxingnan
Fig. 1. Train service network formed with train G1.
The clustering coefficient can describe the compactness degree of the network. The clustering coefficient of node i,
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named Ci, is the ratio of the number of edges connected with the ki nodes to the number of the maximum possible edges.
Ci ¼
2Ei ki ðki 1Þ
ð3Þ
And the network clustering coefficient is the arithmetic means of all the nodes clustering coefficients. That is: n 1X C¼ Ci N i¼1
(2) Average traveling time of the passengers Traveling time includes the train running time and the time consumed in transferring. The train running time is the period of time that consumed in the railway sections, between station i and station j, named T ij . Set t to be the time consumed in a transferring. The whole traveling time T is:
P T¼
ð4Þ
þ lij tÞ M i
i
P
ð7Þ
where N is the number of the nodes in the network.
where lij is the number of transferring from station i to station j.
3.2. Convenience degrees for traveling
3.4. Stations clustering coefficients
The edge between two nodes indicates that a passenger can travel from one of the station to another station without transferring. Then the node degree not only shows what stations we can reach without transferring, but also show how many trains we can take at the station. So the convenience degree for traveling of a specified station can be defined according to the node degree definition.
Clustering coefficients describe the connecting degree of the stations in the railway network. Set Ei to be the number of the existing edges connecting to node i with the zi stations. The maximal number of edges between node i and the zi stations is zi(zi 1)/2. The ratio of Ei to zi(zi 1)/2 is the ratio of direct travels to the maximal possible travels. Set Ci to be the stations clustering coefficient of the train service network. There is no isolated node in the service network. So C i 2 ð0; 1, the smaller Ci is, the lower the connecting degree is defined in Eq. (3). The average station clustering coefficient of the whole train service network C is shown in Eq. (4).
ki ¼
X qij
ð5Þ
i6 j
where qij is the weight of the edge from station i to station j. Then the convenience degree for traveling of the whole service network is defined as the degree distribution, which is described in Eq. (1).
4. Computing case 4.1. Basic data
3.3. Transferring times and the travel time Transferring times and the total traveling time of all the passengers are fundamental indices of a service plan. Transferring times reflect the directness of passenger transportation. And the total travel time shows the consuming time in the travel. For a passenger, the transferring times will be reduced if we add more stops for the trains. Nevertheless, for all the passengers, the more stops the train has, the more travel time will be consumed. So it is a dilemma whether to add more train stops or not. Here we present the transferring times and traveling time calculating method.
4.2. Construction of train service network for the train service plan
(1) Average transferring times The path length in the train service network is the minimum number of edges that are enough to connect two nodes. If a train passes two stations and stops, the path length is 1, which means that it is not necessary to transfer from one train to another to get to the destination. When the path length is 2, there are at least two edges between the two stations, indicating that a passenger must transfer at least once to travel from one station to another. So the number of average transferring times is:
H ¼L1
The data is taken from the Beijing–Shanghai high speed railway and Shanghai–Hangzhou high speed railway. We select 16 stations as the nodes. They are Beijingnan, Tianjinxi, Cangzhouxi, Dezhoudong, Jinanxi, Xuzhoudong, Nanjingnan, Zhenjiangnan, Changzhoubei, Wuxidong, Suzhoubei, Kunshannan, Hongqiao, Jiaxingnan, Hainingxi and Hangzhoudong. Then the relatively high speed trains operating on the Beijing–Shanghai high speed railway and Shanghai–Hangzhou Railway are selected to construct the service plan network. The trains are T109, D311, D313, D321, G3, G13, G21, G35, G113, and G137.
ð6Þ
where L is the average path length of the train service network.
We order the stations according to the graphic position. Beijingnan, Tianjinxi, Cangzhouxi, Dezhoudong, Jinanxi, Xuzhoudong, Nanjingnan, Zhenjiangnan, Changzhoubei, Wuxidong, Suzhoubei, Kunshannan, Hongqiao, Jiaxingnan, Hainingxi and Hangzhoudong are numbered 1,2, . . ., 16. Edge (i, j) indicates that there is a direct train from station i to station j. Since we focus on the down going direction trains, i < j. The information of all the trains in this computing case is given in Table 2. The service network formed according to the data is shown in Fig. 4. We can see from Fig. 4 that there are numerous edges which have the weights bigger than 1. The meaning of the weight of an edge is that the number
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X. Meng et al. / Measurement 58 (2014) 221–229 Table 2 Information of all the trains in this case. No.
Station
T109
D311
D313
D321
G3
G13
G21
G35
G113
G137
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Beijingnan Tianjinxi Canzhouxi Dezhoudong Jinanxi Xuzhoudong Nanjingnan Zhenjiangnan Changzhoubei Wuxidong Suzhoubei Kuanshannan Hongqiao Jiaxingnan Hainingxi Hangzhoudong
G7327
G7393
G7321
Beijingnan Tianjinxi 2 45 4 10
Hangzhoudong 10 333
Cangzhouxi
Hainingxi 2
2
22
2
3
Dezhoudon g
Jiaxingnan
4 3 2 3 2 4 4
Jinanxi
4
3
Hongqiao
5 3
4
Xuzhoudong 4
4 2 3 2
Kuanshannan 3 10
Nanjingnan
3 3
Suzhoubei
4 3 2
Zhenjiangnan
3 Wuxi
Changzhoubei Fig. 4. Train service network formed with all of trains in this case.
of the choices of trains from a station to another station without any transfer. For example, the weight of the edge from Nanjingnan to Hongqiao is 10, indicating that there are 10 trains stopping at both Nanjingnan and Hongqiao and we have 10 choices of trains to travel from Nanjingnan to Hongqiao without transfer. And we can also see that some of the edges have the weight of 1, meaning that the passengers have only train to take from the original station to the destination station of the edge if they do not want to transfer on another station.
4.3. Indices calculation and analysis According to the train service network constructed in Section 4.2, we calculate the indices of the network and analyze them. (1) Convenience degree of traveling We calculate the convenience for all the stations according to Eq. (5) in Section 3.1 and listed them in Table 3.
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X. Meng et al. / Measurement 58 (2014) 221–229
Then the average convenience degree of the whole train service network is:
hki ¼ ¼
16 1 X ki 16 i¼1
1 ð48 þ 9 þ 15 þ 24 þ 16 þ 19 þ 23 þ 4 þ 6 þ 4 16 þ 3 þ 1 þ 7 þ 4 þ 1 þ 0Þ
¼ 11:375 according to data of the station convenience columns in Table 3. Similarly, the average maximal number of reachable stations is:
¼ m
n 1 X mi 16 i¼1
1 ð15 þ 14 þ 13 þ 12 þ þ 3 þ 2 þ 1 þ 0Þ 16 ¼ 7:500
¼
And the average actual number of reachable stations in this case is:
s ¼ ¼
n 1 X si 16 i¼1
1 ð13 þ 9 þ 10 þ 10 þ 8 þ 7 þ 4 þ 3 þ 2 þ 1 þ 1 16 þ 3 þ 2 þ 1Þ
(2) Transferring times and the travel time
¼ 5:186 Hangzhoudong is the terminal station in the down going direction. So the maximal reachable stations number and the actual reachable stations number are both zeros. We can see that the convenience degrees of Dezhoudong, Jinanxi, Xuzhoudong and Hongqiao are higher that of other stations. The convenience degree of Beijingnan k1 ¼ 48 is much bigger that the actual reachable stations number m1 ¼ 15, which means that the passengers have numerous choices of trains when traveling on the railway line, getting to the destination without transfer. It is very convenient to take Table 3 Convenience degrees of each station.
h1i h2i h3i h4i h5i h6i
a
b
c
48 9 15 24 17 19
15 14 13 12 11 10
13 9 10 10 8 8
h7i h8i h9i h10i h11i h12i
trains to travel from Beijingnan station along with the Beijing-Shanghai high speed railway wherever the passengers want to go. Hongqiao station has the same characteristic. The maximal numbers of reachable stations of Hongqiao and Jiaxingnan equal to the numbers of their actual reachable stations. It is to say that the passenger can get to the stations in the down going direction without transferring. But the convenience degrees are lower than that of Beijingnan and Hongqiao, for there are not much choices for the passengers when start their travel from these stations to a certain station. The maximal numbers of reachable stations of other stations are bigger than the numbers of their definite reachable stations, which indicates that the not all the stations are directly reachable. Dezhoudong, Jinanxi, Xuzhoudong and Nanjingnan have the higher convenience degrees. There are relatively more trains for passengers to choose when passengers start their journey from these stations to a specified destination. On the other hand, Tianjinxi, Zhenjiangnan, Kunshannan get the relatively lower convenience degrees, which cannot meet the massive requirements for traveling by trains, especially when the passengers want to get to their destinations without any transfer. They may add more trains, passing by the stations in the service plan to improve the convenience degrees of the stations and provide more choices for the passengers.
The travel time is connected to the transferring times directly according Eq. (6). So the primary step is to calculate the transferring times. In addition, the number of transferring times is related to the path length between the stations. So the paths lengths are calculated and listed in Table 4. From Table 4, we can see that the some of paths lengths are 2, meaning that the passengers travel between the OD stations must transfer once. Other lengths between the stations are 1, indicating that the passengers do not have to transfer when traveling. The average path length is:
L¼ ¼
a
b
c
23 4 6 5 3 1
9 8 7 6 5 4
7 4 3 4 1 1
h13i h14i h15i h16i
a
b
c
7 4 1 0
3 2 1 0
3 2 1 0
Note: In Table 3, the meanings of the characters on the first line and the numbers in parentheses are as follows. a. Convenience degree for traveling. b. Maximal number of reachable stations. c. Actual number of reachable stations in this case. h1i Beijingnan h2i Tianjinxi h3i Cangzhouxi h4i Dezhoudong h5i Jinanxi h6i Xuzhoudong h7i Nanjingnan h8i Zhenjiangnan h9i Changzhoubei h10i Wuxidong h11i Suzhoubei h12i Kuanshannan h13i Hongqiao h14i Jiaxingnan h15i Hainingxi h16i Hangzhoudong.
16 X 16i X X 2 2 dij ¼ dij 16ð16 1Þ i
2 ð16 þ 19 þ 16 þ 14 þ 14 þ 12 þ 11 þ 12 16ð16 1Þ þ 11 þ 10 þ 9 þ 7 þ 3 þ 2 þ 1Þ
¼ 1:308 Then the average transferring times is:
H ¼ L 1 ¼ 1:308 1 ¼ 0:308 Then we can find that all the passengers travel on the Beijing–Shanghai and Shanghai–Hangzhou railway lines will have to transfer 0.308 times averagely to get to their destinations. It reflects the service quality of the service plan on the two railway lines. From this index, we can conclude that the service quality is not so high. We suggest
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X. Meng et al. / Measurement 58 (2014) 221–229 Table 4 Paths lengths between stations.
h1i h2i h3ii h4i h5i h6i h7i h8i h9ii h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
–
1 –
1 1 –
1 1 1 –
1 2 1 1 –
1 1 1 1 1 –
1 1 1 1 1 1 –
1 1 1 1 2 1 1 –
1 1 1 1 1 1 1 1 –
1 1 1 1 1 1 1 1 1 –
1 1 1 1 1 1 1 1 1 1 –
1 2 1 1 1 1 1 2 2 1 2 –
1 1 1 1 1 1 1 1 1 1 1 1 –
2 2 2 2 2 2 2 2 2 2 2 2 1 –
2 2 2 2 2 2 2 2 2 2 2 2 1 1 –
1 2 2 1 1 1 1 2 2 2 2 2 1 1 1 –
Table 5 Passenger OD between Beijingnan and Hongqiao.
h1i h2i h3i h4i h5i h6i h7i h8i h9i h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
0
1026 0
161 75 0
176 81 12 0
586 271 39 42 0
388 180 26 28 97 0
844 391 56 61 212 137 0
196 91 13 14 49 32 73 0
264 122 17 19 66 43 98 21 0
338 156 22 24 85 55 126 27 37 0
383 177 25 28 96 62 143 31 42 54 0
108 50 7 8 27 18 40 9 12 15 17 0
1571 728 104 113 394 255 586 126 171 221 252 69 0
120 87 19 16 51 48 78 23 31 20 21 14 180 0
97 35 13 8 10 8 65 10 11 17 16 10 117 68 0
428 187 45 47 62 130 158 29 31 30 32 21 225 71 52 0
Table 6 The average running time of the trains between the stations (measurement unit: minute).
h1i h2i h3i h4i h5i h6i h7i h8i h9i h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
0
71 0
95 58 0
108 127 49 0
97 272 53 25 0
236 367 212 128 65 0
369 559 349 212 139 106 0
681 600 542 473 300 233 41 0
542 656 598 366 177 201 66 56 0
495 683 434 329 190 189 75 83 27 0
585 724 666 412 201 247 162 124 46 41 0
312 721 261 233 208 143 65 221 165 18 183 0
482 781 502 348 216 223 113 181 87 59 44 19 0
627 926 647 493 361 368 258 326 232 204 189 164 25 0
643 942 663 509 377 384 274 342 248 220 205 180 41 16 0
378 958 679 305 279 214 136 358 264 89 221 196 57 32 20 0
that more trains should be added to the railway, or more stops should be set for the existing trains to reduce the transfer times of the passengers. OD data we got is shown in Table 5.
Then the down going passengers number is calculated out according to the data in Table 5.
X M ij ¼ 15560 i
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X. Meng et al. / Measurement 58 (2014) 221–229
In Eq. (7), T ij þ lij t is the total traveling time from one station to another. According to the data in timetable, we can get T ij . Then the lij can be got according to the data in Table 4. The traveling time between every OD station is calculated out and listed in Table 6. It is should be pointed out that different trains cost different running time in the same railway section. So we average the running time in each section of all different grade trains. When a passenger must transfer from a train to another, we take it for granted that he will transfer as early as possible. That is to say, he will transfer at the first station among all the stations he can choose. Ant it is assumed that each transfer will cost 120 min according to the historical data. For instance, there is no direct train between Beijingnan and Jiaxingnan. So passengers will transfer at Hongqiao station. The running time between Beijingnan and Hongqiao is 482 min and it will cost the train running at least 25 min from Hongqiao to Jiaxingnan. So the total traveling time from Beijingnan to Jiaxingnan is 482 + 120 + 25 = 627 min. According to the passenger OD data in Table 5, the running time in Table 6 and Eq. (7), we calculated out the total traveling time of all the passengers sent from each station, presented in Table 7. The average traveling time of the whole service plan is:
T ¼ ð2325237 þ 1639602 þ 176623 þ 118057 þ 227800 þ 164325 þ 170317 þ 53356 þ 37892 þ 26013
Table 8 Ei and zi values of each station.
h1i h2i h3i h4i
(1) Stations clustering coefficients The number of neighbor stations of station i(zi ) and the actual number of edges connected to station i(Ei ) are listed in Table 8.
h5i h6i h7i h8i
Ci h1i h2i h3i h4i
Ei
zi
60 79 79 57
12 14 14 12
h9i h10i h11i h12i
Ei
zi
64 79 65 33
12 14 12 9
h13i h14i h15i h16i
0.9697 0.8681 0.9848 0.9167
h13i h14i h15i h16i
Ei
zi
84 6 6 35
16 4 4 10
0.8681 1 0.9231 0.8681
Ci h5i h6i h7i h8i
0.9091 0.8681 0.8681 0.8636
Ci h9i h10i h11i h12i
Ci 0.7 1 1 0.7778
According to Eq. (3) and Table 8, the clustering coefficient of Beijingnan station is:
C1 ¼
2E1 2 79 ¼ 0:8681 ¼ z1 ðz1 1Þ 14 ð14 1Þ
In the same manner, the clustering coefficients of all the station are computed out and listed in Table 9. According to Eq. (4), the clustering coefficient of the whole network is:
C¼
It equals to 5.3 h. Its meaning is that the average traveling time of all the OD passengers is 5.3 h, which is determined by the train service plan. We calculate out the index based on the train service network transformed from the train service network. This index of the train service plan reflects the train service plan is with high quality from the point of view of traveling time control. Beijing–Shanghai high speed railway is 1318 km long, and Shanghai– Hanzhou high speed railway is 169 km long. The average traveling time is controlled to be 5.3 h not only because of the high speed of trains, but also the high quality of the train service plan.
zi 14 11 13 14
Table 9 Clustering coefficients of each station.
þ 28520 þ 9523 þ 22122 þ 3360 þ 1040Þ=15560 ¼ 322 min
Ei 79 55 72 79
¼
16 1 X Ci 16 i¼1
1 ð0:8681 þ 1 þ 0:9231 þ 0:8681 þ 0:9091 16 þ 0:8681 þ 0:8681 þ 0:8636 þ 0:9697 þ 0:8681 þ 0:9848 þ 0:9167 þ 0:7 þ 1 þ 1 þ 0:7778
¼ 0:8991 The clustering coefficient of the train service network is 0.8991, which is very close to 1. So the connecting degree of the entire network is prominent. Tianjinxi, Jiaxingnan and Hainingxi have the high connecting degrees, for they have fewer neighbors than other stations, not because they have more connecting edges. Contrary to popular misconception, Hongqiao has the lowest connecting degree. Low connecting degree indicates the station does not connect to the other stations closely. Managers should add more trains or set more stops of trains at Hongqiao station to improve the clustering coefficient to match the great travel demand.
Table 7 Total traveling time of all the passengers sent from each station (measurement unit: minute). M ij ðT ij þ lij tÞ h1i h2i h3i h4i
2,325,337 1,639,602 176,623 118,057
Mij ðT ij þ lij tÞ h5i h6i h7i h8i
227,800 164,325 170,317 53,356
Mij ðT ij þ lij tÞ h9i h10i h11i h12i
37,892 26,013 28,520 9523
M ij ðT ij þ lij tÞ h13i h14i h15i h16i
22,122 3360 1040
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5. Conclusion On the strategic planning level, a railway train service plan evaluation problem is investigated based on the complexity theory. The train service plan is transformed to a complex network according to a constructing method. The key indices of the complex network are listed and analyzed their representativeness of the train service plan. On the basis of the analysis of the train service network, we evaluate the train service plan. The conclusions drawn with the evaluation results of the computing case are as follows: (1) More trains, stopping at Tianjinxi, Zhenjiangnan, Kunshannan should be added into the service plan, or more stops should be set at these three stations to improve the convenience degrees of the stations and provide more choices for the passengers. (2) We suggest that more trains should be added on the railway, or more stops should be set for the existing trains to reduce the transfer times of the passengers. (3) The train service plan is with high quality from the point of view of traveling time control. (4) Managers should add more trains or set more stops of trains at Hongqiao station to improve the clustering coefficient to match the great travel demand. The suggestions above based on the evaluation results indicate that the results can give some essential supporting information for the railway managers. They can take the results as an important reference when design the train service plan. So the evaluation method proposed in this paper has much vale. It is a different viewpoint to evaluate the train service plan, which has the characteristics of briefness practicability. Evaluating results can be attained via the method, which is very practical and valuable for decision makers of the railway operation bureaus. Future research into the train service plan is to optimize the train service plan based on the developing theories on complex networks, such as complex network control and optimization theory. Since the train service plan can be transferred into the train service network, we can control and optimize the train service network. Then we transfer the train service network back into the train service plan, reaching the goal to optimize the train service plan, especially in railway emergencies. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grant: 61263027),
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the New Teacher Project of Research Fund for the Doctoral Program of Higher Education of China (Grant: 20126204120002) and the Natural Science Foundation of Gansu Province – China (Grant: 1310RJZA068). The authors wish to thank anonymous referees and the editor for their comments and suggestions. References [1] M.J. Lighthill, J.B. Whitham, On kinematic waves. I: Flow movement in long rivers; II: a theory of traffic flow on long crowded roads, Proc. Royal Soc. Ser. 22 (1995) 281–345. [2] P.I. Richads, Shockwaves on the highway, Oper. Res. 4 (1956) 42–51. [3] C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transp. Res. Part B 29 (1995) 277–286. [4] C. Bianca, Mathematical modeling of crowds dynamics: complexity and kinetic approach, Nonlinear Stud. 19 (2012) 371–380. [5] C. Bianca, C. Dogbe, A mathematical model for crowd dynamics: multi-scale analysis, fluctuations and random noise, Nonlinear Stud. 20 (2013) 349–373. [6] C. Bianca, V. Coscia, On the coupling of steady and adaptive velocity grids in vehicular traffic modelling, Appl. Math. Lett. 24 (2011) 149– 155. [7] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle, J. Royer, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci. 18 (2008) 1269–1298. [8] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73 (2001) 1067–1141. [9] B. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer, Berlin, New York, 2004. [10] C. Dogbe, Nonlinear pedestrian-flow model: uniform wellposedness and global existence, Appl. Math. Inf. Sci. 7 (2013) 29–40. [11] K. Kottenhoff. Evaluation of passenger train concepts – practical methods for measuring travellers’ preferences in relation to costs, in: 3rd KFB Research Conference in Stockholm, 2000. [12] R. Albert, A.L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47–97. [13] D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 93 (1998) 440–442. [14] A.L. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. [15] L. Benguigui, M. Daoud, Is the suburban railway system a fractal?, Geograph Anal. 23 (1991) 362–368. [16] P. Sen, P. Dagupta, A. Chatterjee, et al., Small-world properties of the Indian railway network, Phys. Rev. 67 (2003) 036106. [17] V. Latora, M. Marchiori, Is the Boston subway a small-world network?, Physica A 314 (2002) 109–113 [18] A.S. Katherine, M.H. Lisa, Stations, trains and small-world networks, Physica A 339 (2004) 635–644. [19] W. Zhao, H. He, Z. Lin, et al., The study of properties of Chinese railway passenger transport network, ACTA Phys. Sin. 55 (2006) 3906–3911. [20] X. Meng, L. Jia, J. Xie et al., Complex characteristic analysis of passenger train flow network, in: Proceeding of CCDC2010, 2010, pp. 2533–2536. [21] X. Meng, L. Jia, Y. Qin, et al., Vulnerability analysis of passenger train flow network in emergency, Trans. Beijing Inst. Technol. 32 (2012) 80–83. [22] T. Kunimatsu, C. Hirai, N. Tomii, Train timetable evaluation from the viewpoint of passengers by microsimulation of train operation and passenger flow, Electr. Eng. Jpn. 181 (2012) 51–62.
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Comprehensive evaluation of passenger train service plan based on complex network theory Xuelei Meng a,b,1, Yong Qin b,c,2, Limin Jia b,c,⇑ a
School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China c School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China b
a r t i c l e
i n f o
Article history: Received 20 January 2014 Received in revised form 7 May 2014 Accepted 20 August 2014 Available online 7 September 2014 Keywords: Comprehensive evaluation Passenger train Service plan Complex network theory
a b s t r a c t Train service plans evaluation remains a longstanding challenge in railway operation. The train service plan evaluation problem based on complex network theory is investigated in this paper. The train service plan is transferred into a train service network, which is a complex network with the typical characteristics of small-world and degree free. Three key indices of a complex network are listed and we relate them with the train service plan. Based on the three indices, we define the convenience degree for traveling, the transferring times, the travel time, and station clustering coefficients to evaluate the train service plan. A computing case is performed to illustrate the applications of the evaluation model and the algorithm. The results show that the evaluating results can be attained via the method proposed effectively, which can offer supporting information for decision makers of the railway operation bureaus. It proposes a novel, practical and valuable tool to evaluate the train service plan. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Complexity phenomena exist in the transportation system and attract studies of the experts from all over the world. LWR (Lighthill and Whitham and Richards) model is to a traffic simulation model based on the fluid kinematics theories, which first describes the shockwaves and complex phenomenon in the traffic system [1,2]. Daganzo pointed out that the high order modifications lead to a fundamentally flawed structure of LWR model based on ⇑ Corresponding author. Tel./fax: +86 10 51683846. E-mail addresses: [email protected] (X. Meng), [email protected] (Y. Qin), [email protected], [email protected]. com (L. Jia). 1 Address: Post Box 405, School of Traffic and Transportation, Lanzhou Jiaotong University, No.88, Anning West Street, Anning District, Lanzhou 730070, Gansu Province, China. Tel.: +86 15117071015; fax: +86 0931 7678976. 2 Tel./fax: +86 10 51683846. http://dx.doi.org/10.1016/j.measurement.2014.08.038 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
the analysis of the logical flaws [3]. And unreasonable predictions were made by all existing models formulated as a quasilinear system of partial differential equations in speed, density, and (sometimes) other variables but not by the LWR model. Bianca dealt with the kinetic theory framework for the modeling of the complex dynamics of crowds constituted by a large number of individuals (pedestrians) that interacted in a nonlinear fashion in a domain with and without obstacles [4], based on the multi-scale mathematical model which reproduces the predominant features of crowd dynamics by taking into account the distance among pedestrians [5]. Bianca and Coscia dealt with the derivation and the analysis of a new mathematical model for vehicular traffic along a one-way road obtained by the coupling of a uniform and an adaptive discretization of the velocity variable in the framework of the kinetic theory [6]. These publications showed that there were complex dynamics in the crowds and traffic flow.
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There are many publications today on the pedestrian, vehicle and the relation of them, which can bring highlights for us to study the train service plan evaluation problem. The essence of the problem is to measure the fit degree of train service plan and the passenger requirements. Berthelin et al. presented a realistic model namely the Second Order Model with Constraints (in short SOMC). An existence result of weak solutions for this model by means of cluster dynamics was proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely [7]. Helbing considered the empirical data and then reviewed the main approaches to modeling pedestrian and vehicle traffic. He studied the vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well [8]. Kerner analyzed complex spatiotemporal behavior of traffic and devoted to the explanation of freeway traffic congestion, a fact of life for many car drivers. Results of empirical observations of freeway congestion, which exhibit diverse complex spatiotemporal patterns including moving traffic jams, are analyzed [9]. Dogbe showed that the monotonicity properties were propagated under certain assumptions. He gained the results by the method of characteristics and a priori estimates under a monotonicity criterion [10]. Kottenhoff presented practical methods in these areas, costs, valuation methods and handling of package effects and finally how to trade off passengers’ valuations for costs [11]. Many complex systems can be described and studied with a complex network [12]. There are two noteworthy achievements, which are the small-world from model Watts and Strogatz [13] and scale-free network model from Barabasi and Albert [14]. A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is L / log n [13]. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes ~c where c is a parameter goes for large values of k as PðkÞk whose value is typically in the range 2 < c < 3, although occasionally it may lie outside these bounds [14]. Numerous researchers have found that there many complex networks in the real world, such as biology network, Internet, research cooperating network, electricity system, a wireless communication system, traffic network and etc. They all have the small-world network or scale-free network characteristics. With the development of the complexity theories of complex networks, researches begin to pay more attention to the implementation of complex network theories in the railway operation. It began with the complex characteristics analysis of the complex networks in the field of railway study, such as the fractal dimension, average distance and average clustering coefficients. The destination is to point out the properties of the virtual
network on railway operation. Benguigui and Daoud studied the possibility that the suburban railway system is fractal [15] Sen et al. studied the train service network in India, concluded that the network is a typical small-world network with the average path length 2.16 and the average clustering coefficient 0.69 [16]. Latora and Marchiori proposed a more universal complex network analysis model, considering the passenger transportation efficiency and used the model to analyze the railway network of Boston (American) [17]. Katherine and Lisa studied the properties of the bipartite graph, constructed with the railway lines of Vienna (American) and Boston (American) and proved that it is a small-world network [18]. Zhao et al. studied the railway passenger service network based on the service plan of Chinese railway of 2005, and they found that the network was a small-world network with the average path length 3.27 and average clustering coefficient 0.83 [19]. Meng et al. already found that the railway service network in China had power law degree distribution character and analyzed the dynamic properties [20]. And Meng et al. analyzed the efficiency of the train service network under the condition of intentional attack and random breakdown respectively to study the vulnerability of the network and concluded that the network was vulnerable under intentional attack while it had a strong resistance to the random breakdown [21]. Kunimatsu et al. developed a micro-simulation system to simulate both train operation and passengers’ train choice behavior. Then the timetable is evaluated, which is a detailed operation file of line plan [22]. In this paper, we build a model for the train service plan evaluation problem based on the complex theory in order to generate valuable supporting information for the railway operators. We list three indices of the complex network, degree distribution, length of the network path and clustering coefficient to map the evaluation indices of train service plan. We apply the evaluation model in the train service plan evaluating of the Beijing–Shanghai and Shanghai–Hangzhou high speed railway. This is a novel method to evaluate the train service plan and it can afford useful information for the train service plan designers. This paper is structured as follows. Section 1 is the introduction of the train service plan evaluation problem. Section 2 presents the method to convert a train service plan to a train service network. In Section 3, to analyze the key indices of a train service plan, the indices of the train service network are defined based on the introduction of three indices of a complex network. Section 4 gives the computing case, based on the data of Beijing–Shanghai and Shanghai– Hangzhou high speed railway. Section 5 draws a conclusion.
2. Construction method of train service network The train service plan determines the origin stations and destination stations of the trains, the paths of the trains, the stops along the railway, the vehicle types and, etc. The paths of the trains and the stops are the key factors of the service plan, which decides the service quality provided to the public. So it is essential to evaluate the quality of the service plan. The service plan can be transferred into as a complex network [19–21].
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We take all the stations as nodes of the network. Then the edges will be added on the network one by one according to the rules in Ref. [19–21]. If a train passes two stations and stops at the two stations, an edge will be added to the network. If the number of edges between two nodes is greater than 1, then we only draw one edge between the two stations and mark the number of the edges as the weight. Thus, a complex network will be constructed when all the trains are processed. The network is a mapping of the service plan, which can represent the service plan. Here we present a sample to construct the train service network. We take train G1, G15 (Beijingnan–Hongqiao) and G7321 (Hongqiao–Hangzhoudong) as an example to show the method to form the service network. The information is listed in Table 1. There are 7 stations in the network, named Beijingnan, Jinanxi, Nanjingnan, Hongqiao, Jiaxingnan, Hainingxi and Hanzhoudong. As showed in Table 1, G1 has 3 stops, which are marked with dark spots. According to the definition of the train service network, three edges should be added to the network, which are Beijingnan–Nanjingnan, Nanjingnan–Hongqiao and Beijingnan–Hongqiao, shown in Fig. 1. Since G15 has the four stops, Beijingnan, Jinanxi, Nanjingnan, Hongqiao. Then we should add 6 edges on the network. They are Beijingnan–Jinanxi, Beijingnan– Nanjingnan, Beijingnan–Hongqiao, Jinanxi–Nanjingnan, Jinanxi–Hongqiao, and Nanjingnan–Hongqiao. If there is already an edge from a node to another node, we add 1 to the weight of the edges. The network is changed into a different service network, as showed in Fig. 2. In the like manner, we add the edges generated from train G7321 on the network, the network is turned into the one shown in Fig. 3. 3. Definition of the indices of the train service network
2
Beijingnan
Hongqiao
Nanjingnan
2 1 1
2
1
Jinanxi
Hangzhoudong
Haining
Jiaxingnan
Fig. 2. Train service network formed with train G1 and G15.
2
Beijingnan
Nanjingnan
2
Hongqiao 2
1
1 Jinanxi
1
1 Hangzhoudong
1 1
1
1
Haining Jiaxingnan
1
Fig. 3. Train service network formed with train G1, G15 and G7321.
of the service plan. So we should map the indices of the service plan to the service plan indices of the service plan. We will introduce the indices of a complex work and define the indices for evaluating the train service plan in this section. 3.1. Key statistics indices of the complex network The fundamental indices of complex networks are the shortest path, clustering coefficients, betweenness, node degree, average distance and density. The definitions of them are as follows: (1) Degrees and degree distribution
Since the train service network is constructed, we can evaluate the service plan with the service network. The indices of the service network stand for some characteristics
The degree ki of a specified node i is defined as the number of edges connecting with node i. The network average degree is defined as the average degree of all the nodes in the network.
Table 1 Information of the train stops.
hki ¼
No.
Station
G1
G15
1 2 3 4 5 6 7
Beijingnan Jinanxi Nanjingnan Hongqiao Jiaxingnan Hainingxi Hanzhoudong
G7321
Nanjingnan
1
ð1Þ
where N is the number of the nodes in the network. (2) Length of the network path
The length of shortest path from node i to node j is defined as the minimal number of edges which can connect node i and node j. The average path length of the whole network is defined as the average length of the path length between all the nodes.
1 Beijingnan
n 1X ki N i¼1
Hongqiao 1
L¼
X 2 dij NðN 1Þ i6j
ð2Þ
where N is the number of the nodes in the network.
Jinanxi
(3) Clustering coefficient Hangzhoudong
Haining
Jiaxingnan
Fig. 1. Train service network formed with train G1.
The clustering coefficient can describe the compactness degree of the network. The clustering coefficient of node i,
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named Ci, is the ratio of the number of edges connected with the ki nodes to the number of the maximum possible edges.
Ci ¼
2Ei ki ðki 1Þ
ð3Þ
And the network clustering coefficient is the arithmetic means of all the nodes clustering coefficients. That is: n 1X C¼ Ci N i¼1
(2) Average traveling time of the passengers Traveling time includes the train running time and the time consumed in transferring. The train running time is the period of time that consumed in the railway sections, between station i and station j, named T ij . Set t to be the time consumed in a transferring. The whole traveling time T is:
P T¼
ð4Þ
þ lij tÞ M i
i
P
ð7Þ
where N is the number of the nodes in the network.
where lij is the number of transferring from station i to station j.
3.2. Convenience degrees for traveling
3.4. Stations clustering coefficients
The edge between two nodes indicates that a passenger can travel from one of the station to another station without transferring. Then the node degree not only shows what stations we can reach without transferring, but also show how many trains we can take at the station. So the convenience degree for traveling of a specified station can be defined according to the node degree definition.
Clustering coefficients describe the connecting degree of the stations in the railway network. Set Ei to be the number of the existing edges connecting to node i with the zi stations. The maximal number of edges between node i and the zi stations is zi(zi 1)/2. The ratio of Ei to zi(zi 1)/2 is the ratio of direct travels to the maximal possible travels. Set Ci to be the stations clustering coefficient of the train service network. There is no isolated node in the service network. So C i 2 ð0; 1, the smaller Ci is, the lower the connecting degree is defined in Eq. (3). The average station clustering coefficient of the whole train service network C is shown in Eq. (4).
ki ¼
X qij
ð5Þ
i6 j
where qij is the weight of the edge from station i to station j. Then the convenience degree for traveling of the whole service network is defined as the degree distribution, which is described in Eq. (1).
4. Computing case 4.1. Basic data
3.3. Transferring times and the travel time Transferring times and the total traveling time of all the passengers are fundamental indices of a service plan. Transferring times reflect the directness of passenger transportation. And the total travel time shows the consuming time in the travel. For a passenger, the transferring times will be reduced if we add more stops for the trains. Nevertheless, for all the passengers, the more stops the train has, the more travel time will be consumed. So it is a dilemma whether to add more train stops or not. Here we present the transferring times and traveling time calculating method.
4.2. Construction of train service network for the train service plan
(1) Average transferring times The path length in the train service network is the minimum number of edges that are enough to connect two nodes. If a train passes two stations and stops, the path length is 1, which means that it is not necessary to transfer from one train to another to get to the destination. When the path length is 2, there are at least two edges between the two stations, indicating that a passenger must transfer at least once to travel from one station to another. So the number of average transferring times is:
H ¼L1
The data is taken from the Beijing–Shanghai high speed railway and Shanghai–Hangzhou high speed railway. We select 16 stations as the nodes. They are Beijingnan, Tianjinxi, Cangzhouxi, Dezhoudong, Jinanxi, Xuzhoudong, Nanjingnan, Zhenjiangnan, Changzhoubei, Wuxidong, Suzhoubei, Kunshannan, Hongqiao, Jiaxingnan, Hainingxi and Hangzhoudong. Then the relatively high speed trains operating on the Beijing–Shanghai high speed railway and Shanghai–Hangzhou Railway are selected to construct the service plan network. The trains are T109, D311, D313, D321, G3, G13, G21, G35, G113, and G137.
ð6Þ
where L is the average path length of the train service network.
We order the stations according to the graphic position. Beijingnan, Tianjinxi, Cangzhouxi, Dezhoudong, Jinanxi, Xuzhoudong, Nanjingnan, Zhenjiangnan, Changzhoubei, Wuxidong, Suzhoubei, Kunshannan, Hongqiao, Jiaxingnan, Hainingxi and Hangzhoudong are numbered 1,2, . . ., 16. Edge (i, j) indicates that there is a direct train from station i to station j. Since we focus on the down going direction trains, i < j. The information of all the trains in this computing case is given in Table 2. The service network formed according to the data is shown in Fig. 4. We can see from Fig. 4 that there are numerous edges which have the weights bigger than 1. The meaning of the weight of an edge is that the number
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X. Meng et al. / Measurement 58 (2014) 221–229 Table 2 Information of all the trains in this case. No.
Station
T109
D311
D313
D321
G3
G13
G21
G35
G113
G137
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Beijingnan Tianjinxi Canzhouxi Dezhoudong Jinanxi Xuzhoudong Nanjingnan Zhenjiangnan Changzhoubei Wuxidong Suzhoubei Kuanshannan Hongqiao Jiaxingnan Hainingxi Hangzhoudong
G7327
G7393
G7321
Beijingnan Tianjinxi 2 45 4 10
Hangzhoudong 10 333
Cangzhouxi
Hainingxi 2
2
22
2
3
Dezhoudon g
Jiaxingnan
4 3 2 3 2 4 4
Jinanxi
4
3
Hongqiao
5 3
4
Xuzhoudong 4
4 2 3 2
Kuanshannan 3 10
Nanjingnan
3 3
Suzhoubei
4 3 2
Zhenjiangnan
3 Wuxi
Changzhoubei Fig. 4. Train service network formed with all of trains in this case.
of the choices of trains from a station to another station without any transfer. For example, the weight of the edge from Nanjingnan to Hongqiao is 10, indicating that there are 10 trains stopping at both Nanjingnan and Hongqiao and we have 10 choices of trains to travel from Nanjingnan to Hongqiao without transfer. And we can also see that some of the edges have the weight of 1, meaning that the passengers have only train to take from the original station to the destination station of the edge if they do not want to transfer on another station.
4.3. Indices calculation and analysis According to the train service network constructed in Section 4.2, we calculate the indices of the network and analyze them. (1) Convenience degree of traveling We calculate the convenience for all the stations according to Eq. (5) in Section 3.1 and listed them in Table 3.
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Then the average convenience degree of the whole train service network is:
hki ¼ ¼
16 1 X ki 16 i¼1
1 ð48 þ 9 þ 15 þ 24 þ 16 þ 19 þ 23 þ 4 þ 6 þ 4 16 þ 3 þ 1 þ 7 þ 4 þ 1 þ 0Þ
¼ 11:375 according to data of the station convenience columns in Table 3. Similarly, the average maximal number of reachable stations is:
¼ m
n 1 X mi 16 i¼1
1 ð15 þ 14 þ 13 þ 12 þ þ 3 þ 2 þ 1 þ 0Þ 16 ¼ 7:500
¼
And the average actual number of reachable stations in this case is:
s ¼ ¼
n 1 X si 16 i¼1
1 ð13 þ 9 þ 10 þ 10 þ 8 þ 7 þ 4 þ 3 þ 2 þ 1 þ 1 16 þ 3 þ 2 þ 1Þ
(2) Transferring times and the travel time
¼ 5:186 Hangzhoudong is the terminal station in the down going direction. So the maximal reachable stations number and the actual reachable stations number are both zeros. We can see that the convenience degrees of Dezhoudong, Jinanxi, Xuzhoudong and Hongqiao are higher that of other stations. The convenience degree of Beijingnan k1 ¼ 48 is much bigger that the actual reachable stations number m1 ¼ 15, which means that the passengers have numerous choices of trains when traveling on the railway line, getting to the destination without transfer. It is very convenient to take Table 3 Convenience degrees of each station.
h1i h2i h3i h4i h5i h6i
a
b
c
48 9 15 24 17 19
15 14 13 12 11 10
13 9 10 10 8 8
h7i h8i h9i h10i h11i h12i
trains to travel from Beijingnan station along with the Beijing-Shanghai high speed railway wherever the passengers want to go. Hongqiao station has the same characteristic. The maximal numbers of reachable stations of Hongqiao and Jiaxingnan equal to the numbers of their actual reachable stations. It is to say that the passenger can get to the stations in the down going direction without transferring. But the convenience degrees are lower than that of Beijingnan and Hongqiao, for there are not much choices for the passengers when start their travel from these stations to a certain station. The maximal numbers of reachable stations of other stations are bigger than the numbers of their definite reachable stations, which indicates that the not all the stations are directly reachable. Dezhoudong, Jinanxi, Xuzhoudong and Nanjingnan have the higher convenience degrees. There are relatively more trains for passengers to choose when passengers start their journey from these stations to a specified destination. On the other hand, Tianjinxi, Zhenjiangnan, Kunshannan get the relatively lower convenience degrees, which cannot meet the massive requirements for traveling by trains, especially when the passengers want to get to their destinations without any transfer. They may add more trains, passing by the stations in the service plan to improve the convenience degrees of the stations and provide more choices for the passengers.
The travel time is connected to the transferring times directly according Eq. (6). So the primary step is to calculate the transferring times. In addition, the number of transferring times is related to the path length between the stations. So the paths lengths are calculated and listed in Table 4. From Table 4, we can see that the some of paths lengths are 2, meaning that the passengers travel between the OD stations must transfer once. Other lengths between the stations are 1, indicating that the passengers do not have to transfer when traveling. The average path length is:
L¼ ¼
a
b
c
23 4 6 5 3 1
9 8 7 6 5 4
7 4 3 4 1 1
h13i h14i h15i h16i
a
b
c
7 4 1 0
3 2 1 0
3 2 1 0
Note: In Table 3, the meanings of the characters on the first line and the numbers in parentheses are as follows. a. Convenience degree for traveling. b. Maximal number of reachable stations. c. Actual number of reachable stations in this case. h1i Beijingnan h2i Tianjinxi h3i Cangzhouxi h4i Dezhoudong h5i Jinanxi h6i Xuzhoudong h7i Nanjingnan h8i Zhenjiangnan h9i Changzhoubei h10i Wuxidong h11i Suzhoubei h12i Kuanshannan h13i Hongqiao h14i Jiaxingnan h15i Hainingxi h16i Hangzhoudong.
16 X 16i X X 2 2 dij ¼ dij 16ð16 1Þ i
2 ð16 þ 19 þ 16 þ 14 þ 14 þ 12 þ 11 þ 12 16ð16 1Þ þ 11 þ 10 þ 9 þ 7 þ 3 þ 2 þ 1Þ
¼ 1:308 Then the average transferring times is:
H ¼ L 1 ¼ 1:308 1 ¼ 0:308 Then we can find that all the passengers travel on the Beijing–Shanghai and Shanghai–Hangzhou railway lines will have to transfer 0.308 times averagely to get to their destinations. It reflects the service quality of the service plan on the two railway lines. From this index, we can conclude that the service quality is not so high. We suggest
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X. Meng et al. / Measurement 58 (2014) 221–229 Table 4 Paths lengths between stations.
h1i h2i h3ii h4i h5i h6i h7i h8i h9ii h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
–
1 –
1 1 –
1 1 1 –
1 2 1 1 –
1 1 1 1 1 –
1 1 1 1 1 1 –
1 1 1 1 2 1 1 –
1 1 1 1 1 1 1 1 –
1 1 1 1 1 1 1 1 1 –
1 1 1 1 1 1 1 1 1 1 –
1 2 1 1 1 1 1 2 2 1 2 –
1 1 1 1 1 1 1 1 1 1 1 1 –
2 2 2 2 2 2 2 2 2 2 2 2 1 –
2 2 2 2 2 2 2 2 2 2 2 2 1 1 –
1 2 2 1 1 1 1 2 2 2 2 2 1 1 1 –
Table 5 Passenger OD between Beijingnan and Hongqiao.
h1i h2i h3i h4i h5i h6i h7i h8i h9i h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
0
1026 0
161 75 0
176 81 12 0
586 271 39 42 0
388 180 26 28 97 0
844 391 56 61 212 137 0
196 91 13 14 49 32 73 0
264 122 17 19 66 43 98 21 0
338 156 22 24 85 55 126 27 37 0
383 177 25 28 96 62 143 31 42 54 0
108 50 7 8 27 18 40 9 12 15 17 0
1571 728 104 113 394 255 586 126 171 221 252 69 0
120 87 19 16 51 48 78 23 31 20 21 14 180 0
97 35 13 8 10 8 65 10 11 17 16 10 117 68 0
428 187 45 47 62 130 158 29 31 30 32 21 225 71 52 0
Table 6 The average running time of the trains between the stations (measurement unit: minute).
h1i h2i h3i h4i h5i h6i h7i h8i h9i h10i h11i h12i h13i h14i h15i h16i
h1i
h2i
h3i
h4i
h5i
h6i
h7i
h8i
h9i
h10i
h11i
h12i
h13i
h14i
h15i
h16i
0
71 0
95 58 0
108 127 49 0
97 272 53 25 0
236 367 212 128 65 0
369 559 349 212 139 106 0
681 600 542 473 300 233 41 0
542 656 598 366 177 201 66 56 0
495 683 434 329 190 189 75 83 27 0
585 724 666 412 201 247 162 124 46 41 0
312 721 261 233 208 143 65 221 165 18 183 0
482 781 502 348 216 223 113 181 87 59 44 19 0
627 926 647 493 361 368 258 326 232 204 189 164 25 0
643 942 663 509 377 384 274 342 248 220 205 180 41 16 0
378 958 679 305 279 214 136 358 264 89 221 196 57 32 20 0
that more trains should be added to the railway, or more stops should be set for the existing trains to reduce the transfer times of the passengers. OD data we got is shown in Table 5.
Then the down going passengers number is calculated out according to the data in Table 5.
X M ij ¼ 15560 i
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X. Meng et al. / Measurement 58 (2014) 221–229
In Eq. (7), T ij þ lij t is the total traveling time from one station to another. According to the data in timetable, we can get T ij . Then the lij can be got according to the data in Table 4. The traveling time between every OD station is calculated out and listed in Table 6. It is should be pointed out that different trains cost different running time in the same railway section. So we average the running time in each section of all different grade trains. When a passenger must transfer from a train to another, we take it for granted that he will transfer as early as possible. That is to say, he will transfer at the first station among all the stations he can choose. Ant it is assumed that each transfer will cost 120 min according to the historical data. For instance, there is no direct train between Beijingnan and Jiaxingnan. So passengers will transfer at Hongqiao station. The running time between Beijingnan and Hongqiao is 482 min and it will cost the train running at least 25 min from Hongqiao to Jiaxingnan. So the total traveling time from Beijingnan to Jiaxingnan is 482 + 120 + 25 = 627 min. According to the passenger OD data in Table 5, the running time in Table 6 and Eq. (7), we calculated out the total traveling time of all the passengers sent from each station, presented in Table 7. The average traveling time of the whole service plan is:
T ¼ ð2325237 þ 1639602 þ 176623 þ 118057 þ 227800 þ 164325 þ 170317 þ 53356 þ 37892 þ 26013
Table 8 Ei and zi values of each station.
h1i h2i h3i h4i
(1) Stations clustering coefficients The number of neighbor stations of station i(zi ) and the actual number of edges connected to station i(Ei ) are listed in Table 8.
h5i h6i h7i h8i
Ci h1i h2i h3i h4i
Ei
zi
60 79 79 57
12 14 14 12
h9i h10i h11i h12i
Ei
zi
64 79 65 33
12 14 12 9
h13i h14i h15i h16i
0.9697 0.8681 0.9848 0.9167
h13i h14i h15i h16i
Ei
zi
84 6 6 35
16 4 4 10
0.8681 1 0.9231 0.8681
Ci h5i h6i h7i h8i
0.9091 0.8681 0.8681 0.8636
Ci h9i h10i h11i h12i
Ci 0.7 1 1 0.7778
According to Eq. (3) and Table 8, the clustering coefficient of Beijingnan station is:
C1 ¼
2E1 2 79 ¼ 0:8681 ¼ z1 ðz1 1Þ 14 ð14 1Þ
In the same manner, the clustering coefficients of all the station are computed out and listed in Table 9. According to Eq. (4), the clustering coefficient of the whole network is:
C¼
It equals to 5.3 h. Its meaning is that the average traveling time of all the OD passengers is 5.3 h, which is determined by the train service plan. We calculate out the index based on the train service network transformed from the train service network. This index of the train service plan reflects the train service plan is with high quality from the point of view of traveling time control. Beijing–Shanghai high speed railway is 1318 km long, and Shanghai– Hanzhou high speed railway is 169 km long. The average traveling time is controlled to be 5.3 h not only because of the high speed of trains, but also the high quality of the train service plan.
zi 14 11 13 14
Table 9 Clustering coefficients of each station.
þ 28520 þ 9523 þ 22122 þ 3360 þ 1040Þ=15560 ¼ 322 min
Ei 79 55 72 79
¼
16 1 X Ci 16 i¼1
1 ð0:8681 þ 1 þ 0:9231 þ 0:8681 þ 0:9091 16 þ 0:8681 þ 0:8681 þ 0:8636 þ 0:9697 þ 0:8681 þ 0:9848 þ 0:9167 þ 0:7 þ 1 þ 1 þ 0:7778
¼ 0:8991 The clustering coefficient of the train service network is 0.8991, which is very close to 1. So the connecting degree of the entire network is prominent. Tianjinxi, Jiaxingnan and Hainingxi have the high connecting degrees, for they have fewer neighbors than other stations, not because they have more connecting edges. Contrary to popular misconception, Hongqiao has the lowest connecting degree. Low connecting degree indicates the station does not connect to the other stations closely. Managers should add more trains or set more stops of trains at Hongqiao station to improve the clustering coefficient to match the great travel demand.
Table 7 Total traveling time of all the passengers sent from each station (measurement unit: minute). M ij ðT ij þ lij tÞ h1i h2i h3i h4i
2,325,337 1,639,602 176,623 118,057
Mij ðT ij þ lij tÞ h5i h6i h7i h8i
227,800 164,325 170,317 53,356
Mij ðT ij þ lij tÞ h9i h10i h11i h12i
37,892 26,013 28,520 9523
M ij ðT ij þ lij tÞ h13i h14i h15i h16i
22,122 3360 1040
X. Meng et al. / Measurement 58 (2014) 221–229
5. Conclusion On the strategic planning level, a railway train service plan evaluation problem is investigated based on the complexity theory. The train service plan is transformed to a complex network according to a constructing method. The key indices of the complex network are listed and analyzed their representativeness of the train service plan. On the basis of the analysis of the train service network, we evaluate the train service plan. The conclusions drawn with the evaluation results of the computing case are as follows: (1) More trains, stopping at Tianjinxi, Zhenjiangnan, Kunshannan should be added into the service plan, or more stops should be set at these three stations to improve the convenience degrees of the stations and provide more choices for the passengers. (2) We suggest that more trains should be added on the railway, or more stops should be set for the existing trains to reduce the transfer times of the passengers. (3) The train service plan is with high quality from the point of view of traveling time control. (4) Managers should add more trains or set more stops of trains at Hongqiao station to improve the clustering coefficient to match the great travel demand. The suggestions above based on the evaluation results indicate that the results can give some essential supporting information for the railway managers. They can take the results as an important reference when design the train service plan. So the evaluation method proposed in this paper has much vale. It is a different viewpoint to evaluate the train service plan, which has the characteristics of briefness practicability. Evaluating results can be attained via the method, which is very practical and valuable for decision makers of the railway operation bureaus. Future research into the train service plan is to optimize the train service plan based on the developing theories on complex networks, such as complex network control and optimization theory. Since the train service plan can be transferred into the train service network, we can control and optimize the train service network. Then we transfer the train service network back into the train service plan, reaching the goal to optimize the train service plan, especially in railway emergencies. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grant: 61263027),
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the New Teacher Project of Research Fund for the Doctoral Program of Higher Education of China (Grant: 20126204120002) and the Natural Science Foundation of Gansu Province – China (Grant: 1310RJZA068). The authors wish to thank anonymous referees and the editor for their comments and suggestions. References [1] M.J. Lighthill, J.B. Whitham, On kinematic waves. I: Flow movement in long rivers; II: a theory of traffic flow on long crowded roads, Proc. Royal Soc. Ser. 22 (1995) 281–345. [2] P.I. Richads, Shockwaves on the highway, Oper. Res. 4 (1956) 42–51. [3] C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transp. Res. Part B 29 (1995) 277–286. [4] C. Bianca, Mathematical modeling of crowds dynamics: complexity and kinetic approach, Nonlinear Stud. 19 (2012) 371–380. [5] C. Bianca, C. Dogbe, A mathematical model for crowd dynamics: multi-scale analysis, fluctuations and random noise, Nonlinear Stud. 20 (2013) 349–373. [6] C. Bianca, V. Coscia, On the coupling of steady and adaptive velocity grids in vehicular traffic modelling, Appl. Math. Lett. 24 (2011) 149– 155. [7] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle, J. Royer, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci. 18 (2008) 1269–1298. [8] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73 (2001) 1067–1141. [9] B. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer, Berlin, New York, 2004. [10] C. Dogbe, Nonlinear pedestrian-flow model: uniform wellposedness and global existence, Appl. Math. Inf. Sci. 7 (2013) 29–40. [11] K. Kottenhoff. Evaluation of passenger train concepts – practical methods for measuring travellers’ preferences in relation to costs, in: 3rd KFB Research Conference in Stockholm, 2000. [12] R. Albert, A.L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47–97. [13] D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 93 (1998) 440–442. [14] A.L. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. [15] L. Benguigui, M. Daoud, Is the suburban railway system a fractal?, Geograph Anal. 23 (1991) 362–368. [16] P. Sen, P. Dagupta, A. Chatterjee, et al., Small-world properties of the Indian railway network, Phys. Rev. 67 (2003) 036106. [17] V. Latora, M. Marchiori, Is the Boston subway a small-world network?, Physica A 314 (2002) 109–113 [18] A.S. Katherine, M.H. Lisa, Stations, trains and small-world networks, Physica A 339 (2004) 635–644. [19] W. Zhao, H. He, Z. Lin, et al., The study of properties of Chinese railway passenger transport network, ACTA Phys. Sin. 55 (2006) 3906–3911. [20] X. Meng, L. Jia, J. Xie et al., Complex characteristic analysis of passenger train flow network, in: Proceeding of CCDC2010, 2010, pp. 2533–2536. [21] X. Meng, L. Jia, Y. Qin, et al., Vulnerability analysis of passenger train flow network in emergency, Trans. Beijing Inst. Technol. 32 (2012) 80–83. [22] T. Kunimatsu, C. Hirai, N. Tomii, Train timetable evaluation from the viewpoint of passengers by microsimulation of train operation and passenger flow, Electr. Eng. Jpn. 181 (2012) 51–62.