Xiaobing Ding，Hua
Hu，Zhigang Liu
College of Urban Rail
Transportation, Shanghai University of Engineering Science, Shanghai, 201620,
China
Correspondence should be
addressed to Xiaobing Ding; dxbsuda@163.com
Abstract: The suburban line connects the suburbs and the city centre; it is of
huge advantage to attempt the expressslow mode. The passengers’ average travel
time is the key factor to reflect the level of rail transport services,
especially under the expressslow mode. So it is important to study the
passengers’ average travel time under expressslow, which can get benefit on
the optimization of operation scheme. First analyses the main factor that
affects passengers’ travel time, and then mines the dynamic interactive
relationship among the factors. Second, a new passengers’ travel time evolution
algorithm is proposed after studying the stop schedule and the proportion of
express/slow train, and then membrane computing theory algorithm is introduced
to solve the model. Finally, Shanghai metro line 22 is set as an example to
apply the optimization model to calculate the total passengers’ travel time;
the result shows that the total average travel time under the expressslow mode
can save 1 minute 38 second; the social influence and value of it are very
huge. The proposed calculation model is of great help for the decision of stop
schedule, and provides theoretical and methodological support to determine the
proportion of express/slow trains, improves the service level, enriches and
complements the rail transit operation scheme optimization theory system.
1. Introduction
Starting from the beginning of the 21st century, the
urban rail transit in China entered a rapid period of development, and the
urban development grows faster, the urban traffic congestion of metropolises
has become more and more serious, people's living costs continue to increase,
so the advantage of suburban is very obvious, and the suburban passenger flow
surges, it is of great market demand to attempt expressslow mode between city
and suburbs. The suburban line is longer than the ordinary one, and the
passenger flow has more obviously spatial and temporal distribution. W. Lin et
al. [1] found that more
flexible stop schemes can be determined according to the passenger flow under
this mode, including the proportion of express/slow trains. Jin et al. [2] In this mode, the travel time of express passenger decreases, while the
slow ones will prolong because of the express trains’ overtakes. The total
travel time of passengers is an important index to reflect the services level
of rail transport, so it is of good practical significance to study the
optimization modelling of the travel time under expressslow mode.
Liu et al. [3] established a multiobjective
01 programming model, in which three
optimization goals are considered. The construction of the model laid the
foundation for the development of the train plan, which has a good reference
value for the study of this paper. Dundar
et al. [4] introduced the Dijkstra algorithm to solve the shortest path problem, and add angle
increasing function in the cost matrix to determine priorities.The minimal complexities of decision trees representing
rules defined for features and functionalities of online learning community, the feasible
space for complexities of decision trees can only be implicitly reflected by
relationships of sub tables and associated objective functions due to the great
number of possible trees.
Shi et al. [5] The
railway transport market was
subdivided from the
passenger travel time, travel expenses, convenience and comfort, and several
other aspects, the optimized the model was established and the operation scheme
was developed at last, this study was of good academic value. Casteili [6] studied the passenger flow,
passenger flow characteristics and direction characteristics between the lines,
an optimization method of train schedules was proposed based on features of each
passenger line. Rachel C [7] discussed the passenger transfer
problem in the urban rail network, assumed the minimum waiting time as the
goal, established a mixed integer programming model, and obtained the coordination
train timetable.
The basic problems such as studying the formulation
rail transit line planning, establishing the operation scheme of multigoal programming model
according to the general situation of dedicated passenger line and highspeed
rail line planning from passengers’ travel time and cost benefits
were studied. It
involves some new concepts, such as the calculation model of passenger transfer
fatigue recovery time, the consumption choice model of passenger lines, the
calculation of the number of trains etc.. Through the analysis and calculation above, programming model,
integrating the railway operation scheme optimization problem was determined, and at last, a high feasibility optimization method of
operation scheme was proposed.
This paper intends to study the dynamic quantitative
relation between the passengers’ total travel time and train stop scheme and
express/slow trains ratio in the
expressslow mode, then aim to establish the optimization model of passengers’
travel time.
2. The
analysis of the factors that affect the passengers’ travel time
2.1The analysis of the factors affect the travel time
The passengers on suburban lines can be divided into
two categories: the slow passengers, who at least get down at one common
station, and the other one is express passengers, in which the OD is usually
large passenger stations. The slow train passengers can get more satisfied
service with single station stop scheme, while it will affect the travel time
of express passengers[8]; when under the
expressslow mode, it will increase the waiting time of slow trains, but for
the express passengers, they can reach their destination station more quickly.
According to the actual situation, the
passengers of the same direction continued to reach the station one after another in section.there are kinds of trains for passengers to select. We can assume that the amount of passengers
arriving at the station obeys the Poisson process which is subject to the
intensity of, the probability of taking the trainis, using to indicate the passenger flow
that taking in trainin section.
，in whichare units
independent to each other. So using to express the arrival time of the nth passenger
to reach the train, the
cumulative distribution function and probability density are shown in formula（1）and（2）respectively.
（1）
（2）
Assuming that the amount of passengers
arriving at the station is n, time
that need to wait at the station can be expressed as a mathematical expectation
shown in formula（3）:
（3）
So let the departure time for train to be, the reasonable train departure interval can be expressed in formula（4）:
（4）
We can assume the amount of passengers getting on the trainisin the period, the time for the passenger i to
arrive at the station is, so the sum of the waiting time for traincan be described as in formula（5）:
（5）
So the mathematical expectation of waiting time can be shown in formula（6）:
（6）
Through the analysis above, the train
departure interval is very important, according to which the reasonable waiting
time can be obtained, so the mathematical expectation value can be expressed as. This calculation
method of waiting time is of great value for travel time optimizes research, the other main factors that also affect the passengers’ travel time are as follows:
（1）Station passenger flow
Obtaining the accurate passenger flow of rail transit
line especially prediction is very complex, which will influence stop scheme,
etc. Besides, passenger flow changes dynamically, the stop scheme and ratio of
express/slow trains will react to
passenger flow, forms a dynamic interaction relationship. Compared with
highspeed railway, the traffic density is still relatively large; passengers’
arrival can be regarded as independent of the train schedule, random
distribution [9]. When the traffic interval is smaller, the average
waiting time of the passengers is close to half of the interval. In
expressslow mode, the average waiting time by express trains can be expressed as formula (7):
(7)
In which：—operation
time，min；
—the quantity of trains that involved in
operation.
While the
waiting time by slow Trans can be expressed as formula (8):
(8)
In which： —the sharing rate of passengers whose original
plan was to take the slow trains, its value：
—Passenger flow of section i.
（2）The stop scheme
When the train enters the station and stop, it
includes the process of acceleration and deceleration, from the theoretical
analysis; the more stations the Tran stops, the more additional time
consumption due to stop and start [10]. But when
considering stop more stations, the express passengers can reach their
destinations with transferring less slow trans by saving some time, in fact,
the more stations the trans stop the better, so suitable stop scheme should be
made according to the passengers’ flow, aiming to shorten the travel time. The
common stop scheme can be shown as figure 1[11].
Figure1: the stop scheme of Shanghai Metro line 16
Passengers’ travel time is an important indicator of
the services level of urban rail transport, so how to save the total
passengers’ travel time is an important goal to study the optimization of stop
scheme. The passenger’ travel time can be divided into: waiting time, train
running time, stop time at station, if the transfer is included at the same
line[12], transferring time also needs to be considered
between the express and slow trains.
①when only the slow trains are used, the passengers’
total travel time can be expressed as formula (9):
（9）
In which：——the
passenger flow between station and station,
person;
——the pure running time between station and station, minute;
——the stop time at station, minute;
——the additional time of start and stop consumption at
station, minute;
——Train turnaround time, minute。
②while under the express/slow mode，usingas the
quantity of passengers of express trans，using as the quantity of passengers of slow trans，so the total
waiting time can be expressed as formula (10):
(10)
③the total time passengers
on rail transit trans, in which the additional time of start and stop are
included as formula(11).
(11)
④so the total time of passengers can be
expressed as formula(12):
(12)
2.2 Analysis of passengers’ total
travel time under expressslow mode
Through the study in
section 1.1, the passengers’ total travel time can be divided into four parts:
waiting time of passengers, the transit time, and the time trains stop at
stations, transfer time (if exists), the actual situation should also contain a
passenger station time, because it is related to actual distance, so it is
usually regarded as a constant. We can establish the time amusement model
with the base on the four parts.
3. Optimization model of
total passengers’ travel time
The total time of
passengers mainly consists of all passengers’ time on the train and the time
waiting at the station, while travel time of express and slow trains differs much, it is too hard to calculate
any one separately; besides, it is not easy to compare and solve [13]. The average total travel
time of rail transit passengers is studied, and the transit time is changed in
the transit time compared with the existing station.
Firstly, the time
consumption matrix is studied under the expressslow mode:
Suppose there are five
stations numbered from A to E, including A, C, D three stations express trains stop, and the range of research is limited
from A to E which can be shown in table 1. The proportion of train and the stop
scheme have an effect on the T value, while if the stop scheme is determined
first, the proportion will also affect T value; the three parameters have
dynamic interactions.
Table.1 The running time matrix in certain conditions

A

B

C

D

E

A

T_{AA}

T_{AB}

T_{AC}

T_{AD}

T_{AE}

B


T_{BB}

T_{BC}

T_{BD}

T_{BE}

C



T_{CC}

T_{CD}

T_{CE}

D




T_{DD}

T_{DE}

E





T_{EE}

To obtain accurate T_{ij}, traction calculation is called for according to the vehicle and the related
technical parameters, combining with the rail transit line of horizontal and
vertical curve data. It involves many parameters and related quantities. During
the course of the calculation of train running time, it is approximately
divided into accelerating time, uniform operation time, deceleration time, and
the running time of each interval can be calculated by equation (13).
(13)
In which: ——the
maximum speed;——the
length from section i to j;
——the acceleration
of trains leaving stations;
——the
deceleration of trains entering the
stations.
The spacing of suburban
lines between stations is usually larger than the ordinary ones; the train
speed can reach the maximum value in each interval, so each running time of
section can be calculated from (13).
The total passengers’ travel
time can be expressed as the weighted mean product of and by formula (14) and minus
time can be shown in formula
(15).
(14)
(15)
The subjects to the model are as formula (16) to formula (19):
(16)
(17)
(18)
(19)
In which： r—— the amount of stations left except the section from station i to station j;
——the
amount of passengers from station i to
station j;
——transport capacity;
——Time of
cycle operation diagram;——amount of trains from station i to station j;
——the total
amount of cycle;
——the amount of passengers who get on at the former station p and get off at the next station q;
——all types
of trains;——the
amount times of stop of trains from station i to station j;
——the frequency of corresponding trains；
、、——average stop
time , stop time, and the additional time due to start/stop.
Formula (16) ensures that the transport capacity is greater than the passenger
demand of various categories of trains; Formula (17) ensures that the train passengers must not exceed the total load ratio
of the train; Formula (18) represents the lower and upper bound of amount of train stops; Formula (19) ensures the average attendance rate to meet a certain threshold.
4. Verification and
analysis of the optimization passengers’ total travel time
4.1 Analysis of an
example
The length of Shanghai Metro
Line 22 is 56.4 kilometers from Shanghai South Railway Station to Jinshanwei
station, passing by Xuhui District, Minhang District, Songjiang District and
Jinshan District [14], with a total
of eight stations, the stations and route map are shown in figure 2:
Figure.2 The route map of Shanghai metro line 22
During the
course of modelling, it is assumed that each station is of skip ability, and
each station has the potential as an express stop station. The steps of the
algorithm are as follows:
Step 1
Obtain the related information of Shanghai Metro line 22, as shown in table 2:
Table.2 The related information of Shanghai Metro line
22
section

length（km）

Running
time（s）

stopping
time（s）

South shanghai ~chunshen

8.90

239.0667

65

chunshen~xinqiao

5.70

191.869

45

xinqiao~chedun

6.70

206.618

40

chedun~yexie

8.50

233.167

45

yexie~tingling

5.10

183.019

50

tingling~jinshanyuanqu

8.10

227.267

50

jinshanyuanqu~jinshanwei

11.20

272.989

55

Step 2 calculating the running time interval matrix,
based on the operation data of each station. The acceleration and deceleration
of the trains are about 0.41m/s^{2} in normal condition, the maximum
speed is 27.8m/s. it is required 67.8s to the maximum speed. The section
running time of South Shanghai Railway Station to Chun shen:
Step 3 after calculating the time matrix in Step 2,
the corresponding OD matrix is required for next study, the OD matrix is shown
in table 3:
Table 3 the OD
matrix of Metro line 22
Station

South shanghai

Chun
shen

Xin
qiao

Che
dun

Ye
xie

Ting
ling

Jinshan
yuanqu

Jinshan
wei

Total

South shanghai

0

24

98

154

18

24

15

147

480

Chun shen

67

0

31

57

8

11

34

14

222

Xin qiao

59

31

0

54

2

0

3

27

176

Che dun

117

53

11

0

8

20

27

38

274

Ye xie

47

8

9

34

0

2

5

1

106

Ting ling

68

12

21

17

0

0

3

0

121

Jinshan yuanqu

34

17

14

38

6

19

0

14

142

Jin shan wei

106

61

96

118

51

36

94

0

562

Step 4
calculating the average of O_{i}D_{j}T_{ij}
When calculating the interval time matrix, the
stopping time in each station is also needed, during weekday peak time, the
operation interval is 32 minutes [15]. If properly
divided, the passengers entering stations can be assumed that it obeys negative
exponential distribution. The waiting time of passengers can be approximated to
be half of the interval, just 16 minutes.
The corresponding interval running time matrix can be
drawn as shown in table 4. Using the same calculation method as step (2), the
running time matrix of each section can be obtained under the 1:1, and the
corresponding ODT matrix can be obtained at the same:
Table 4 The corresponding interval running time matrix
ODT(s)

South shanghai

Chun
shen

Xin
qiao

Che
dun

Ye
xie

Ting
ling

Jinshan
yuan

Jinshan
wei

South shanghai

0

7297.68

53012.12

121284.2

19183.14

31170

23640.3

279889.5

Chun shen

0

0

7342.97

27558.93

6093.28

10941.48

43246.3

22399.16

Xin qiao

0

0

0

13317.48

1049.58

0

3105.24

36802.89

Che dun

0

0

0

0

2225.36

10223.4

21287.88

42424.34

Ye xie

0

0

0

0

0

466.04

2551.45

838.28

Ting ling

0

0

0

0

0

0

831.81

0

Jinshanyuanqu

0

0

0

0

0

0

0

4591.86

Jin shan wei

0

0

0

0

0

0

0

0

Sum all elements of ODT matrix and the OD, so
the average travel time is 2335 s，when facing the same passenger flow, the average
spending time is 2335s in express/slow mode, that is 39min 2s, compared to the
ordinary mode 40min 28s, per capita reduced time by 1min 26s.
4.2 The analysis of optimization results
For single passenger, 1min26s is not of great
importance, but more important to the peak period, the total passengers’ travel
time is reduced, the main advantage reflected can be described as follows:
①
From the GDP calculation: According to the 2015 GDP data of Shanghai: 2015
Shanghai it has reached 24964.99 billion yuan [16], according to the resident population, the average GDP reached 10.31
million yuan. Due to the express/slow mode, when one train is used, the average time 1min26s (86s), can be converted into
economic benefits:
Setting
the time interval 6 minutes and the daily total operating time is 5:5022:00,
the total time is 970 minutes, and there are 320 trips a day, the total
economic benefits of conversion [17]:
②If
passengers cannot get their company on time, there may be subject to penalty
units or absenteeism in the workplace, the professional evaluation and
promotion, etc.
③For
business people, if late, it will affect signature of major contract, and
directly result in the loss, or greater economic benefits etc..
From the stop scheme calculation, it can save
2min59s, if the stop plan can be further optimized, it will attract more passengers
OD; and cause more influence on the stop time, and the average travel time can
be further reduced to 4min 38s. It is of great importance for the peak time
goingwork passengers, for the overall rail transport system, the passengers’
total travel time is reduced. Through the calculation and analysis, the train stopping
at the express station will have more effect to the travel time of passengers. Because
of the passenger flow of the express station is large, when sum all the passengers’
travel time, calculate the average value, the travel time is less than the ordinary
stop scheme [18, 19]. If change the ordinary stations of which the passenger flow
is large into express station, the reduced time will be more obvious. If taking
the ratio of express/slow trains and position of overtaking into consideration, the optimization results will be more excellent.
5. Conclusions
The next 1020 years is an important development
period in which China's economy will reach a higher stage, the 13th FiveYear
plan and the 14th FiveYear plan will more focus on the construction of the
city, urbanization will further accelerate, and promote the passengers. In this
paper, a method of travel time optimization model is proposed, and the optimization
object is defined as the minimum mean value of ODT. The smaller the average
value of ODT, the less time passengers spend on the trip, and better benefits
the passengers. Under the condition of imbalance flow, the express/slow mode
can reduce passengers’ total travel time, and improve the satisfaction of
passengers. While during peak stage, it is not suitable to use express/slow mode. The demand for transport capacity and
service level will be improved with the rapid increase of the passenger flow in
the suburbs. The optimization result will be better if the passengers’ travel time,
the proportion of trains and the carry capacity are considered. It is of great
help to the operation practice of urban rail enterprise.
Competing Interests
The authors declare that
they have no competing interests.
Acknowledgments
This work was supported in part by the National
Natural Science Foundation under Grants 71601110 and 61672002, the Key Technologies of Integrated Test
and Authentication Platform for Urban Rail Transit under Grant 2015BAG19B0228,
and the Detection and Management of Risk and Control Technology of Disaster
Prevention System for Urban Rail Transit under Grant 2016YFC0802505.
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