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Transportmetrica A: Transport Science
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Measuring and enhancing the connectivity
reliability of a rail transit network
Jie Liu, Paul M. Schonfeld, Shuguang Zhan, Qiyuan Peng & Yuhong Liu
To cite this article: Jie Liu, Paul M. Schonfeld, Shuguang Zhan, Qiyuan Peng & Yuhong
Liu (2021): Measuring and enhancing the connectivity reliability of a rail transit network,
Transportmetrica A: Transport Science, DOI: 10.1080/23249935.2021.1965241
To link to this article: https://doi.org/10.1080/23249935.2021.1965241
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TRANSPORTMETRICA A: TRANSPORT SCIENCE
https://doi.org/10.1080/23249935.2021.1965241
Measuring and enhancing the connectivity reliability of a rail
transit network
Jie Liu a,b, Paul M. Schonfeldc, Shuguang Zhana,QiyuanPeng
aand Yuhong Liua
aSchool of Transportation and Logistics, Southwest Jiaotong University, National United Engineering
Laboratory of Integrated and Intelligent Transportation, Chengdu, People’s Republic of China; bFaculty of
Transportation Engineering, Kunming University of Science and Technology, Kunming, People’s Republic of
China; cDepartment of Civil and Environmental Engineering, University of Maryland, College Park, MD, USA
ABSTRACT
A model for estimating the connectivity reliability (CR) of a rail tran-
sit network (RTN) is proposed that considers the passengers’ travel
behavior. Passengers choose acceptable paths whose trip times are
below the passengers’ acceptable trip time. An origin-destination
station (OD) pair’s CR is defined as the probability that at least one
acceptable path is connected between that OD pair. The RTN’s CR
is defined as the average value of CR for each passenger on the
RTN. A model is proposed to maximize an RTN’s CR by adding trains,
subject to constraints on operational cost, allowable track capac-
ity and available vehicles on each line. The model is solved with
a multi-population genetic algorithm (MPGA). The model applica-
tion in Chengdu’s RTN shows that adding trains corresponding to
the optimized solution with an operational cost constraint, not only
increases the net benefit of RTN operations, but also enhances the
RTN’s CR.
ARTICLE HISTORY
Received 18 March 2021
Accepted 2 August 2021
KEYWORDS
Rail transit network;
connectivity reliability;
passengers’ travel behavior;
binary decision diagram
1. Introduction
Travelers expect transportation to be reliable as well as efficient. However, uncertainties
in a transportation system on both the demand and supply sides affect the system’s reli-
ability and performance in daily operations (Szeto and Wang 2016; Liu et al. 2020). Thus,
the unreliability of a transportation system can be a major problem. Operators manag-
ing transportation networks must deal with uncertainties, such as disruptions and demand
fluctuations in daily operations (Li et al. 2015;Zhanetal.2021a,2021b). Therefore, they
expect to improve the reliability of the existing transportation networks and establish reli-
able transportation networks in network planning (Koulakezian et al. 2015; Szeto and Wang
2016).
To analyze the reliability of transportation networks, connectivity reliability (CR), travel
time reliability and capacity reliability have been analyzed (Chen et al. 1999; Iida 1999;Kato
and Uchida 2018). Although the reliability of transportation networks has interested many
CONTACT Shuguang Zhan shuguangzhan@my.swjtu.edu.cn School of Transportation and Logistics,
Southwest Jiaotong University, National United Engineering Laboratory of Integrated and Intelligent Transportation,
Chengdu 610031, Sichuan, China
© 2021 Hong Kong Society for Transportation Studies Limited
2J. LIU ET AL.
researchers, most of the relevant studies to date have analyzed the reliability of road net-
works (Zhang, Zhong, and Luo 2019). Only about 8% of transportation network reliability
studies involved RTNs (Wang et al. 2015).
The connectivity of transportation networks is crucial in meeting demands (Mishra,
Welch, and Jha 2012). Only if a transportation network is connected can persons and vehi-
cles move on it. In addition, network connectivity has been used to study the system
performance of transportation networks (Hadas and Ceder 2010). Therefore, CR is very
important for transportation network operations and passenger travel.
Rail transit is the main mode of public transportation in some cities because of its high
speed, low carbon emissions, and high capacity. Many passengers tend to travel on rail
transit in crowded cities due to its advantages. However, the connectivity of a rail transit
network (RTN) is affected by some disturbances, such as equipment failures, severe weather
and operational accidents, which decrease the service quality of RTN and cause rail tran-
sit passengers to shift to other transportation modes. Therefore, an RTN’s CR should be
measured and enhanced to cope with these disturbances.
Our contribution is to develop a model for measuring and enhancing an RTN’s CR by
considering the passengers’ travel behavior. Passengers choose acceptable paths (Akgün,
Erkut, and Batta 2000; Tan et al. 2007; Milakis et al. 2015; Liu et al. 2020) to travel between
origin-destination station (OD) pairs, since the trip times of acceptable paths are below the
passengers’ acceptable trip time. As long as there is at least one connected acceptable path
between an OD pair, that OD pair is connected. An OD pair’s CR is defined as the probabil-
ity that at least one acceptable path is connected between that OD pair. The RTN’s CR is
defined as the average value of CR for each passenger. A model is developed for maximiz-
ing an RTN’s CR by adding trains. It is constrained by the operational cost of adding trains,
the allowable track capacity and available vehicles on each line. The allowable track capac-
ity is indicated by the lines’ maximum train frequencies. The available vehicles on each line
constraints the maximum number of trains that can be added on lines. Optimized solutions
for maximizing an RTN’s CR with different operational cost constraints are obtained. This
paper is the first to estimate the benefit of adding trains from both the passengers’ and
operator’s perspectives. The benefit of adding trains from the operator’s perspective is the
benefit of CR enhancement, which equals the increase in fare revenue due to an increase in
passengers who can travel on the RTN. The passengers’ generalized travel cost (GTC), which
is the sum of the fare and monetary value of perceived trip time, is reduced by adding trains.
Thus, the benefit of adding trains from the passengers’ perspective is the reduction in aver-
age GTC per passenger multiplied by passengers who can travel on the RTN during the
analysis period. The results show that adding trains corresponding to the optimized solu-
tion can increase the net benefit of RTN operations when the operational cost constraint is
considered.
The remainder of this paper is organized as follows: research on the CR of transportation
networks is reviewed in section 2. The methodology in section 3 presents the model for
measuring an RTN’s CR, the model for maximizing an RTN’s CR by adding trains on routes.
The algorithm for measuring an RTN’s CR and obtaining the optimized solution for max-
imizing the CR is introduced in section 4. The proposed model and method are applied
to Chengdu’s RTN in section 5. Finally, some conclusions of the study are presented in
section 6.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 3
2. Literature review
The existing studies on road networks’ reliability were mainly limited to three aspects, CR,
travel time reliability, and capacity reliability. The concept of time reliability was commonly
associated with the variability of time (de Oliveira, da Silva Portugal, and Junior 2016; Chen
et al. 2018). The capacity reliability of transportation networks was defined as the probabil-
ity that a certain volume of traffic could be accommodated at a required service level (Chen
et al. 1999; Chen et al. 2000). The original intent for researching the transportation networks’
CRs was to solve the problem of network paralysis due to natural disasters. The network’s CR
was first proposed in Mine and Kawai (1982). They assessed the probability that the network
could maintain connectivity when it was damaged. Thus, the CR of transportation networks
was usually defined as the probability that the nodes of the transportation network were
connected (Wakabayashi and Iida 1992; Iida 1999;MaandZhou2015; Hosseini and Wad-
bro 2016). Travelers could travel from the origin to the destination only when the origin and
destination pair was connected and vehicles could run on it. Therefore, CR represented the
basic reliability of transportation networks and the connectivity of a network was used to
reflect its performance (Mishra, Welch, and Jha 2012).
Different methods, such as statistical methods, simulation methods were used for com-
puting network reliability (Hosseini and Wadbro 2016; Muriel-Villegas et al. 2016; Feng, He,
and Li 2019). Time reliability on transportation networks was evaluated using statistical
methods according to travelers’ trip time obtained from Automatic Fare Collection data,
Global Positioning Systems data and Automatic Vehicle Location data (Rakha, El-Shawarby,
and Arafeh 2010; Woodard et al. 2017; Liu et al. 2020). Simulation methods were commonly
used to measure CR and capacity reliability of transportation networks (Chen et al. 2002;
Jiang et al. 2013;MaandZhou2015; Guidotti, Gardoni, and Chen 2017). The connectiv-
ity state of an OD pair was represented as ‘0’ or ‘1’ in the early studies. ‘0’ or ‘1’ represented
that an OD pair was in the disconnected or connected state, respectively. Researchers found
that the connectivity state of OD pairs could be easily and effectively expressed by apply-
ing complex network theory (Barabási and Albert 1999). Therefore, simulation methods
based on complex network theory were widely applied in many studies to analyze the CR of
transportation networks. However, it was common that the probabilities of network inter-
ruptions were ignored when measuring CR with simulation methods based on complex
network theory. The reason was that it was difficult to obtain interruptions or disturbances
data from transportation operators.
Networks for urban rail transit (Wang and Xu 2009), road (Jenelius, Petersen, and
Mattsson 2006) and high-speed rail (Zhang et al. 2016) were represented as graphs to
analyze their CR. The network efficiency (the mean of the reciprocal of the shortest dis-
tance between all nodes), the maximal connected subgraph, clustering coefficient and
other derivative indicators were used by researchers to measure the connectivity of trans-
portation networks (Jiang et al. 2013; Liu et al. 2020). The CR of transportation networks
was evaluated with the changes in network connectivity indicators when the networks
were damaged. Network damage simulations removed stations or links. The effects of the
removal of stations on Guangzhou’s subway network were analyzed by simulating random
and deliberate attacks on stations (Liu and Song 2010). The resilience of 17 transporta-
tion networks in terms of connectivity was analyzed by simulating the removal of nodes
and links (Jiang et al. 2013). Other similar studies were also conducted: the robustness
4J. LIU ET AL.
of Beijing’s subway was assessed when removing stations (Yang et al. 2015). Zhang et al.
(2018) assessed the reliability of a bus network and Zhou, Wang, and Sheu (2019) evalu-
ated the connectivity of road networks when those networks were attacked deliberately
and randomly.
The above studies thoroughly analyzed the influence of removing links and nodes on the
topologies of transportation networks. They focused on connectivity issues when networks
became disconnected due to the removal of links or nodes. The critical nodes and links
which have major effects on the topologies of transportation networks were identified in
the above studies. Some indicators, such as the degree of nodes, as well as the betweenness
of links and nodes, were widely used to identify the critical links and nodes in a transporta-
tion network (Scott et al. 2006). Although the above studies yielded valuable guidelines on
measuring the transportation networks’ CR as well as identifying critical links and stations
for network structures, they did not propose models for improving the networks’ CR, while
the probabilities of link interruptions and node interruptions were neglected.
In the literature on reliability improvement for transportation networks, new link con-
struction (Jenelius and Cats 2015) and capacity expansion (Shariat Mohaymany and Babaei
2013), as well as network protection (Miller-Hooks, Zhang, and Faturechi 2012), were
applied. Lubis et al. (2010) evaluated the CR enhancement for a road network after apply-
ing several capacity expansion and maintenance scenarios. They found that periodic and
efficient maintenance on road links might yield a greater improvement in the CR of a road
network than a moderate capacity expansion of links. Chu and Chen (2016) proposed a two-
stage stochastic programming model to identify the optimal protection plan which could
maximize the CR for highway networks. A stochastic programming model was also applied
by Miller-Hooks, Zhang, and Faturechi (2012) to optimize the resilience of a freight trans-
portation network. The results showed that the stochastic programming model had the
potential for improving the reliability of highway networks that were subject to disasters.
Some methods and models were proposed for improving the reliability of transportation
networks. Nevertheless, few studies researched CR enhancement for RTNs.
The studies on transportation networks’ CR assume that as long as there is at least one
connected path between an OD pair, the OD pair is in a connected state and passengers
travel between the OD pair. This assumption ignores the passengers’ travel behavior, which
is important for analyzing an RTN in terms of transporting passengers (Zhan, Wong, and
Lo 2020). Passengers choose acceptable paths whose trip time is below the passengers’
acceptable trip time. If the acceptable paths between an OD pair are interrupted in an
RTN, then passengers shift to other transportation modes even though other connected
paths still exist between that OD pair. Therefore, the connection state of OD pairs in an RTN
is determined by the connection state of acceptable paths rather than the state of con-
nected paths. The methods and models for enhancing road networks’ CR cannot be used
directly for enhancing an RTN’s CR due to the differences between RTNs and road networks
in terms of travelers’ travel characteristics, network connectivity and network operations.
For instance, travelers plan their paths according to road traffic flow and train schedules,
respectively,onaroadnetworkandanRTN.Inagivenarea,thenumberoflinksandnodes
on an RTN is much smaller than that on a road network. Thus, an RTN usually has worse
connectivity than a road network. Vehicles change their routes flexibly on a road network.
However, the trains cannot flexibly change their routes due to the lack of switching tracks
and differences among the signal system on lines. Connectivity interruptions on an RTN
TRANSPORTMETRICA A: TRANSPORT SCIENCE 5
network cause higher negative impacts on passengers’ travel due to low connectivity of the
network and difficulty of changing the train routes. Therefore, measuring and enhancing
the CR of an RTN is important for ensuring an RTN’s operations and passenger service.
3. Methodology
3.1. Assumptions and notations
The assumptions of the proposed model and their justifications are as follows:
1) Adding trains does not affect the rail transit operator’s disturbance handling capacity
and efficiency. Therefore, we assumed that the average number of canceled trains due
to disturbances is the same during the analysis period with and without added trains;
2) Although adding trains may attract more passengers to travel on an RTN, the num-
ber of attracted passengers is difficult to estimate, especially if trains are added on
short notice. Moreover, estimating the attracted passengers is not the focus of this
study. Therefore, new passengers attracted by added trains are not considered and the
passenger trips with and without added trains are assumed to stay the same;
3) Passengers’ acceptable trip times affect the number of tolerable paths among OD pairs
and thus affect the CR of an RTN. The CR enhancement is measured under the condition
that passengers’ acceptable trip times stay the same. Therefore, passengers’ tolerable
trip times among OD pairs stay the same with and without added trains;
4) A common assumption that links are independent of each other in many studies
(Wang and Xu 2009; Lubis et al. 2010; Günneç and Salman 2011; Jiang et al. 2013;
Kuang, Tang, and Shan 2013; Muriel-Villegas et al. 2016; Guidotti, Gardoni, and Chen
2017), is also used here. The reason is that it is difficult to quantify the relations
among different links and considering the relations among links greatly complicates
the problem.
The notations used in the model formulation are listed in Table 1.
3.2. RTN denition
An RTN consists of lines and stations of different rail transit modes, such as light rail, metro,
suburban railway, commuter rail and high-speed rail. An RTN is represented here as a
directed and weighted graph G=(S,E).Sand Erepresent the station set and link set,
respectively, on the RTN. The set of operating lines on the RTN are represented as L.The
set of links on a line is represented as El⊂E,l∈L. There is a transfer walking link when pas-
sengers transfer from one line to another. Therefore, when representing an RTN, the transfer
stations are divided and transfer walking links are added, as shown in Figure 1.
3.3. Trip time computation and acceptable path determination
Passengers will not select paths whose trip times exceed the acceptable trip time. The trip
time components include waiting time, walking time (access, egress and transfer walking
time) and in-vehicle time. The trip time tod
mfrom station oto station don path mis computed
6J. LIU ET AL.
Tab le 1 . Notation used in the model formulation.
Set Definition
ESet of links on an RTN.
ElSet of links on linel,El⊂E,l∈L.
Eod
mSet of links on path mfrom station oto station d,Eod
m⊂E.
LSet of operating lines.
Pod Set of acceptable paths from station oto station d.
SSet of stations on an RTN.
Sland SlSet of stations on line land line l, respectively.
Sod
mSet of transfer stations on path mfrom station oto station d.
Element Definition
eLink e,e∈E.
land lLine land l,l,l∈L.
oand dOrigin station oand destination station d,o,d∈S.
pod
mPath mfrom station oto station d.Iftod
m≤λ·tod
∗,thenpod
m∈Pod.
Parameters Definition
bThe number of train routes on an RTN.
CAverage GTC per passenger without adding trains ().
cod
mGTC on path mfrom station oto station dwithout adding trains ().
djThe round-trip distance of trains on route j(kilometers).
fjThe operational cost of adding a train on route j,j=1, 2, ...,b().
F∗A certain operational cost which is used to limit the operational cost of adding trains ().
fod
mTravel fare from station oto station don path m().
fod Travel fare from station oto station d().
hlLine l’s allowable minimum headway (hour).
jRoute jon an RTN, j=1, 2, ...,b.
mjThe number of cars per train on route j,j=1, 2, ...,b.
nThe number of acceptable paths without added trains from station oto station d.
ne,n
eand n∗
eThe actual, planned running trains and the average number of canceled trains, respectively,
passing through link eduring the analysis period.
nl,max The maximum number of trains that can be added by available vehicles on line l,l∈L.
nlExisting trains running on line lduring the analysis period.
RAn RTN’s CR without adding trains during the analysis period.
r(e)The CR of link ewithout adding trains during the analysis period.
r(pod
m)The CR of the acceptable path mfrom station oto station dduring the analysis period.
rod TheCRfromstationoto station dduring the analysis period.
Rl,lTheCRfromlinelto line l.
Tod
mPerceivedtriptimeonpathmfrom station oto station d(hours).
tod
∗The lowest trip time among all connected paths from station oto station d(hours).
tod
mTrip time from station oto station don path m(hours).
to
1and to
2Average walking access time and average waiting time at station o, respectively (hours).
ts
1and ts
2Average transfer walking time and average waiting time at station s,s∈Sod
m, respectively
(hours).
td
1Average walking egress time at station d(hours).
te
3Average in-vehicle time on link e(hours).
tjTheround-triptimeoftrainsonroutej(hours).
Vand vod Passengers traveling on the RTN and traveling from station oto station d, respectively, during
the analysis period (trips per hour).
yj,eA 0–1 parameter, if the route jcontains link e,thenyj,e=1; otherwise, yj,e=0.
αod
mProbability of acceptable path mbeing selected by passengers (%).
αThe monetary value per person/hour ().
βeThe weight of in-vehicle time on link e.
λPassengers’ acceptable coefficient of the trip time.
μ1and μ2The maintenance cost per vehicle-hour and maintenance cost per vehicle-kilometer,
respectively ().
μ3,μ4and μ5The depreciation fee per vehicle-hour, electricity cost per vehicle-kilometer and labor cost per
vehicle-hour, respectively ().
(continued)
TRANSPORTMETRICA A: TRANSPORT SCIENCE 7
Tab le 1 . Continued.
Set Definition
Variables Definition
Ba,1 and Ba,2 The benefit of CR enhancement by adding trains according to afrom the operator’sperspec tive
and the passengers’ perspective, respectively, during the analysis period ().
CaAverage GTC per passenger with added trains according to a().
cod
m,aGTC on path mfrom station oto station dwith added trains according to a().
FaThe operational cost of adding trains according to aduring the analysis period ().
naThe number of acceptable paths with added trains according to afrom station oto station d.
RaAn RTN’s CR with added trains according to aduring the analysis period.
ra(e)The CR enhancement for link eduring the analysis period after trains are added according to a.
ra(e),ra(pod
m), and rod
aThe CR of link e,CR of the acceptable path mfrom station oto station dand CR from station o
to station d, respectively, with added trains according to aduring the analysis period.
Decision variables Definition
a=[x1,...,xj,...,xb]xjis the number of trains added on route j,j=1, 2, ...,b.
Figure 1. Dividing a transfer station and adding a transfer walking link.
with Eq. (1):
tod
m=to
1+to
2+
e∈Eod
m
te
3+
s∈Sod
m
(ts
1+ts
2)+td
1(1)
where to
1and to
2are average walking access time and average waiting time at station o,
respectively. te
3and Eod
mareaveragein-vehicletimeonlinkeand the set of links on path m,
respectively. ts
1and ts
2are average transfer walking time and average waiting time at station
s,s∈Sod
m, respectively. Sod
mis the set of transfer stations on path mfrom station oto station
d.td
1is average walking egress time at station d.
There are several paths among OD pairs in an RTN. However, only the acceptable paths
whose trip times are below the passengers’ acceptable trip time are selected by passengers.
Figure 2provides an example to illustrate the determination of acceptable paths. There are
four connected paths from station O to station D, as shown in Figure 2. The trip times of
paths 1–4 are 0.2, 0.25, 0.45 and 0.25 h, respectively. The passengers’ acceptable trip time
for traveling from station O to station D is assumed to be twice the minimum trip time of
paths from station O to station D (0.4 h). Rail transit passengers shift to path 4 on another
mode when rail transit paths 1 and 2 are interrupted, even if rail transit path 3 is connected.
The reason is that the trip time of path 3 exceeds passengers’ acceptable trip time.
8J. LIU ET AL.
Figure 2. The paths between an OD pair.
The acceptable paths from station oto station dare determined with constraint (2):
pod
m∈Pod;tod
m≤λ·tod
∗
/∈Pod;tod
m>λ·tod
∗
(2)
where pod
mand Pod are the path mand the set of acceptable paths from station oto station
d, respectively. λis the passengers’ acceptable coefficient of the trip time. tod
∗is the lowest
trip time among all connected paths from station oto station d.
3.4. The CR metric of RTN
3.4.1. CR of links
Passengers board trains and travel through links to get to their destination stations. They
can only travel through a link passable by trains. Some disruptions, such as facility failures,
natural disasters or operational accidents cause train cancelations (Zhan et al. 2015). These
disruptions lead to a decrease in trains passing through links. A link’s CR is defined as the
fulfillment probability of the planned trains running through that link, which is the ratio of
actual trains to planned trains running through the link during an analysis period:
r(e)=ne
n
e
=n
e−n∗
e
n
e
(3)
where r(e)is the CR of link eduring the analysis period. neand n
eare the actual and planned
running trains, respectively, passing through link eduring the analysis period. n∗
eis the
average number of canceled trains that pass through link eduring the analysis period.
3.4.2. A acceptable path’s CR
An acceptable path consists of a series of links. An acceptable path is connected only if all
the links belonging to it are connected. Therefore, an acceptable path’s CR is the product
of the links’ CRs that belong to that path:
r(pod
m)=
e∈Eod
m
r(e)(4)
where r(pod
m)is the CR of the acceptable path mfrom station oto station d.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 9
3.4.3. An OD pair’s CR
As long as at least one acceptable path between an OD pair is connected, the OD pair is
connected. An OD pair’s CR is defined as the probability that there is at least one con-
nected acceptable path between the OD pair. The acceptable paths between an OD pair
may have common links. Therefore, an OD pair’s CR is computed based on the sum-of-
disjoint products of acceptable paths between that OD pair. The sum-of-disjoint products
rely on Boolean algebra manipulation to convert a set of paths polynomials into a set of
exclusive and mutually disjoint terms (Chaturvedi and Misra 2002). The algebraic sum of
exclusive and mutually disjoint terms is the reliability expression, i.e. an OD pair’s CR is given
by Eq. (5):
rod =r∪n
m=1pod
m=r(pod
1)+r(pod
1∩pod
2)+...+r(pod
1∩pod
2∩...∩pod
n−1∩pod
n)(5)
where rod is CR from station oto station d.nis the number of acceptable paths from sta-
tion oto station d.pod
m,m=1, 2, ...,nrepresents that the acceptable path mfrom station
oto station dis not connected. r(pod
1∩pod
2)is the probability that the acceptable path 1 is
disconnected and the acceptable path 2 is connected from station oto station d.
3.4.4. An RTN’s CR
The number of passenger trips among OD pairs varies greatly and thus the importance of
OD pairs is different. The ratio of passenger trips between an OD pair to total passenger trips
on the RTN is defined as the weight of that OD pair. The CR of an RTN is the weighted sum
of OD pairs’ CRs, which is computed with Eq. (6). It equals the average value of CR for each
passenger. Only if the OD pair is connected can passengers travel between that OD pair.
Therefore, Eq. (6) also shows the fraction of passengers who can travel on an RTN during
the analysis period:
R=o∈Sd∈S,o=dvod ·rod
V(6)
where Ris the RTN’s CR. Vand vod are passengers traveling on the RTN and traveling from
station oto station d, respectively, during the analysis period.
3.5. Enhancing an RTN’s CR by adding trains
Assume that stations S1 to S3 in Figure 3are turn-back stations on a line. The three train
routesexistbetweenstationsS1andS2,stationsS2andS3,aswellasstationsS1andS3.The
CR of links between stations S1 and S2 is enhanced by adding trains on the routes between
stations S1 and S2 as well as between stations S1 and S3, which is proved as follows.
A decision variable set a,a=[x1,...,xj,...,xb] is used in the paper to represent
adding trains on broutes in an RTN according to it during an analysis period. xj,j=
1, 2, ...,bin ais a decision variable representing that adding xjtrains on routej. Eq. (7)
computes a link’s CR after adding trains according to aduring an analysis period:
ra(e)=n
e−n∗
e+b
j=1xj·yj,e
n
e+b
j=1xj·yj,e
(7)
where ra(e)is the CR of link eafter adding trains during the analysis period. yj,eis a 0–1
parameter; if the route jcontains link e, then yj,e=1; otherwise, yj,e=0.
10 J. LIU ET AL.
Figure 3. Trainroutesonaline.
The CR enhancement for link eduring the analysis period equals the CR of link ewith
added trains according to aminus the CR of link ewithout any added train, which is
computed with Eq. (8):
ra(e)=ra(e)−r(e)=n
e−n∗
e+b
j=1xj·yj,e
n
e+b
j=1xj·yj,e
−n
e−n∗
e
n
e
=n∗
e·b
j=1xj·yj,e
n
e·n
e+b
j=1xj·yj,e(8)
where ra(e)is the CR enhancement for link eduring the analysis period. If any xj,j=
1, 2, ...,bexceeds 0 and route jcontains link e, then ra(e)>0 and thus demonstrates
that the CRs of the links can be enhanced when the added trains run through them.
3.5.1. The benefit of adding trains on an RTN
The passengers intended to travel on an RTN stay the same because the passengers
attracted by added trains are not considered. Adding trains on an RTN enhances the RTN’s
CR. Therefore, adding trains increases the fraction of passengers who can travel on an
RTN, and thus increases the fare revenue. The benefit of adding trains from the opera-
tor’s perspective can be measured by the increased fare revenue. In addition, adding trains
on an RTN decreases the passengers’ travel time and increases travel comfort from the
passengers’ perspective, which is measured with the passengers’ total GTC reduction.
The GTC is the sum of the monetary and non-monetary costs of a trip (Bruzelius 1981).
The monetary cost is the price paid to operators, i.e. the fare. The non-monetary cost is the
monetary value of passengers’ perceived trip time on the path. Passengers’ perceived trip
time considers crowding in the train and seat availability. The perceived trip time and the
GTC on path mfrom station oto station drepresented with Tod
mand cod
mare computed with
Eqs. (9) and (10), respectively:
Tod
m=to
1+to
2+
e∈Eod
m
te
3·βe+
s∈Sod
m
(ts
1+ts
2)+td
1(9)
cod
m=fod
m+α·Tod
m(10)
where βeis the weight of in-vehicle time which is related to crowding in trains and seat
availability. The crowding in trains is measured with load factors. The values of βeat differ-
ent load factors when passengers sit or stand are shown in Table 8(Wardman and Whelan
2011). fod
mis travel fare from station oto station don path m.αis the monetary value per per-
son/hour that converts perceived trip time into money, which is related to the passengers’
income (Litman 2008).
TRANSPORTMETRICA A: TRANSPORT SCIENCE 11
The benefit of CR enhancement by adding trains is the increased fare revenue due to
the increase in the fraction of passengers who can travel on an RTN, from the operator’s
perspective:
Ba,1 =
o∈S
d∈S,o=d
vod ·(rod
a−rod)·fod (11)
where Ba,1 is the benefit of CR enhancement by adding trains according to afrom the opera-
tor’s perspective during the analysis period. rod
ais CR from station oto station dafter adding
trains according to a.fod is the fare for traveling from station oto station d.fod is estimated
with Eq. (12):
fod =
na
m=1
αod
m·fod
m(12)
where αod
m,fod
mand naare the probability of acceptable path mbeing selected by passen-
gers, travel fare on path m, and the number of acceptable paths, respectively, from station
oto station dafter adding trains according to a.αod
mis computed after passenger trips are
assigned to the RTN according to GTCs of acceptable paths among OD pairs.
The benefit of adding trains from the passengers’ perspective is the reduced average
GTC per passenger multiplied by passengers who can travel on the RTN after adding trains
during an analysis period:
Ba,2 =(C−Ca)·
o∈S
d∈S,o=d
vod ·rod
a(13)
Ca=o∈Sd∈S,o=dvod ·rod
ana
m=1αod
m·cod
m,a
o∈Sd∈S,o=dvod ·rod
a
(14)
C=o∈Sd∈S,o=dvod ·rod n
m=1αod
m·cod
m
o∈Sd∈S,o=dvod ·rod (15)
where Ba,2 is the benefit of adding trains according to afrom the passengers’ perspective.
Cand Caare average GTCs per passenger without adding any train and with added trains
according to a, respectively. Passengers who can travel from station oto station dequal vod ·
rod
aand vod ·rod, respectively, with added trains according to aand without any added train.
cod
m,aand naare the GTC of path mand the number of acceptable paths, respectively, after
adding trains according to a.cod
mand naretheGTCofpathmand the number of acceptable
paths, respectively, without adding any train.
3.5.2. The operational cost of adding trains
Adding trains on an RTN increases its operational cost, which includes the maintenance
cost, depreciation fee, electricity cost and labor cost for added trains. The cost of adding
a train on route j,j=1, 2, ...,bis estimated with Eq. (16) and the cost of adding trains
according to aon an RTN is estimated with Eq. (17).
fj=mj·(μ1·tj+μ2·dj+μ3·tj+μ4·dj+μ5·tj)(16)
Fa=
b
j=1
xj·fj(17)
12 J. LIU ET AL.
where Faand fjare the operational cost of adding trains according to aon an RTN and the
operational cost of adding one train on route j, respectively, during the analysis period.
mjis the number of cars per train on route j.μ1and μ2are the maintenance cost per
vehicle-hour and maintenance cost per vehicle-kilometer, respectively. μ3,μ4and μ5are
the depreciation fee per vehicle-hour, electricity cost per vehicle-kilometer and labor cost
per vehicle-hour, respectively. tjand djare the round-trip time and round-trip distance of
trains on route j, respectively.
3.5.3. Maximizing an RTN’s CR by adding trains
A model is developed for maximizing an RTN’s CR with constraints on the operational cost
of adding trains and the maximum train service frequencies on routes as well as the maxi-
mum trains that can be added by available vehicles on each line, which is shown in Eqs. (18)
to (24).
maximizeRa=
o∈S
d∈S,o=d
vod
V·rod
a(18)
subject to:
ra(e)=n
e−n∗
e+b
j=1xj·yj,e
n
e+b
j=1xj·yj,e
,∀e∈E(19)
ra(pod
m)=
e∈Eod
m
ra(e),∀m=1, 2, ...,na(20)
rod
a=ra∪na
m=1pod
m,∀o,d∈N(21)
j∈l
xj=nl,max
,∀l∈L(22)
j∈l
xj+nl=1/hl,∀l∈L(23)
Fa≤F∗(24)
Eq. (18) aims to maximize an RTN’s CR in the model. The CR of links, paths, and OD pairs
are computed with Eqs. (19) to (21), respectively, after adding trains according to aon
the RTN during the analysis period. Constraint (22) requires that the sum of trains added
on routes belonging to line l,l∈Lcannot exceed the maximum number of trains (nl,max
)
that can be added by available vehicles on line l,l∈L. Constraint (23) requires that the
sum of added trains and existing trains running on line lrepresented as nlis below the
line’s allowable track capacity. That capacity on line lis the maximum train frequency on
that line, which is the inverse of linel’s allowable minimum headway hl. Constraint (24)
requires that the operational cost of adding atrains cannot exceed a certain operational
cost F∗.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 13
4. Solution procedure
4.1. Acceptable paths generation
4.1.1. A Depth-First-Search algorithm with a branch search reducing strategy
A Depth-First-Search algorithm with a branch search reducing strategy is proposed for
generating the acceptable paths between OD pairs accurately. This algorithm generates
acceptable paths more correctly than the heuristic search algorithm used for generating
K-shortest paths such as the A∗algorithm (Feng 2013), since all acceptable paths can be
generated by the algorithm used here; In addition, this algorithm generates acceptable
paths faster than a simple Depth-First-Search algorithm, since fewer branches must be
searched. The steps in applying the algorithm to generate acceptable paths from station
oto station dare as follows:
Step 1: Let the acceptable path Pod =[]; let the station search sequence =[] and com-
pute the shortest trip time using Dijkstra’s algorithm (Golden 1976) from any station to the
destination station d.
Step 2: Place origin station ointo the station search sequence .
Step 3: Search the next station zaccording to the last station in the station search
sequence . Station zshould be adjacent to the last station in and not in . If station
zmeets these conditions, place it in and go to Step 5; otherwise, remove the last station
in and go to Step 4.
Step 4: Algorithm stops evaluating. If =[], then stop the algorithm; otherwise, return
to Step 3.
Step 5: Determine the searched path pcorresponding to station search sequence .
The searched path pcorresponding to station search sequence and the links among
the station search sequence can be determined. Thus, trip time tof path pis com-
puted according to the in-vehicle times on links among the stations in search sequence
. Determine whether to keep searching for the adjacent station of the last station in .
If t+t∼d
∗≤λ·tod
∗(t∼d
∗is the shortest trip time from the last station in to the destina-
tion station d), then go to Step 6; otherwise, remove the last station in and return to
Step 3.
Step 6: If the last station in is the destination station d, then path pis an accept-
able path and pbelongs to the acceptable path set (i.e. Pod =Pod ∪p, where Pod is the
acceptable path set from station oto station d); otherwise, return to Step 3.
The acceptable paths on the RTN network in Figure 2from station O to station D are
generated and shown in Figure 4when the acceptable trip time from station O to station D
is assumed to be 0.4 h. The search for the left branch in Figure 4will not continue after
station S3 is searched. The reason is that the sum of the trip time of the searched path
from station O to station S3 (i.e. 0.2 h) and the shortest trip time from station S3 to station
D (i.e. 0.25 h) is 0.45 h which exceeds 0.4 h. Therefore, applying the algorithm reduces the
number of branches that must be searched, and thus the effectiveness of the algorithm is
enhanced.
4.1.2. Acceptable paths generation after adding trains
Adding trains increases the train frequency on lines and reduces waiting time at sta-
tions, thus the number of acceptable paths increases. To obtain the optimized solution for
maximizing the CR of an RTN by adding trains, the acceptable paths should be generated
14 J. LIU ET AL.
Figure 4. Search for acceptable paths in Figure 2from station O to station D.
after adding trains corresponding to different solutions. To reduce the time for generating
acceptable paths when adding trains corresponding to different solutions, we propose a
way as follows to avoid multiple generations:
Step 1: The acceptable path sets among OD pairs without adding any trains are gener-
ated using the algorithm introduced in section 4.1.1.
Step 2: The acceptable path sets among OD pairs are generated using the algorithm
introduced in section 4.1.1 when the maximum number of trains constrained by the
allowable track capacity are added on routes.
Step 3: The possible acceptable paths are determined. The paths in acceptable path sets
generated in step 2, although not in acceptable path sets generated in step 1, become pos-
sible acceptable paths when adding trains, since the added trains corresponding to any
solution cannot exceed the maximum number of trains on routes.
Step 4: The acceptable paths when adding trains on routes corresponding to a solu-
tion are determined. The trip times on possible acceptable paths are recomputed when
adding trains corresponding to that solution and the paths whose trip times are below
acceptable trip time are selected. The selected paths and the paths in the acceptable path
set generated in step 1 are acceptable paths when adding trains corresponding to that
solution.
The acceptable paths corresponding to a solution can be generated using steps 1–4
and then the acceptable paths corresponding to another solution can be obtained by
implementing steps 3 and 4, which reduces the time for generating acceptable paths. We
use the acceptable path generation from station 27 to station 7 in Figure 10 as an example
to illustrate the above steps. The acceptable trip time from station 27 to station 7 equals
TRANSPORTMETRICA A: TRANSPORT SCIENCE 15
Tab le 2. Acceptable path set generated in step 1.
Path number Path (via stations)
127-26-25-24-23-22-21-20-19-18-17-16-15-14-13-12-11-10-9-8-7
227-26-25-24-23-22-21-20-19-18-17-16-15-14-13-12-11-10-86-85-50-7
Tab le 3 . Acceptable path set generated in step 2.
Path number Path (via stations)
1 27-26-25-24-23-22-21-20-19-18-17-16-15-14-13-12-11-10-9-8-7
2 27-26-25-24-23-22-21-20-19-18-17-16-15-14-13-12-11-10-86-85-50-7
3 27-26-25-24-23-22-21-20-19-18-17-16-15-14-13-12 −11-10-86-85-50-84-111-6-7
4 27-26-25-24-23-22-21-20-19-18-17-16-15-14-13-141-142-90-89-88-87-10-9-8-7
the passengers’ acceptable coefficient of the trip time (λ,λ=1.58 in Chengdu’s RTN) mul-
tiplied by the lowest trip time among all connected paths from station 27 to station 7 (19.02
min), i.e. 30.05 min. The steps for generating acceptable paths from station 27 to station 7
after adding trains corresponding to different solutions are as follows:
Step 1: The acceptable path set from station 27 to station 7 without adding any trains is
generated and shown in Table 2.
Step 2: The acceptable path set from station 27 to station 7 is generated and shown in
Table 3when the maximum number of trains constrained by the allowable track capacity
are added on routes.
Step3:Paths3and4inTable3are possible acceptable paths, since they are in acceptable
path sets generated in step 2 but not in acceptable path sets generated in step 1.
Step 4: The trip times on paths 3 and 4 are recomputed with added trains corresponding
to a solution. The path whose trip times are below the acceptable trip time from station
27 to station 7 (i.e. 30.05 min) are selected from paths 3 and 4. The selected path and the
generated path in step 1 (paths 1 and 2) are acceptable paths after adding trains on routes
corresponding to that solution.
4.2. CR computation based on a binary decision diagram (BDD)
BDD is a directed acyclic graph, which consists of terminal nodes and non-terminal nodes
(Xing 2007). The nodes are connected by links whose states are ‘1’ or ‘0’. Each terminal node
has two disjoint logical states ‘1’ and ‘0’, respectively, representing the normal state and the
failure state of the system. The paths traversing from the root node to terminal nodes on a
BDD are disjoint. Therefore, constructing a BDD is an efficient approach to obtain the sum-
of-disjoint products for minimal paths (Pan, Xing, and Mo 2017; Kawahara et al. 2019). Here,
the acceptable paths are minimal paths (a path with no repeating nodes), since passengers
do not return to the stations which they have passed. Thus, the sum-of-disjoint products for
acceptable paths between an OD pair are obtained by constructing a BDD. Here, the states
‘1’ and ‘0’ of the terminal nodes indicate that the OD pair is connected and unconnected,
respectively. An OD pair’s CR is computed according to the links’ state and the probabilities
of links state, thus the CR of RTN is computed. The steps for measuring an OD pair’s CR based
on BDD are as follows:
16 J. LIU ET AL.
Figure 5. A bridge network.
Step 1: Generate the acceptable paths between the OD pair with the algorithm intro-
duced in section 4.1.
Step 2: Generate the BDD according to the acceptable paths.
Step 2.1: Select the link to branch. A link is selected from an acceptable path which has
the fewest links according to the number of times that the link appears in all acceptable
paths (from large to small). If more than one link satisfies the above condition, then select
a link randomly.
Step 2.2: Branch the selected link. The selected link is branched according to Shannon’s
decomposition theorem (Bryant 1986) which is shown as:
f=e·fe=1+¯
e·fe=0(25)
where fis the Boolean expression for all variables (links).eand ¯
erepresent the connected
state ‘1’ and unconnected state ‘0’ of the link e, respectively. fe=1and fe=0represent the
Boolean expression when link eis connected or not, respectively. The links are branched
according to Shannon’s decomposition theorem until the branch determines the state of a
terminal node (‘0’ or ‘1’).
Step 3: Compute the CR between the OD pair
The sum-of-disjoint products for acceptable paths are obtained by searching the paths
from the root node to the terminal nodes whose value is ‘1’ in the BDD. Then, the CR
between the OD pair is computed according to the sum-of-disjoint products for acceptable
paths.
Taking a small case (Figure 5) as an example, the above steps are applied to compute the
CR from nodes 1–3.
We assume that four acceptable paths exist from nodes 1–3, which are listed in Table 4.
The BDD from nodes 1–3 is generated as Figure 6by applying step 2. The sum-of-
disjoint products for acceptable paths are determined by searching the paths from the
root node to the terminal nodes whose value is 1 in Figure 6(i.e. e1e4+e2e5e4e1+
e3e2e5e4e1+e5e2e1+e4e3e5e2e1). The CR from station 1–3 equals r(e1)·r(e4)+r(e2)·
r(e5)·(1−r(e4)) ·r(e1)+r(e3)·(1−r(e2)) ·r(e5)·(1−r(e4)) ·r(e1)+r(e5)·r(e2)·(1−
r(e1)) +r(e4)·r(e3)·(1−r(e5)) ·r(e2)·(1−r(e1)) according to the sum-of-disjoint prod-
ucts for acceptable paths.
The OD pairs’ CRs are computed using the above steps and then the RTN’s CR is
computed with Eq. (6).
TRANSPORTMETRICA A: TRANSPORT SCIENCE 17
Figure 6. The BDD for acceptable paths from nodes 1 to –3.
Tab le 4. Attributes of lines in Chengdu’s RTN.
Acceptable paths Links belong to the path
1e1,e4
2e1,e3,e5
3e2,e3,e4
4e2,e5
4.3. Multi-population genetic algorithm for solving the CR maximization model
The objective function (i.e. Eq. (18)) computed with Eqs. (19) to (21) for maximizing an RTN’s
CR is nonlinear and the decision variables are integers greater than or equal to 0. Therefore,
the model is an integer nonlinear problem, which is generally very difficult to solve with
an exact algorithm (You and Grossmann 2008; Lee, Modiano, and Lee 2010). In addition, to
compute CRs of OD pairs, the acceptable paths must be generated and the sum-of-disjoint
products for acceptable paths must be obtained, which makes the objective function com-
putation complex and increases the difficulty of linearizing the model. Genetic algorithms
are often used to solve an integer nonlinear model where the decision variables are integers
(Zhang et al. 2017; Kosari and Teshnehlab 2018). In addition, genetic algorithms can be used
for optimizing a complex continuous or discrete objective function. Therefore, a genetic
algorithm can be used here to maximize an RTN’s CR by adding trains. However, premature
convergence, i.e. getting trapped in a locally optimal solution, is a weakness of a simple
genetic algorithm (SGA). The multi-population genetic algorithm (MPGA) overcomes this
shortcoming. It is used to solve scheduling problems (Zegordi and Beheshti Nia 2009;Shi
et al. 2020) and dynamic facility layout problems (Pourvaziri and Naderi 2014), since it is
18 J. LIU ET AL.
Figure 7. A sample of a chromosome.
effective in both speed and solution quality (Gao et al. 2015). Therefore, the MPGA is used
here to solve the model for maximizing an RTN’s CR.
The steps for applying the MPGA to solve the model for maximizing an RTN’s CR by
adding trains are as follows:
Step 1: Initialize the populations.
Generate Msubpopulations and each subpopulation has Hchromosomes. Each chromo-
some is composed of bgenerated real numbers, as shown in Figure 7.Thekj,j=1, 2, ...,b
in the chromosome means adding kjtrains on route j. The real numbers on each chromo-
some have the following constraints: (a) The sum of trains added on routes belonging to a
line cannot exceed the maximum number of trains that can be added by available vehicles
on that line in constraint (22). (b) the sum of added trains and existing trains is below the
allowable track capacity on the line in constraint (23); (c) the cost of adding trains must not
exceed the operational cost F∗in constraint (24).
We use a line to illustrate constraints (a) and (b). Routes 1–3 are on this line. The maxi-
mum number of trains that can be added by available vehicles on this line is assumed to
be 8. The allowable track capacity (i.e. the maximum train frequency) and operational train
frequency on this line are 40 trains per hour and 30 trains per hour, respectively. Although
the allowable track capacity (40 trains per hour) allows adding 10 trains on routes, the sum
of added trains on routes 1–3 (i.e. k1+k2+k3) cannot exceed the maximum number of
trains that can be added by available vehicles (i.e. 8).
Constraint (c) requires that the sum of operational cost of adding trains on routes (i.e.
stays below F∗. The operational cost of adding one train on route j=1, 2, ...,bis computed
with Eq. (16) and the cost of adding trains on the RTN is estimated with Eq. (17).
Step 2: Compute individuals’ fitness and identify elite individuals in subpopulations.
The numbers of added trains on routes corresponding to each chromosome are the
values of kj,j=1, 2, ...,bin the chromosome. The links’ CRs with added trains correspond-
ing to each chromosome are computed with Eq (7). The acceptable paths with added
trains corresponding to each chromosome are generated with the method proposed in
section 4.1. Then, the CRs of the RTN with added trains corresponding to each chromosome
are computed applying the method introduced in section 4.2.
The fitness of a chromosome in a subpopulation is determined with Eq. (26):
fs,g=1
1−Rs,g
,s=1, 2, ...,M;g=1, 2, ...,H(26)
where fs,gis the fitness of the gth chromosome in the sth subpopulation. Rs,gis the CR of the
RTN after adding trains on routes corresponding to gth chromosome in sth subpopulation.
Solutions with higher fitness of their chromosomes are better, and thus the elite individ-
uals with the highest fitness in each subpopulation are identified based on the fitness of
chromosomes.
Step 3: Perform crossover, mutation and selection for chromosomes in subpopulations.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 19
Figure 8. Crossover and mutation on chromosomes.
The chromosomes’ crossover or mutation probabilities are generated randomly,
between 0.7 and 0.9 or between 0.001 and 0.05, respectively, which improves the global and
local search capabilities of the algorithm. The single point crossover is applied for chromo-
some crossover. The cross point is determined according to the routes on lines, assuming
that, routes 1–3 and routes 4–6 are on lines 1 and 2, respectively. Thus, a cross point is
between 3th and 4th real numbers on a chromosome. An example of a crossover between
two chromosomes is shown in Figure 8(a). During mutations of chromosomes, the real
numbers on chromosomes are changed and constrained by the maximum number of trains
that can be added on lines. It assumes that routes 1–3 belongs to a line and the sum of
added trains on routes 1–3 cannot exceed 10, as well as the maximum number of trains
that can be added by available vehicles on line 1 is 8. Thus, Figure 8(b) shows that the sec-
ond real number in a chromosome is changed from 4 to x.xis an integer between 0 and
6, provided that x= 4, since the sum of added trains on routes 1–3 (i.e. the sum of the first
three real numbers on a chromosome) cannot exceed the maximum number of trains that
can be added by available vehicles (i.e. 8).
Chromosomes in subpopulations are selected according to roulette wheel selection. If
the fitness of the gth chromosome in sth subpopulation satisfies Eq. (27), then it is selected
for the next generation:
g−1
d=1fs,d
H
d=1fs,d
<ξ
k<g
d=1fs,d
H
d=1fs,d
(27)
where ξk∈[0, 1] is a uniformly distributed random number.
Step 4: Compute individual fitness and insert the elite individuals into the new subpop-
ulations.
The costs of adding trains corresponding to chromosomes are computed according to
Eqs. (16) and (17), which are compared with the operational cost F∗. If those costs exceed F∗,
then the solutions corresponding to the chromosomes are not satisfied with constraint (24)
and thus the fitness of the chromosomes is set to be a small value (e.g. 0.5 or 1); otherwise,
the CRs of Chengdu’s RTN with added trains corresponding to chromosomes are computed
using the method introduced in sections 4.1 and 4.2. The fitness of the chromosomes is then
computed with Eq. (26).
20 J. LIU ET AL.
To avoid the disappearance of good individuals during the selection, crossover and
mutation for chromosomes, the sth,s=1, 2, ...,M, elite individual identified in each sub-
population at step 2, is inserted into the sth,s=1, 2, ...,M, new subpopulation.
Step 5: Migrate subpopulations.
The best and worst chromosomes in each subpopulation whose fitness are the high-
est and lowest, respectively, are identified. The best and worst chromosomes are randomly
selected when the fitness of multiple chromosomes equals the highest or lowest fitness
in a subpopulation. The migration operator is used to replace the worst chromosome in
each subpopulation with the best chromosome in another subpopulation. The matching
relation between the worst chromosome in a subpopulation and the best chromosome in
another subpopulation is randomly generated.
Step 6: Obtain the optimal individual and generate a subpopulation to replace the worst
subpopulation.
The optimal individual is the chromosome whose fitness is the highest among all
chromosomes in subpopulations. After 5 generations, a new subpopulation with Hchro-
mosomes is generated to replace the worst subpopulation. The worst subpopulation is
determined according to the harmonic mean of individuals’ fitness in subpopulations.
Step 7: Stop the iterations.
If the optimal individual stays the same for at least 10 generations, then the iterations
end; otherwise, return to step 2.
The algorithm structure of MPGA is shown in Figure 9.
5. Case studies
In this section, the CR of Chengdu’s RTN is measured during morning peak periods. The opti-
mized solutions with different operational cost constraints are obtained by applying the
MPGA to maximize the CR of Chengdu’s RTN. The benefits of adding trains corresponding
to the optimized solutions are computed.
5.1. Chengdu’s RTN
Chengdu is the capital of Sichuan province and one of the largest cities in southwestern
China. Chengdu’s RTN consisted of 12 rail transit lines and 174 stations in May 2019, which
is shown in Figure 10. Lines 1–6, lines 7– 9 and lines 10–12 are metro lines, suburban railway
lines and intercity high-speed rail lines, respectively.
5.2. RTN operation data, parameter values and train routes
5.2.1. RTN operation data and parameter values
The related data for computing trip time on paths during morning peak hours (from 7:30 am
to 9:30 am) are obtained from Chengdu’s RTN operator. The operational headway, allow-
able minimum headway, provided seats per hour and capacity per hour on Chengdu’s rail
transit lines during morning peak periods are shown in Table 5. The average waiting time
at stations is estimated as half of the operational headways (De-Los-Santos et al. 2012; Dixit
et al. 2019). The average transfer walking time at all transfer stations, as well as train running
TRANSPORTMETRICA A: TRANSPORT SCIENCE 21
Figure 9. Algorithm structure of MPGA.
time on all links during morning peak periods, are obtained from the operator and a survey
(Liu et al. 2020).
We only list the transfer walking times at some transfer stations and the train running
time on some links during morning peak periods, which are shown in Tables 6and 7,respec-
tively. It is difficult to determine the walking egress and access time for 174 stations in
Chengdu’s RTN. Therefore, the average walking egress and access time are estimated to
be 5 min. The average OD trip distribution during morning peak periods for 28 workdays is
obtained and by processing Automatic Fare Collection data and conducting a survey. The
results are shown in Figure 11.
To compute the GTC on paths, the values of βeat different load factors when passen-
gers sit or stand are shown in Table 8(Wardman and Whelan 2011). The monetary value
of per person/hour αin Eq. (10) equals 30% of household income per hour according to
economist Gwilliam’s recommendation (Litman 2009). The monetary value of per person
hour αis 16.83 /hour based on average household income (134,187 /per year) which
22 J. LIU ET AL.
Figure 10. The Chengdu RTN network in May 2019.
Figure 11. Average OD trip distribution during morning peak periods.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 23
Tab le 5 . Attributes of lines in Chengdu’s RTN.
Line Operational headway (min) Allowable minimum headway (min) Seats (seats per hour)
1 2.00 1.50 348×30
2 2.73 2.00 348×22
3 3.00 2.00 348×20
4 3.00 2.00 348×20
5 4.00 2.00 348×15
6 6.00 3.00 348×10
7 6.00 3.00 250×10
8 10.00 7.50 250×6
9 10.00 7.50 250×6
10 10.00 7.50 610×6
11 7.50 6.00 610×8
12 7.50 6.00 610×8
Seats =seats per train ×frequency of trains.
Tab le 6 . Average transfer walking times at selected stations.
Station Transfer direction Walking time (min) Transfer direction Walking time(min)
50 lines 2–3 2.8 lines 3–2 3.2
56 lines 2–5 2.0 lines 5–2 2.9
80 lines 3–5 2.2 lines 5–3 1.8
84 lines 3–4 2.7 lines 4–3 2.5
Tab le 7 . Train running times (including dwell time) on selected links.
Time (min) Time (min)
Link (station-station) Ds Us Link (station-station) Ds Us
1–2 1.87 1.88 5–6 1.22 1.25
2–3 2.08 2.07 6–7 1.35 1.33
3–4 1.47 1.45 7–8 1.15 1.20
4–5 1.57 1.58 8–9 1.15 1.17
Ds =Downstream; Us =Upstream.
Tab le 8 . Value of β3at different load factors.
Load factor (%) Sitting Standing
0–75 0.86 –
75–100 0.95 –
100–125 1.05 1.62
125–150 1.16 1.79
150–175 1.27 1.99
175–200 1.40 2.20
>200 1.55 2.44
Load factor is the ratio of passengers to seats in trains.
is obtained from Chengdu Bureau of Statistics (2018). An in-depth questionnaire survey of
passengers in Chengdu’s RTN system was conducted by our team and indicated that the
passengers’ acceptable coefficient of the trip time is 1.58.
The pricing standard for Chengdu metro system is that the fare between two stations is
determined by the shortest distance between that OD pair. The relation between fare and
distance is a piecewise function, and thus fares corresponding to the shortest distances
24 J. LIU ET AL.
Tab le 9 . Fare corresponding to the shortest distance.
Distance (km) (0, 4] (4,8] (8, 12] (12, 18] (18, 24] (24, 32] (32, 40] (40, 50] >50
Fare () 2345678910
Table 10. Operational cost parameters.
Lines ni(vpt) μ1(/pvh) μ2(/pvk) μ3(/pvh) μ4(/pvk) μ5(/pvh)
1–6 6 50.50 7.69 42.47 2.03 56.25
7–9 4 46.80 8.09 355.75 3.07 56.25
10–12 8 46.80 8.09 365.75 3.07 56.25
vpt =vehicles per train; pvh =per vehicle-hour; pvk =per vehicle-kilometer.
Table 11. Average canceled trains, actual trains, and planned trains for
each morning peak period.
Line Average canceled trains Actual trains Planned trains Ratio (%)
1 0.51 29.49 30 98.31
2 0.41 21.59 22 98.15
3 0.33 19.67 20 98.35
4 0.33 19.67 20 98.36
5 0.24 14.76 15 98.43
6 0.13 9.87 10 98.72
7 0.11 9.89 10 98.89
8 0.06 5.94 6 98.95
9 0.03 5.97 6 99.48
10 0.03 5.97 6 99.57
11 0.03 7.97 8 99.60
12 0.04 7.96 8 99.54
Ratio =ratio of actual trains to planned trains
among OD pairs are shown in Table 9. Therefore, the fare is the same for passengers trav-
eling between an OD pair in the metro system, regardless of which path they choose. The
relation between fare and distance is linear for travel by suburban railway and intercity high-
speed rail. The pricing standards for suburban railway and intercity high-speed rail are 0.31
and 0.46 /km, respectively. The parameters for estimating the operational cost of adding
trains in Eq. (16) are listed in Table 10 (Yang et al. 2017).
Data for 13 months of train operation during morning peak periods on Chengdu’s RTN
are obtained from its rail transit operator. The number of planned trains and their routes
during morning peak periods are obtained from the train schedule map. The number of
actual trains equals the planned trains minus the average number of canceled trains during
the analysis period. The data for the average number of canceled trains, actual trains and
planned trains for each morning peak period are shown in Table 11. The average numbers
of canceled trains on lines for each morning peak period are below 1, because trains are
rarely canceled. The trains run between the two end stations on lines and thus the links’
CRs are determined with Table 11.
5.2.2. The train routes on Chengdu’s RTN
Trains cannot switch between lines on Chengdu’s RTN but can run between turn-back sta-
tions. The routes on every line in Chengdu’s RTN are determined and shown in Figure 12.
Trains are added on routes during morning peak periods to enhance the CR of Chengdu’s
TRANSPORTMETRICA A: TRANSPORT SCIENCE 25
Figure 12. Train routes on Chengdu’s RTN.
RTN. The maximum service frequencies are determined according to the allowable mini-
mum headways of lines which are shown in Table 5.
5.3. The CR of Chengdu’s RTN without adding any train
5.3.1. CRs of OD pairs
The CRs of OD pairs are computed during morning peak periods and their contours are
shown in Figure 13. The OD pairs’ CRs on line 1 (stations 1–35) are shown in the dotted
rectangle 1
. The OD pairs’ CRs between line 1 and line 2 (stations 37–66) are shown in the
dotted rectangle 2
. The OD pairs’ CRs on line 1 are higher than OD pairs’ CRs between line
1 and line 2. Figure 13 also shows that the OD pairs’ CRs on the same line are higher than the
OD pairs’ CRs on different lines. In addition, the closer to the middle of the dotted rectangles
1
and 2
are, the greater the CR value is. Therefore, the shorter distance between the OD
pair on the same line, the higher is the OD pair’s CR.
The fractions of OD pairs in different CR intervals during morning peak periods are shown
in Figure 14. It shows that 73.38% of OD pairs’ CRs are between 90% and 100%, which
demonstrates that these OD pairs have high CRs. Only 6.74% OD pairs’ CRs are below 80%.
5.3.2. CRs among lines
To analyze the CRs of OD pairs on the same line and the CRs of OD pairs on different lines,
Eq. (28) is used to measure the CRs among lines. The average value of CR for each passenger
who travels from stations on line l,l∈Lto stations on line l,l∈Lis used to measure the
CR from line lto line l. Eq. (28) measures the CR on the same line, when l=l.
Rl,l=o∈Sld∈Sl,o=d(rod ·vod +rdo ·vdo)
o∈Sld∈Sl,o=d(vod +vdo)(28)
where Rl,lis the CR from line lto line l.Sland Slare sets of stations on line land line l,
respectively.
26 J. LIU ET AL.
Figure 13. Contours of OD pairs’ CRs during morning peak periods.
Figure 14. Fractions of OD pairs in different CR intervals.
The CRs among lines are shown in Figure 15 (a). CRs on the same line and among lines
canbeanalyzedinFigure15 (a). To analyze the CRs among lines belonging to the same
rail transit mode and CRs among lines belonging to different rail transit modes, the average
values of CRs among metro lines (in black boxes 1
), among suburban railway lines (in black
boxes 2
), among intercity high-speed rail lines (in black boxes 3
), among metro lines and
suburban railway lines (in black boxes 4
), among metro lines and intercity high-speed rail
TRANSPORTMETRICA A: TRANSPORT SCIENCE 27
lines (in black boxes 5
), and among suburban railway lines and intercity high-speed rail
lines (in black boxes 6
), are computed and shown in Figure 15 (b).
Some conclusions may be obtained from Figure 15: Compared with the average values
of CRs among lines on the same rail transit mode ( 1
,2
and 3
in Figure 15 (b)), the average
value of CRs among metro lines is the lowest and the average value of CRs among intercity
high-speed rail lines is the highest. The reason is that, compared with the fulfillment proba-
bility of the planned trains running on different rail transit modes, the fulfillment probability
of the planned trains on metro lines is low, but the fulfillment probability of the planned
trains on intercity high-speed rail lines is high. In addition, the three intercity high-speed
rail lines are connected at station 56, as shown in Figure 10. Thus, passengers can trans-
fer to any intercity high-speed rail line conveniently at station 56 when traveling among
intercity high-speed rail lines. Compared with the average values of CRs among lines on
different rail transit modes ( 4
,5
and 6
in Figure 15 (b)), the average value of CRs among
metro lines and suburban railway lines ( 4
in Figure 15 (b)) is the lowest. The reason is that
the suburban railway lines are located in suburban areas where the connections among
suburban railway lines and metro lines are weak. In addition, the fulfillment probabilities of
the planned trains on metro lines and suburban railway lines are not high.
5.3.3. CR of the RTN
According to the OD pairs’ CRs and passenger trips among OD pairs during morning peak
periods, the CR of Chengdu’s RTN during morning peak periods is computed to be 92.24%.
Therefore, the fraction of passengers who can travel on Chengdu’s RTN is 92.24% during
morning peak periods.
5.4. CR enhancement for Chengdu’s RTN
5.4.1. CR enhancement
The operational cost constraints of 100,000 and 200,000 , the maximum number of
trains that can run on routes and the maximum number of trains that can be added by
available vehicles on lines are considered for enhancing the CR of Chengdu’s RTN by adding
trains. Here, the maximum number of trains that can be added by available vehicles on a
line is assumed to equal the maximum frequency of that line minus the number of trains
already on that line. The SGA and MPGA are applied on a personal computer with a 2.80 GHz
i7-7700HQ central processing unit and eight cores and 8GB RAM to obtain the solutions
with operational cost constraints. The CR curves over successive generations for operational
cost constraints of 100,000 and 200,000 are shown in Figure 16 (a) and (b), respectively.
Figure 16 shows that the MPGA converges in fewer generations to obtain better solutions
than the SGA. The MPGA and SGA computation times for optimizing solutions with an oper-
ational cost constraint of 100,000 are 6.37 and 8.94 min, respectively. The MPGA and SGA
computation times for optimizing solutions with an operation cost constraint of 200,000
are 6.53 and 7.84 min, respectively. Thus, the MPGA computes faster than SGA and the
number of generations is smaller for the MPGA than SGA when their computation results
converge. The optimized solutions for operational cost constraints can be obtained using
the MPGA within 50 generations.
The optimized solutions obtained with the MPGA for cost constraints of 100,000 and
200,000 are shown in Table 12. It shows the network’s CR and increased operational cost,
28 J. LIU ET AL.
Figure 15. (a) CRs among lines; (b) average values of CRs among lines on the same rail transit mode and
the average values of CRs among lines on different rail transit modes.
as well as the percentage increase in CR and train operational cost after adding trains on
routes.
Table 12 shows that the percentage increase in CR is smaller than the percentage
increase in train operational cost when adding trains corresponding to the optimized solu-
tions. It seems that adding trains on routes is unwise. However, if the benefit of adding
trains is considered from both the passengers’ and operator’s perspectives, which is intro-
duced in section 5.4.2, then we find that the benefit of adding trains corresponding to
TRANSPORTMETRICA A: TRANSPORT SCIENCE 29
Figure 16. The CR curve over successive generations when applying SGA and MPGA.
Table 12. The optimized solutions obtained with MPGA.
After adding trains
Cost constraints () Trains added on routes CR (%)
Percentage
increase in
CR (%)
Increased
operational
cost ()
Percentage
increase in
operational
cost (%)
100,000 R2 +1, R4 +2, R6 +2,
R7 +1, R9 +2, R10 +2,
R12 +1, R13+2,
R15 +1, R19+1,
R22 +1, R25+1,
R27 +1, R29 +1
93.72 1.48 98,835 6.43
200,000 R1 +5, R3 +1, R4 +3,
R5 +2, R7 +2, R8 +3,
R10 +2, R11+4,
R13 +2, R14+3,
R16 +3, R19+3,
R20 +1, R21+2,
R23 +1, R25+1,
R27 +1, R28+1,
R30 +1
94.31 2.23 196,960 12.82
R1 to R30 is shown in Fig. 12 and Rx +y represents that y trains are added on route Rx.
optimized solutions exceeds the cost of adding trains when the operational cost constraints
are 100,000 or 200,000 .
The solutions for enhancing the CR of Chengdu’s RTN with different operational cost
constraints (specified as 50,000 , 100,000 , . .. , 500,000 ) are obtained by applying
MPGA. The CR enhancements corresponding to the optimized solutions with different cost
constraints are shown in Figure 17. It shows that as the operational cost constraint values
increase, the CR of Chengdu’s RTN continues to increase due to adding trains corresponding
to the optimized solutions. However, the rate of increase decreases.
5.4.2. The benefit of adding trains
Adding trains on routes enhances the CR of Chengdu’s RTN. The benefit of CR enhancement
is estimated from the operator’s perspective, or the passengers’ perspective, or both. To
compute the benefit from the passengers’ perspective, a Stochastic User Equilibrium model
30 J. LIU ET AL.
Figure 17. The CR enhancement corresponding to optimized solutions with different cost constraints.
Figure 18. The cost of adding trains corresponding to the optimized solutions compared with the
operator benefit, user benefit and total benefit.
is applied to assign passenger trips to the network which is solved by the method of suc-
cessive weighted averages (Qian and Zhang 2013). Then, the passengers’ average GTC with
and without added trains is computed. The costs of adding trains on routes corresponding
to the optimized solutions with different cost constraints (50,000 , 100,000 , . .. , 500,000
) are computed. The cost of adding trains corresponding to the optimized solutions and
the benefit of adding trains from the operator’s perspective or the passengers’ perspective,
or both, are estimated and shown in Figure 18. It shows that as the cost constraints values
increase, the benefits of adding trains on routes corresponding to the optimized solutions
with different cost constraints keep increasing, but at a decreasing rate.
Figure 18 shows that the benefit of adding trains corresponding to the optimized solu-
tions from the operator’s perspective is smaller than from the passengers’ perspective.
However, the benefit of adding trains corresponding to the optimized solutions from the
passengers’ perspective is higher than the cost of adding trains when the cost constraint is
below 250,000 , and the benefit of adding trains corresponding to the optimized solutions
from both the operator’s and passengers’ perspectives is higher than the cost of adding
TRANSPORTMETRICA A: TRANSPORT SCIENCE 31
them when the cost constraint is below 300,000 . The shaded area in Figure 18 repre-
sents the net benefit of adding trains which equals the benefit of adding trains minus the
operational cost of adding trains. Therefore, from both the operator’s and passengers’ per-
spectives, adding trains can not only increase the RTN’s net benefits, but also increase the
network’s CR.
6. Conclusions
The passengers’ travel behavior is considered here for measuring the RTN’s CR and a model
for maximizing an RTN’s CR by adding trains is proposed and solved with an MPGA. The
proposed method and model are applied to Chengdu’s RTN. The results show that the
measured CR of Chengdu’s RTN during morning peak periods is 92.24%. The optimized
solutions for enhancing the CR of Chengdu’s RTN with different operational cost constraints
are obtained by the MPGA. The benefit of adding trains is measured from both the opera-
tor’s and passengers’ perspectives, which equals the sum of CR enhancement benefit and
the reduction in average GTC for per passenger multiplied by passengers who can travel on
the RTN. The result shows that as values of operational cost constraints increase, the ben-
efits of adding trains on routes corresponding to the optimized solutions keep increasing
in Chengdu’s RTN during morning peak periods, but, at a decreasing rate. The benefit of
adding trains on routes corresponding to the optimized solutions from both the operator’s
and passengers’ perspectives is higher than the operational cost of adding trains during
morning peak periods when the operational cost constraint is below 300,000 .
The proposed model can support operators in adding trains on routes to enhance the
CR of the RTN and decrease passengers’ total GTC. Potentially, the application of the model
can be extended to bus networks.
The acceptable paths are determined according to trip times on paths. However, pas-
sengers may be concerned with travel time uncertainty when selecting travel paths. To
improve the realism of our model, we will try to consider the time reliability in further studies
when measuring the CR of RTNs. The RTN’s CR can be improved by adding links and adding
lines. However, planners and managers mainly introduce new links and lines for satisfying
transportation demand rather than improving the network’s CR. Therefore, it seems desir-
able to study the effects of adding links or adding lines on the RTN’s CR, and to develop a
model for maximizing the benefit of constructing new links or lines while considering the
benefit to the RTN’s CR.
Acknowledgements
The authors thank the Chengdu’s rail transit manager for providing relevant data. We also acknowl-
edge the support of China’s National Key R&D Programmes (2017YFB1200700), and the National
Natural Science Foundation of China (NSFC) (71701174), and the Fundamental Research Funds for
the Central Universities (2682021ZTPY072). The first author is supported by China Scholarship Council
(201907000071).
Disclosure statement
No potential conflict of interest was reported by the author(s).
32 J. LIU ET AL.
Funding
This work was supported by China Scholarship Council [Grant Number 201907000071]; National Sci-
ence Foundation of China [Grant Number 71701174]; the Fundamental Research Funds for the Central
Universities [Grant Number 2682021ZTPY072]; China’s National Key R&D Programmes [Grant Number
2017YFB1200700].
ORCID
Jie Liu http://orcid.org/0000-0002-1920-1043
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