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Transportmetrica A: Transport Science
ISSN: 2324-9935 (Print) 2324-9943 (Online) Journal homepage: https://www.tandfonline.com/loi/ttra21
Customized bus routing problem with time
window restrictions: model and case study
Rongge Guo, Wei Guan, Wenyi Zhang, Fanting Meng & Zixian Zhang
To cite this article: Rongge Guo, Wei Guan, Wenyi Zhang, Fanting Meng & Zixian Zhang
(2019) Customized bus routing problem with time window restrictions: model and case study,
Transportmetrica A: Transport Science, 15:2, 1804-1824, DOI: 10.1080/23249935.2019.1644566
To link to this article: https://doi.org/10.1080/23249935.2019.1644566
Published online: 25 Jul 2019.
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TRANSPORTMETRICA A: TRANSPORT SCIENCE
2019, VOL. 15, NO. 2, 1804–1824
https://doi.org/10.1080/23249935.2019.1644566
Customized bus routing problem with time window
restrictions: model and case study
Rongge Guoa,WeiGuan
a, Wenyi Zhanga, Fanting Mengband Zixian Zhanga
aKey Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport,
Ministry of Transport, Beijing Jiaotong University, Beijing, P. R. People’s Republic of China; bState Key
Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, People’s Republic of China
ABSTRACT
Considering that, in reality, passengers usually hold preferred time
windows when waiting for a bus at a station, this study develops
a mixed integer programming model for the customized bus rout-
ing problem (CBRP) with full spatial–temporal constraints based on
one of our previous studies. Specifically, bus routing and passenger
assignment are simultaneously optimized with better vehicle capac-
ity utilization and more realistic considerations of partial service,
characteristics of customized bus service, and a range of operational
constraints. To solve the formulated model, an exact algorithm (i.e.
the branch-and-cut algorithm) and two heuristics (i.e. the genetic
and tabu search algorithms) are numerically compared through an
illustrative example, and subsequently, a case study in Beijing is con-
ducted to assess the proposed approach. A comparison with the
practical customized bus system shows that the proposed approach
realizes effective vehicle usage on several routes.
ARTICLE HISTORY
Received 10 December 2018
Accepted 13 July 2019
KEYWORDS
Customized bus routing
problem; time windows;
mixed integer programming
Introduction
The sharp increase in mobility demands has posed great challenges for the public transit
(PT) system. The new generation PT system aims to provide personalized services to pas-
sengers, especially in areas with dense road networks and large populations, which has
tremendous advantages in terms of service convenience, punctuality, and high comfort-
ability. However, driven by fixed routes and pre-given timetables, a regular bus service
cannot achieve service innovation, accessibility, and individual goals, and therefore, has
suffered from passenger loss. The annual number of passengers for conventional buses
has decreased significantly since 2012. Specifically, the data dropped to 369,019 million
by the end of 2016 (Ji 2017). To satisfy the fast-growing urban travel demands, the trans-
portation policymakers and PT system operators have made significant efforts over several
decades to deal with such issues. Recently, a new type of public transport, named cus-
tomized bus (CB), was launched in Beijing. Such PT service enjoys high popularity and
CONTACT Wei Guan weig@bjtu.edu.cn Key Laboratory of Transport Industry of Big Data Application
Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, People’s
Republic of China
© 2019 Hong Kong Society for Transportation Studies Limited
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1805
holds promise to provide greater advanced, attractive, and demand-responsive service to
clienteles, especially commuters (Liu and Ceder 2015).
Compared to the limited research on CB, the demand-responsive transit (DRT) system
has been extensively studied (Wong, Han, and Yuen 2014;Hoetal.2014; Charisis, Iliopoulou,
and Kepaptsoglou 2018). Considering the on-demand transportation service, the dial-a-ride
problem (DARP) is similar to that studied here, and to the best of our knowledge, it is a theo-
retical challenge associated with routing planning and dispatching vehicles (Cordeau 2006;
Cordeau and Laporte 2007). Masmoudi et al. (2017) proposed a hybrid genetic algorithm
(GA) to solve the heterogeneous dial-a-ride problem. To increase the possibility to pool
routes between Medico-Social Institutions (MSI), Tellez et al. (2018) considered the recon-
figurable vehicle capacity in the mixed DARP. Theoretically, DARP mainly offers ‘door-to
door’ service to individuals, instead of a group. Another similar DRT system is a feeder bus,
which merges the flexibility of DRT and the low operation cost of fixed-route transit (Quadri-
foglio and Li 2009;LiandQuadrifoglio2010; Tabassum et al. 2017; Drakoulis et al. 2018).
Chandra and Quadrifoglio (2013) formulated an analytical model to implement an efficient
feeder transit service, by maximizing the service quality. Yu, Machemehl, and Xie (2015)
defined a circulator service network design problem to determine a feeder bus route and
stopping sequence, which sought to minimize the tour cost and walking time of each pas-
senger. However, the feeder bus system is less applicable to serve passengers with similar
trips compared to CB.
CB has evolved from the ‘car sharing’ phenomenon proposed by the Sefage (Harms and
Truffer 1998), and has been springing up with the advantages of alleviating congestion
and reducing pollution (Kirby and Bhatt 1975). The CB system has been formally investi-
gated in China since 2013. It has been emphasized that CB may exert a positive effect on
the PT system as it offers a friendly environmental transit service with superior user experi-
ence, compared to alternatives such as conventional buses (Xu et al. 2013; Liu et al. 2016).
In addition, it is an adaptive management solution for the ‘last mile problem’ (Wang and
Wang 2015). Presently, the CB system is being operated in two popular ways, the com-
muting bus and express transit, where the former shows higher efficiency owing to more
obvious advantages in reducing private and public system costs.
To the best of our knowledge, the high-quality travel information is a vital input param-
eter for designing the transit service. Some researchers have foreseen the opportunity
to investigate the personalized travel data of CB by employing various data sources and
methodologies. Existing studies have utilized smartcard data or taxi GPS trajectory data
to estimate the origin and destination (OD) distribution for designing the CB service. For
instance, some studies have used the density-based spatial clustering of applications with
noise to capture the passenger travel pattern from smartcard data (Ren et al. 2016;Lietal.
2018a; Qiu et al. 2018), while others have used a holistic framework based on taxi GPS trajec-
tory data (Lyu et al. 2016). In terms of passengers’ travel mode choice, a contingent valuable
method was proposed to determine the effect of the distance between home and work
place on passenger choices of CB (Cao and Wang 2016), while the mode shift performance
was driven by the fare and out-of-vehicle time (Zhang, Wang, and Meng 2017).
In the literature, the CB operation has been discussed, in terms of, but not limited to,
timetable development and vehicle scheduling. Ma, Zhao et al. (2017) investigated the
stop planning and timetables with the objective of minimizing the interests of passengers,
operators, and society. Cao and Wang (2017) characterized passenger assignment on CBs,
1806 R. GUO ET AL.
and found that vehicles with smaller busloads could serve passengers flexibly and offer a
greater level of benefits. Based on the vehicle routing problem with time windows (VRPTW),
Zhou et al. (2017) conducted two dispatching strategies on the level of dynamic traffic
information and validated their effectiveness through an example.
Generally, a typical CB operation-planning process contains five basic activities, while the
fundamental problem that the operators encounter is the method of designing the routes
based on passenger-demand data. Ma, Yang et al. (2017) exhibited a route selection model
for CBRP by considering environmental costs, operating costs and traffic congestion costs.
However, they failed to assign passengers to vehicles. Tong et al. (2017) formulated a flow-
based optimization model where a time geography modelling framework was employed
to analyse passengers’ behaviours. Another CB routing design problem was proposed by
Li et al. (2018b), who formulated a mixed load routing model that incorporated multiple
depots regarding the actual situation. The above-mentioned studies viewed CBRP as a
generalized version of the vehicle routing problem with pick-up and delivery, where the
modelling framework is insufficient in describing the CBRP, as it cannot characterize the
number of passengers that are picked up or delivered by the specific vehicle and the in-
vehicle passengers at the station. To resolve this issue, a mixed integer programming, rather
than a VRP-like pure integer programming, was introduced by us (Guo, Guan, and Zhang
2018), where a new decision variable was proposed to address the multi-vehicle routing
problem.
In reality, passengers desire to be served on time or within their preferred time windows;
the behind-schedule buses are considered unfavourable. Considering their behavioural
preferences, in this paper, a CBRP with full spatial–temporal constraints has been presented,
as an extension of our previous study (Guo, Guan, and Zhang 2018). We introduce a math-
ematical programming formulation to simultaneously optimize vehicle routes and deter-
mine passenger assignment, which incorporates a partial service in the form of penalty
depending on the unserved passengers, as well as a realistic constraint concerning essential
characteristics of the CB service. Specifically, the proposed method aims at increasing the
occupancy rate per vehicle, i.e. maximizing the number of in-vehicle passengers. Addition-
ally, we design two experiments for the proposed problem: (1) an illustrative example on a
small-scale network that can assess the optimal gap among three algorithms (branch-and-
cut, genetic, and tabu search algorithms), and (2) a large-scale case study, in which we study
the effectiveness of the introduced methodology compared to the practical CB service.
The remainder of this paper is organized as follows. Section 2 contains a precise math-
ematical description and the proposed new integer programming model for CBRP. Sub-
sequently, three algorithms are developed in Section 3. Section 4 provides a simplified
example to assess the performances of the algorithms, followed by a large-sized experi-
ment to demonstrate the performance of the proposed method. Finally, we conclude the
study in Section 5 with a discussion on possible extensions.
Problem definition and formulation
Problem description and modelling assumption
The model introduced in this paper incorporates important characteristics of CB, which can
typically be formulated as VRPTW with known passenger demands beforehand in the field
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1807
of vehicle network programming. Despite the similarity and continuity in the modelling
framework to that proposed by Guo, Guan, and Zhang (2018), the current study makes two
key extensions: addition of service time windows and clarification of visiting order between
stations. The first extension equips the current model with enhanced behaviourism, while
the second strengthen its strictness.
The CBRP with time windows is formulated as a mixed-integer problem, which can be
used in the numerical experiments to determine optimal solutions. The CB route design
problem is defined on a complete graph G=(V,A), where V={0, 1, ...,N,N+1}is a com-
bination of stations and depots. Vertices 0 and N+1 denote the same depot, which means
that each vehicle should depart from 0 and return to N+1. Additionally, let S={1, ...,N}
be the set of stations with S=O∪D, where Oand Ddenote the origin and destination sets,
respectively. A positive travel demand qrs is associated with an origin r∈Oand a destina-
tion s∈D. Each element of the set A={(i,j)|i,j∈V,i= j}connects two vertices in Vand is
associated with two non-negative values, i.e. travel time tij and distance dij.
A set of homogeneous vehicles with maximal capacity of Cap is positioned at the depot
0. Vehicles are routed to transport passengers before returning to the depot within specific
time intervals. Here, each vertex i∈Vhas a time window [ArrTi,DepT
i], which denotes that
the vehicle must arrive at the station after ArrTiand leave the station before DepTi. In addi-
tion, each vertex i∈Vhas a service time ti(t0=tN+1=0). Note that the vehicle departure
time from each station is represented by the arrival time and service time, i.e. Tk
i+ti,while
the vehicle departure time from the depot 0 is represented by the arrival time as t0=0.
To simplify the problem, some hypothetical assumptions are made beforehand: (1) The
travel demand from any origin to corresponding destination is known; (2) The time win-
dows and service time of each vertex are pre-given; (3) The travel time and distance of any
arc are given before; (4) The capacity and average speed of vehicles are constant and given;
(5) The CB transit network is a two-way physical network.
Relevant parameter variable denition
For facilitating the statement, as well as helping readers to understand the context better,
the notations used throughout this paper are listed in Table 1.
Formulation
The mathematical model of customized service design problem is formulated as a mixed-
integer programming as follows:
Min
ω1μf
k∈K
j∈S
xk
0j+ω2μr
k∈K
i∈V\{N+1}
j∈V\{0},i=j
xk
ij ·(dij/v+tj)
+ω3μp
r∈O
s∈D(qrs −
k∈K
yk
rs)(1)
s.t.
k∈K
i∈S,i=j
xk
ij ≥1∀j∈S(2)
1808 R. GUO ET AL.
Tab le 1. Notation.
Notation Definition
Sets and indices
0, N+1 Depot instances
VSet of vertices
SSet of stations
i,jIndices of vertices
OSet of origins
DSet of destinations
rIndex of origins
sIndex of destinations
ASet of arcs connecting pairs of vertices
(i,j)Index of arcs
KSet of vehicles
kIndex of vehicles
Parameters
ω1,ω2,ω3Weight factors of fixed cost, running cost and penalty
μfFixed cost per vehicle
μrRunning cost per minute
μpPenalty per unserved passenger
dij Distance of arc (i,j)
tij Trav el time of arc (i,j)
tiService time at vertex i
qrs Travel demand from origin rto destination s
ArrTiEarliest arrival time at station i
DepTiLatest departure time at station i
ArrTN+1Earliest arrival time at depot N+1
DepT0Latest departure time at depot 0
vVehicle average speed
λMinimum load requirement
Cap Vehicle capacity
lmin Lower limit of route length
lmax Upper limit of route length
MMaximum station quantity for the route
BA large number
Intermediate variable
Pk
iNumber of passengers in vehicle kat vertex i
Tk
iArrival time of vehicle kat vertex i
Decision variable
xk
ij Equal to 1 if arc (i,j)is on the optimal route of vehicle k
yk
rs Number of passengers served by vehicle kfrom origin rto destination s
i∈S,i=j
xk
ij ≤1∀k∈K,j∈S(3)
j∈S
xk
ij =
j∈S
xk
ji =1∀i∈{0, N+1},k∈K(4)
i∈V\{N+1},i=j
xk
ij =
i∈V\{0},i=j
xk
ji ∀j∈S,k∈K(5)
yk
rs >0⇒
j∈S
xk
rj =
i∈S
xk
is =1∀r∈O,s∈D,k∈K(6)
r∈O
s∈D
yk
rs ≥λ∀k∈K(7)
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1809
k∈K
yk
rs ≤qrs ∀r∈O,s∈D(8)
Pk
j=Pk
i+
s∈D
yk
js −
r∈O
yk
rj·xk
ij ∀i∈V\{N+1},j∈S,k∈K(9)
Pk
i≤Cap ∀i∈V,k∈K(10)
Pk
0=Pk
N+1=0∀k∈K(11)
Tk
j−Tk
i−B·xk
ij ≥tij +ti−B∀i∈V\{N+1},j∈V\{0},i= j,k∈K(12)
yk
rs >0⇒Tk
s≥Tk
r∀r∈O,s∈D,k∈K(13)
Tk
i+ti≤DepTi∀i∈S,k∈K(14)
Tk
i≥ArrTi∀i∈S,k∈K(15)
Tk
0≤DepT0∀k∈K(16)
Tk
N+1≥DepTN+1∀k∈K(17)
lmin ≤
i∈V\{N+1}
j∈V\{0},i=j
xk
ij dij ≤lmax ∀k∈K(18)
i∈V\{N+1}
j∈S,i=j
xk
ij ≤M∀k∈K(19)
The objective of minimizing the total cost is defined in Equation (1), which comprises three
parts. The first part is the sum of the fixed costs of all vehicles; i.e. if a vehicle departs from
the depot for service, a fixed cost μfis generated, the second part counts the total running
cost of each vehicle, and the last part computes the penalty of the unserved passengers.
In this study, vehicles may not serve all passengers; in other words, the CB system provides
a partial service, as ‘no-show’ passengers may change their minds towards other transport
means or cancel their travel plans.
Constraints (2)–(5) are the route constraints of the multi-vehicle routing problem. Con-
straints (2)–(3) indicate that a station can be visited multiple times, while the number of
visits by the same vehicle is not more than once. Constraint (4) states that each vehicle
starts and ends its route at the depot. Constraint (5) is the flow balance constraint.
Constraint (6) shows the relationship between the two variables, i.e. if an OD pair (say r,
s) is served by a vehicle k, the vertices rand sneed to be visited by k.
To guarantee the modest profit goals for a long-term development of CB, operation con-
straint (7) states that the number of passengers served by the assigned vehicle exceeds the
minimum load requirement. Constraint (8) indicates that the number of served passengers
with travel OD (say r,s) is less than or equal to the total passengers of this demand.
Constraints (9)–(11) are capacity constraints. Constraint (9) computes the in-vehicle pas-
sengers of the station j, which depends on the in-vehicle passengers of the previous station
1810 R. GUO ET AL.
Pk
iand pick-up and delivery passengers at the station j. Constraint (10) guarantees that the
capacity of each vehicle is not exceeded. Constraint (11) specifies that all vehicles leaving
and returning to the depot are unloaded.
The time window constraints are covered by (12)–(17). Equation (12) is the constraint of
vehicle visiting order, showing that the vehicle arrival time at vertex jis at least equal to
the sum of travel time from the preceding vertex i, service time, and vehicle arrival time at
the preceding vertex i. Constraint (13) restricts the visiting order of the paired origin and
destination. If an OD pair (say r,s) is served by a vehicle k, the vehicle arrival time at the
destination sis greater than that of origin r. Constraints (14)–(17) ensure that the vehicle
must arrive at and leave the vertices within each defined time interval.
Finally, (18)–(19) are customized service constraints. Constraint (18) ensures that each
route length is within a certain range. Constraint (19) limits the number of stations of
each route, because passengers expect few intermediate stations between the starting and
destination areas, to have a better experience.
Solution algorithm
The model developed in this study can be viewed as a variant of VRPTW, for which finding
the exact solution within a reasonable time has been commonly considered challeng-
ing (Bräysy and Gendreau 2005). Here, three approaches, including the branch-and-cut
algorithm, the genetic algorithm (GA) and the tabu search (TS) algorithm, are adopted to
solve the problem, conduct analyses, and compare with each other. The performances of
these three algorithms have already proven in related studies (Garcia, Potvin, and Rousseau
1994; Thangiah, Nygard, and Juell 1991; Fukasawa et al. 2006).
Branch-and-cut algorithm
The branch-and-cut algorithm is the same approach introduced for the travelling salesman
problem (Padberg and Rinaldi 1991). As a combination of branch-and-bound and cutting
planes, the branch-and-cut algorithm addresses the problem by solving a series of relax-
ation problems of integer linear programming (Fischetti et al. 1997; Lysgaard, Letchford,
and Eglese 2004). The branch-and-cut algorithm contains several steps: (i) pre-processing
initial integer programming problem and setting upper and lower bounds; (ii) solving the
relaxation problem to determine whether it is feasible; (iii) cutting the unfeasible part and
adjusting the two bounds to find a feasible solution; and (iv) obtaining the optimal solution
(Mitchell and John 2011).
The solution results of the customized bus routing problem with time windows are found
by using the software ILOG CPLEX, which was introduced by IBM. CPLEX solves the mixed
integer programming with accurate and effective solutions. In the current model, constraint
(6) of the decision variables is regarded as a hard constraint that is difficult to transform
into a computational form in CPLEX. To combat this issue, we introduce a new binary vari-
able zk
rs and propose constraints (20) and (21), which are equal to the constraint (6), as
below:
zk
rs =1⇒yk
rs >0∀r∈O,s∈D,k∈K(20)
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1811
zk
rs ≤
j∈S
xk
rj +
i∈S
xk
is −1≤zk
rs ·B∀r∈O,s∈D,k∈K(21)
where zk
rs equals to 1 if an OD pair (say r,s) is served by a vehicle k. Constraint (20) states
that if an OD pair (say r,s) is served by a vehicle k, the number of served passengers should
be strictly positive. Constraint (21) shows the relationship between the two variables.
Genetic algorithm
The genetic algorithm (GA) was proposed by Holland (1992), and has been widely used for
solving complex problems (Bräysy and Gendreau 2005). It begins from an initial population
and encodes individuals’ characteristics as string chromosomes, each of which is associated
with a fitness function value that refers to a measure of the objective. To evolve a new popu-
lation of individuals, GA generates a set of individuals of the offspring through an iterative
process and the algorithm ends when the process reaches the stopping principles. Next,
the best chromosome is generated by comparing the fitness, and subsequently decoded
to provide the corresponding solution (Thangiah 1993).
The CBRP solutions in this paper are encoded by decimal digits. In our given chromo-
some, every individual with the same length comprises two parts: a vehicle route including
the stations assigned to the vehicle and the number of boarding and alighting passengers
at corresponding stations. Specifically, the vehicle route dictates the order in which these
stations are visited, and can enable the vehicle to arrive at and depart from stations within
the corresponding time windows.
In the given routing problem, passengers can be served by multiple vehicles, between
paired origins and destinations. To ensure that each pair (origin rand its associated desti-
nation s) belongs to the same route, we need to restrict that the vertices rand sare visited
in order. Knowing the number of vehicles, depots, and spatial–temporal travel demands
of each OD, the initial population (population size δ) is created randomly according to the
following steps: find the shortest path between two vertices by Dijkstra’s algorithm, assign
passengers to vehicles with available capacity, and calculate the vehicle arrival time at each
vertex to ensure that it satisfies the given time intervals.
The creation of the new generation for individuals contains three major steps: selec-
tion, crossover and mutation. The roulette method is used to select the parent to generate
offspring, while the one-point crossover is applied to adjust the routes according to the
crossover rate α. For mutation operation, a new route is generated to replace the original
one according to the mutation rate β. Here, the stopping condition of the iterative process
is the maximum generation θ.
Tabu search algorithm
Tabu search (TS) algorithm, as an extension of the local search algorithm, is a powerful meta-
heuristic for solving combinatorial optimization problems (Glover and Laguna 1997). TS can
avoid the paths already investigated and ensure that the new search spaces of the solu-
tion will be investigated. To avoid the local minima, a memory structure, named tabu list,
is introduced to save the explored local optimal solution; these solutions are marked with
1812 R. GUO ET AL.
attributes that remain a tabu (Nguyen, Crainic, and Toulouse 2013). Generally, some key
elements may influence the search performance of TS:
Tabu length: it can affect the memory occupation and search range.
Aspiration criterion: if the following conditions occur, the tabu length of the current
object is set to 0: (i) if there is a solution whose value is better than that of any of the pre-
vious best candidate solutions, (ii) if there is a solution with the smallest value when all
objectives are forbidden, and (iii) if there is a solution that has a large impact on the target
value.
Candidate set: The size of the candidate set may influence the memory, calculation and
optimal solution.
Measuring function: The objective function can be adopted as a measuring function.
Stop condition: (i) the algorithm reaches the iterative step, (ii) an objective value exceeds
a certain value, and (iii) the current optimal value does not change within an iterative step.
In the customized bus routing problem with time windows, the main steps of the
algorithm are given below.
Algorithm
Step 1 Initial solution Generate initial solution saccording to the generation principle mentioned
in the GA and calculate the function Equation (1), set the function value
fas optimal objective value, and sis the current optimal solution x∗.
Generate tabu list A=null and the length of the list is l.
Step 2 Neighbourhood searching Searcha set of its neighbourhood N(s)randomly and calculate the function
value f∗of every solution s∗of N(s).
Step 3 Update tabu list Evaluate the neighbourhood N(s)of current solution by aspiration
criterion. When f∗is better than f, judge whether s∗is forbidden, if it
has been forbidden, choose the suboptimal solution sand continue;
otherwise, update x∗and A. Once all the solutions of N(s)are forbidden,
eliminate the forbidden attribute of the header solution in Aaccording
to first-in, first-out (FIFO). The duration of the forbidden attribute of the
tabu objects minus 1 with the update ofA.
Step 4 Stop condition Repeat the Step 2 to Step 3 until the iteration reaches the determined
iterative step σ
Numerical experiments
This section conducts a small-scale numerical example and a real-world case study of the
transport network in Beijing, to explore the effectiveness of the proposed model and algo-
rithms. All algorithms are coded and performed in MATLAB R2014b and ILOG CPLEX on a
PC with the following configuration: intel i5-6500 CPU, 3.20 GHz, and 4 GB memory.
Small-scale example
This example contains a simple network with 17 vertices and 48 arcs; the distance between
any two adjacent vertices is shown in Figure 1. Four vehicles are originally assigned at the
depot, namely vertex 1, to satisfy the travel demands. The vehicle capacity is assumed to
be 55 (unit: person) with the minimum load requirement λ=40 person, and the average
speed of each vehicle is set as 25 km/h. The route length range is estimated as [5, 40] (unit:
km) and the number of stations visited by the vehicle is less than or equal to 8. The objective
function should be computed considering the proportions of different expenses. To mini-
mize the impact of ‘no-show’ passengers on the results, the weight factor of the penalty is
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1813
Figure 1. Simple network.
assumed to be 0.2, while that of the other two factors is 0.4. The values of the other involved
parameters are given below: μf=600Y−/veh, μr=540Y−/h, and μp=100Y−/person.
As each vehicle should depart from and return to the depot within the defined tempo-
ral interval, we set the time window of the depot as the time interval [0, 80] (unit: min). In
addition, Table 2exhibits the spatial and temporal constraints of 140 passengers, as well as
the service time of each station. Taking the first row as an example, the number of travel
demands from vertex 2 to vertex 4 is 15. The earliest arrival time and latest departure time
of two vertices are in the time intervals [5, 15] and [20, 35] (unit: minute), respectively.
Parameter testing of GA and TS
We first conduct the robustness test of GA by recording the objective values of 10 opera-
tions in the case of fixed parameters. (δ,θ,α,β) are set to (50, 100, 0.9, 0.4) in the algorithm.
The values of 10 operations are not significantly different from each other, which shows
that the GA proposed in the paper is robust (Figure 2(a)).
1814 R. GUO ET AL.
Tab le 2. Passenger demands of example.
Origin Service time Time window Destination Ser vice time Time window Demand
2 1 [5,15] 4 0.7 [20,35] 15
2 1 [5,15] 14 0.9 [55,70] 15
3 0.5 [3,10] 9 1 [20,40] 19
3 0.5 [3,10] 12 0.8 [25,55] 17
5 0.5 [8,18] 11 1 [40,60] 13
6 0.5 [8,18] 13 0.8 [45,60] 18
8 0.7 [15,40] 16 1 [50,65] 9
9 1 [20,40] 14 0.9 [55,70] 16
10 0.5 [30,45] 16 1 [50,65] 18
Figure 2. Results of robustness tests (a) and different parameter combinations (b).
Tab le 3. Results of different parameter combination.
Tabu list length 30 40 50
Iterative step 50 100 150 50 100 150 50 100 150
Total cost 2622 2586 2588 2588 2622 2522 2708 2588 2550
We implement several preliminary tests to set the parameters used in the GA and TS
algorithm. In these tests, the results are obtained by employing the different values of a
parameter, while the other parameters are kept constant. δ,θ,andN(s)are set to 100, 200,
and 50; α,β,l,andσare set to (0.7, 0.8, 0.9), (0.3, 0.4, 0.5), (30, 40, 50), and (50, 100, 150),
respectively. According to the tests performed for the GA and TS algorithm, (α,β)and(l,σ)
are set to (0.9, 0.3) and (40, 150), respectively (Figure 2(b) and Table 3).
Solutions obtained by three algorithms
This part discusses the different aspects of the results obtained for the small-scale example
by the three algorithms employed in the study. The solution results of the branch-and-cut
algorithm are found using ILOG CPLEX, while those of the heuristic algorithms are acquired
using MATLAB. Figures 3–5provide the optimal routes of three algorithms, respectively.
Passengers’ pick-up and drop-off locations are marked in orange and the visiting order of
the vehicle is represented by arrows with numbers. The computational results of the three
algorithms show that 140 passengers can be served by four customized vehicles, namely
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1815
Figure 3. Results of branch-and-cut algorithm.
Figure 4. ResultsofGA.
vehicles 1, 2, 3, and 4. Notably, vehicles 1 and 3 serve the same route in the branch-and-cut
algorithm. Likewise, the route of vehicle 1 is same as that of vehicle 4 in GA.
Table 3reports the gap of the solutions found by each heuristic method for the CPLEX
solution. The results show that the branch-and-cut algorithm performs best with total cost
gaps of 190.4 and 45.6 for the GA and TS algorithm, respectively. Specifically, the branch-
and-cut method decreases the number of unserved passengers to zero with respect to
other solutions. By contrast, this reduction mostly increases the travel time for vehicles
(5.16 vs. 4.56 and 4.26, unit: h) and CPU time (67 vs. 30 and 57.8, unit: s). Therefore, this
method cannot improve the solutions for the large-scale experiment due to the long calcu-
lation time. Comparing the results of TS with those of GA, we can see that the TS algorithm
improves all solutions, but at the expense of a significant increase in CPU time (57.8 VS.
30, unit: s). Note that the TS method reduces the number of unserved passengers (12 vs.
16), with more significantly in total cost (2120.16 vs. 2264.96, unit: Y−) and vehicle travel
time (4.56 vs. 4.26, unit: h). These results clearly indicate that the TS method leads to better
results for a large-scale experiment with respect to GA Table 4.
Case study based on the Beijing transport network
To further demonstrate the performance of the proposed approach for a large-scale prob-
lem, a more realistic case study is devised using the real-world transport network of
Beijing.
1816 R. GUO ET AL.
Figure 5. ResultsofTS.
Tab le 4. Comparison of solution results.
GA TS
Results Branch-and-cut Value Gap Value Gap
CPU time (s) 67.00 30.00 −37 57.80 −9.8
Total cost (Y−) 2074.56 2264.96 −190.4 2120.16 −45.6
Number of vehicles 4 4 0 4 0
Number of unserved passengers 0 16 +16 12 +12
Travel time (h) 5.16 4.56 −0.6 4.26 −0.9
A detailed information of travel requests (e.g. passengers’ spatial and temporal con-
straints) is generated based on historical smartcard data in five workdays in April 2017 (Guo
et al. 2019). During the morning period, passengers’ travel data show a strong commut-
ing pattern (Ma et al. 2013). Typically, they commute from large residential areas to main
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1817
Figure 6. The spatial location distribution of passengers’ travel demands (red and blue markers denot-
ing origins and destinations, respectively).
business districts, namely Guomao, Zhongguancun, Deshengmen and Dongzhimen. Here,
35 stations are selected as a large-scale experiment network with 54 trip groups and 1228
home-to-work trip requests, which covers large communities and four business areas. The
spatial distribution of the travel demands is exhibited in Figure 6, where origins and des-
tinations are marked by blue and red dots, respectively. Some passenger travel demands
are listed in Table 5, including the spatial and temporal constraints as well as service time.
Theoretically, the time window of each station is defined by two timestamps, as given
in Equations (14) and (15). However, the smartcard data only contain each passenger’s
boarding and alighting time; in other words, each origin or destination corresponds to one
timestamp. We thereby assume these two timestamps to be the time window of this trip, i.e.
the vehicle needs to leave the origin after the first timestamp and arrive at the destination
before the second timestamp.
In the implementation, we assume a fleet of vehicles with a 50-seat capacity and a mini-
mum load requirement of 40 persons. Each route contains 6 intermediate stations, and the
route length range is set in the interval [5, 35] (unit: km). Similar to related studies (Li and
1818 R. GUO ET AL.
Tab le 5. Passengers’ travel demands information.
Origin Longitude Latitude Destination Longitude Latitude Passenger Time window
10 116.3436 39.96613 7 116.3104 39.98196 8 [7:00,8:00]
11 116.3598 39.96636 7 116.3104 39.98196 8 [7:00,8:00]
1 116.2621 40.13031 8 116.3108 39.97673 9 [7:00,8:00]
1 116.2621 40.13031 7 116.3104 39.98196 11 [7:00,8:00]
11 116.3598 39.96636 7 116.3104 39.98196 13 [8:00,9:00]
... ... ... ... ... ... ... ...
16 116.3949 39.96713 8 116.3108 39.97673 13 [7:00,8:00]
Quadrifoglio 2010; Xiong et al. 2013; Ma, Yang et al. 2017), we assume the weight factors of
fixed cost, running cost, and penalty to be 0.4, 0.4, and 0.2, respectively. Other parameters
are given below: μf=500Y−/veh, μr=15Y−/min, and μp=60Y−/per.
The following section discusses the optimal solutions obtained by the TS algorithm and
analyses the results of two cases. The TS algorithm parameters are set as follows: tabu list
length l=40, iterative step σ=150, candidate set size N(s)=40.
Solution analysis for cases of partial and complete services
We investigate the different aspects of the results for the cases of partial service and com-
plete service. Here, the first case corresponds to the proposed model, while the second
case is similar to the practical commuting bus system, as commuting buses are supposed to
serve all passengers who book the seats in advance and ensure that each user gets to his/her
destination. To represent the case of complete service, we modify the objective function
and constraint (8) of the introduced model, with the remaining essentially unchanged. The
penalty is removed from the objective function and constraint (8) is modified to ensure that
the number of served passengers from the origin to the corresponding destination is equal
to the total passengers of this OD, as in the following expression:
k∈K
yk
rs =qrs ∀r∈O,s∈D(22)
Figure 7shows the optimal routes for the case of partial service, where the travel
demands are divided into three areas according to the station locations. Table 6compares
the results generated for different areas of two cases and Table 7lists the different aspects
of the results. Notably, the total cost (Cos., unit: Y−), number of passengers (Pass.), number of
vehicles (Veh.), number of served passengers (Ser.), number of unserved passengers (Uns.),
travel time (Tim., unit: h) and route length (Leg., unit: km) represent the values obtained.
In Table 6, we observe that the total cost for the complete service increases compared to
those of the partial service, while the number of unserved passengers in the complete ser-
vice reduces to zero due to the increase in the number of vehicles. Table 7shows that the
objective function value, travel time, and route length of the complete service increase at
about the same rate, and the number of served passengers increases by 2.85%. Specifi-
cally, the number of vehicles increases by 12%, while that of unserved passengers reduces
significantly.
Sensitivity analysis with respect to parameters for two cases
To further compare the different aspects of results for the cases of partial service (I) and
complete service (II), we implement more experiments with different values of parameters.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1819
Figure 7. Customized service network.
Tab le 6. Optimal results of case assuming all passengers served.
Results of case with partial service Results of case with complete service
Area no. Cos. Pass. Veh. Uns. Cos. Pass. Veh. Uns.
1 4335.37 460 9 10 4578.89 460 10 0
2 5076.02 510 10 10 5266.19 510 11 0
3 2287.19 258 6 14 2462.47 258 7 0
Theoretically, the computational result is closely related to the vehicle average speed v,as
shown in Equation (1). We use the vehicle average speed over six scenarios to demonstrate
the influence of this parameter; the corresponding computational results are presented in
Figure 8. It is clear that with an increase in v, all costs and travel time decrease continuously,
while the route length reveals a trend of fluctuation. This result is reasonable as the route
length is affected by several parameters simultaneously (e.g. lower and upper limits of route
length). Specifically, the optimal values of case II are greater than those of case I, because
more vehicles are assigned in case II to guarantee that all passengers are served.
1820 R. GUO ET AL.
Tab le 7. Details of results obtained for two cases.
Solutions for case with partial service Solutions for case with complete service Gap (%)
Cos. 11698.58 12307.55 5.21%
Veh. 25 28 12.00%
Ser. 1194 1228 2.85%
Uns. 34 0 −100%
Tim. 17.47 18.63 6.64%
Leg. 524.21 558.96 6.63%
Figure 8. Costs (a), route length and travel time (b) versus speed for the case of partial service (I) and
case of complete service (II).
In practice, route planning can be influenced by the vehicle capacity Cap. To demon-
strate this characteristic, we implement more numerical experiments with different vehicle
capacities, where six scenarios are considered. By pre-setting different parameters, the
computational results of the two cases are displayed in Figure 9. We observe that both
Figure 9. Costs (a), route length and travel time (b) versus vehicle capacity for the case of partial service
(I) and case of complete service (II).
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1821
subfigures exhibit the decreasing tendency. In particular, the solution results of case I
decline dramatically with the expansion of vehicle capacity. Notably, when the vehicle
capacity is enhanced to a certain value (i.e. Cap =50, unit: person), the data of caseIstart
increasing over those of case II. The reason behind this outcome is as follows: the increasing
vehicle capacity is beneficial to reduce the number of vehicles in case II, which can decrease
the related values.
Conclusions
Considering the spatial–temporal needs of different passengers, this study explicitly exam-
ines CBRP with time window restrictions. A mixed integer programming model is formu-
lated to determine routes and passenger-to-vehicle assignment simultaneously, combined
with passengers’ time window satisfaction. A series of constraints, such as load requirement
and intermediate station limitation, are incorporated into the model to represent real-world
requirements of customized bus services. Two computational experiments are conducted
by the branch-and-cut algorithm, GA and TS algorithm. A small-scale example is applied
to validate the effectiveness of the TS algorithm, while the large-scale experimental results
demonstrate the benefits of the proposed model. Compared to the practical customized
bus system, the introduced methodology results in significant savings in terms of the total
cost, travel time, route length and vehicle quantity, by explicitly incorporating the partial
service.
In the future, several relevant practical aspects need to be systematically formulated
for practical customized bus systems, and this study will focus on the development of the
model for the following cases: (i) the traffic congestion significantly influences the travel
time of vehicles, and therefore, it is important to consider the time variation of each path, (ii)
the specific paths are restricted during a certain period in Beijing, and hence, the path selec-
tion will be an integrated decision in CBRP. We will also concentrate on finding solutions
for dynamic travel demands and devote more efforts on improving the quality of results
in terms of vehicle routes, travel time, and operation costs by adopting hybrid heuristic
algorithms.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This paper is sponsored by National Natural Science Foundation of China Project [grant numbers
71621001, 91746201, 71601015].
References
Bräysy, O., and M. Gendreau. 2005. “Vehicle Routing Problem with Time Windows, Part ii: Metaheuris-
tics.” Transportation Science 39 (1): 119–139. doi:10.1287/trsc.1030.0057.
Cao, Y., and J. Wang. 2016. “The Key Contributing Factors of Customized Shuttle Bus in Rush
Hour: A Case Study in Harbin City.” Procedia Engineering 137: 478–486. doi:10.1016/j.proeng.2016.
01.283.
1822 R. GUO ET AL.
Cao, Y., and J. Wang. 2017. “An Optimization Method of Passenger Assignment for Customized Bus.”
Mathematical Problems in Engineering 2017: 1–9. doi:10.1155/2017/7914753.
Chandra, S., and L. Quadrifoglio. 2013. “A Model for Estimating the Optimal Cycle Length of Demand
Responsive Feeder Transit Services.” Transportation Research Part B: Methodological 51: 1– 16.
doi:10.1016/j.trb.2013.01.008.
Charisis, A., C. Iliopoulou, and K. Kepaptsoglou. 2018. “DRT Route Design for the First/Last
Mile Problem: Model and Application to Athens, Greece.” Public Transport 10: 499–527.
doi:10.1007/s12469-018-0188-0.
Cordeau, J. F. 2006. “A Branch-and-cut Algorithm for the Dial-a-Ride Problem.” Operations Research 54
(3): 573–586. doi:10.1287/opre.1060.0283.
Cordeau, J. F., and G. Laporte. 2007. “The Dial-a-Ride Problem: Models and Algorithms.” Annals of
Operations Research 153 (1): 29–46. doi:10.1007/s10479-007-0170-8.
Drakoulis, R., F. Bellotti, I. Bakas, R. Berta, P. K. Paranthaman, G. R. Dange, et al. 2018. “A Gami-
fied Flexible Transportation Service for On-Demand Public Transport.” IEEE Transactions Intelligent
Transportation Systems 19 (3): 921–933. doi:10.1109/TITS.2018.2791643.
Fischetti, M., Salazar González, Juan José, and P. Toth. 1997. “A Branch-and-cut Algorithm for
the Symmetric Generalized Traveling Salesman Problem.” Operations Research 45 (3): 378–394.
doi:10.1287/opre.45.3.378.
Fukasawa, R., H. Longo, J. Lysgaard, M. P. de Aragão, M. Reis, E. Uchoa, and R. F. Werneck. 2006.
“Robust Branch-and-cut-and-Price for the Capacitated Vehicle Routing Problem.” Mathematical
Programming 106 (3): 491–511. doi:10.1007/s10107-005-0644-x.
Garcia, B. L., J. Y. Potvin, and J. M. Rousseau. 1994. “A Parallel Implementation of the Tabu Search
Heuristic for Vehicle Routing Problems with Time Window Constraints.” Computers & Operations
Research 21 (9): 1025–1033. doi:10.1016/0305-0548(94)90073-6.
Glover, F., and M. Laguna. 1997.Tabu Search. Boston, MA: Kluwer Academic Publishers.
Guo, R., W. Guan, and W. Zhang. 2018. “Route Design Problem of Customized Bus: Mixed Integer Pro-
gramming Model and Case Study.” Journal of Transportation Engineering, Part A: Systems 144 (11).
doi:10.1061/JTEPBS.0000185.
Guo, R., W. Guan, W. Zhang, Z. Zhang, and A. Huang. 2019. “Extracting Potential Customized Service
from Smartcard Data.” Submitted.
Harms, S., and B. Truffer. 1998. “The Emergence of a Nationwide Carsharing Cooperative in Switzer-
land.” In a Case-study for the EC-supported Research Project “Strategic Niche Management as a
Tool for Transition to a Sustainable Transport System”. Zürich: EAWAG.
Ho, S. C., W. Y. Szeto, Y. H. Kuo, J. M. Y. Leung, M. Petering, and T. W. H. Tou. 2018. “A Survey of
Dial-a-Ride Problems: Literature Review and Recent Developments.” Transportation Research Part
B: Methodological 111: 395–421. doi:10.1016/j.trb.2018.02.001.
Holland, J. H. 1992. “Adaptation in Natural and Artificial Systems.” Ann Arbor 6 (2): 126–137.
Ji, L. 2017. “Reasons and Control Measures for the Decline of Bus Passenger Flow.” Urban Public
Transport 2017 (11): 13–16.
Kirby, R., and K. Bhatt. 1975. “An Analysis of Subscription Bus Experience.” Traffic Quarterly 29 (3):
403–425.
Li, J., Y. Lv, J. Ma, and Q. Ouyang. 2018a. “Methodology for Extracting Potential Customized Bus Routes
Based on Bus Smart Card Data.” Energies 11 (9): 2224. doi:10.3390/en11092224.
Li, X., and L. Quadrifoglio. 2010. “Feeder Transit Services: Choosing Between Fixed and Demand
Responsive Policy.” Transportation Research Part C: Emerging Technologies 18 (5): 770–780.
doi:10.1016/j.trc.2009.05.015.
Li, Z., R. Song, S. He, and M. Bi. 2018b. “Methodology of Mixed Load Customized Bus Lines and
Adjustment Based on Time Windows.” Plos one 13 (1): e0189763.
Liu, T., and A. Ceder. 2015. “Analysis of a New Public-Transport-Service Concept: Customized Bus in
China.” Transport Policy 39: 63–76. doi:10.1016/j.tranpol.2015.02.004.
Liu, T., A. Ceder, R. Bologna, and B. Cabantous. 2016. “Commuting by Customized Bus: A Compara-
tive Analysis with Private Car and Conventional Public Transport in Two Cities.” Journal of Public
Transportation 19 (2): 55–74. doi:10.5038/2375-0901.19.2.4.
TRANSPORTMETRICA A: TRANSPORT SCIENCE 1823
Lysgaard, J., A. N. Letchford, and R. W. Eglese. 2004. “A New Branch-and-cut Algorithm for the Capaci-
tated Vehicle Routing Problem.” Mathematical Programming 100 (2): 423–445. doi:10.1007/s10107-
003-0481-8.
Lyu, Y., C. Y. Chow, V. C. Lee, Y. Li, and J. Zeng. 2016. “T2CBS: Mining Taxi Trajectories for Cus-
tomized Bus Systems.” In Computer Communications Workshops (INFOCOM WKSHPS), 2016 IEEE
Conference on. IEEE, 441–446. doi:10.1109/INFCOMW.2016.7562117.
Ma, X., Y. J. Wu, Y. Wang, F. Chen, and J. Liu. 2013. “Mining Smart Card Data for Tran-
sit Riders’ Travel Patterns.” Transportation Research Part C: Emerging Technologies 36: 1– 12.
doi:10.1016/j.trc.2013.07.010.
Ma, J., Y. Yang, W. Guan, F. Wang, T. Liu, W. Tu, and C. Song. 2017. “Large-Scale Demand Driven Design
of a Customized Bus Network: A Methodological Framework and Beijing Case Study.” Journal of
Advanced Transportation 2017 (3): 1–14. doi:10.1155/2017/3865701.
Ma, J., Y. Zhao, Y. Yang, T. Liu, W. Guan, J. Wang, et al. 2017. “A Model for the Stop Planning and
Timetables of Customized Buses.” Plos One 12 (1). doi:10.1371/journal.pone.0168762.
Masmoudi, M. A., K. Braekers, M. Masmoudi, and A. Dammak. 2017. “A Hybrid Genetic Algorithm
for the Heterogeneous Dial-a-Ride Problem.” Computers & Operations Research 81: 1–13.
doi:10.1016/j.cor.2016.12.008.
Mitchell, and E.John. 2011.Branch and Cut. Wiley Encyclopedia of Operations Research and Manage-
ment Science.
Nguyen, P. K., T. G. Crainic, and M. Toulouse. 2013. “A Tabu Search for Time-Dependent Multi-Zone
Multi-Trip Vehicle Routing Problem with Time Windows.” European Journal of Operational Research
231 (1): 43–56. doi:10.1016/j.ejor.2013.05.026.
Padberg, M., and G. Rinaldi. 1991. “A Branch-and-cut Algorithm for the Resolution of Large-Scale
Symmetric Traveling Salesman Problems.” SIAM Review 33 (1): 60–100. doi:10.1137/1033004.
Qiu, G., R. Song, S. He, W. Xu, and M. Jiang. 2018. “Clustering Passenger Trip Data for the Poten-
tial Passenger Investigation and Line Design of Customized Commuter Bus.” IEEE Transactions on
Intelligent Transportation Systems, 1–10. doi:10.1109/tits.2018.2875466.
Quadrifoglio, L., and X. Li. 2009. “A Methodology to Derive the Critical Demand Density for Design-
ing and Operating Feeder Transit Services.” Transportation Research Part B: Methodological 43 (10):
922–935. doi:10.1016/j.trb.2009.04.003.
Ren, Y., G. Chen, Y. Han, and H. Zheng. 2016. “Extracting Potential Bus Lines of Customized City Bus
Service Based on Public Transport Big Data.” IOP Conference Series: Earth and Environmental Science
46: 1. doi:10.1088/1755-1315/46/1/012017.
Tabassum, S., S. Tanaka, F. Nakamura, and A. Ryo. 2017. “Feeder Network Design for Mass Transit Sys-
tem in Developing Countries (Case Study of Lahore, Pakistan).” Transportation Research Procedia
25: 3129–3146. doi:10.1016/j.trpro.2017.05.343.
Tellez, O., S. Vercraene, Fabien Lehuédé, Olivier Péton, and T. Monteiro. 2018. “The Fleet Size and
Mix Dial-a-Ride Problem with Reconfigurable Vehicle Capacity.” Transportation Research Part C:
Emerging Technologies 91: 99–123. doi:10.1016/j.trc.2018.03.020.
Thangiah, S. R. 1993.Vehicle Routing with Time Windows using Genetic Algorithms. Artificial Intelligence
Lab., Slippery Rock University.
Thangiah, S. R., K. E. Nygard, and P. L. Juell. 1991. “Gideon: A Genetic Algorithm System for Vehicle
Routing with Time Windows.” In Artificial Intelligence Applications, 1991. Proceedings., Seventh
IEEE Conference on. IEEE. Vol. 1. 322-328.doi:10.1109/CAIA.1991.120888.
Tong, L. C., L. Zhou, J. Liu, and X. Zhou. 2017. “Customized Bus Service Design for Jointly Optimizing
Passenger-to-Vehicle Assignment and Vehicle Routing.” Transportation Research Part C: Emerging
Technologies 85: 451–475. doi:10.1016/j.trc.2017.09.022.
Wang, Z., and R. Wang. 2015. “Adaptive Management of Beijing Urban Traffic: A Case Study of the
Customized Transit Bus Service.” Modern Urban Research 3: 1–8.
Wong, K. I., A. F. Han, and C. W. Yuen. 2014. “On Dynamic Demand Responsive Transport
Services with Degree of Dynamism.” Transportmetrica A: Transport Science 10 (1): 55–73.
doi:10.1080/18128602.2012.694491.
Xiong, J., W. Guan, L. Song, A. Huang, and C. Shao. 2013. “Optimal Routing Design of a Com-
munity Shuttle for Metro Stops.” Journal of Transportation Engineering 139 (12): 1211– 1223.
doi:10.1061/(ASCE)TE.1943-5436. 0000608.
1824 R. GUO ET AL.
Xu, K. M., J. L. Li, J. Feng, and Z. Y. Meng. 2013. “Discussion on Subscription Bus Service.” Urban
Transport of China 11 (5): 24–27.
Yu, Y., R. B. Machemehl, and C. Xie. 2015. “Demand-responsive Transit Circulator Service Net-
work Design.” Transportation Research Part E: Logistics and Transportation Review 76: 160– 175.
doi:10.1016/j.tre.2015.02.009.
Zhang, J., D. Z. Wang, and M. Meng. 2017. “Analysing Customized Bus Service on a Multimodal Travel
Corridor: An Analytical Modelling Approach.” Journal of Transportation Engineering, Part A: Systems
143 (11): 04017057. doi:10.1061/JTEPBS.0000087.
Zhou, C., Wei Guan, J. Xiong, and S. Jiang. 2017. “A Study on Dynamic Dispatching Strategy of Cus-
tomized Bus.” In Control Science and Systems Engineering (ICCSSE), 2017 3rd IEEE International
Conference on. IEEE, 751–755. doi:10.1109/CCSSE.2017.8088034.
