Ticket Allocation Optimization of Fuxing Train Based on Overcrowding Control: An Empirical Study from China - MDPI
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sustainability
Article
Ticket Allocation Optimization of Fuxing Train Based on
Overcrowding Control: An Empirical Study from China
Yu Wang *, Xinghua Shan, Hongye Wang, Junfeng Zhang, Xiaoyan Lv and Jinfei Wu
Institute of Computing Technology, China Academy of Railway Sciences Corporation Limited,
Beijing 100081, China; zhangxuegykj@163.com (X.S.); 18810540511@139.com (H.W.); 15210936532@139.com (J.Z.);
hnxxwyk123@163.com (X.L.); 13426481699@139.com (J.W.)
* Correspondence: luoyu167@163.com
Abstract: At the peak of passenger flow, some passengers extend travel sections, which will be
likely to lead to overcrowding of high-speed railway (HSR) trains. Therefore, the problem of train
overcrowding control needs to be considered in ticket allocation. Firstly, by simulating the passenger
demand function and utility function, an optimization model of ticket allocation for multiple trains
and multiple stops with the goal of maximizing revenue is constructed. Secondly, the concepts of the
travel extension coefficient and risk coefficient are introduced, the number of passengers is estimated
under the risk coefficient as the probability, and the total number of passengers on the train arriving
at any station is obtained. Thus, preventing the number of passengers on the train from exceeding
the train capacity is introduced to the ticket allocation optimization model of multiple trains and
multiple stops as a constraint. Finally, this model is solved by the particle swarm optimization
algorithm (PSO). The research results show that the idea of controlling passenger numbers so as
not to exceed train capacity based on ticket allocation proposed in this paper has strong practical
feasibility. By reasonably and accurately allocating the tickets to the departure terminal section and
long-distance terminal sections, it can ensure that, even if there are some passengers extending their
Citation: Wang, Y.; Shan, X.; Wang, travel section, the train will not be overcrowded under a certain probability, improving the train
H.; Zhang, J.; Lv, X.; Wu, J. Ticket safety and passenger travel experiences.
Allocation Optimization of Fuxing
Train Based on Overcrowding Keywords: railway transportation; ticket allocation; train overcrowding control; particle swarm
Control: An Empirical Study from optimization; sustainable development of railway transportation
China. Sustainability 2022, 14, 7055.
https://doi.org/10.3390/
su14127055
Academic Editor: Luca D’Acierno 1. Introduction
Received: 9 May 2022
High-speed railway is a kind of green, environmentally friendly, and sustainable
Accepted: 6 June 2022
transportation, which is favored by passengers. The factors that impact the sustainability of
Published: 9 June 2022
high-speed railways are mainly economic and safety. High-speed railways can operate well
only by ensuring the revenue of tickets is greater than the cost. Therefore, the goal of ticket
Publisher’s Note: MDPI stays neutral
allocation is to maximize revenue. On the other hand, the safety of HSR trains is important
with regard to jurisdictional claims in
for sustainability as well. However, during the peak period of railway transportation such
published maps and institutional affil-
as holidays, passengers cannot buy tickets for the section they want to take. As a response,
iations.
passengers will purchase any origin–destination (OD) pair ticket from their current station
in order to get on the train first and then buy another continuing OD pair ticket, which
could allow them to extend their travel sections to the target station (terminal station). If
Copyright: © 2022 by the authors.
there are too many passengers extending their travel section (OD pair) in the above manner,
Licensee MDPI, Basel, Switzerland. the number of passengers on the train will be likely to exceed the train capacity when the
This article is an open access article train arrives at some stations. This phenomenon is called “overcrowding”, which will
distributed under the terms and affect the safety of train operation and cause the train to be delayed or even shut down [1].
conditions of the Creative Commons Therefore, overcrowding especially for the Fuxing train is not conducive to the sustainable
Attribution (CC BY) license (https:// development of high-speed railway (HSR) in China.
creativecommons.org/licenses/by/ When the train experiences “overcrowding”, the conductor should let the passengers
4.0/). who extend their travel section through buying tickets after getting on the train get off;
Sustainability 2022, 14, 7055. https://doi.org/10.3390/su14127055 https://www.mdpi.com/journal/sustainabilitySustainability 2022, 14, 7055 2 of 12
however, the cost is too high in practice, and it is easy to risk negative public opinion.
One possible solution to solve the train overcrowding problem is to consider how to avoid
train overcrowding when allocating tickets. This means allocating tickets reasonably and
accurately under the constraint of controlling passenger numbers so as not to exceed train
capacity. Therefore, this paper studies how to avoid train overcrowding through ticket
allocation, so as to ensure the safety of train operation.
Literature Review and Innovation of This Paper
Ticket allocation of HSR trains is mainly borrowed from the aviation field, because
these two fields both take revenue maximization as the objective function. However,
there are still many differences between them [2,3]. What is more important is that ticket
allocation in HSR is more complex than aviation seat allocation. Thus, according to the
difference in complexity and constraints of the application scenarios, ticket allocation in
HSR presents a different research focus:
(1) Single-train ticket allocation is the most basic research entry. Ciancimino et al. [4]
first proposed treating different classes of carriages as different train products, so the
problem of train ticket allocation can be transformed into a seat control problem in
single trains, with multiple sections and a single ticket price. Some studies [5] assumed
that the passenger demand obeys an independent normal distribution and used the
particle swarm optimization algorithm (PSO) to solve the final optimization model.
Shan et al. [6] predicted the passenger flow by adopting time series firstly. Then, they
obtained a ticket allocation method through the formulation of rules such as long
distance before short distance, seat before no seat, and allocating tickets in advance
by quantity and then by proportion. Gopalakrishnan et al. [7] mainly considered
the long-distance passenger demand of Indian railways and studied the utilization
method of single-train seat capacity.
(2) Ticket allocation involving multiple trains is further complicated on the basis of single-
train ticket allocation. Jiang et al. [8] integrated the short-term passenger flow demand
forecasting method into the railway multi-train ticket allocation model, which solved
the situation that some stations were short of tickets and other stations were rich in
tickets. Yan et al. [9] developed a seat allocation model for multiple HSR trains with
flexible train formation. Jiang et al. [10] proposed a dynamic adjustment method for
ticket allocation. Zhao et al. [11] proposed a probabilistic nonlinear programming
model for the problem of railway passenger ticket allocation. Deng et al. [12] focused
on the joint pricing and ticket allocation problem for multiple HSR trains with different
stop patterns. Luo et al. [13] developed a multi-train seat inventory control model
based on revenue management theory. The above was discussed with respect to the
complexity of ticket allocation.
(3) Customer choice behavior plays an important role in estimating customer demand,
which has also been more discussed in ticket allocation in recent years. It is an obvious
research conclusion that there are many factors influencing customer choice behavior,
and these factors can be divided mainly into two aspects, personal attributes and trip
attributes, in a great deal of the literature [14–17]. As for the method describing pas-
senger choice behavior, Wang et al. [18] described different passenger needs through
the Logit model and established a multi-stage random ticket allocation model under
passenger selection behavior.
(4) In the two most important sub-studies of revenue management, ticket allocation is
always constrained by ticket prices. However, it is unfortunate that in the past few
years, pricing and ticket allocation issues have always been treated separately, and
there is a gap in the research regarding joint pricing and ticket allocation models [2,19].
Weatherford [20] first stressed the importance of considering prices and suggested
them as decision variables for the ticket allocation problem. After Weatherford’s
contribution, we can see more and more research that has optimized ticket allocation
and ticket prices at the same time. Hetrakul et al. [19] put forward a comprehensiveSustainability 2022, 14, 7055 3 of 12
optimization model of dynamic ticket pricing and ticket allocation based on the impact
of passenger heterogeneity on ticket allocation. Some of the literature regards ticket
allocation as the basis of dynamic pricing and gives priority to solving the problem of
ticket allocation while solving dynamic pricing [21–24].
In summary, we found that there are many scholars who have researched ticket allo-
cation based on revenue maximization, but few scholars have studied how to avoid train
overcrowding through ticket allocation. As Table 1 shows, the above studies about ticket
allocation only take revenue maximization as the objective function of ticket allocation,
but lack the consideration of combining the problem of “train overcrowding” with ticket
allocation. Therefore, as the most significant innovation, this paper integrates the problem
of “train overcrowding” into the ticket allocation model as a constraint. Our research intro-
duces a “risk coefficient” as the probability boundary and a “travel extension coefficient”
to describe the proportion of passengers who extend their travel section out of the total
number of passengers, then constructs and solves the ticket allocation optimization model
by taking into account the goal of maximizing the expected revenue of multiple trains
and multiple stops under the constraint that the train will not experience “overcrowding”
under a certain probability. This is exactly the innovation of this paper and the original
intention of writing this paper. It also enriches the research dimension of ticket allocation
and sustainable operations and management of railway systems at the same time.
Table 1. Comparison of our research with the literature in ticket allocation.
Revenue Maximization Overcrowding
References
Single Train Multiple Trains Pricing Control
√
[4–7] ×
√ × ×
[8–13] × ×
√ ×
[2,19–24] × ×
√ ×
√
This paper × ×
This paper proceeds as follows: Part 2 lists the model assumptions and the variable
definition and then constructs a mathematical model formulation; Part 3 provides the
algorithms to solve the model; Part 4 shows the numerical experiments; Part 5 concludes
the paper.
2. Research Methodology and Process
2.1. Model Assumptions and Notations
Suppose a high-speed railway line contains l stations and involves K trains, k = 1, 2 . . . K.
Each train has different stopping plans and serves the travel demand of a set of OD pairs
on the line. Two adjacent stations on the line form a section, named the h section, and in
total, h = 1, 2 . . . l − 1.
To simplify the problem, we make the following reasonable assumptions:
(1) Passengers whose requests are fulfilled will not cancel their reservations or change
their ticket.
(2) Take the second-class seat of HSR as the research object.
(3) If passengers need to extend their travel section after getting on the train, they only
make up the tickets to the terminal. The travel section that extends from the boarding
station to the terminal is called the “target travel section (OD pair)”. In order to
get on the train first, passengers buy a short OD pair ticket, and the travel section
corresponding to these tickets is called the “short travel section (OD pair)”. A target
travel section corresponds to several short travel sections.
To simplify the description of the problem, the notations in Table 2 will be used in
the formulation.Sustainability 2022, 14, 7055 4 of 12
Table 2. Notations.
Symbols Definition Unit
Variable or Parameters
(r, s) Origin–destination pair in the transportation service, which is composed of the train stops, from station r to station s /
t Station serial number, the train arrives at the tth station /
h Section serial number, h = 1, 2 . . . l − 1 /
k Train serial number, k = 1, 2 . . . K /
(k)
prs Price of train k between (r, s) CNY
(k)
ηrs The 0–1 binary parameter denoting whether train k can provide services between (r, s) /
(k)
trs Travel time of train k between (r, s) min
α1 , α2 Parameters of utility function /
(k)
Urs Total utility of train k for passengers between (r, s) /
(k)
wrs Probability of train k chosen by passengers between (r, s) %
Qrs Expected sales volume between (r, s) /
Ck Capacity of train k passengers
(k)
qt Total number of passengers on train k arriving at station t passengers
(k)
qt· The number of passengers boarding train k when arriving at station t passengers
(k)
q̂t The number of passengers who get on train k before station t and get off the train after station t passengers
(k)
q·t For train k, the number of passengers who should get off before station t, but are still on the train after extending their travel section passengers
grs The number of passengers who buy tickets between (r, s), but extend the travel section to the terminal passengers
β rs Parameter, travel extension coefficient /
γ Parameter, risk coefficient /
xrl Passenger demand under γ for target travel sec tion (r, l ) /
Frl−1 ( x ) Inverse function of cumulative probability function of passenger demand for target travel sec tion (r, l ) /
Decision Variable
(k)
brs The number of tickets allocated to train k between (r, s) /
2.2. Model Formulation
For any given OD pair, the probability function of passenger demand can be fitted by
normal distribution [25]. Then, passenger demand between (r, s) obeys an independent
normal distribution as follows:
( x − µ )2
exp(− 2σ2rs )
rs
f rs ( x ) = √ ,1 ≤ r < s ≤ l (1)
2πσrs
µrs is the expectation of passenger demand, while σrs is the standard deviation of passenger
(k)
demand. Frs ( x ) is the cumulative probability function of passenger demand. ηrs is the
(k)
service parameter of train k. brs is the number of tickets allocated to train k. If train k
(k) (k) (k)
provides a transportation service between (r, s), ηrs = 1, brs ≥ 0. Otherwise, ηrs = 0,
(k) K (k) (k) (k)
brs = 0. Set ∑ ηrs ·brs = Brs . Finally, the expected ticket sales volume between (r, s)
k =1
can be described as below:
(k)
Brs
Z Z ∞
(k)
Qrs = x f rs ( x )dx + Brs · (k)
f rs ( x )dx (2)
Brs
0
When passengers choose a train between (r, s), they will mainly focus on two factors:
ticket price and travel time. The utility obtained by passengers choosing train k can be
expressed as the following formula:
(k)
(k) prs (k)
Urs = α1 + α2 · Trs , α1 < 0, α2 < 0 (3)
λ
(k) (k)
α1 and α2 are parameters of the utility function. prs and Trs are ticket price and
travel time for train k between (r, s). λ is revised average monthly passenger income [19].Sustainability 2022, 14, 7055 5 of 12
Passengers will choose each train based on utility. Then, we can define the probability of
passengers choosing train k between (r, s) by the Logit model as
(k) (k)
(k) ηrs exp(Urs )
wrs = (4)
K (k) (k)
∑ ηrs exp(Urs )
k =1
With the goal of maximizing the overall expected revenue, the following objective
function is obtained for the ticket allocation of HSR:
K
R = max ∑ ∑ Qrs (brs
(k) (k) (k)
)·wrs · prs , 1 ≤ r < s ≤ l (5)
k =1 (r,s)
The following constraints shall also be taken into account for ticket allocation:
(1) Train capacity constraints: the ticket number allocated for each train between (r, s)
cannot exceed their capacity constraint. Ck is the capacity of train k
h l
∑ ∑
(k)
brs ≤ Ck , h = 1, 2 . . . l − 1 (6)
r = 1s = h + 1
(2) Train service constraints:
(
(k) (k) (k)
ηrs = 1, brs > 0, brs ∈ Z +
(k) (k) (7)
ηrs = 0, brs =0
2.3. Train Overcrowding Control
In the peak period of the railway, due to the lack of sufficient train capacity, some
of the passengers who want to reach the terminal will first buy tickets that can allow
them to get on the train from the current station and then make up tickets on the train
to extend their travel section to the terminal. Through the above alternative approach,
passengers can achieve the goal of getting off at the terminal without buying the target
travel section tickets. Although such passengers’ behavior can increase train revenue for
railway transport enterprises, it will also bring a problem: overcrowding. This paper
estimates the number of passengers on the train when arriving at each station to ensure that
the passenger number does not exceed the train capacity for a certain probability. When
train k arrives at station t, 1 < t < l, the number of passengers on the train at this time
(k)
is qt .
(k) (k) _(k) (k)
qt = qt· + q t + q·t (8)
(k)
In Formula (8), qt· indicates passengers boarding train k at station t:
∑ min[Qrs ·wrs
(k) (k) (k)
qt· = , brs ], ∀(r, s)|t = r < s ≤ l (9)
(r,s)
(k)
q̂t represents the number of passengers who get on train k before station t and get off the
train after station t. These passengers are still on the train at this time.
∑ min[Qrs ·wrs
(k) (k) (k)
q̂t = , brs ], ∀(r, s)|1 ≤ r < t < s ≤ l (10)
(r,s)
(k)
q·t means the number of passengers who should have disembarked at station t or before
station t, but are still on the train due to the extension of the travel section through buyingSustainability 2022, 14, 7055 6 of 12
a continued OD pair of tickets. This is this segment of passengers that may cause the
train overcrowding.
∑ min[ grs ·wrs
(k) (k) (k)
q·t = , brs ], ∀(r, s)|1 ≤ r < s ≤ t (11)
(r,s)
K
∑ brl
(k)
grs = β rs ·( xrl − ), ∀(r, s)|1 ≤ r < s ≤ t (12)
k =1
β rl = ∑ βrs , ∀(r, s) ∈ (r, l ) (13)
(r,s)
xrl = Frl−1 (γ), ∀(r, s) 1 ≤ r < s ≤ t, 0 < γ < 1 (14)
Equation (12) shows that when the passenger demand xrl in the target section (r, l )
K (k)
exceeds the supply capacity ∑ brl , the phenomenon of extending the travel section will
k =1
occur in all short travel sections (r, s). β rs is the travel extension coefficient, which is used
to describe the proportion between the passengers extending their travel section and the
K (k)
excess demand x1l − ∑ brl in the target travel section (r, l ). Equation (14) is used to
k =1
calculate the corresponding number of passengers under the risk coefficient γ. Frl−1 ( x )
is the inverse function of the cumulative probability function of passenger demand (the
inverse function of the cumulative probability function of the normal distribution in this
paper). For the Fuxing train as a representative product of HSR, it is more important to
ensure that it is not overcrowded rather than increasing revenue in the actual operation
process, which will not affect the train’s operation safety and weaken the passenger riding
experience. For a given scheme of ticket allocation, when train k arrives at station t, it needs
to satisfy:
(k)
qt ≤ Ck , 1 < ∀t < l (15)
Finally, we construct the ticket allocation optimization model of the Fuxing train
considering solving the problem of train overcrowding caused by passengers extending
their travel sections:
K (k) (k) (k)
R = max ∑ ∑ Qrs (brs )·wrs · prs , 1 ≤ r < s ≤ l
k = 1 (r,s)
s.t.
(k)
qt ≤ Ck , 1 < ∀t < l
h l (16)
(k)
∑ ∑ brs ≤ Ck , h = 1, 2 . . . l − 1
r(= 1 s = h + 1
(k) (k) (k)
ηrs = 1, brs > 0, brs ∈ Z +
(k) (k)
ηrs = 0, brs =0
3. Solution Algorithm
Because the optimization model proposed in this paper is a mixed-integer nonlinear
programming model (MINLP) with several constraints (Equation (16)), it cannot calculate
the maximum value directly by derivation. Therefore, we chose the heuristic algorithm
to solve Model (16). The PSO algorithm is one of the heuristic algorithms that is simple
and easy to implement and has fewer parameters, all of which lead to faster convergence
speed, and it requires only small computer memory. Moreover, the leap of the PSO
algorithm makes it easier to find the global optimal value without being trapped in the
local optimal value.Sustainability 2022, 14, 7055 7 of 12
(k)
Setting the initial solution vector A, if train k provides service for ∑ ηrs OD pairs, A is
(r,s)
K (k)
the ∑ ∑ ηrs dimension vector. We convert Equation (16) into the following equivalent
k = 1 (r,s)
model [2,26]:
K (k) (k)
max ∑ ∑ brs · prs , 1 ≤ r < s ≤ l
k = 1 (r,s)
s.t.
qkt ≤ Ck , 1 < ∀t ≤ l
h l (17)
(k)
∑ ∑ brs ≤ Ck , h = 1, 2 . . . l − 1
r(= 1 s = h + 1
(k) (k) (k)
ηrs = 1, brs > 0, brs ∈ Z +
(k) (k)
ηrs = 0, brs = 0
(k)
Equation (17) is a linear programming model, and brs can be solved directly. Setting
(1) (1) (k)
A = (b12 , b13 , . . . , brs ), this is the initial solution. Psize is the particle size, and the initial
position of the hth particle is recorded as x0h , h = 1, 2, 3 . . . Psize .
x0h = A + gh (18)
In the above formula, gh is a vector with the same dimension of A, and each component
obeys the uniform distribution of (−d, d).
xlh is the position at the lth iteration. lmax represents the maximum number of iterations,
1 ≤ l ≤ lmax . t(xlh ) can reflect the fitness of xlh , and then:
t(xlh ) = T (Alh ) (19)
Alh is the vector for which each component of xlh is tested by Constraints (6), (7), and (15)
and rounded. T (Alh ) represents the output that inputs Alh into the objective function (16).
We set vl+1 l
h as the velocity of the hth particle at the lth iteration. xh is the updated
l
position of the hth particle after the lth iteration. yh is described as the best historical
position at the lth iteration of the hth particle. ylg represents the historical position of all
particles at the lth iteration. The above process can be expressed as follows [19]:
vl+1 l l l l l l l
h = δvh + ϕ1 rh1 (yh − xh ) + ϕ2 rh2 (yg − xh ) (20)
xl+1 l l+1
h = xh + vh (21)
yl+1 l+1 l+1 l+1
h = Ah , Th = t (yh ), ∀ t (xh ) ≤ Th (22)
yl+1 l+1 l+1 l+1
g = Ah , T = t (yg ), ∀ t (xh ) ≤ T, h = 1, 2 . . . Psize (23)
Th is the fitness of the best historical position of the hth particle. T is the fitness of the
best historical position of all particles. rlh1 and rlh2 have the same dimensions as vector A
and obey the uniform distribution of (0, 1). δ, ϕ1 , ϕ2 are control parameters. Finally, the
solving process of this paper is as follows (Figure 1).Sustainability
Sustainability 14,14,
2022,
2022, 7055 PEER REVIEW
x FOR 9 of8 of
1312
Begin
Generate initial solution vector
input Psize , , 1 , 2
Calculate particle fitness
Initializtion l = 0, h = 1 ,input lmax
Update particle position by formula(22)(23)
x hl +1
whether satisfied N
constraints(6)(7)(15)
Y
Update position of xhl +1 and calculate fitness t ( xhl +1 )
Update particle position by formula(22)(23)
N
h Psize
Y
h = h +1
N
l lmax
Y
l = l + 1, h = 1
Output A, T
End
Figure 1.1.
Figure Solving process.
Solving process.
4.4.Results
Results andAnalysis
and Analysis
4.1. Basic Data
4.1. Basic Data
We chose the Shanghai–Beijing line in China as an example to verify the validity and
We chose the Shanghai–Beijing line in China as an example to verify the validity and
applicability of the model in this paper. For the situation of multiple trains and multiple
applicability of the model in this paper. For the situation of multiple trains and multiple
stops, there are two trains, G2 and G22, providing service between Shanghai Hongqiao
stops, there are two trains, G2 and G22, providing service between Shanghai Hongqiao
station and Beijing South station. The total second-class seat capacity of each of the trains
station and Beijing South station. The total second-class seat capacity of each of the trains
is set at 1113. Like the Fuxing train, G2 and G22 have few stops and a short travel time
is set at 1113. Like the Fuxing train, G2 and G22 have few stops and a short travel time
(Figure 2), which are deeply favored by passengers. The seat occupation rate of these two
(Figure 2), which are deeply favored by passengers. The seat occupation rate of these two
trains is higher than the average level of all the trains on the same line. During the peak
trains is higher than the average level of all the trains on the same line. During the peak
period of passenger flow, such as holidays, many passengers take these two trains and
period of passenger flow, such as holidays, many passengers take these two trains and
extend their travel section by purchasing continued OD pair tickets, which is likely to
extend their
increase thetravel
risk ofsection by purchasing
overcrowding. continued
The train ODtravel
stops and pair tickets,
times arewhich
shownis likely to 3.
in Table
increase the risk of overcrowding. The train stops and travel times are shown in Table 3.Sustainability 2022, 14, x FOR PEER REVIEW 1
Sustainability 2022, 14, 7055 9 of 12
“/” means the train does not provide service between the OD pair. There are four st
on“/”
the line,the
means namely, Shanghai
train does Hongqiao
not provide (SH), the
service between Nanjing South
OD pair. There (NS), Jinan
are four West (JW
stations
Beijing South (BS).
on the line, namely, Shanghai Hongqiao (SH), Nanjing South (NS), Jinan West (JW), and
Beijing South (BS).
SH NS JW BS
Departure station Terminal
G22
G2
The train
Figure2.2. The
Figure trainstop schemes.
stop schemes.
Table 3. Train stops and travel times.
Table 3. Train stops and travel times.
OD
Train Factors of Passengers Choice
Factors of Passengers OD
Train (SH, BS) (SH, NS) (SH, JW) (NS, BS) (NS, JW) (JW, BS)
Price (CNY)
Choice
553
(SH,BS)
134.5
(SH,NS)
398.5
(SH,JW)
443.5
(NS,BS)
279
(NS,JW) (JW
184.5
G2
Travel time (min) Price (CNY)
268 60 553 179 134.5 206 398.5 117443.5 87 279 18
G2
Price (CNY) 553 134.5 / 443.5 / /
G22
Travel time (min)
Travel time258
(min) 60 268 / 60 196 179 /
206 /
117
Price (CNY) 553 134.5 / 443.5 /
G22
Travel
We take time number
the actual (min) of passengers
258 of the60two trains from
/ 30 September
196 2019 to/ 7
October 2019 and from 30 September 2020 to 7 October 2020 as a case for empirical analysis.
Then, the estimated values of the main parameters of the passenger demand function are
We take the actual number of passengers of the two trains from 30 Septembe
obtained as in Table 4.
to 7 October 2019 and from 30 September 2020 to 7 October 2020 as a case for emp
Table 4. Main
analysis. parameter
Then, values of passenger
the estimated values demand function.
of the main parameters of the passenger de
function are obtained as in Table 4. OD
Parameter
(SH, BS) (SH, NS) (SH, JW) (NS, BS) (NS, JW) (JW, BS)
Table 4. Main parameter values of passenger demand function.
µrs 1701.4 170 89.6 277 13.4 96.6
σrs 222.5 92.4 68.7 142.4 14 70.3
OD
Parameter
The parameter(SH,BS) (SH,NS)
value of the utility (SH,JW)
function is (NS,BS)
obtained according (NS,JW)
to the literature [25]: (JW,
α1 = −2.24,
rs
α2 = −0.0113,
1701.4and λ = 190.170 Referring89.6to the finance277
risk control theory,
13.4 the 96
confidence interval of the VaR method is generally 95%, and then, γ = 0.95. The values
222.5
of otherrsparameters for the example92.4
are as follows:68.7
P = 25, l 142.4 = 100, δ = 0.5,14 ϕ = 2, 70
size max 1
and ϕ2 = 2.
The parameter
The behaviors of value
extending travel
of the sections
utility occur inisall
function the corresponding
obtained accordingshort
to travel
the literatur
1 =-2.24 , 2 =-0.0113 , and = 190 . Referring to the finance risk control theory, the
sections. It can be seen from Table 2 that both two trains will terminate at BS. For train G2,
the target travel sections are (SH, BS) and (NS, BS). Train G22 only has one target travel
dence interval
section (SH, BS).ofTherefore,
the VaR the
method is generally
short travel 95%,NS)
sections (SH, and then,
and =0.95
(SH, JW) . The
belong values of
to the
size =25 , lmax = 100 , =0.5 , 1 =2 , and 2
target travel for
parameters section
the(SH, BS). Similarly,
example (NS, BS) hasPonly
are as follows: one short travel section (NS, JW).
All travel extension coefficients of each of the short travel sections are shown in Table 5:
The behaviors of extending travel sections occur in all the corresponding short
Table 5. Travel
sections. It canextension
be seen coefficient
from Tablevalues.2 that both two trains will terminate at BS. For tra
the target
Arrivetravel sections
at the Station t are (SH,BS)
NS and (NS,BS). TrainJWG22 only has one target
section Target
(SH,BS).
travel Therefore,
section the(SH,
short
BS) travel sections
(NS, BS) (SH,NS)(SH,andBS)(SH,JW) belong
target travel section (SH,BS).
Short travel section Similarly,
(SH, BS) (NS,BS) has
(NS, JW) only one
(SH, NS)short travel
(SH, JW)section (N
All Travel
travel extension
extension coefficients
coefficient β rs of each of the0.1short travel
0.065 sections are
0.065 shown in Tab
0.035
Table 5. Travel extension coefficient values.
Arrive at the Station t NS JW
Target travel section (SH,BS) (NS,BS) (SH,BS)Sustainability 2022, 14, 7055 10 of 12
4.2. Computational Results
We used the Python language programming to solve the model. The final ticket
allocation details (Scheme I) are shown in Table 6, and the total expected revenue of the two
trains is CNY 1,084,120. As can be seen from Table 5, most tickets are allocated to departure
terminal section (SH, BS) for both G2 and G22.
Table 6. Ticket allocation results (Scheme I).
Train OD Number of Tickets Finally Allocated
G2 (SH, BS) 833
G2 (SH, NS) 110
G2 (SH, JW) 117
G2 (NS, BS) 153
G2 (NS, JW) 43
G2 (JW, BS) 160
G22 (SH, BS) 919
G22 (SH, NS) 184
G22 (NS, BS) 184
If the aim is not train overcrowding control, just allocating tickets with the goal of
maximizing overall revenue (remove Constraints (15)), we can obtain the ticket allocation
optimization model under general constraints as a comparison scheme, and the results of
the comparison scheme (Scheme II) by solving the model are shown in Table 7.
Table 7. Ticket allocation results of comparison scheme (Scheme II).
Train OD Number of Tickets Finally Allocated
G2 (SH, BS) 833
G2 (SH, NS) 75
G2 (SH, JW) 131
G2 (NS, BS) 139
G2 (NS, JW) 64
G2 (JW, BS) 195
G22 (SH, BS) 888
G22 (SH, NS) 225
G22 (NS, BS) 225
Through the comparative analysis of the ticket allocation results of the two schemes,
it can be summarized that, in order to prevent the train from overcrowding, the goal of
maximizing the revenue of the trains will be weakened. At the same time, 1752 tickets
were allocated for the departure terminal OD pair (SH, BS) in Scheme I, 31 more than
the comparison scheme, and more tickets were allocated to the long-distance terminal
sections such as (NS, BS) compared with Scheme II. In contrast to long-distance terminal
sections, midway short sections such as (NS, JW) or the short departure section (SH, NS)
are given more tickets in Scheme II, which will give passengers more opportunities to
extend their travel section. It is indicated that, for the sake of controlling the passengers not
exceeding the train capacity, the tickets should be given to the departure terminal sections
or long-distance terminal sections instead of allocating a large number of tickets to the
short sections or short departure sections.
5. Conclusions
Different from the general research on ticket allocation, this paper took the risk coeffi-
cient as the lower limit of probability and reasonably and scientifically allocated the tickets
to each travel section of the train to prevent the number of passengers from exceeding
the train capacity resulting from too many passengers extending their travel sections, so
as to solve the problem of train overcrowding. After numerical verification, the conclu-Sustainability 2022, 14, 7055 11 of 12
sions of this paper demonstrated that in the peak period of passenger flow, such as some
holidays, in order to control the number of passengers on the train so as not to exceed
the capacity, railway operation departments should allocate more tickets to the departure
terminal and the long-distance terminal sections, while reducing the number of tickets
for the short-distance or the short departure sections and, finally, achieve a balance be-
tween revenue maximization and passenger riding experiences, which is crucial for the
sustainable development and operation of the HSR [27]
Compared with those ticket allocation literature works that only aim to maximize rev-
enue, this paper provided a new way of thinking about the railway operation department
to reasonably allocate tickets so as to solve the problem of train overcrowding, which is an
innovation for and supplement to ticket allocation theory. At the same time, preventing
train overcrowding can effectively improve the safety of train operation, which is beneficial
to the promotion of passenger experiences. It is of great significance to the management and
operation of railway enterprises and to enhance their market competitiveness. However,
the research conclusions of this paper are based on the assumption that all passengers only
extend their travel to the terminal station and there are no passengers extending travel in
other sections. The situation in real life is obviously more complex than that. In addition,
the optimization model proposed in this paper is a mixed-integer nonlinear programming
model (MINLP) with several constraints. This model is complex and difficult to solve
directly by derivation. We chose the PSO algorithm to solve the model. The iterative
process of the PSO algorithm takes much more time, and there is a small probability of
falling into local optimization.
Author Contributions: Y.W. conceived of this research and completed this paper. X.S. contributed
to all aspects of this work. H.W. conducted an analysis of the case experiment. J.Z. gave critical
comments on the manuscript. X.L. and J.W. collected the data. All authors have read and agreed to
the published version of the manuscript.
Funding: This research was funded by the research plan of China Railway, Grant Number 2021F017.
Conflicts of Interest: The authors declare no conflict of interest.
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