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Collaborative optimisation of resource capacity allocation and fare rate for high-speed railway passenger transport
Journal of Rail Transport Planning & Management 10 (2019) 23–33
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Journal of Rail Transport Planning & Management journal homepage: www.elsevier.com/locate/jrtpm
Collaborative optimisation of resource capacity allocation and fare rate for high-speed railway passenger transport
T
Zhen-ying Yana,b, Xiao-juan Lib, Bao-ming Hana,∗ a b
School of Traffic and Transportation, Bei Jing Jiao Tong University, Beijing, 100044, China Transportation Institute of Inner Mongolia University, Hohhot, 010070, China
ARTICLE INFO
ABSTRACT
Keywords: High-speed railway Resource capacity allocation Pricing Particle swarm optimisation
The reasonable pricing of high-speed railway tickets and the optimal allocation of transport resource capacity can not only enhance competitiveness in the transport market, but also reasonably coordinate the revenue of the enterprise and utilities to passengers. This study uses price signals to adjust resource capacity allocation; and develops a co-optimisation model of resource capacity allocation and fare rates of high-speed trains in different train operation routes. The developed model aims at the comprehensive optimisation of railway enterprise's revenue and passengers' travel benefits, with the ratio of supply-demand and the floating rate of the fare as the main constraints. The Particle Swarm Optimisation (PSO) algorithm is applied to obtain the seat resource allocation scheme and the optimal fare rate for each train operation route. Finally, the case analysis is carried out to test the model and the algorithm. Based on a statistical analysis of actual ticket sale data of the Beijing-Shanghai high-speed railway for a certain month, the optimal unit fare and optimal seat resource allocation scheme are obtained to meet the corresponding passenger demand. The case analysis shows that after optimisation by the proposed method, the total value of the objective function is 2.04% higher than that before optimisation.
1. Introduction Over the past decade, high-speed railways have developed rapidly across the world. Especially in China, by the end of 2017, operations reached 25 000 km, accounting for two-thirds of the world total. With increasing high-speed rail network construction, more attention has focused on high-speed railway transport operation. There is a general consensus on the hierarchical formulation of railway passenger transport organizational plans. Macro-planning (strategic level) to micro-implementation (operational level) lead to network design, line planning, timetabling, rolling stock, crew scheduling and operational control (Goossens et al., 2004; Fu et al., 2009). The characteristics of different transport plans lead to corresponding optimisation theories and methods. Line planning belongs to the “strategy level”. It forms the basis for the follow-up “operational level” implementation plan. According to the characteristics of passenger flow, line planning determines the starting and ending points, routes, stop schedule, service frequencies, train grades and train composition. Research on high-speed railway line planning in China is scattered and focuses on the optimisation of different elements, such as optimising train orgin and destination stations, stopping schemes, service frequencies, train types, train composition and transfer schemes (Fu et al., 2009). The resource capacity allocation plays an important role in line planning. Service frequencies, train grades, train types and train composition are all based on the resource capacity allocation. The resource capacity allocation here refers to the allocation of transportation products in the transport corridor. The efficiency of resource allocation
∗
Corresponding author. E-mail address: [email protected] (B.-m. Han).
https://doi.org/10.1016/j.jrtpm.2019.05.001 Received 21 September 2018; Received in revised form 5 May 2019; Accepted 13 May 2019 Available online 18 May 2019 2210-9706/ © 2019 Elsevier Ltd. All rights reserved.
Journal of Rail Transport Planning & Management 10 (2019) 23–33
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directly affects the revenue of operation enterprises and the utility to passengers. At present, there are few studies on the allocation of high-speed rail transport resources. In the line planning optimisation, resource capacity allocation is based on the train unit. For example, the service frequencies are determined based on the unit of train. So the seat resources are not classified, which results in seat resource allocation not being sufficiently elaborate. In operation, often some types of resources are in short supply while other resources are surplus. For example, the second-class seats of high-speed trains are in short supply on a certain line, while the first-class seats are in surplus. Therefore, resources should be classified and allocated to deal with such problems. At the same time, an operating enterprise is permitted to formulate fare rates for high-speed railway independently in China. Obviously, an operating enterprise desires to adjust fare rates to improve its revenue. Based on the basic law of supply and price in any market, we link resource allocation with fare rates to propose a co-optimisation method. Based on the economic allocation principle of grid resources (Huang, 2010), this paper proposes a more refined allocation method for seat resources. The types of high-speed rail resources are divided into six categories, and the price signal is applied to guide the allocation of all types of resources. The model takes into account the benefits of both enterprises and passengers and aims at a comprehensive optimisation of enterprise revenue and passenger utility. Then we design a particle swarm optimisation (PSO) algorithm (Kennedy and Eberhart, 1995) to solve the model, and obtain a refined optimal resource allocation scheme based on the seat unit, which can be used to determine the service frequencies, vehicle configuration and fleet size. At the same time, optimal fare rates are obtained to improve an enterprise's revenue. The remainder of this paper is structured as follows. Section 2 reviews related literature. Section 3 presents the model. Section 3.1 describes economic environment and the principles pertinent to resource allocation, and then Section 3.2 presents the formulation for the collaborative optimisation problem. In Section 4, we develop PSO algorithm to solve the model efficiently. Section 5 describes the case analysis based on actual operational data and provides discussions. Section 6 presents our conclusions. 2. Literature review The problems studied in this paper involve two aspects: resource allocation and pricing. The resource allocation here refers to the allocation of seat resources for each train operation route based on given line routes. The results can be used to optimise the service frequency, vehicle configuration and train composition. Thus it refers to an optimisation problem for line planning. Line planning usually includes line route design, stop schedule planning, service frequency and fleet size. From the point of view of optimisation, the objectives of the travel plan include maximising enterprise profits, minimising costs, minimising passenger travel time and so on. For the overall optimisation scope, it includes the comprehensive optimisation of all factors and the optimisation of individual factors. Goossens et al. (2004) formulated the line-planning problem with the objective of minimising total operating costs and solved the model by an improved branch-and-cut approach for large operations. Goossens et al. (2006) formulated multi-type line planning problems by integer programming with the objective of minimising operating costs. Borndörfer et al. (2007) formulated the lineplanning problem with a new multi-commodity flow model to minimise the combined objective of total operating costs and passenger traveling time. Chang et al. (2000) proposed an allocation model to determine the train stop schedule plan, service frequency and fleet size. The goal of the model is to minimise total operating costs and passenger's total travel time loss. Wang et al. (2011) developed a two-layer model to determine stop-schedules and service frequencies. An upper-layer model was applied to minimise total operating costs and unserved passenger volumes, while a bottom model assigned the passenger flow to maximise the served passenger volume and minimise total travel time. Lin and Ku (2014) focused on the stop schedule plan of line planning and applied the integer programming to maximise operator profit. Fu et al. (2015) proposed a hierarchical line planning approach for a largescale high-speed railway network in China, which can incorporate cost-oriented and/or customer-oriented objectives. Zhu and Zhang, (2000) put forward an optimisation model for the organisation mode and system resource allocation through a quantitative study. However, the above literature focuses on resource allocation based on train units and does not subdivide the resources into different types. In fact, demands for different seat types and different speeds should be served by their corresponding resources. For example, if a passenger expecting a first-class seat with higher speed is served by a second- class seat with higher speed in the planning, the plan will not meet the passenger's expectations. So the passenger may choose other modes to travel or cancel the travel. Dividing resources into several types and allocating resources based on the seat level can better meet the needs of passengers and improve the efficiency of resource allocation. Pricing is primarily implemented at the marketing design stage. Shi et al, (2002), Zhang et al, (2016) and Zhang et al, (2017), established various multi-level dynamic pricing models to study the optimal strategy for dynamic pricing of railway passenger tickets. Zhu Y (2015) studied the optimisation of differential pricing based on shifts in demand. Zheng et al. (2016, 2017) proposed dynamic pricing methods considering passenger choice behaviours for high-speed rail-ways in China. Whelan and Johnson (2004) applied the PRAISE rail operations model to explore efficient fare structures and ticketing restrictions to reduce overcrowding in Great Britain. The above studies primarily consider the relationship between the price and demand, and apply a dynamic pricing strategy in the marketing stage. Considering the relationship between price and supply, this study uses differential pricing in the planning stage to guide resource allocation and thus optimise resource allocation efficiency and improve enterprise revenue. Based on the relationship between price and demand, some literature has studied the comprehensive optimisation of pricing and other factors. Hetrakul and Cirillo, (2014) used a multinomial logit model and latent class model to characterize passenger choice behaviours and built a collaborative optimisation model for pricing and seat allocation under a given capacity. Shi et al, (2009) optimised operating frequency and ticket prices collaboratively. To address the gaps in the literature discussed above, we link resource allocation and differential pricing, and propose a more 24
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refined resource allocation optimisation method. The comprehensive utility offered to passengers is described by train speed, seat class and fare. The fare rate is used to guide resource allocation, and the optimal allocation of various resources is sought to make both enterprise revenue and passenger utility synthetically optimal. The specific contributions to the existing literature are shown as follow. (1) We propose a theoretical approach which optimises capacity allocation and fare rates for the high-speed rail transport. We apply the idea of economic allocation model of grid resources in high-speed rail transport capacity allocation. Through differentiated pricing to guide the allocation of resources, a comprehensive optimisation model of resource capacity and fare rates is established to achieve the comprehensive optimisation of enterprise revenue and passenger utility. (2) We obtain classified and seat-based resource allocation that meets passengers' demand. Compared with the existing train-based resource allocation in line planning, our method refines resources into six types and allocates seat-based resources. The output results of seat-based resource allocation can be used to evaluate and optimise the number of running trains and the configuration of vehicle types. (3) We apply differential pricing among train operation routes. Compared with the existing situation of fare rates, the method we proposed allocates resources according to bidding mode, makes differentiated pricing on each train operation route in the line, and improves enterprise's revenue by considering passenger travel utility. 3. Model 3.1. Problem statement The problem of resource capacity allocation in high-speed railways can be defined as the allocation of multiple types of resources to multiple users. Drawing on the principle of grid resource allocation (Huang, 2010), the price signal is used to regulate the allocation of transportation resources. In the process of allocation, resources are shared by multiple users, and users acquire resource usage rights by bidding. The resource supplier allocates resources according to bidding prices and demand. To maximise operator revenue and passenger utility, the model is proposed and the PSO algorithm is applied to obtain the optimal resource allocation scheme and fare rates for the resource supplier and resources users. The program flow is shown in Fig. 1. According to the operational characteristics of high-speed railways, the parameters needed for the economic environment of resource allocation can be descripted as follows. Resource supplier: This is the railway transportation system. The resources available are classified by speed level and seat class, which can be divided into six types: business class seats, first class seats, and second class seats for the higher-speed grade, and business class seats, first class seats, second class seats for the lower-speed grade. Resource user: According to the line route design, a train's operation route and stop schedule are known. The train lines with same originating station and destination station are merged into a train operation route. Train operation routes are selected as applicants and users of resources, which are expressed by the set ofF = {f1 , f2 , ...fn } . The train operation route fk includesN fk stations. Resource set: According to the description of a resource supplier, resources are classified by train speed level and seat class. The resource set is represented by the setR = {r1, r2, ...rm} . For any type of resourceri , the train speed level and seat class are simultaneously {vri, z ri} , wherevri and z ri represent the train speed level and seat class, respectively. attributed, which can be formulated asri qri : Represents the passenger flow corresponding to resourceri , which describes the ability to select the resourceri . This flow must be predicted and converted. q fri : Represents the passenger flow of train operation route fk corresponding to resourceri , andqri = f F q fri . q fri is calculated by k k k k origin-destination (OD) passenger flow served by train operation route fk from historical data or a prediction; thus, N 1 N fk rf q fri = o =fk1 q ik d = o + 1 od . k ri p f : Represents the unit fare/price of resourceri needed to train operation route fk , which describes the bid price of the applicant fk k
Fig. 1. The program flow of the collaborative optimization. 25
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for the resourceri . Its units are yuan/km. Tbri : Represents the total bid for resource ri received fromF . prif : Represents the total bid by all other applicants except the resource applicant fk for resourceri , namely: k
prif = k
p fri j
f j F , f j fi
(1)
The higher the user's bid, the more resources are used. Assume that all resource applicants have perfect information on the price states of various resources, i.e. prif is known. p fri does not depend on prif , and p fri + prif can replaceTbri . The ratio of the number of k k k k k resources allocated to resource applicant fk to the entire resources available is equal to the ratio of its bid to the bid of all resource applicants. Therefore, the number of resources ri allocated to resource applicants fk is:
u frki =
p fri L fk q fri
qri
ri fk
k
¯fri k
k
fk F
=
Tbri
p fri L fk q fri
qri
ri fk
k
¯fri k
fk
k
p ri L fk q fri F f k
(2)
k
Here, is the average seat reuse rate of the trains running on train operation route fk , and ¯fk is the average load factor of the trains running on train operation route fk . Therefore, based on the above settings, the resource allocation model is developed. 3.2. Formulation 3.2.1. Objective function The model aims to maximise the total revenue of the operating enterprise and the total travel utility to all passengers. Therefore, the objective function consists of two parts, as follows:
(z1 g1fk + z2 g2fk )
max p, u
(3)
fk F
Here, z1 and z2 are the weight values of the two parts of the objective function. f g1 k represents the total revenue of train operation route fk , as defined by (4). In the seat-based resource allocation, total cost is related to the total amount of each type of resources, and the total amount of each type of resources is determined by demand. Therefore, the cost is basically unchanged under given demand. Here we choose revenue maximization instead of profit maximization as the goal. f
g1 k = ri R
u frki
ri fk
¯fri p fri L fk k
(4)
k
f route fk .g2 k represents
L fk represents the distance for train operation the total passenger travel utility on train operation route fk , as defined by (5). The passenger travel utility is described by comfort and economy. Comfort is influenced by the speed of the train and the seat class. Economy is the reciprocal of fare rate. Thus, the total passenger travel utility of train operation route fk is formulated as: f
ri ri u fk
g2 k = ri R
+
ri ri u fk
+
1 ri u p fri fk
0<
ri ,
ri
<1
(5)
k
In (5), ri and ri are the weight values for train speed and seat type based on the effect of resourceri on passenger travel comfort. Maximising total passenger travel utility not only considers the public welfare of the railway transportation service, but also effectively balances fare rates to prevent excessively high fare rates due to seeking to maximise the total revenue of the enterprise. 3.2.2. Constraints Formula (2) applied as a constraint guarantees that the fare rate serves as a signal to guide resource allocation. Constraint (6) ensures that the quantity of resource ri allocated to train operation route fk matches the actual passenger flow (demand) to a certain extent. min
u frki
ri fk
q fri
¯fri
k
max
fk
F,
ri
R
(6)
k
Here, min and max are the minimum and maximum ratios of supply and demand, respectively. Constraint (7) ensures that the fare rate fluctuates within appropriate limits. ri down ps fk
p fri
k
ri up ps fk
(7)
In (7), ps frji is the original fare rate of resource ri on train operation route fk , whilst
down
and
up are the floating rates of the fare.
3.2.3. Definitions for optimal solution The definitions of the feasible solution and the optimal solution of the model are given below. Definition 1. Feasible solution set. Suppose that u frki is the quantity of resource ri requested and received by the resource user fk , such 26
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that u = {(u11, u21, …, um1 ), (u12, u 22, …, um2 ), …, (u1n , u 2n, …, umn)|i = 1,2, …, m ; k = 1,2, …, n} .Suppose that p fri is the bid from resource k
user fk for resourceri , such thatp = {(p11 , p21 , …, pm1 ), (p12 , p22 , …, pm2 ), …, (p1n , p2n , …, pmn ) . If for anyk = 1,2, …, n and i = 1,2, …, m , p fri satisfies the constraints imposed by (7) and the correspondingu frki satisfies the constraints imposed by(6). The set of feasible k solutions is then formulated asU = {u1, u2, ...,u a} , where a is the number of feasible solutions.
Definition 2. According to the set of feasible solutionsU = {u1, u2, ...,u a} , the solution that satisfies the following conditions is the optimal solutionu :
U; a. u b. There is an optimal fare vectorp , which optimises the overall efficiency of the enterprise and the travel utility to the passenger. c. The balance between the number of resources requested by each resource applicant in solution u and the actual passenger demand meets certain constraints. 4. Algorithm The PSO algorithm was developed by Kennedy J and Eberhart RC (1995). It is easy to implement with few parameters and requires low computational memory. Thus, the PSO algorithm is used to find a feasible and high-quality solution for the above nonlinear model described by (2)–(7). Each particle represents a bid matrix, where the rows of the matrix represent the train operation routes, and the columns of the matrix represent different resource types. Therefore, each element in the matrix represents the average fare offered by a certain train operation route for a certain seat class/speed level. The particle is used to compare the fitness function values after each iteration and determine whether to regenerate the encoded information or keep the last iterated particle coding information. 4.1. Basic parameter settings for the algorithm 4.1.1. Inertia factor In this paper, we adopt a linearly decreasing dynamic inertia factor, where (t ) represents the value of the inertia factor in the t -th iteration. The initial value s , and the final value, t , are given. tmax represents the maximum number of iterations. After tmax iterations, the initial value is transformed to the final value as:
(t ) = Here,
s
s
+
= 0.9,
t
t
t max
s
t (t = 1,2,...,t max )
(8)
= 0.4 .
4.1.2. Learning factors 1and 2 are the learning factor of a particle's own optimal experience and the learning factor of the group's optimal experience, respectively. Through empirical analysis, 1 = 1.56, 2 = 2. 4.1.3. Control equation for particle flight velocity The flight velocities of particles represent the performance of the particles learning from each other and cause the particles to move towards the optimal values gradually. Let si (t )be the velocity for particle i in the t -th iteration, updated by (9).
si (t + 1) =
(t ) s i (t ) +
1
(yi (t )
x i (t )) +
2
(y g (t )
(9)
x i (t ))
In (9), x i (t ) is the position of particle i after t iterations; yi (t ) is the best position of particle i aftert iterations; the global optimal position after t iterations; and and are random numbers distributed uniformly in the interval [0,1]. We set the particle's flight velocity range to [ vmax , vmax ], and usuallyvmax = Zmax . Based on empirical analysis from the literature, = 0.1. According to the fitness function in this paper, let the upper limit of the search spaceZmax = 1500 , so thatvmax = 150 .
y g (t ) is
4.1.4. Equation of motion The particles move to the optimal solution according to the position of the previous iteration and the flight velocity. The equation of motion is expressed as follows: (10)
x i (t + 1) = xi (t ) + µ si (t + 1) Here, µ is the probability rule of particle flight velocity and we letµ = 0.01.
4.1.5. Termination condition for the algorithm The termination condition of the algorithm is the maximum number of iterations. Let the maximum iteration number tmax = 80 in this study. 4.1.6. Fitness function The fitness function is an indicator used to evaluate the current position of a particle. In this study, an objective function 27
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Fig. 2. PSO algorithm for the model.
composed of two parts is used as the fitness function of the algorithm, as given by (3). To unify the order of magnitude for the two objective values, letz1 = 0.01, z2 = 1. 4.2. Solution procedure In the following, we provide an overview of the PSO algorithm designed for the model as shown in Fig. 2. 4.2.1. Generating the initial particles A particle represents a fare rate matrix p . Randomly generate M initial particles in the interval [ down ps frki , velocity matrix s(0) is set to zero. Finally, set yi (0) = x i (0) andy g (0) = max { yi (0)} .
ri up ps fk ].
Particle initial
i = 1,2, ..., M
4.2.2. Updating position and velocity Update particle fight velocitysi (t ) and x i (t ) according to (9) and (10) respectively. 4.2.3. Checking constraints and evaluating fitness For each particle x i (t ) represents one fare rate matrixp . Calculateu by substituting p into (2). Then check constraints (6) and (7). If they are satisfied, calculate the fitness function according to (3). Otherwise, it is regarded as an invalid particle, and the position value of the particle i at the iteration t is set as negative infinity. Finally evaluate the fitness and update yi (t ) and y g (t ) . The best position means the fare rates corresponding to the best value of the fitness function. 1 2 3 4 5 6 7 8 9 10 11
Initialise t max = 80 for i = 1to M do Generate the initial particles x i (0) from [ down ps frki , yi (0) = x i (0) , si (0) = 0 end for g (0) = max { yi (0)}
ri up ps fk ]
i = 1,2, ..., M
for t = 1to t max do for i = 1to M do Update particle fight velocity si (t ) according to (9) Update particle position x i (t ) according to (10) p = x i (t ) and calculate ui (t ) according to (2) 28
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Fig. 3. Beijing-Shanghai high-speed railway line.
12 13 14 15 16 17 18 19 20 21
if constraint (6) is satisfied and constraint (7) is satisfied then calculate fitness function according to (3) else particlei is invalid, sox i (t ) = end if evaluate fitness function, updateyi (t ) andy g (t ) end for end for p = y g (t max ) , calculate u according to (2) output u andp
4.2.4. Output At the end of tmax -th iteration, the optimal p is recorded by y g (t max ) . The optimal u is obtained by substituting p into (2). Output the optimal particle position (the corresponding optimal fare rates), and the optimal resource allocation results.
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Fig. 4. Train operation routes of the Beijing-Shanghai high-speed railway.
5. Case analysis 5.1. Basic parameter settings 5.1.1. Background and operating data The Beijing-Shanghai high-speed railway, shown in Fig. 3, was officially opened on June 30, 2011. There are 24 stations on the line. At its inauguration, there were two types of trains, with speeds of 300 km/h and 250 km/h, running on the line. According to the line planning at that time, there were six train operation routes, as shown in Fig. 4. Thus, there are six resource users in this case. The basic train operation information includes running distance and number of trains as shown in Table 1. The Beijing-Shanghai highspeed railway system is considered as the resource supplier in the proposed model. Based on the actual operating data, the model is applied to optimise resource capacity allocation and fare rates. Then, the optimisation result is compared to actual operations. The resources available are divided into six types according to seat type and speed, i.e. business class, first class, and second class seats for the higher-speed grade, and business class, first class, and second class seats for the lower-speed grade. The passenger flow for each train operation route q fri is described as the average daily passenger flow. We collected the ticket sales k data for the Beijing-Shanghai high-speed railway in a certain month, and then calculated the daily average passenger flow as presented in Table 2. The fare rates of various resources in each train operation route are shown in Table 3. The data are obtained by dividing the ticket price by the operating distance at that time. According to operational experience, the average seat reuse rate is frki = 1.3 and the average load factor is ¯fri = 0.8 . The minimum k and maximum ratios of supply and demand are min = 0.9 and max = 1.1 respectively. The fare fluctuation rates are down = 0.95and up = 1.05 respectively. 5.1.2. Weight values for different resource types The value of ri is determined by the average travel speed. The value of ri is determined by the actual area occupied by a seat. The average travel speed is calculated as the maximum operating speed multiplied by the speed coefficient. According to operational experience, let the speed coefficient of higher speed (300 km/h) trains be 0.8, and the speed coefficient of lower speed (250 km/h) trains be 0.7. The speed ratio of the higher speed grade and lower speed grade is thus 240:175. Therefore, the value of ri is 0.24 for a higher-speed grade train, and the value of ri is 0.175 for the lower-speed grade train. According to the seating capacity of CRH380A trains, the ratio of the average area occupied by each type of seat can be calculated as business class seats: first class seats: second class seats = 0.0685:0.0179:0.0125. The seat weights for the respective seat types are 0.685, 0.179 and 0.125. 5.2. Results and discussion The results including the optimal fare rates and seat resource allocation scheme were obtained by applying the algorithm through MATLAB 2016 running on a computer with fourth generation intelligent Intel Core i7, 8 G RAM and 64-bit Windows 7 system. The processing time was 37.56s. The seat resource allocation scheme based on the actual passenger flow and the actual fare rates is used Table 1 Train information of the Beijing-Shanghai high-speed railway. Train operation route
Origin-Destination
Length (km)
Higher-speed Trains (trains/ day)
Lower-speed Trains (trains/ day)
Total (trains/day)
1 2 3 4
Beijing South-Jinan West Beijing South-Xuzhou East Beijing South-Nanjing South Beijing South-Shanghai Hongqiao Tianjin West-Shanghai Hongqiao Jinan West-Shanghai Hongqiao
419 688 1018 1302
1 1 1 25
2 1 1 3
3 2 2 28
1171 883
1 1 30
1 1 9
2 2 39
5 6 Total
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Table 2 Actual passenger flow. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train (persons/day)
Lower-speed Train (persons/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
19 25 38 62 17 14 175
129 180 293 508 176 157 1443
567 798 1441 3158 1021 930 7915
5 3 4 5 2 1 20
55 64 76 129 71 66 461
437 598 823 1413 770 690 4731
Table 3 Actual fare rates. Train operation route
1 2 3 4 5 6
Higher-speed Train (yuan/km)
Lower-speed Train (yuan/km)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
1.41 1.42 1.38 1.34 1.37 1.43
0.75 0.76 0.74 0.72 0.73 0.76
0.44 0.45 0.44 0.43 0.44 0.45
0.89 0.93 0.93 0.94 0.95 0.96
0.48 0.49 0.50 0.50 0.50 0.51
0.30 0.31 0.31 0.31 0.32 0.32
for comparison. After optimisation using the proposed method, the total value of the objective function is 2.04% higher than that before optimisation. This indicates that our model and algorithm can improve the combination of company revenue and passenger travel utility. The optimal fare rates and seat-based resource allocation obtained from the model are analysed as follows. 5.2.1. Optimal fare rate vector Table 4 shows the optimal unit fare for different speed grades and different types of seats in each train operation route. According to the optimisation results, when the speed is constant, fare rates of different seat types decrease with comfort, and when the seat type is the same, fare rates of different resources increase with speed. Compared to the actual fare rates, which strictly decline with operating distance, the optimal fare rates obtained from our model are more flexible. Floating within a 5% range, the optimal fare rates produce an increase by 4.15% in operating revenue. 5.2.2. Optimal seat-based resource allocation Based on the optimal fare rate vector, the corresponding optimal result of seat-based resource allocation is obtained, as shown in Table 5. According to the average seat reuse rate and the average loading factor, the seat-based resource allocation scheme can serve the passenger flow shown in Table 6. Comparing the total passenger flow in Table 2 with that in Table 6, it can be concluded that the allocation scheme obtained by the proposed model can meet the total demand of all resources. Comparing the allocation results of Table 6 with actual passenger flow of Table 2, there is little difference. A small amount of passenger demand is transferred to other train operation routes including the corresponding OD to achieve the overall optimum. The optimisation of fare rates achieves differential pricing amongst various train operation routes and resources. The results can be used to determine basic fare rates, and on this basis, revenue management technology can be implemented to improve total revenue. Combined with vehicle configuration, the resource allocation results can be used to evaluate whether the service frequency in line planning can meet the demand of each type of passengers (corresponding to seat class and speed). So, the results of seat-based resource allocation can be used to adjust and optimise line planning, including service frequency optimisation, vehicle type Table 4 Optimal fare rates. Train operation route
1 2 3 4 5 6
Higher-speed Train(yuan/km)
Lower-speed Train(yuan/km)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
1.34 1.49 1.45 1.41 1.44 1.50
0.71 0.8 0.78 0.76 0.77 0.80
0.42 0.43 0.46 0.45 0.46 0.47
0.86 0.98 0.98 0.99 1.00 1.01
0.46 0.47 0.53 0.53 0.53 0.54
0.29 0.29 0.29 0.33 0.34 0.30
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Table 5 Optimal seat-based resource allocation. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train (seats/day)
Lower-speed Train (seats/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
17 25 37 59 16 14 168
115 179 285 481 169 157 1386
503 725 1415 3030 1003 934 7610
4 3 4 5 2 1 19
48 56 75 128 70 67 444
385 544 749 1422 800 648 4548
Table 6 Passenger flow served by the optimal allocation. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train(persons/day)
Lower-speed Train(persons/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
18 26 38 61 17 15 175
120 186 296 500 176 163 1441
523 754 1472 3151 1043 971 7914
4 3 4 5 2 1 20
50 58 78 133 73 70 462
400 566 779 1479 832 674 4730
allocation, design of Electric Multiple Unit (EMU) seat configuration and fleet size. According to the actual case analysis, it can be concluded that: (1) this study establishes a collaborative optimisation method of fare rates and resource capacity allocation. The results obtained are consistent with real-world case and the method is feasible. (2) For the case presented in this paper, there is a certain discrepancy between the actual supply and demand. The supply can be adjusted by adjusting fare rates to achieve a balance between supply and demand. 6. Conclusion This study develops a comprehensive optimisation model of fare rates and seat resource allocation. The results of the model can be used to optimise service frequency, vehicle type allocation and train formation in the line planning. First, under the premise of a certain degree of supply and demand matching, the optimal fare rates for seats of different train operation routes can be obtained. Compared with the existing pricing method used for the Beijing-Shanghai high-speed railway, this paper applies differential pricing among train operation routes, which achieves the optimal combination of company revenue and passenger travel utility. Second, with the optimal fare rates as the guiding mechanism, and by considering the enterprise's efficiency and the passenger's demand for each type of resources, the optimal quantity of each type of resources can be obtained. Compared with the current allocation method to meet the total demand, this method refines the allocation of resources by considering various types of resources. The case analysis shows that the method proposed can improve the combined enterprise revenue and passenger utility by 2.04%. In the future, to enhance the optimisation results, the train stopping scheme will be taken into account, and the demand will consider OD to obtain more refined optimisation results. Declaration of conflicting interests The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This work is supported by the Scientific Research Projects in Universities of Inner Mongolia(No. NJZY18012); the National Natural Science Foundation of China (No. U1434207 and No. 51668048); and the Inner Mongolia Natural Science Foundation (No. 2017BS0501). The authors deeply appreciate the support. References Borndörfer, R., Grötschel, M., Pfetsch, M.E., 2007. A column-generation approach to line planning in public transport [J]. Transport. Sci. 41 (1), 123–132. Chang, Y.H., Yeh, C.H., Shen, C.C., 2000. A multiobjective model for passenger train services planning: application to Taiwan's high-speed rail line [J]. Transport. Res. Part B 34 (2), 91–106.
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Z.-y. Yan, et al.
Fu, H., Lei, N., Meng, L., et al., 2015. A hierarchical line planning approach for a large-scale high speed rail network: the China case [J]. Transport. Res. Part A 75, 61–83. Fu, H., Lei, N., Yang, H., et al., 2009. Analysis of compilation flow of train line planning for high speed railway [J]. Railway Transport and Economy 31 (10), 4–7. Goossens, J.W., Hoesel, S.V., Kroon, L., 2004. A branch-and-cut approach for solving railway line-planning problems [J]. Transport. Sci. 38 (3), 379–393. Goossens, J.W., Hoesel, S.V., Kroon, L., 2006. On solving multi-type railway line planning problems [J]. Eur. J. Oper. Res. 168 (2), 403–424. Hetrakul, P., Cirillo, C., 2014. A latent class choice based model system for railway optimal pricing and seat allocation [J]. Transport. Res. E Logist. Transport. Rev. 61, 68–83. Huang, F., Li, Z., 2010. Economic Allocation Model of Grid Resources. Science Publishing Company, Beijing, pp. 51–53. Kennedy, J., Eberhart, R., 1995. Particle swarm optimisation. In: Proceedings of the IEEE International Conference on Neural Networks. ICNN 95, Perth, Australia, pp. 1942–1948. Lin, D., Ku, Y., 2014. An implicit enumeration algorithm for the passenger service planning problem: application to the Taiwan Railways Administration line [J]. Eur. J. Oper. Res. 238 (3), 863–875. Shi, F., Zheng, G., Gu, Q., 2002. Optimal dynamic pricing of railway passenger ticket [J]. J. China Railw. Soc. 24 (1), 1–4. Shi, F., Luo, D., Wang, Y., et al., 2009. Optimisation of operating frequency and ticket price of intercity bus with elastic demand [J]. J. Jilin Univ. (Eng. Technol. Ed.) 39 (6), 1475–1479. Wang, L., Jia, L.M., Qin, Y., et al., 2011. A two-layer optimisation model for high-speed railway line planning [J]. J. Zhejiang Univ. - Sci. 12 (12), 902–912. Whelan, G., Johnson, D., 2004. Modelling the impact of alternative fare structures on train overcrowding [J]. Int. J. Transp. Manag. 2 (1), 51–58. Zhang, X., Ma, L., Zhang, J., 2017. Dynamic pricing for passenger groups of high-speed rail transportation [J]. J. Rail Transport. Plann. Manag. 6, 346–356. Zhang, X., LI, Y., Huang, S., et al., 2016. Maximum concave envelope theory based dynamic pricing for railway passenger transportation [J]. J. Transport. Syst. Eng. Inf. Technol. 16 (6), 1–8. Zheng, J., Liu, J., 2016. The research on ticket fare optimisation for China's High-speed train. Math. Probl. Eng. 2016, 1–8. Zheng, J., Liu, J., Clarke, D.B., 2017. Ticket fare optimisation for China's High-speed railway based on passenger choice behaviours. Discrete Dynam Nat. Soc. 2017, 1–6. Zhu, S., Zhang, Q., 2000. Study on the modes of the resource allocation and the arrangement of the railway transportation network [J]. J. Syst. Eng. 10 (3), 238–242.
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Journal of Rail Transport Planning & Management journal homepage: www.elsevier.com/locate/jrtpm
Collaborative optimisation of resource capacity allocation and fare rate for high-speed railway passenger transport
T
Zhen-ying Yana,b, Xiao-juan Lib, Bao-ming Hana,∗ a b
School of Traffic and Transportation, Bei Jing Jiao Tong University, Beijing, 100044, China Transportation Institute of Inner Mongolia University, Hohhot, 010070, China
ARTICLE INFO
ABSTRACT
Keywords: High-speed railway Resource capacity allocation Pricing Particle swarm optimisation
The reasonable pricing of high-speed railway tickets and the optimal allocation of transport resource capacity can not only enhance competitiveness in the transport market, but also reasonably coordinate the revenue of the enterprise and utilities to passengers. This study uses price signals to adjust resource capacity allocation; and develops a co-optimisation model of resource capacity allocation and fare rates of high-speed trains in different train operation routes. The developed model aims at the comprehensive optimisation of railway enterprise's revenue and passengers' travel benefits, with the ratio of supply-demand and the floating rate of the fare as the main constraints. The Particle Swarm Optimisation (PSO) algorithm is applied to obtain the seat resource allocation scheme and the optimal fare rate for each train operation route. Finally, the case analysis is carried out to test the model and the algorithm. Based on a statistical analysis of actual ticket sale data of the Beijing-Shanghai high-speed railway for a certain month, the optimal unit fare and optimal seat resource allocation scheme are obtained to meet the corresponding passenger demand. The case analysis shows that after optimisation by the proposed method, the total value of the objective function is 2.04% higher than that before optimisation.
1. Introduction Over the past decade, high-speed railways have developed rapidly across the world. Especially in China, by the end of 2017, operations reached 25 000 km, accounting for two-thirds of the world total. With increasing high-speed rail network construction, more attention has focused on high-speed railway transport operation. There is a general consensus on the hierarchical formulation of railway passenger transport organizational plans. Macro-planning (strategic level) to micro-implementation (operational level) lead to network design, line planning, timetabling, rolling stock, crew scheduling and operational control (Goossens et al., 2004; Fu et al., 2009). The characteristics of different transport plans lead to corresponding optimisation theories and methods. Line planning belongs to the “strategy level”. It forms the basis for the follow-up “operational level” implementation plan. According to the characteristics of passenger flow, line planning determines the starting and ending points, routes, stop schedule, service frequencies, train grades and train composition. Research on high-speed railway line planning in China is scattered and focuses on the optimisation of different elements, such as optimising train orgin and destination stations, stopping schemes, service frequencies, train types, train composition and transfer schemes (Fu et al., 2009). The resource capacity allocation plays an important role in line planning. Service frequencies, train grades, train types and train composition are all based on the resource capacity allocation. The resource capacity allocation here refers to the allocation of transportation products in the transport corridor. The efficiency of resource allocation
∗
Corresponding author. E-mail address: [email protected] (B.-m. Han).
https://doi.org/10.1016/j.jrtpm.2019.05.001 Received 21 September 2018; Received in revised form 5 May 2019; Accepted 13 May 2019 Available online 18 May 2019 2210-9706/ © 2019 Elsevier Ltd. All rights reserved.
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directly affects the revenue of operation enterprises and the utility to passengers. At present, there are few studies on the allocation of high-speed rail transport resources. In the line planning optimisation, resource capacity allocation is based on the train unit. For example, the service frequencies are determined based on the unit of train. So the seat resources are not classified, which results in seat resource allocation not being sufficiently elaborate. In operation, often some types of resources are in short supply while other resources are surplus. For example, the second-class seats of high-speed trains are in short supply on a certain line, while the first-class seats are in surplus. Therefore, resources should be classified and allocated to deal with such problems. At the same time, an operating enterprise is permitted to formulate fare rates for high-speed railway independently in China. Obviously, an operating enterprise desires to adjust fare rates to improve its revenue. Based on the basic law of supply and price in any market, we link resource allocation with fare rates to propose a co-optimisation method. Based on the economic allocation principle of grid resources (Huang, 2010), this paper proposes a more refined allocation method for seat resources. The types of high-speed rail resources are divided into six categories, and the price signal is applied to guide the allocation of all types of resources. The model takes into account the benefits of both enterprises and passengers and aims at a comprehensive optimisation of enterprise revenue and passenger utility. Then we design a particle swarm optimisation (PSO) algorithm (Kennedy and Eberhart, 1995) to solve the model, and obtain a refined optimal resource allocation scheme based on the seat unit, which can be used to determine the service frequencies, vehicle configuration and fleet size. At the same time, optimal fare rates are obtained to improve an enterprise's revenue. The remainder of this paper is structured as follows. Section 2 reviews related literature. Section 3 presents the model. Section 3.1 describes economic environment and the principles pertinent to resource allocation, and then Section 3.2 presents the formulation for the collaborative optimisation problem. In Section 4, we develop PSO algorithm to solve the model efficiently. Section 5 describes the case analysis based on actual operational data and provides discussions. Section 6 presents our conclusions. 2. Literature review The problems studied in this paper involve two aspects: resource allocation and pricing. The resource allocation here refers to the allocation of seat resources for each train operation route based on given line routes. The results can be used to optimise the service frequency, vehicle configuration and train composition. Thus it refers to an optimisation problem for line planning. Line planning usually includes line route design, stop schedule planning, service frequency and fleet size. From the point of view of optimisation, the objectives of the travel plan include maximising enterprise profits, minimising costs, minimising passenger travel time and so on. For the overall optimisation scope, it includes the comprehensive optimisation of all factors and the optimisation of individual factors. Goossens et al. (2004) formulated the line-planning problem with the objective of minimising total operating costs and solved the model by an improved branch-and-cut approach for large operations. Goossens et al. (2006) formulated multi-type line planning problems by integer programming with the objective of minimising operating costs. Borndörfer et al. (2007) formulated the lineplanning problem with a new multi-commodity flow model to minimise the combined objective of total operating costs and passenger traveling time. Chang et al. (2000) proposed an allocation model to determine the train stop schedule plan, service frequency and fleet size. The goal of the model is to minimise total operating costs and passenger's total travel time loss. Wang et al. (2011) developed a two-layer model to determine stop-schedules and service frequencies. An upper-layer model was applied to minimise total operating costs and unserved passenger volumes, while a bottom model assigned the passenger flow to maximise the served passenger volume and minimise total travel time. Lin and Ku (2014) focused on the stop schedule plan of line planning and applied the integer programming to maximise operator profit. Fu et al. (2015) proposed a hierarchical line planning approach for a largescale high-speed railway network in China, which can incorporate cost-oriented and/or customer-oriented objectives. Zhu and Zhang, (2000) put forward an optimisation model for the organisation mode and system resource allocation through a quantitative study. However, the above literature focuses on resource allocation based on train units and does not subdivide the resources into different types. In fact, demands for different seat types and different speeds should be served by their corresponding resources. For example, if a passenger expecting a first-class seat with higher speed is served by a second- class seat with higher speed in the planning, the plan will not meet the passenger's expectations. So the passenger may choose other modes to travel or cancel the travel. Dividing resources into several types and allocating resources based on the seat level can better meet the needs of passengers and improve the efficiency of resource allocation. Pricing is primarily implemented at the marketing design stage. Shi et al, (2002), Zhang et al, (2016) and Zhang et al, (2017), established various multi-level dynamic pricing models to study the optimal strategy for dynamic pricing of railway passenger tickets. Zhu Y (2015) studied the optimisation of differential pricing based on shifts in demand. Zheng et al. (2016, 2017) proposed dynamic pricing methods considering passenger choice behaviours for high-speed rail-ways in China. Whelan and Johnson (2004) applied the PRAISE rail operations model to explore efficient fare structures and ticketing restrictions to reduce overcrowding in Great Britain. The above studies primarily consider the relationship between the price and demand, and apply a dynamic pricing strategy in the marketing stage. Considering the relationship between price and supply, this study uses differential pricing in the planning stage to guide resource allocation and thus optimise resource allocation efficiency and improve enterprise revenue. Based on the relationship between price and demand, some literature has studied the comprehensive optimisation of pricing and other factors. Hetrakul and Cirillo, (2014) used a multinomial logit model and latent class model to characterize passenger choice behaviours and built a collaborative optimisation model for pricing and seat allocation under a given capacity. Shi et al, (2009) optimised operating frequency and ticket prices collaboratively. To address the gaps in the literature discussed above, we link resource allocation and differential pricing, and propose a more 24
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refined resource allocation optimisation method. The comprehensive utility offered to passengers is described by train speed, seat class and fare. The fare rate is used to guide resource allocation, and the optimal allocation of various resources is sought to make both enterprise revenue and passenger utility synthetically optimal. The specific contributions to the existing literature are shown as follow. (1) We propose a theoretical approach which optimises capacity allocation and fare rates for the high-speed rail transport. We apply the idea of economic allocation model of grid resources in high-speed rail transport capacity allocation. Through differentiated pricing to guide the allocation of resources, a comprehensive optimisation model of resource capacity and fare rates is established to achieve the comprehensive optimisation of enterprise revenue and passenger utility. (2) We obtain classified and seat-based resource allocation that meets passengers' demand. Compared with the existing train-based resource allocation in line planning, our method refines resources into six types and allocates seat-based resources. The output results of seat-based resource allocation can be used to evaluate and optimise the number of running trains and the configuration of vehicle types. (3) We apply differential pricing among train operation routes. Compared with the existing situation of fare rates, the method we proposed allocates resources according to bidding mode, makes differentiated pricing on each train operation route in the line, and improves enterprise's revenue by considering passenger travel utility. 3. Model 3.1. Problem statement The problem of resource capacity allocation in high-speed railways can be defined as the allocation of multiple types of resources to multiple users. Drawing on the principle of grid resource allocation (Huang, 2010), the price signal is used to regulate the allocation of transportation resources. In the process of allocation, resources are shared by multiple users, and users acquire resource usage rights by bidding. The resource supplier allocates resources according to bidding prices and demand. To maximise operator revenue and passenger utility, the model is proposed and the PSO algorithm is applied to obtain the optimal resource allocation scheme and fare rates for the resource supplier and resources users. The program flow is shown in Fig. 1. According to the operational characteristics of high-speed railways, the parameters needed for the economic environment of resource allocation can be descripted as follows. Resource supplier: This is the railway transportation system. The resources available are classified by speed level and seat class, which can be divided into six types: business class seats, first class seats, and second class seats for the higher-speed grade, and business class seats, first class seats, second class seats for the lower-speed grade. Resource user: According to the line route design, a train's operation route and stop schedule are known. The train lines with same originating station and destination station are merged into a train operation route. Train operation routes are selected as applicants and users of resources, which are expressed by the set ofF = {f1 , f2 , ...fn } . The train operation route fk includesN fk stations. Resource set: According to the description of a resource supplier, resources are classified by train speed level and seat class. The resource set is represented by the setR = {r1, r2, ...rm} . For any type of resourceri , the train speed level and seat class are simultaneously {vri, z ri} , wherevri and z ri represent the train speed level and seat class, respectively. attributed, which can be formulated asri qri : Represents the passenger flow corresponding to resourceri , which describes the ability to select the resourceri . This flow must be predicted and converted. q fri : Represents the passenger flow of train operation route fk corresponding to resourceri , andqri = f F q fri . q fri is calculated by k k k k origin-destination (OD) passenger flow served by train operation route fk from historical data or a prediction; thus, N 1 N fk rf q fri = o =fk1 q ik d = o + 1 od . k ri p f : Represents the unit fare/price of resourceri needed to train operation route fk , which describes the bid price of the applicant fk k
Fig. 1. The program flow of the collaborative optimization. 25
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for the resourceri . Its units are yuan/km. Tbri : Represents the total bid for resource ri received fromF . prif : Represents the total bid by all other applicants except the resource applicant fk for resourceri , namely: k
prif = k
p fri j
f j F , f j fi
(1)
The higher the user's bid, the more resources are used. Assume that all resource applicants have perfect information on the price states of various resources, i.e. prif is known. p fri does not depend on prif , and p fri + prif can replaceTbri . The ratio of the number of k k k k k resources allocated to resource applicant fk to the entire resources available is equal to the ratio of its bid to the bid of all resource applicants. Therefore, the number of resources ri allocated to resource applicants fk is:
u frki =
p fri L fk q fri
qri
ri fk
k
¯fri k
k
fk F
=
Tbri
p fri L fk q fri
qri
ri fk
k
¯fri k
fk
k
p ri L fk q fri F f k
(2)
k
Here, is the average seat reuse rate of the trains running on train operation route fk , and ¯fk is the average load factor of the trains running on train operation route fk . Therefore, based on the above settings, the resource allocation model is developed. 3.2. Formulation 3.2.1. Objective function The model aims to maximise the total revenue of the operating enterprise and the total travel utility to all passengers. Therefore, the objective function consists of two parts, as follows:
(z1 g1fk + z2 g2fk )
max p, u
(3)
fk F
Here, z1 and z2 are the weight values of the two parts of the objective function. f g1 k represents the total revenue of train operation route fk , as defined by (4). In the seat-based resource allocation, total cost is related to the total amount of each type of resources, and the total amount of each type of resources is determined by demand. Therefore, the cost is basically unchanged under given demand. Here we choose revenue maximization instead of profit maximization as the goal. f
g1 k = ri R
u frki
ri fk
¯fri p fri L fk k
(4)
k
f route fk .g2 k represents
L fk represents the distance for train operation the total passenger travel utility on train operation route fk , as defined by (5). The passenger travel utility is described by comfort and economy. Comfort is influenced by the speed of the train and the seat class. Economy is the reciprocal of fare rate. Thus, the total passenger travel utility of train operation route fk is formulated as: f
ri ri u fk
g2 k = ri R
+
ri ri u fk
+
1 ri u p fri fk
0<
ri ,
ri
<1
(5)
k
In (5), ri and ri are the weight values for train speed and seat type based on the effect of resourceri on passenger travel comfort. Maximising total passenger travel utility not only considers the public welfare of the railway transportation service, but also effectively balances fare rates to prevent excessively high fare rates due to seeking to maximise the total revenue of the enterprise. 3.2.2. Constraints Formula (2) applied as a constraint guarantees that the fare rate serves as a signal to guide resource allocation. Constraint (6) ensures that the quantity of resource ri allocated to train operation route fk matches the actual passenger flow (demand) to a certain extent. min
u frki
ri fk
q fri
¯fri
k
max
fk
F,
ri
R
(6)
k
Here, min and max are the minimum and maximum ratios of supply and demand, respectively. Constraint (7) ensures that the fare rate fluctuates within appropriate limits. ri down ps fk
p fri
k
ri up ps fk
(7)
In (7), ps frji is the original fare rate of resource ri on train operation route fk , whilst
down
and
up are the floating rates of the fare.
3.2.3. Definitions for optimal solution The definitions of the feasible solution and the optimal solution of the model are given below. Definition 1. Feasible solution set. Suppose that u frki is the quantity of resource ri requested and received by the resource user fk , such 26
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that u = {(u11, u21, …, um1 ), (u12, u 22, …, um2 ), …, (u1n , u 2n, …, umn)|i = 1,2, …, m ; k = 1,2, …, n} .Suppose that p fri is the bid from resource k
user fk for resourceri , such thatp = {(p11 , p21 , …, pm1 ), (p12 , p22 , …, pm2 ), …, (p1n , p2n , …, pmn ) . If for anyk = 1,2, …, n and i = 1,2, …, m , p fri satisfies the constraints imposed by (7) and the correspondingu frki satisfies the constraints imposed by(6). The set of feasible k solutions is then formulated asU = {u1, u2, ...,u a} , where a is the number of feasible solutions.
Definition 2. According to the set of feasible solutionsU = {u1, u2, ...,u a} , the solution that satisfies the following conditions is the optimal solutionu :
U; a. u b. There is an optimal fare vectorp , which optimises the overall efficiency of the enterprise and the travel utility to the passenger. c. The balance between the number of resources requested by each resource applicant in solution u and the actual passenger demand meets certain constraints. 4. Algorithm The PSO algorithm was developed by Kennedy J and Eberhart RC (1995). It is easy to implement with few parameters and requires low computational memory. Thus, the PSO algorithm is used to find a feasible and high-quality solution for the above nonlinear model described by (2)–(7). Each particle represents a bid matrix, where the rows of the matrix represent the train operation routes, and the columns of the matrix represent different resource types. Therefore, each element in the matrix represents the average fare offered by a certain train operation route for a certain seat class/speed level. The particle is used to compare the fitness function values after each iteration and determine whether to regenerate the encoded information or keep the last iterated particle coding information. 4.1. Basic parameter settings for the algorithm 4.1.1. Inertia factor In this paper, we adopt a linearly decreasing dynamic inertia factor, where (t ) represents the value of the inertia factor in the t -th iteration. The initial value s , and the final value, t , are given. tmax represents the maximum number of iterations. After tmax iterations, the initial value is transformed to the final value as:
(t ) = Here,
s
s
+
= 0.9,
t
t
t max
s
t (t = 1,2,...,t max )
(8)
= 0.4 .
4.1.2. Learning factors 1and 2 are the learning factor of a particle's own optimal experience and the learning factor of the group's optimal experience, respectively. Through empirical analysis, 1 = 1.56, 2 = 2. 4.1.3. Control equation for particle flight velocity The flight velocities of particles represent the performance of the particles learning from each other and cause the particles to move towards the optimal values gradually. Let si (t )be the velocity for particle i in the t -th iteration, updated by (9).
si (t + 1) =
(t ) s i (t ) +
1
(yi (t )
x i (t )) +
2
(y g (t )
(9)
x i (t ))
In (9), x i (t ) is the position of particle i after t iterations; yi (t ) is the best position of particle i aftert iterations; the global optimal position after t iterations; and and are random numbers distributed uniformly in the interval [0,1]. We set the particle's flight velocity range to [ vmax , vmax ], and usuallyvmax = Zmax . Based on empirical analysis from the literature, = 0.1. According to the fitness function in this paper, let the upper limit of the search spaceZmax = 1500 , so thatvmax = 150 .
y g (t ) is
4.1.4. Equation of motion The particles move to the optimal solution according to the position of the previous iteration and the flight velocity. The equation of motion is expressed as follows: (10)
x i (t + 1) = xi (t ) + µ si (t + 1) Here, µ is the probability rule of particle flight velocity and we letµ = 0.01.
4.1.5. Termination condition for the algorithm The termination condition of the algorithm is the maximum number of iterations. Let the maximum iteration number tmax = 80 in this study. 4.1.6. Fitness function The fitness function is an indicator used to evaluate the current position of a particle. In this study, an objective function 27
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Fig. 2. PSO algorithm for the model.
composed of two parts is used as the fitness function of the algorithm, as given by (3). To unify the order of magnitude for the two objective values, letz1 = 0.01, z2 = 1. 4.2. Solution procedure In the following, we provide an overview of the PSO algorithm designed for the model as shown in Fig. 2. 4.2.1. Generating the initial particles A particle represents a fare rate matrix p . Randomly generate M initial particles in the interval [ down ps frki , velocity matrix s(0) is set to zero. Finally, set yi (0) = x i (0) andy g (0) = max { yi (0)} .
ri up ps fk ].
Particle initial
i = 1,2, ..., M
4.2.2. Updating position and velocity Update particle fight velocitysi (t ) and x i (t ) according to (9) and (10) respectively. 4.2.3. Checking constraints and evaluating fitness For each particle x i (t ) represents one fare rate matrixp . Calculateu by substituting p into (2). Then check constraints (6) and (7). If they are satisfied, calculate the fitness function according to (3). Otherwise, it is regarded as an invalid particle, and the position value of the particle i at the iteration t is set as negative infinity. Finally evaluate the fitness and update yi (t ) and y g (t ) . The best position means the fare rates corresponding to the best value of the fitness function. 1 2 3 4 5 6 7 8 9 10 11
Initialise t max = 80 for i = 1to M do Generate the initial particles x i (0) from [ down ps frki , yi (0) = x i (0) , si (0) = 0 end for g (0) = max { yi (0)}
ri up ps fk ]
i = 1,2, ..., M
for t = 1to t max do for i = 1to M do Update particle fight velocity si (t ) according to (9) Update particle position x i (t ) according to (10) p = x i (t ) and calculate ui (t ) according to (2) 28
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Fig. 3. Beijing-Shanghai high-speed railway line.
12 13 14 15 16 17 18 19 20 21
if constraint (6) is satisfied and constraint (7) is satisfied then calculate fitness function according to (3) else particlei is invalid, sox i (t ) = end if evaluate fitness function, updateyi (t ) andy g (t ) end for end for p = y g (t max ) , calculate u according to (2) output u andp
4.2.4. Output At the end of tmax -th iteration, the optimal p is recorded by y g (t max ) . The optimal u is obtained by substituting p into (2). Output the optimal particle position (the corresponding optimal fare rates), and the optimal resource allocation results.
29
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Fig. 4. Train operation routes of the Beijing-Shanghai high-speed railway.
5. Case analysis 5.1. Basic parameter settings 5.1.1. Background and operating data The Beijing-Shanghai high-speed railway, shown in Fig. 3, was officially opened on June 30, 2011. There are 24 stations on the line. At its inauguration, there were two types of trains, with speeds of 300 km/h and 250 km/h, running on the line. According to the line planning at that time, there were six train operation routes, as shown in Fig. 4. Thus, there are six resource users in this case. The basic train operation information includes running distance and number of trains as shown in Table 1. The Beijing-Shanghai highspeed railway system is considered as the resource supplier in the proposed model. Based on the actual operating data, the model is applied to optimise resource capacity allocation and fare rates. Then, the optimisation result is compared to actual operations. The resources available are divided into six types according to seat type and speed, i.e. business class, first class, and second class seats for the higher-speed grade, and business class, first class, and second class seats for the lower-speed grade. The passenger flow for each train operation route q fri is described as the average daily passenger flow. We collected the ticket sales k data for the Beijing-Shanghai high-speed railway in a certain month, and then calculated the daily average passenger flow as presented in Table 2. The fare rates of various resources in each train operation route are shown in Table 3. The data are obtained by dividing the ticket price by the operating distance at that time. According to operational experience, the average seat reuse rate is frki = 1.3 and the average load factor is ¯fri = 0.8 . The minimum k and maximum ratios of supply and demand are min = 0.9 and max = 1.1 respectively. The fare fluctuation rates are down = 0.95and up = 1.05 respectively. 5.1.2. Weight values for different resource types The value of ri is determined by the average travel speed. The value of ri is determined by the actual area occupied by a seat. The average travel speed is calculated as the maximum operating speed multiplied by the speed coefficient. According to operational experience, let the speed coefficient of higher speed (300 km/h) trains be 0.8, and the speed coefficient of lower speed (250 km/h) trains be 0.7. The speed ratio of the higher speed grade and lower speed grade is thus 240:175. Therefore, the value of ri is 0.24 for a higher-speed grade train, and the value of ri is 0.175 for the lower-speed grade train. According to the seating capacity of CRH380A trains, the ratio of the average area occupied by each type of seat can be calculated as business class seats: first class seats: second class seats = 0.0685:0.0179:0.0125. The seat weights for the respective seat types are 0.685, 0.179 and 0.125. 5.2. Results and discussion The results including the optimal fare rates and seat resource allocation scheme were obtained by applying the algorithm through MATLAB 2016 running on a computer with fourth generation intelligent Intel Core i7, 8 G RAM and 64-bit Windows 7 system. The processing time was 37.56s. The seat resource allocation scheme based on the actual passenger flow and the actual fare rates is used Table 1 Train information of the Beijing-Shanghai high-speed railway. Train operation route
Origin-Destination
Length (km)
Higher-speed Trains (trains/ day)
Lower-speed Trains (trains/ day)
Total (trains/day)
1 2 3 4
Beijing South-Jinan West Beijing South-Xuzhou East Beijing South-Nanjing South Beijing South-Shanghai Hongqiao Tianjin West-Shanghai Hongqiao Jinan West-Shanghai Hongqiao
419 688 1018 1302
1 1 1 25
2 1 1 3
3 2 2 28
1171 883
1 1 30
1 1 9
2 2 39
5 6 Total
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Table 2 Actual passenger flow. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train (persons/day)
Lower-speed Train (persons/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
19 25 38 62 17 14 175
129 180 293 508 176 157 1443
567 798 1441 3158 1021 930 7915
5 3 4 5 2 1 20
55 64 76 129 71 66 461
437 598 823 1413 770 690 4731
Table 3 Actual fare rates. Train operation route
1 2 3 4 5 6
Higher-speed Train (yuan/km)
Lower-speed Train (yuan/km)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
1.41 1.42 1.38 1.34 1.37 1.43
0.75 0.76 0.74 0.72 0.73 0.76
0.44 0.45 0.44 0.43 0.44 0.45
0.89 0.93 0.93 0.94 0.95 0.96
0.48 0.49 0.50 0.50 0.50 0.51
0.30 0.31 0.31 0.31 0.32 0.32
for comparison. After optimisation using the proposed method, the total value of the objective function is 2.04% higher than that before optimisation. This indicates that our model and algorithm can improve the combination of company revenue and passenger travel utility. The optimal fare rates and seat-based resource allocation obtained from the model are analysed as follows. 5.2.1. Optimal fare rate vector Table 4 shows the optimal unit fare for different speed grades and different types of seats in each train operation route. According to the optimisation results, when the speed is constant, fare rates of different seat types decrease with comfort, and when the seat type is the same, fare rates of different resources increase with speed. Compared to the actual fare rates, which strictly decline with operating distance, the optimal fare rates obtained from our model are more flexible. Floating within a 5% range, the optimal fare rates produce an increase by 4.15% in operating revenue. 5.2.2. Optimal seat-based resource allocation Based on the optimal fare rate vector, the corresponding optimal result of seat-based resource allocation is obtained, as shown in Table 5. According to the average seat reuse rate and the average loading factor, the seat-based resource allocation scheme can serve the passenger flow shown in Table 6. Comparing the total passenger flow in Table 2 with that in Table 6, it can be concluded that the allocation scheme obtained by the proposed model can meet the total demand of all resources. Comparing the allocation results of Table 6 with actual passenger flow of Table 2, there is little difference. A small amount of passenger demand is transferred to other train operation routes including the corresponding OD to achieve the overall optimum. The optimisation of fare rates achieves differential pricing amongst various train operation routes and resources. The results can be used to determine basic fare rates, and on this basis, revenue management technology can be implemented to improve total revenue. Combined with vehicle configuration, the resource allocation results can be used to evaluate whether the service frequency in line planning can meet the demand of each type of passengers (corresponding to seat class and speed). So, the results of seat-based resource allocation can be used to adjust and optimise line planning, including service frequency optimisation, vehicle type Table 4 Optimal fare rates. Train operation route
1 2 3 4 5 6
Higher-speed Train(yuan/km)
Lower-speed Train(yuan/km)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
1.34 1.49 1.45 1.41 1.44 1.50
0.71 0.8 0.78 0.76 0.77 0.80
0.42 0.43 0.46 0.45 0.46 0.47
0.86 0.98 0.98 0.99 1.00 1.01
0.46 0.47 0.53 0.53 0.53 0.54
0.29 0.29 0.29 0.33 0.34 0.30
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Table 5 Optimal seat-based resource allocation. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train (seats/day)
Lower-speed Train (seats/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
17 25 37 59 16 14 168
115 179 285 481 169 157 1386
503 725 1415 3030 1003 934 7610
4 3 4 5 2 1 19
48 56 75 128 70 67 444
385 544 749 1422 800 648 4548
Table 6 Passenger flow served by the optimal allocation. Train operation route
1 2 3 4 5 6 Total
Higher-speed Train(persons/day)
Lower-speed Train(persons/day)
Business Class Seat
1st Class Seat
2nd Class Seat
Business Class Seat
1st Class Seat
2nd Class Seat
18 26 38 61 17 15 175
120 186 296 500 176 163 1441
523 754 1472 3151 1043 971 7914
4 3 4 5 2 1 20
50 58 78 133 73 70 462
400 566 779 1479 832 674 4730
allocation, design of Electric Multiple Unit (EMU) seat configuration and fleet size. According to the actual case analysis, it can be concluded that: (1) this study establishes a collaborative optimisation method of fare rates and resource capacity allocation. The results obtained are consistent with real-world case and the method is feasible. (2) For the case presented in this paper, there is a certain discrepancy between the actual supply and demand. The supply can be adjusted by adjusting fare rates to achieve a balance between supply and demand. 6. Conclusion This study develops a comprehensive optimisation model of fare rates and seat resource allocation. The results of the model can be used to optimise service frequency, vehicle type allocation and train formation in the line planning. First, under the premise of a certain degree of supply and demand matching, the optimal fare rates for seats of different train operation routes can be obtained. Compared with the existing pricing method used for the Beijing-Shanghai high-speed railway, this paper applies differential pricing among train operation routes, which achieves the optimal combination of company revenue and passenger travel utility. Second, with the optimal fare rates as the guiding mechanism, and by considering the enterprise's efficiency and the passenger's demand for each type of resources, the optimal quantity of each type of resources can be obtained. Compared with the current allocation method to meet the total demand, this method refines the allocation of resources by considering various types of resources. The case analysis shows that the method proposed can improve the combined enterprise revenue and passenger utility by 2.04%. In the future, to enhance the optimisation results, the train stopping scheme will be taken into account, and the demand will consider OD to obtain more refined optimisation results. Declaration of conflicting interests The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This work is supported by the Scientific Research Projects in Universities of Inner Mongolia(No. NJZY18012); the National Natural Science Foundation of China (No. U1434207 and No. 51668048); and the Inner Mongolia Natural Science Foundation (No. 2017BS0501). The authors deeply appreciate the support. References Borndörfer, R., Grötschel, M., Pfetsch, M.E., 2007. A column-generation approach to line planning in public transport [J]. Transport. Sci. 41 (1), 123–132. Chang, Y.H., Yeh, C.H., Shen, C.C., 2000. A multiobjective model for passenger train services planning: application to Taiwan's high-speed rail line [J]. Transport. Res. Part B 34 (2), 91–106.
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Fu, H., Lei, N., Meng, L., et al., 2015. A hierarchical line planning approach for a large-scale high speed rail network: the China case [J]. Transport. Res. Part A 75, 61–83. Fu, H., Lei, N., Yang, H., et al., 2009. Analysis of compilation flow of train line planning for high speed railway [J]. Railway Transport and Economy 31 (10), 4–7. Goossens, J.W., Hoesel, S.V., Kroon, L., 2004. A branch-and-cut approach for solving railway line-planning problems [J]. Transport. Sci. 38 (3), 379–393. Goossens, J.W., Hoesel, S.V., Kroon, L., 2006. On solving multi-type railway line planning problems [J]. Eur. J. Oper. Res. 168 (2), 403–424. Hetrakul, P., Cirillo, C., 2014. A latent class choice based model system for railway optimal pricing and seat allocation [J]. Transport. Res. E Logist. Transport. Rev. 61, 68–83. Huang, F., Li, Z., 2010. Economic Allocation Model of Grid Resources. Science Publishing Company, Beijing, pp. 51–53. Kennedy, J., Eberhart, R., 1995. Particle swarm optimisation. In: Proceedings of the IEEE International Conference on Neural Networks. ICNN 95, Perth, Australia, pp. 1942–1948. Lin, D., Ku, Y., 2014. An implicit enumeration algorithm for the passenger service planning problem: application to the Taiwan Railways Administration line [J]. Eur. J. Oper. Res. 238 (3), 863–875. Shi, F., Zheng, G., Gu, Q., 2002. Optimal dynamic pricing of railway passenger ticket [J]. J. China Railw. Soc. 24 (1), 1–4. Shi, F., Luo, D., Wang, Y., et al., 2009. Optimisation of operating frequency and ticket price of intercity bus with elastic demand [J]. J. Jilin Univ. (Eng. Technol. Ed.) 39 (6), 1475–1479. Wang, L., Jia, L.M., Qin, Y., et al., 2011. A two-layer optimisation model for high-speed railway line planning [J]. J. Zhejiang Univ. - Sci. 12 (12), 902–912. Whelan, G., Johnson, D., 2004. Modelling the impact of alternative fare structures on train overcrowding [J]. Int. J. Transp. Manag. 2 (1), 51–58. Zhang, X., Ma, L., Zhang, J., 2017. Dynamic pricing for passenger groups of high-speed rail transportation [J]. J. Rail Transport. Plann. Manag. 6, 346–356. Zhang, X., LI, Y., Huang, S., et al., 2016. Maximum concave envelope theory based dynamic pricing for railway passenger transportation [J]. J. Transport. Syst. Eng. Inf. Technol. 16 (6), 1–8. Zheng, J., Liu, J., 2016. The research on ticket fare optimisation for China's High-speed train. Math. Probl. Eng. 2016, 1–8. Zheng, J., Liu, J., Clarke, D.B., 2017. Ticket fare optimisation for China's High-speed railway based on passenger choice behaviours. Discrete Dynam Nat. Soc. 2017, 1–6. Zhu, S., Zhang, Q., 2000. Study on the modes of the resource allocation and the arrangement of the railway transportation network [J]. J. Syst. Eng. 10 (3), 238–242.
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