Integrated Train Timetabling and Rolling Stock Scheduling Model Based on 1

Time-Dependent Demand for Urban Rail Transit 2

3

Yixiang Yue (Corresponding author) 4

School of Traffic and Transportation, Beijing Jiaotong University, 5

No.3 Shang Yuan Cun, Hai Dian District, Beijing, China. 100044 6

Phone: 8613511054454; Fax: 86-10-51683124 7

Email: yixiangyue@yahoo.com 8

9

Juntao Han 10

School of Traffic and Transportation, Beijing Jiaotong University, 11

No.3 Shang Yuan Cun, Hai Dian District, Beijing, China. 100044 12

Phone: 8613810437903; 13

Email: 15120813@bjtu.edu.cn 14

15

Shifeng Wang 16 China Railway Electrification Survey Design & Research Institute Co. Ltd., 17

No.33 Jiangdu Road, He Dong District, Tianjin, China. 300250 18

Phone: 8615210568355 19

Email: wangshifeng@tjedi.com.cn 20

21

Xiang Liu 22

Department of Civil and Environmental Engineering 23

Rutgers, The State University of New Jersey 24

CoRE 606, 96 Frelinghuysen Road, Piscataway, NJ 08854-8018 25

Email: xiang.liu@rutgers.edu 26

27

28

29

Word count: 4,796words text + (6 figures + 4 tables) x 250 words (each) = 6,796 words 30

31

Submission Date: August 1st, 2016 32

33

Paper submitted for peer review at the 96th

Transportation Research Board Annual Meeting. 34

35

2

ABSTRACT 1

The congestion problem of urban transportation is becoming increasingly critical for many 2

metropolises. The Urban Rail Transit (URT) system has attracted substantial attention due to its 3

safety, high speed, high capacity and sustainability. With a focus on providing a holistic 4

modelling framework for train scheduling problems, this paper proposes a novel optimization 5

methodology that integrates both train timetabling and rolling stock scheduling based on time-6

dependent passenger flow demands. We particularly consider the trade-off between waiting times 7

of passengers and operation costs of the URT system. Using train paths and rolling stock 8

indicators as decision variables, this problem is formulated as a bi-level programming model. The 9

simulated-annealing (SA) based heuristic algorithm is employed to solve the proposed model and 10

generate approximate optimal solutions. Numerical examples are developed to demonstrate the 11

performance of the proposed approaches. The calculation results and comparisons indicate that, 12

even for the large-scale Beijing rail transit operations, the SA-based algorithm can efficiently 13

produce the approximate optimal scheduling strategies within acceptable computational limits, 14

demonstrating the practical value of our proposed approaches. 15

16

Keywords: Urban rail transit, Passenger flow, Train timetable, Rolling stock schedule, Bi-level 17

programming, Simulated annealing algorithm 18

19

20

3

INTRODUCTION 1

With the emergence of a more social economy, many rail transit networks have recently been put 2

into operation or are under construction in many metropolises to satisfy large passenger travel 3

demands. Therefore, the efficient operation of rail transit lines has become an important issue for 4

all URT systems. 5

The planning process for public transportation usually consists of several consecutive phases. 6

The process begins with network design, which is typically followed by line planning, 7

timetabling, and vehicle and crew scheduling (1). Obtaining a high quality railway operation plan 8

will take several iterations. Although there are many available models and algorithms to address 9

each step, the entire multi-step process is still very challenging and computationally cumbersome. 10

Therefore, formulating and solving a large-scale integrated railway planning problem has long 11

been a pursuit for academics and practitioners. 12

In compiling the train diagram for urban rail system, it is important to consider the interval 13

time of trains, the circulation and the running route of trains (2). Min et al. (2011) consider the 14

train-conflict resolution problem for which the optimal conflict-free timetable and use a column-15

generation method to compute it (3). Corman et al. (2012) consider a bi-objective problem of 16

minimizing train delays and missed connections to provide a set of feasible non-dominated 17

schedules (4). Kecman et al. (2013) propose underlying algorithms automatically identify route 18

conflicts with conflicting trains, determine accurate arrival and departure times/delays at stations 19

(5). These papers focus on arranging trains depending on conflicts and use some useful 20

algorithms to solve it. 21

Line planning is the first important strategic element in the railway operation planning 22

process. Goossens et al. (2004, 2006) consider a model formulation of the line-planning problem 23

where total operating costs are to be minimized and the model is solved with a branch-and-cut 24

approach (6) (7). Kaspi and Raviv (2013) formulate an integrated line planning and timetabling 25

model with the objective of minimizing both user inconvenience and operational costs (8). These 26

researches lay the foundation for our study of timetable and rolling stock. 27

Developing optimization models for constructing periodic transit timetables and synchronized 28

schedules is another research direction in the field. Carey and Crawford (2007) design a series of 29

heuristics for finding and resolving trains conflicts so to satisfy various operational constraints 30

and objectives (9). Zhou and Zhong (2005, 2006) formulate train scheduling models which 31

consider segment and station headway capacities as limited resources, and developed algorithms 32

to minimize both passenger waiting times and total train travel times (10) (11). Goverde (2007) 33

describe a railway timetable stability measure by using a max-plus system theory and analyzed 34

train delay propagation processes (12). Focusing on reducing passenger waiting time at stops and 35

transfers, Liebchen (2008) adapts a periodic event-scheduling approach and a well-established 36

graph model to optimize the Berlin subway timetable (13). Wong et al. (2008) concentrate on the 37

synchronization between the different lines of an URT network to minimize passengers’ transfer 38

times (14). 39

There are also a number of studies related to rolling stock scheduling. Schlake et al. (2011) 40

conduct an analysis of the effect of lean production methods on main-line railway operations (15). 41

Nielsen et al. (2012) deal with real-time disruption management of railway rolling stock (16). 42

Thorlacius et al. (2015) propose an integrated rolling stock planning model that simultaneously 43

takes into account all practical requirements for rolling stock planning (17). 44

A number of recent studies have put more attention on developing integrated optimization 45

models for timetable and passenger flow or rolling stock schedule. Niu and Zhou (2013) focus on 46

4

optimizing a passenger train timetable in a heavily congested urban rail corridor (18). Another 1

paper of them (2015) focuses on the demand-oriented passenger train-timetable optimization for a 2

rail corridor under time-dependent demand and skip-stop patterns (19). Yang et al. (2015) 3

propose a new collaborative optimization method for both train stop planning and train 4

scheduling problems on the tactic level (20). Yue et al. (2016) propose an innovative 5

methodology using a column-generation-based heuristic algorithm to simultaneously account for 6

both passenger service demands and train scheduling (21). 7

Although a few researchers have attempted to explore integrated railway planning, 8

simultaneously accounting for line planning, timetabling and rolling stock scheduling is rarely 9

seen in the literature. Line plan, timetable and rolling stock usage of integrated optimization is the 10

high-priority problem that must be solved on most busy URT lines. In this paper, we propose a 11

new methodology to simultaneously account for total passengers’ waiting times, train timetabling 12

cost and rolling stock usage for URT lines. The framework of our proposed methodology is 13

illustrated in Figure 1. In our model, the inputs are URT line data and section-specific passenger 14

flow. The decision variables correspond to the train path and rolling stock trajectory. In both the 15

model and algorithm, the first step is to optimize train frequency while guaranteeing passenger 16

flow constraints. Following this, the model simultaneously schedules train timetables and the 17

rolling stock usage while guaranteeing constraints related to passenger flow, train flow and 18

rolling stock size. The model output is a near-optimal train timetable rolling stock scheduling. 19

20

Input 1:

Urban rail transit network

(stations, sections, tracks)

Input 2:

Timetable parameters

(travel time, stopping time)

Train timetabling Rolling stock usage

Real time management

Network planningStrategical Level

Tactic Level

Operational Level

Our integrated model

Crew schedules

Line Planning

Input 3:

Time dependent passenger flow

Output :

Near optimal timetable and

rolling stock usage plan

21

FIGURE 1 Framework of integrated URT planning methodology 22

23

5

We intend to address the following specific objectives: 1

1. Analyze time-dependent passenger flow, train timetabling and rolling stock scheduling in 2

URT systems and discusses their interactions among them. The paper develops an 3

optimization methodology that enables the integration of train timetabling, line planning, 4

and rolling stock scheduling. 5

2. Include both loop and linear URT lines. In order to formulate an integrated optimization 6

model, this paper proposes a general train flow model that can be applied to all types of 7

URT lines. 8

3. Formulate a bi-level programming model for integrated train timetabling and rolling stock 9

scheduling. The upper-level model optimizes train frequency and train timetables; 10

minimize passengers’ waiting times and operation costs. The lower-level model schedules 11

rolling stock to minimize the number of infeasible train paths and proposes a SA-based 12

heuristic algorithm to solve the model. 13

4. Illustrates the use of an integrated optimization model and the SA-based algorithm to 14

improve the timetable of typical real-world URT lines in Beijing, China. 15

The remainder of this paper proceeds as follows. First, we present detailed problem 16

descriptions and assumptions for URT lines. Second, we develop an integrated train frequency, 17

train timetabling and rolling stock scheduling model. Third, a SA-based algorithm to solve the 18

model is proposed. Subsequently, we generate computational results from real-world instances of 19

the Beijing URT, and demonstrate the effectiveness of our proposed model. Finally are principal 20

conclusions and suggest possible future research directions. 21

22

PROBLEM DESCRIPTION 23

Key elements in URT 24

A train timetable defines train departure and arrival times at each station, which is an essential 25

plan for the operation of a railway system. Each train departs from a depot when service begins 26

and returns to a depot when service terminates. Stations that directly connect to depots are of 27

greater importance and are named “D-stations” in the model. For example, in Figure 2, Xizhimen 28

and Jishuitan are D-stations for Line 2, Tiantongyuanbei and Songjiazhuang are D-stations for 29

Line 5, Bagou, Chedaogou and Songjiazhuang are D-stations for Line 10. 30

JJ

J

J

J

Dongzhimen

Guomao

Fengtai

Chedaogou

Bagou

Xizhimen

Jishuitan

D

D

D

Line 5

Line 2

Line 10

J

D

Tiantongyuanbei

Huixinxijienankou

Yonghegong

Chongwenmen

Other stationD Depot J D-station

…

…

…

…

…

Songjiazhuang

D

J

31 FIGURE 2 Illustration of rail transit network in our cases 32

33

6

There are three key elements in URT lines: passenger flow, train flow and rolling stock. 1

Passenger flow is easy to understand. And we illustrate relationships between train flow and 2

rolling stock in Figure 3. Each solid line represents a train and the lines of the same color denote 3

rolling stock trajectories. Assume there are three rolling stocks in depot 1. The number of trains 4

between station 1 and station 2 are determined by train intervals (passenger flow). Due to 5

constraints on the number of available rolling stocks, some trains (denoted by solid lines) can be 6

placed into service, whereas other trains (denoted by dashed lines cannot. Dashed lines need to be 7

deleted in the final output train timetable. 8

Depot1

Station1

Station2

………

Time direction

Feasible train Infeasible trainRolling stock trajectory

9

FIGURE 3 Relationship between train flow and rolling stocks 10 11

MATHEMATICAL FORMULATION AND ALGORITHM 12

Notations 13

The general subscripts, input parameters and decision variables that are employed in our 14

mathematical formulations are listed as follows. 15

General subscripts： 16

C Set of stations: 1, ,c

D Set of D-stations: 1, ,d ， D C

E Set of time intervals: 1, ,e

F Set of trains

G Set of rolling stocks

, ', , 'i i j j Station indices， , ', , 'i i j j C

, 'k k D-station indices, , 'k k D

, ,t s r Time interval indices, , ,t s r E

f Trains index, f F

g Rolling stocks index, g G

17

7

Parameters： 1

,

t

i jo Number of passengers who leave station i for station j at time t

,

t

i jq Passengers volume between station i and station j at time t

maxL The maximum number of passengers for a train

exptL Expected number of passengers for a train

( )d f The origin station of train path, ( )d f D

h Minimum headway (time interval) between two consecutive train paths at a station

num

kG The maximum rolling stocks of depot which connected to D-station k

( )d g Depot which connected to origin station that rolling stock g starts to carry out a task

,i je Running time between station i and station j

max

rune The maximum running time for a rolling stock in a day

max

dwelle The maximum waiting time in a D-station for a rolling stock

min

stope The minimum stopping time in the depot for a rolling stock

t

iw Waiting passengers of station i at time t

2

Decision variables： 3

( )t

iu f 1, if train f departs from station i at time t

0, otherwise

( )t

iv f 1, if train f arrives at station i at time t

0, otherwise

,

, '( )t s

k kx g 1, if rolling stock g departs from D-station k at time t and arrives at D-station 'k at time s

0, otherwise

,

, ( )t s

k ky g

1, if rolling stock g waits at D-station k from time t to time s

0, otherwise

and 1s t

,

, ( )t s

k kz g 1, if rolling stock g stops at depot connected to D-station k from time t to time s

0, otherwise

4

General train flow model for urban transit lines 5

Each rail passenger trip consists of an origin station, a destination station and travel time. For 6

URT systems, the number of passengers who travel between two stations is important. Thus, we 7

need to calculate this number using the number of passengers for each origin-destination matrix. 8 ',

, 1 ', '

' [ 1, ] ' [1, ]

,i it et

i i i j

j i c i i

q o t E i C

(1) 9

8

Using equation 1, we can obtain the passenger volume between two adjacent stations. The 1

passenger volume may differ among various sections. We use a reference value to replace the real 2

value, as shown in Equation 2. Based on the section-specific passenger flow between two 3

adjacent stations, we can obtain the maximum passenger volume of some successive sections. 4

, 1, 1

, , 1 1, 2 1,=max{ , , , } , ,i ji i t et et t

i j i i i i j jq q q q t E i j C

(2) 5

In a URT network, the line topology structure can be divided into two main categories: linear 6

lines and loop lines. Different types of lines have different rolling stock trajectories. Figure 4(a) 7

shows the topologies of six common types of rail transit lines. Each line may have a single depot, 8

such as type 1, type 3 and type 5, or multiple depots, such as type 2, type 4 and type 6. In types 1 9

and 2, depots are connected to origin or terminal stations; in types 3 and 4, depots are connected 10

to intermediate stations. For mathematical modelling purposes, we modify linear lines to loop 11

lines by regarding tracks in different directions as virtually different stations. Thus, we can 12

transfer linear lines into a looped line topology with one or more depots. A rolling stock departs 13

from a depot, takes some loops along the stations and returns to the origin depot, as shown in 14

Figure 4(b). In this example, type 1 has one depot, type 2 has two depots and type 3 has three 15

depots. Multiple-depot lines trajectories must be coordinated to satisfy minimum train headway 16

constraints at D-stations. 17

Topology 3 Topology 4

Origin

Topology 2

Topology 5 Topology 6

1 c

JD

c -12

1 c

D

c-12

J

i

1

J

c

JD

c-12

D

1c

D

c-12

J

i

J

D

j

J

1 2 i

c-1c

D

c-1c

1 2 iJ JD

D

Other stationD Depot J D-station

One depot multiple depots

Liner

line

Loop

line

Looped line

2D

c-1

c-1'

J

2'

1c

2D

c-1

c-1'

JJ

2'

1c

D

2

D

c-1J

i

c-1'

J

i'

D2'

1 c

2 c-1

c-1'2'

1 c

D

J

i

Ji'

D

D

J

j

Jj'

D

Topology 1

Origin

Origin

Origin

Looped line

Looped lineLooped line

18

(a) Origin topologies and modified topologies of URT lines 19

J

1

c

D J D

i

Type 2

J

1

c

D

Type 1

Other stationD Depot Trajectory of rolling stockJ D-station

J

j

D

J

1

c

D J D

i

Type 3

20

(b) Trajectories of rolling stocks 21

FIGURE 4 Illustration for topological structure of URT lines and rolling stock arrangement 22

9

Bi-Level Mathematical Model 1

We use a bi-level programming model to describe the problem for URT. The upper level model is 2

used to optimize train timetables, including minimizing waiting times for passengers and 3

reducing operating costs for URT system. The objective of the lower level model is to schedule 4

rolling stock and minimize infeasible trains. When the solutions of the upper level model and the 5

lower level model are feasible, the problem will be solved. 6

The upper level model 7

In the upper level model, we replace the waiting times of each passenger with the number of 8

queuing passengers in each time interval. , are weights for the cost of passengers and the 9

operation cost. If decision makers want to save passengers waiting time and improve service 10

levels, the value of / should be larger; if decision makers want to reduce the train operating 11

costs and improve operating income, the value of / should be smaller. 12

_ ( _ _ )obj up Min obj w obj u (3) 13

_ t

k

k D t E

obj w w

(4) 14

_ ( )t

k

f F k D t E

obj u u f

(5) 15

1) “Passengers flow” constraints 16

The number of passengers who wait at time t is the number of waiting passengers at the prior 17

time 1t plus the number of arrival passengers at time t minus the number of departure 18

passengers at time t . During saturation, the number of departing passengers is equal to maxL , and 19

the number of departing passengers in non-saturation is less than maxL . Considering cases of 20

saturation and non-saturation, we use “greater than” in equation (6), equation (7) and equation (8). 21

0 ,t

kw t E k D (6) 22 1 1 1 max

, ' ( ) , 'k k k k

f F

w q u f L k k D

(7) 23

1 max

, ' ( ) / {1}, , 't t t t

k k k k k

f F

w w q u f L t E k k D

(8) 24

2) “Train flow” constraints 25

Two trains that travel in the same direction cannot depart from a station at the same time, and 26

a reasonable time interval between the two trains is needed (typically, the minimum headway is 27

based on the shortest braking distance between the trains given the train types and the signaling 28

systems). The interval between two consecutive train departures from the same station i must be 29

greater than or equal to the minimum departure headway h . 30

[ ,

( ) 1t

k

f F t r r h

u f r E k D

）

， (9) 31

[ ,

( ) 1t

k

f F t r r h

v f r E k D

）

， (10) 32

Train flow f departs from D-station k at time t and it arrives at D-station 'k at time , 'k kt e . If 33

' ( )k d f , it departs from D-station 'k at time , 'k kt e 34

, '

'( ) ( ) , , ' ,k kt et

k ku f v f t E k k D f F

(11) 35

( ) ( ) , / { ( )},t t

k kv f u f t E k D d f f F (12) 36

For all train paths, the number of departure times and arrival times are equal. 37 ( ) ( )t t

k k

k D t E k D t E

v f u f f F

(13) 38

10

The lower level model 1

The lower level model is used to schedule rolling stock based on train flows. The objective 2

function is to minimize the number of infeasible train paths. 3 _ _obj low Min obj x (14) 4

,

, '

'

_ [ ( ) ( )] ( ) 1, ,t t s t

k k k k

k D t E k D s E

obj x u f x g u f f F g G

(15) 5

1) “Flow conservation” constraint 6

For any space-time node ( ,k t ), the number of outflows is no more than one. 7 , , ,

, ' , ,

'

[ ( ) ( ) ( )] 1 , ,t s t s t s

k k k k k k

s E k D

x g y g z g t E k D g G

(16) 8

For any space-time node ( ,k t ), the subtracted difference between outflows and inflows should be 9

equal to 1 or -1 if the node is a source or a sink node, respectively; otherwise, it should be equal 10

to 0. 11 , , , , , ,

, ' , , ', , ,

' '

[ ( ) ( ) ( )] [ ( ) ( ) ( )]

1 1, ( )

1 , ( ) , ,

0 otherwise

t s t s t s s t s t s t

k k k k k k k k k k k k

s E k D s E k D

x g y g z g x g y g z g

t k d g

t e k d g t E k D g G

(17) 12

2) “Train flow” constraints 13

For any space-time node ( ,k t ), the number of train trajectories is less than the number of train 14

paths. 15 ,

, '

'

( ) ( ) , , ( ) ( )t s t

k k k

k D s E

x g u f t E k D d f d g

(18) 16

3) “Rolling stock” constraints 17

For any space-time node ( ,k t ), trains cannot be overtaken in a station, and only one train at a time 18

may remain in a station. 19 , ,

, ' ,

'

( ) ( ) 1 ,r s r t

k k k k

g G k D s E g G t E

x g y g r E k D

(19) 20

For any space-time node ( ,k t ), a train cannot run more than max

rune per day。 21

, max

, ' , '

'

[ ( )]t s

k k k k run

k D k D t E s E

r x g e g G

(20) 22

For any space-time node ( ,k t ), a train cannot wait in a D-station for more than max

dwelle each time. 23

max

, max

,

[ , ]

( ) , ,

dwell

t s

k k dwell

t E s r r G

y g e r E k D g G

(21) 24

For any space-time node ( ,k t ), a train cannot stop in a depot for less than min

stope for one time. 25

min

,

,

[ , )

( ) 0 ,

stop

t s

k k

t E s t t e

z g k D g G

(22) 26

SIMULATED ANNEALING ALGORITHM 27

To solve the model, we tried commercial solvers such as GAMS and CPLEX, and we also 28

designed our own algorithm. Because of the scale of the problem, we select a simulated-29

annealing-based algorithm to solve our proposed models. Its evaluation function is 30

_ _ _ _obj A obj w obj u obj x , where is the weight for the number of infeasible train paths. 31

In the early stage of iteration, _obj x has little influence on the objective function and reduces the 32

limitation of feasible solution. In the late stage of iteration, _obj x has great influence on the 33

objective function, this can ensure the rolling stock usage plan. Figure 5 shows the framework for 34

the simulated annealing algorithm. 35

11

lower level modelupper level model

Initialize

Simulated Annealing Algorithm

Integrated train timetable and rolling stocks usage plan

Train timetable Rolling stock scheduling

1

FIGURE 5 Framework of the simulated annealing algorithm 2 3

We provide detailed procedures for the simulated annealing algorithm to solve the model: 4

Step 1. Initialize parameters and variables. 5

Input date of line and passenger volume, initialize parameters of the algorithm, and 6

initialize variables: ( ), ( )t t

i iu f v f and t

iw . 7

Step 2. Obtain an initial feasible solution. 8

We use a blank train timetable as the initial feasible solution: A . 9

Step 3. Update the current feasible solution, 'A . 10

We add and reduce trains based on a small probability to obtain a new feasible solution. 11

Step 4. Compute an evaluation function for the new feasible solution, 'A . 12

Compute _ 'obj w , _ 'obj u . 13

Compute _ 'obj x . We use a neighborhood search method to solve _ 'obj x . 14

Calculate the evaluation function: _ ' _ ' _ ' _ 'obj A obj w obj u obj x . 15

Calculate the difference: _ ' _obj obj A obj A 16

Step 5. Update the current best solution: 17

Determine whether the new solution 'A is accepted based on the Metropolis-Hastings 18

algorithm. If accepted, 'A A 19

Step 6. Stop or continue. 20

Two rules can end the algorithm: 1) Iteration number reaches the maximum limit, 2) 21

Stable iteration number reaches the maximum limit and the number of infeasible train 22

paths equals “0”. If one rule is satisfied, we proceed to Step 8; otherwise, we proceed to 23

Step 7. 24

Step 7. Update temperature 25

We update the temperature using a method of “two stage-exponential decline,” which 26

can ensure efficiency and accuracy. 27

Step 8. Schedule rolling stock. 28

We use a neighborhood search method to schedule rolling stock. 29

Step 9. Present the output. 30 31

CASE STUDY: LINE 2 OF BEIJING RAIL TRANSIT NETWORK 32

We use a real-world instance (Subway Line 2) from Beijing’s rail transit network to demonstrate 33

the application of our optimization model and algorithm. The scheduling algorithms are 34

12

implemented in Microsoft Visual Studio 2010 on Windows 7 OS. All experiments are conducted 1

on a PC with an Intel Core Duo 2.93 GHz CPU and 4 GB RAM. 2

Table 1 shows the parameters of the algorithm. d is the number of depots and b represents the 3

number of time intervals of 1 min.( For example, when 1b , a time interval is 60 seconds; when 4

4b , a time interval is 15 seconds.) The time span for the train operation considered in this paper 5

is 20 hours (1,200 mins) from 5:00 am to 1:00 am the next day. The time interval of passenger 6

flow is from 5:00 am to 11:00 pm for a total time of 18 hours (1,080 mins). 7

TABLE 1 Parameters definition in our case 8

Parameters Value Meaning

Number of time intervals 1,200×b Number of time intervals

1 Weight of passenger waiting time

exp t2 b L Weight of train frequency

500,000×d Weight of infeasible trains

0M 500,000×b×d Initial temperature

rise 0.3 Heating coefficient

0.95 Cooling coefficient

0N 30,000×b×(d+3) Maximum number of iterations

stopN 0 / 200N Iteration number of terminating algorithm for

no longer improvement

1 2N → 0 /10,000N Iteration number in initial stage

2N →3 0 / 500N Iteration number in medium stage

upateN 0 / 200N Iteration number for temperature to update

riseN 0 / 4N Iteration number for temperature to increase

1

changep 0.02/b/d Update probability in initial stage

2

changep 0.005/b/d Update probability in medium stage

3

changep 0.001/b/d Update probability in later stage

max

rune 720×b Maximum running time

max

dwelle 3×b Maximum dwelling time in D-stations

min

stope 90×b Minimum stopping time in depots

9

Table 2(a) lists the data of Beijing rail transit lines 2. Line 2 of the Beijing railway transit 10

network is a loop line. There is only one depot (refer to green line in Figure 2) that connects with 11

Xizhimen station and Jishuitan station. We consider Xizhimen station as a D-station in this case. 12

The minimum departure headway between two consecutive train paths is 2 minutes. In this case, 13

13

1, 1b d . We consider the clockwise direction of this line in our (model/parameter- use one). 1

Table 2(b) shows passenger demand in workdays. 2

TABLE 2 Input data of Beijing rail transit Line 2 3

(a) Basic data of line 2 4

Line Number of

depots

Number of

rolling stocks

Number of

stations

Running time

（minutes/lap）

Line length

（km/lap）

2 1 25 18 46 23.1

(b) Passenger demand of line 2 5

Scenarios 5:00~

7:00

7:00~

9:00

9:00~

11:00

11:00~

17:00

17:00~

19:00

19:00~

22:00

22:00~

23:00

Workday（persons/min） 50 750 500 400 600 400 50

Weekend（persons/min） 20 300 450 400 500 400 50

6

Results of Line 2 and analysis. 7

Figure 6(a) shows the results of passenger flow during workdays. We counted the average 8

number and the maximum number of waiting passengers in 30 minutes. Operators can make 9

reasonable decisions based on passenger data. The number of waiting passengers reaches the 10

maximum value of 1550 between 7:30am and 8:00am. The train timetable of Beijing rail transit 11

Line 2 for workdays is shown in Figure 6(b). Due to larger passenger flows, the train paths on the 12

train diagram are more intensive, and train operating frequency reaches its maximum limit during 13

peak hours (7:00-9:00, and 17:00-19:00). Fewer trains operate during off-peak hours. The 14

solution not only avoids wasting resources and underutilization of transport capacity, but also 15

decreases the cost of train operation. The density of the train paths adequately reflects the change 16

in traffic flow. 17

18

(a) Passenger flow of Beijing rail transit Line 2 in workdays 19

20

14

1

(b) Train timetable of Beijing rail transit Line 2 in workdays 2

FIGURE 6 Outputs of Beijing rail transit Line 2 in workdays 3

4

Table 3 shows the rolling stock schedule of the depot that is connected to Xizhimen station 5

for workdays. The first column represents the rolling stocks number. The second column shows 6

total running time for each rolling stock. The third column represents the timetable of every 7

rolling stock. For example, [8 54] means that for the first rolling stock, it begins its first service at 8

the 8th

minute, and ends at the 54th

minute (service time interval is 46 minutes). Similarly, its last 9

service starts at the 962nd

minute and ends at the 1008th

minute. As mentioned before, the time 10

span of rolling stock operation is 1200min. The results reveal that the proposed model and 11

algorithm applies to rail transit ring line of multi-depots and can get an approximate optimal 12

solution. 13

TABLE 3 Rolling stock schedule of Beijing rail transit Line 2 in workdays 14

NO. Running Time

(min) Rolling stocks scheduling

1 690 [8 54][145 191][191 237][238 284][285 331][331 377][378 424][424

470][472 518][519 565][685 731][731 777][777 823][824 870][962 1008]

2 644 [17 63][158 204][204 250][250 296][297 343][344 390][496 542][542

588][590 636][636 682][779 825][919 965][965 1011][1012 1058]

3 644 [26 72][72 118][258 304][430 476][478 524][526 572][574 620][621

667][667 713][714 760][760 806][906 952][952 998][999 1045]

4 690

[35 81][82 128][128 174][174 220][221 267][535 581][581 627][627

673][674 720][721 767][767 813][814 860][861 907][909 955][1046

1092]

5 644 [43 89][323 369][371 417][418 464][466 512][512 558][558 604][606

652][653 699][805 851][851 897][899 945][945 991][993 1039]

6 690

[52 98][100 146][147 193][193 239][240 286][288 334][554 600][602

648][650 696][698 744][744 790][791 837][838 884][886 932][1079

1125]

7 690 [62 108][108 154][155 201][202 248][248 294][295 341][341 387][389

435][436 482][482 528][529 575][723 769][770 816][816 862][955 1001]

8 598 [92 138][138 184][185 231][231 277][278 324][326 372][564 610][610

656][656 702][702 748][749 795][796 842][934 980]

9 598 [117 163][164 210][210 256][256 302][303 349][350 396][397 443][443

15

489][490 536][729 775][870 916][1009 1055][1056 1102]

10 552 [121 167][168 214][215 261][568 614][736 782][783 829][829 875][877

923][923 969][971 1017][1019 1065][1067 1113]

11 644 [124 170][170 216][217 263][263 309][309 355][356 402][584 630][630

676][678 724][725 771][772 818][819 865][867 913][915 961]

12 690 [126 172][172 218][219 265][266 312][312 358][359 405][405 451][451

497][499 545][546 592][593 639][640 686][688 734][734 780][912 958]

13 690 [130 176][176 222][223 269][269 315][315 361][362 408][408 454][454

500][502 548][550 596][596 642][643 689][691 737][739 785][959 1005]

14 644 [132 178][179 225][225 271][272 318][318 364][365 411][411 457][458

504][505 551][747 793][794 840][841 887][889 935][937 983]

15 598 [134 180][181 227][227 273][599 645][646 692][694 740][742 788][789

835][836 882][883 929][931 977][979 1025][1027 1073]

16 644 [136 182][183 229][229 275][275 321][321 367][368 414][415 461][462

508][509 555][755 801][801 847][848 894][895 941][941 987]

17 644 [141 187][187 233][233 279][281 327][328 374][374 420][421 467][469

515][515 561][561 607][762 808][808 854][854 900][1002 1048]

18 506 [143 189][189 235][236 282][283 329][613 659][659 705][706 752][752

798][799 845][845 891][892 938]

19 414 [149 195][196 242][243 289][290 336][336 382][781 827][827 873][874

920][1015 1061]

20 460 [151 197][198 244][245 291][292 338][338 384][385 431][433 479][785

831][832 878][983 1029]

21 322 [153 199][200 246][787 833][834 880][880 926][927 973][975 1021]

22 690 [160 206][206 252][253 299][300 346][347 393][393 439][439 485][486

532][532 578][578 624][624 670][670 716][718 764][765 811][968 1014]

23 322 [162 208][208 254][810 856][857 903][903 949][949 995][996 1042]

24 690 [166 212][213 259][260 306][306 352][353 399][401 447][447 493][493

539][539 585][587 633][633 679][681 727][727 773][774 820][986 1032]

25 598 [333 379][381 427][427 473][475 521][523 569][571 617][617 663][663

709][710 756][757 803][803 849][989 1035][1036 1082]

1

Efficiency analysis of the algorithm 2

We also apply a similar optimization approach to Line 5 and Line 10 in Beijing. To evaluate the 3

efficiency of the proposed algorithm, we try to use two service packages (GAMS and ILOG 4

CPLEX) to solve the model and compare the results of different methods. Due to a large number 5

of variables, we figure out the upper model using GAMS because CPLEX is not suitable to solve 6

the complete model. We only list the calculation results of the upper model by GAMS, CPLEX 7

and SA ( =0 ). We employ the calculation results of GAMS as a benchmark and make 8

comparative analysis. Table 4 lists the comparison of the different solving methods. Three cases 9

16

are employed : Line 2 on workdays, Line 5 on workdays and Line 10 on workdays. For a large-1

sized problem, our proposed algorithm outperforms the commercial software. 2

TABLE 4 Comparison of different solving methods 3

Case Number

of Depot Solver _obj A

Comparison of

object value

Computing

time(s)

Comparison of

computational time

Line 2,

workday 1

Upper level

model/GAMS 1,267,350 1 12 1

Upper level

model/CPLEX 1,267,350 1 124 10.33

Upper level

model/SA 1,286,750 1.01 12 1

Bi-level model

/SA+ neighbor-

hood searching

1,306,215 1.03 15 1.25

Line 5,

workday 2

Upper level

model/GAMS 3,967,667 1 4,239 1

Upper level

model/CPLEX unavailable unavailable unavailable unavailable

Upper level

model/SA 3,788,936 0.95 95 0.02

Bi-level model

/SA+ neighbor-

hood searching

3,808,960 0.96 126 0.03

Line 10,

workday 3

Upper level

model/GAMS 15,533,020 1 8,769 1

Upper level

model/CPLEX unavailable unavailable unavailable unavailable

Upper level

model/SA 13,119,341 0.84 814 0.09

Bi-level model

/SA+ neighbor-

hood searching

13,358,397 0.86 1,004 0.11

4

The following are several observations based on the results presented in Table 4: 5

1) For the case of Line 2 on workdays, the results of the upper level model using GAMS, 6

CPLEX and SA are similar, but the computing time by CPLEX is ten times faster than the 7

computing time by GAMS, and SA is also much faster than GAMS. The computing time of the 8

bi-level model by SA and neighborhood searching is slightly longer than the computing time of 9

the bi-level model by GAMS and the results are worse than GAMS by 2% considering the lower 10

level of the rolling stock schedule. 11

2) For the large-scale problems, such as cases 2 and 3, CPLEX cannot obtain a solution 12

within a reasonable computing time. In the case of Line 5 with a time interval of 30s and two 13

depots, GAMS can solve the model; however, the computing time exceeds1 hour. SA and 14

neighborhood searching can obtain a better solution within 3 minutes. 15

3) In the case of Line 10 with a time interval of 15s and 3 depots, GAMS can barely obtain a 16

solution within 2 hours, whereas SA and neighborhood searching can obtain the solution within 17

16 minutes and 44 seconds. The objective function value is greater than 14%. 18

17

From these observations, it appears that SA, as a meta-heuristics, is fast but cannot 1

guarantee global optimality. For a practical, large-scale problem, SA may be a promising 2

approach to yield adequate enough solutions (may not be perfect) given a reasonable time span. 3

CONCLUSIONS 4

This paper proposes a new mathematical model for the optimization of train service plans, train 5

timetables and rolling stock schedules for URT. This model can simultaneously reduce passenger 6

waiting time and train operation costs while also improving the utilization of train sets. 7

First, we introduce three key elements in URT lines and propose a general train flow model, 8

which can be employed across all topologies of train lines. Then we apply a bi-level 9

programming model to formulate the scheduling problem for URT. The upper level model 10

optimizes train timetables by minimizing waiting times for passengers and operation costs for 11

URT systems. The lower level model is used to schedule rolling stock by minimizing the number 12

of infeasible train paths. We use a simulated-annealing-based heuristic algorithm to solve the 13

large-scale model. In a case study of the Beijing rail transit network, we test our optimization 14

mathematical model and algorithm. The calculation results indicate that the proposed model and 15

algorithm can obtain reasonable schedule planning. In particular, our new, integrated algorithm 16

can rapidly obtain a near-optimal solution and best utilize rolling stocks, which renders it useful 17

for complex real-world applications. A similar optimization methodology can also be adapted for 18

high-speed and freight rail lines. 19

Our future research will focus on three major areas. First, the neighborhood searching method 20

for vehicle scheduling is not a global optimization method; we aim to improve the algorithm 21

performance. Second, the model can be extended to account for train skip-stop patterns and 22

long/short trains. Last, we will refine and validate the proposed model with observed data from 23

URT systems in Beijing. 24

25

18

REFERENCES 1

1. Schöbel, A., Line planning in public transportation: models and methods. OR 2

Spectrum Vol. 34, 2012, pp. 491–510. 3

2. Xu, R., Jiang, Z., Zhu, X., 2005. Key problems in computer compilation of train 4

diagram in UMT. Urban Mass Transit. 5

3. Min, Y. H., Park, M. J., Hong, S. P., Hong, S. H., An appraisal of a column-6

generation-based algorithm for centralized train-conflict resolution on a metropolitan 7

railway network. Transportation Research Part B, Vol. 45, No. 2, 2011, pp. 409-429 8

4. Corman, F., D’Ariano, A., Pacciarelli, D., Pranzo, M., Bi-objective conflict detection 9

and resolution in railway traffic management. Transportation Research Part C, Vol. 20, 10

No. 1, 2012, pp. 79-94. 11

5. Kecman, P., Goverde, R. M. P., (2013). Process mining approach for recovery of 12

realized train paths and route conflict identification. Transportation Research Board 13

92nd Annual Meeting. 14

6. Goossens, J. W., Van Hoesel, S., & Kroon, L. A branch-and-cut approach for 15

soling railway line-planning problems. Transportation Science, Vol. 38, No. 3, 16

2004, pp. 379-393. 17

7. Goossens, J. W., Hoesel, S. V., & Kroon, L. On solving multi-type railway line 18

planning problems. European Journal of Operational Research, Vol. 168, No. 2, 19

2006, pp. 403-424. 20

8. Kaspi, M., & Raviv, T. Service-oriented line planning and timetabling for passenger 21

trains. Transportation Science, Vol. 47, No. 3, 2013, pp. 295-311. 22

9. Carey, M., Crawford, I., Scheduling trains on a network of busy complex stations. 23

Transportation Research Part B, Vol. 41, No. 2, 2007, pp. 159–178. 24

10. Zhou, X., Zhong, M., 2005. Train scheduling for high-speed passenger railroad 25

planning applications. European Journal of Operational Research, Vol. 167, No. 3, 26

2005, pp. 752–771. 27

11. Zhou, X., Zhong, M., Single-track train timetabling with guaranteed optimality: 28

branch and bound algorithms with enhanced lower bounds. Transportation Research 29

Part B, Vol. 41, No. 3, 2006, pp. 320–341. 30

12. Goverde, R.M.P., Railway timetable stability analysis using max-plus system theory. 31

Transportation Research Part B, Vol. 41, No. 2, 2007, pp. 179–201. 32

13. Liebchen, C., The first optimized railway timetable in practice. Transportation 33

Science, Vol. 42, No. 4, 2008, pp. 420–435. 34

14. Wong, C.W., Yuen, T.W.Y., Fung, K.W., Leung, M.Y., Optimizing timetable 35

synchronization for rail mass transit. Transportation Science, Vol. 42, No. 1, 2008, pp. 36

57–69. 37

15. Schlake, B. W., Barkan, C. P. L., & Edwards, J. R. (2011). Train delay and economic 38

impact of in-service failures of railroad rolling stock. In Transportation Research 39

Record: Journal of the Transportation Research Board, No. 2261, Transportation 40

Research Board of the National Academies, Washington, D.C., 2011, pp. 124–133. 41

16. Nielsen, L. K., Kroon, L., & Maróti, G. A rolling horizon approach for disruption 42

management of railway rolling stock. European Journal of Operational Research, Vol. 43

220, No. 2, 2012, pp. 496-509. 44

19

17. Thorlacius, P., Larsen, J., & Laumanns, M., An integrated rolling stock planning 1

model for the Copenhagen suburban passenger railway. Journal of Rail Transport 2

Planning & Management, Vol. 5, No. 4, 2015, pp. 240-262. 3

18. Niu, H., Zhou, X., Optimizing urban rail timetable under time-dependent demand and 4

oversaturated conditions. Transportation Research Part C, Vol. 36, No. 11, 2013, pp. 5

212–230. 6

19. Niu, H., Zhou, X., Gao, R. Train scheduling for minimizing passenger waiting time 7

with time-dependent demand and skip-stop patterns: nonlinear integer programming 8

models with linear constraints. Transportation Research Part B, Vol. 76, 2015, pp. 9

117-135. 10

20. Yang, L., Qi, J., Li, S., Gao, Y. 2015. Collaborative optimization for train 11

scheduling and train stop planning on high-speed railways. Omega. 12

21. Yue, Y., Wang, S., Zhou, L., Tong, L., & Saat, M. R. Optimizing train stopping 13

patterns and schedules for high-speed passenger rail corridors. Transportation 14

Research Part C, Vol. 63, No. 2, 2016, pp. 126-146. 15