PREDICTIVE MODEL OF RAIL CONSUMPTION FOR

BEIJING SUBWAY LINE 2

Lin-rong Pang, Yi-hong Ru

School of Economics

＆

Management, Beijing Jiaotong University, Beijing 100044, China

Zi-kui Lin

School of Economics

＆

Management, Beijing Jiaotong University, Beijing 100044, China

Keywords: Beijing Subway Line 2, Steel Rail Consumption, Fuzzy Time Series Prediction Model.

Abstract: This paper focus on the rare researched project, the consumption of subway steel rail based on quantitative

analysis; make a Fuzzy Time Series Prediction Model for the aggregate consumption of the steel rail

expended in subway. Set Beijing Subway Line.2 as a case object, make an analysis and give a prediction,

conclude the rule of steel rail consumption, the result provides a scientific basis for management of Beijing

subway steel rail maintenance.

1 INTRODUCTION

As the development of science and urbanization, the

effective rail transportation plays more and more

important roles in modem cities nowadays. The

transportation development history in western

countries shows us that the only way to better the

urban traffic fundamentally is to adopt urban track

transportation or so called mass transit (made up of

subway and light rail). Great traffic volume, fast,

safe, punctual, eco-friendly, energy saving, subway

release great pressure of urban traffic, accelerating

the city development as an essential part of mass

transit.

From domestic and international practice, as the

improvement of traffic volume and vehicles there

exist many problems that is exigent to be solved in

urban subway Operation. One of the most important

problem is how to manage the consumption of

materials in subway scientifically.Subway materials

consumption increase year by year, requires a large

amount of money on subway line maintenance. In

such a realistic condition, metro lines material costs

budget is becoming more and more important

2 REVIEWS

As the main bearing parts of urban rail transit, rail

bares the reciprocal action of train wheels directly.

The statues of the steel rail affect the whole urban

transit, and the relations between the two had been

studied for a long time. But most of the researches

are based on main line railway. Even though there

are many similarities between subway and main line

railway, there do exist some differences that can’t be

ignored. The main line railway is fast, low traffic

volume, heavy loaded while the subway run high

traffic volume, light loaded and mostly concentrated

on safety.

The last decade saw the start of studies on urban

rail transit; the studies endure a period that is

focused on main line railway. Starting with

quantitative analysis on steel rail consumption(Cao

Minghua, Chen Yonggui, 2008; Matsumoto K, Suda

Y, et al., 2006), several scholars are looking into the

cause of the consumption(YU Chunhua, 2007).

And some scholars turn to the area of the

management steel rail maintenance. It aims to lower

the consumption and cost, while the urban transit

runs smoothly. This project has been divided into

two levels, on one hand is to replace the badly

fatigue damaged steel rail, maintain the rail with oil

on time (Liu Canlong, 2008; Matsumoto K, Suda Y

489

Pang L., Ru Y. and Lin Z..

PREDICTIVE MODEL OF RAIL CONSUMPTION FOR BEIJING SUBWAY LINE 2.

DOI: 10.5220/0003550904890492

In Proceedings of the 13th International Conference on Enterprise Information Systems (ICEIS-2011), pages 489-492

ISBN: 978-989-8425-56-0

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

et al., 2008

[

), one the other hand is to look into the

management skills (ZhouYu, XuYude, 2008;

Bozyslaw et al., 2003) ,technology and method (YU

Chunhua, 2007) adopted in rail maintenance.

We can draw a conclusion that, the current

domestic and overseas scholars are mainly focused

on two aspects: the causes of the rail consumption

and the maintenance of the rails and related

management work. Few have been tried to discuss it

based on quantitative analysis, which is of great

value from the materials management. Realized this

we will make a Fuzzy Time Series Prediction Model

for the aggregate consumption of the steel rail

expended in subway, Set Beijing Subway Line 2 as a

case object, make an analysis and give a prediction,

conclude the rule of steel rail consumption, and test

it in the end.

3 THE FORECAST OF RAIL

CONSUMPTION ON BEIJING

SUBWAY LINE 2

3.1 Fuzzy Time Series Prediction

Model

3.1.1 Source Data into the Fuzzy

Source data of the Steel rail consumption is a real

number set, x

1

, x

2

,…,x

T

. Use the set defined in A

fuzzy set, SV

1

, SV

2

,…,SV

T

, to restore the original

data’s uncertainty.

Preset: U

t

=max(x

t-1

,x

t

,x

t+1

), V

t

=min(x

t-1

,x

t

,x

t+1

),

t=2,3, … T-1, V

1

=min(x

1

,x

2

) ， U

1

=max(x

1

,x

2

),

V

T

=min(x

T-1

,x

T

), U

T

=max(x

T-1

,x

T

).

Define SV

t

(x) as follow:

If x belongs to [V

t

,U

t

], then SV

t

(x)=1-∣x-a

t

∣/c

t

;

otherwise, SV

t

(x)=0, And c

t

=(U

t

-V

t

)/2, a

t

=(U

t

+V

t

)/2,

t=1,2,…,T.

3.1.2 The Order of the Fuzzy Time Series

According to the figure of the Steel rail consumption

scatterplot, it was observed that the curve of these

consumption numbers approximate to a linear

diagram or a conic diagram. So we could determine

that the function of the fuzzy time series is first

order or second order.

3.1.3 Fuzzy Coefficient

Define: p

i

is Triangular Fuzzy Number and p

i

` is p

i

’s

estimated value.

Then we determine (ß

i

,S

i

) With the fuzziness of

the tendency equation as small as possible. The

ambiguity S of the tendency equation is S=∑

i

k

w

i

S

i

,

including w

i

is S

i

’s weight. The ambiguity S can be

determined by linear regression method.

We can suppose that the linear regression

equation of the original sequence is like this:

SV

t

`=a

0

`+a

1

`*t+a

2

`*t

2

+…+a

k

`*t

k

(1)

And a

i

` is real number, for i=0,1,2,…,k.

Define: w

i

=∣a

i

`∣/∑∣a

i

`∣. The closeness between

SV

t

and SV

t

` can be expressed by h

t

, h

t

=(SV

t

, SV

t

`)

，t=1,2,3,…,T. suppose h

t

is not less than a given

number h

0

. So the question to make the ambiguity S

of the tendency equation minimum can be changed

into a linear programming problem as follows:

Min S=∑w

i

S

i

(2)

s.t. h

t

≥h

0

, t=0, 1, 2,…T

For SV

t

`=p

0

`+p

1

`*t+p

2

`*t

2

+…+p

k

`*t

k

, SV

t

` is the

triangular fuzzy number by (∑ß

i

t

i

,∑S

i

t

i

). So h

t

can

be expressed by:

h

t

=(SV

t

，SV

t

`)=1-∣a

t

-∑ß

i

t

i

∣/(c

t

+∑S

i

t

i

)

(3)

And if h

t

is not less than h

0

, the only way it can

happen is as follow:

∑ß

i

t

i

-(1-h

0

)∑S

i

t

i

≤ a

t

+c

t

(1-h

0

), t=1,2,3,…T

∑ß

i

t

i

-(1-h

0

)∑S

i

t

i

≥ a

t

-c

t

(1-h

0

), t=1,2,3,…T

S

i

≥0,i=0,1,2,…k

After these operations, we can get p

i

’s estimated

value p

i

`, for i=0, 1, 2, … , k, and the tendency

equation:

SV`(t)=p

0

`+p

1

`*t+p

2

`*t

2

+…+p

k

`*t

k

(4)

If t is bigger than T, the value of SV`(t) is a

Triangular Fuzzy Number too. When time changes,

the Equation graphic of SV`(t) is not only one curve

but also curve clusters with border curves f

1

(t) and

f

2

(t) and a central curve f

0

(t), for

f

1

(t)=ß(t)+S(t) (5)

f

2

(t)=ß(t)-S(t)

and

(6)

f

0

(t)=ß(t) (7)

3.1.4 Error Regulation

Using the tendency equation from above, we can get

a Series value x

t

` which can compare with the Actual

value x

t

and calculate the mean error ð by the

ICEIS 2011 - 13th International Conference on Enterprise Information Systems

490

function: ð= ∑ (x

i

-x

i

`)^2/n. based on the central

curve f

0

(t), it can be regulated to two forecast

curves:

g

1

(t)=ß(t)+ð (8)

g

2

(t)=ß(t)-ð (9)

3.2 Forecasting Results

Through the investigation into Beijing Subway

Group, we got a Series data of Rail Consumption

shown in the table 1.

Table 1: Rail Consumption (Unit: ton).

year Rail Consumption

2003 0.00

2004 23.44

2005 3.86

2006 7.73

2007 0.00

2008 78.30

2009 62.26

From consumption data curve, it was observed

that it presented upswing change tendency but not

around centre line fluctuating. For this kind of data,

we can use Fuzzy Time Series Prediction Model

predict its change.

In This paper, MATLAB has been used to

program for the proposed model. Finally got the

predicting curves shown as below:

Figure 1: 2003-2009’s Rail consumption fitting curve.

Functions of fitting curves are as follow:

The top curve: f

1

(t)= -4.2087+13.8521*t.

The central curve: f

0

(t)= -4.2087+6.2847*t.

The under curve: f

2

(t)= -4.2087+1.2827*t.

The conic function: f

3

(t)=21.6786-15.1486*t+3.2*t

2

.

Shown in the Fig.1, The changing trend of the

conic is closer to the real one. So use the conic

function forecast rail consumption. as a result we

can get a Series value and the mean error ð. through

calculation, the value of ð is 15.2325. Based on the

central curve f

3

(t), it can be regulated to two forecast

curves:

The top curve: g

1

(t)=36.9111-15.1486*t+3.2*t

2

;

The central curve: g

2

(t)=f

3

(t)=21.6786-

15.1486*t+3.2*t

2

;

The under curve: g

3

(t)=6.4461-15.1486*t+3.2*t

2

;

And the value of t takes 2003 as the starting

point, namely t was equal to 1 in 2003.

According to the final prediction curve function,

we can forecast rail consumption in the next five

years as the follow table.2.

Table 2: forecasting rail consumption in (unit: ton).

year

the top

curve

the central

curve

the under

curve

2010 120.5223 105.2898 90.0573

2011 159.7737 144.5412 129.3087

2012 205.4251 190.1926 174.9601

2013 257.4765 242.244 227.0115

2014 315.9279 300.6954 285.4629

According to Beijing Subway Group's operation

Management, it needs to make a budget plan of

materials consumption for the next year at the end of

each year. Therefore, the model in this article is

important to the manager.

3.3 Optimize the Model

As all the method we adopt to predict can no escape

from relative error, and the sample data is so limited,

it is quite important to apply operational change

management. Having proved the method is effective,

we can dig into a new round of consumption rule by

enlarge the sample data volume. To make the

prediction more reliable, the Beijing subway group

may take measures to collect the specific data along

the whole rail line, with which the model will work

better.

4 CONCLUSIONS

This paper gives three prediction line based on

Fuzzy Time Series Prediction Model, Get an annual

interval consumption of rails. On one hand, this

interval provides the material manager a way to

check whether the rail purchase plan is reasonable.

On the other hand, The Purchasing Department can

1 2 3 4 5 6 7

-20

0

20

40

60

80

100

time

年消耗 量总

PREDICTIVE MODEL OF RAIL CONSUMPTION FOR BEIJING SUBWAY LINE 2

491

just make ends meet as been more acknowledged of

the sum amount rail that is needed, which, cut the

cost as a result.

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Cao Minghua Chen Yonggui, 2008. Causes of Rail

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Matsumoto K, Suda Y, et al., 2006. The optimum design of

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rail in a railway system for improved curving

negotiation, Vehicle System Dynamics. Taylor &

Francis Inc.

Yu Chunhua, 2007.An Assumption of Standardization of

Track Facilities for Urban Rail Transit. Journal of

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Liu Canlong, 2008. A Study on Metro Wheel-Rail s Wear.

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Matsumoto K, Suda Y et al., 2008. Wheel/rail friction

control with feedback system detecting yaw moment

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Francis Inc.

ZhouYu，XuYude, 2008. 11le Life Cycle Maintenance

Strategy of Rail in Urban Mass Transit. Urban Mass

Transit.

Bozyslaw Bogdaniuk, Andrzej Massel, Rafal Radomski,

2003. Increasing rail life by forecasting fatigue

failure, NDT&E Internationa. Elsevier SCI LTD.

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