Masashi MIWA
1
Modeling the Optimal Decision-Making
for Multiple Tie Tamper Operations
Railway Technical Research Institute
2-8-38 Hikari-cho, Kokubunji-shi, Tokyo
185-8540, Japan
Tel: +81-42-573-7278
Fax: +81-42-573-7296
e-mail: [email protected]
+
ex-National Graduate Research Institute for Policy Studies
2-2 Wakamatsu-cho, Shinjuku-ku, Tokyo 162-8677, Japan
++
ex-
2-2-2 Hiromachi, Shinagawa-ku, Tokyo
140-0005, Japan
Tel: +81-3-5709-3665
Fax: +81-3-5709-3666
e-mail: [email protected]
+++
National Graduate Research Institute for Policy Studies
2-2 Wakamatsu-cho, Shinjuku-ku, Tokyo 162-8677, Japan
Tel & Fax: +81-3-3341-0454
e-mail: [email protected]
Summary
Through appropriate maintenance activities, railway track irregularities must be
kept at a satisfactory level with the optimal cost. In practice, however, most of
maintenance strategies focus only on sustaining railway track irregularities at a
necessary level, as they are difficult to treat quantitatively.
In order to determine the maintenance strategies effectively, we develop a
mathematical programming model for the optimal decision-making for railway
maintenance strategies based upon both maintenance costs and level of surface
irregularity for riding quality and safety. Using the model, we can obtain the optimal
annual tamping schedule with a multiple tie tamper (MTT) shared by several track
depots, which gives an optimal monthly MTT operation and optimal tamping schedule
with the MTT. The schedule indicates the division for which tamping must be executed
in a term (ten days). Then, we apply the model to the actual data in the field and
confirm that our model is effective and useful from the results.
Keywords
Surface irregularity, Degradation model, Restoration model, Optimal maintenance,
Mathematical programming
Masashi MIWA
1 Introduction
2
In this paper, we describe the modeling of decision making for optimal railway
track maintenance strategies. To determine effective strategies, we develop a decision
supporting system (DSS). In the process of developing the system, we analyze the
actual data using the model shown in Figure 1. We focus on the model, decisionmaking for maintenance strategies.
First, we show a transition model for predicting changes in the surface
irregularity, which is applied to an optimal maintenance scheduling model. The model
is composed of a degradation model and a restoration model.
Next, we build a mathematical programming model for making an optimal
maintenance strategies for the annual tamping schedule. The scheduling model is for
tamping with a Multiple Tie Tamper (MTT) shared by several track maintenance
depots. This model allows us to decide which month the MTT should be allocated to a
particular depot, and which lot should be provided with maintenance work with the
MTT in consideration of various constraints.
Analysis for developing DSS
2
Transition model of surface irregularity
In this section, we describe a model of transition process of surface irregularity,
which is applied to the optimal maintenance scheduling model. From the results for
testing the goodness of fit to surface irregularity data, we can express the condition of
surface irregularity well with the parameter of the logisitic probability distribution.
The probability density function of the logistic distribution is expressed by the
following formula.
f ( x)
exp{(x
[1 exp{(x
: Mean,
)/ }
2
)/ }]
(1)
/30.5 : Standard deviation
In the logistic distribution, the parameter corresponds to the standard deviation.
Thus, we focus on the parameter
In Figure 2, the transition process of the surface
irregularity is composed of degradation and recovery processes. Therefore, we apply
the following models to predict changes in the surface irregularity for the optimal
scheduling model.
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Image of transition process
2.1 Degradation model
A degradation model is built in each lot (100m) by applying the exponential
smoothing method2). We can predict the growth of the parameter with the model by
using historical data in each lot. We can express the model as follows.
( t)
Tt
s {
s
(t)
(t)
(1 s)
(t 1) 1 s s Tt
( t 1)} (1
( t L)
( t) L Tt
t, L : Term (1term = 90days)
: Growth rate of
(t) : Actual
(2)
s)Tt 1
(t) : Estimated
at t
at t s : Smoothing constant Tt : Tendency of increase
Using the actual data, we confirm that the estimated
actual data.
180
agrees well with the
2.2 Restoration model
By comparing the pre-tamping parameter mbef and the quantity of improvement
( = mbef - post-tamping parameter maft), and using the actual data, we can see a
imp
tendency that the
increases in proportion to the mbef. Therefore, we build a
imp
restoration model with a regression line by the following formula.
imp=a mbef+ b
3
a, b : constant
(3)
Modeling the optimal maintenance schedule
In this section, we describe a modeling of decision making for annual tamping
schedule with an MTT shared by several track maintenance depots.
3.1 Input data and assumption
The input data of the model is layout of depots & lots, historical data of surface
irregularity, total length of maintenance with an MTT in a year and so on. In order to
make the schedule of tamping with the MTT, we use a historical data set of surface
irregularity measured for two years. To control the irregularity, we divide the
irregularity data along the railway track into the data set of “LOT” and “UNIT”. As
shown in Figure 3, the difference between “LOT” and “UNIT” is as follows.
LOT : 100m in length. We predict the transition process with the transition model
in each lot.
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UNIT : N lot sets and tamping unit. If a UNIT is scheduled to receive tamping
with the MTT, we execute tamping all lots included in the UNIT at the
scheduled time.
“Lot” and “UNIT”
To obtain the optimal schedule by, while taking into account the constraints that
can be provided, we develop an integer programming model to solve the problem.
Then, the model provides us with an optimal monthly MTT operation and optimal
tamping schedule with the MTT. The schedule indicates the division for which tamping
must be executed in a term (ten days).
3.2 Structure of the mathematical programming model
1. Sets
i) Months M={1, 2, 3, ..., 12}
ii) Terms
K={1, 2, 3}
iii) Depots D={1, 2, 3, ... , Dmax}
iv) Units
Ui={1, 2, 3, ... , Uimax} [Depot No. i]
v) Lots
Li={1, 2, 3, ... , Limax} [Depot No. i]
2. Decision variables
i) zmd
0-1 type integer
m M, d D
=1 MTT allocated to the depot d on the month m.
=0 MTT not allocated to the depot d on the month m.
ii) wmkj
0-1 type integer
m M , k K (ten days), j Ui
=1 Maintenance executed to the unit j on the month m, term k.
=0 Maintenance not executed to the unit j on the month m, term k.
3. Constraints
i) Selection of a depot
One MTT is available. The MTT is allocated to each depot every month.
(4)
zmd 1
m M
d
ii) Upper limit for the number of UNITs for which tamping is to be executed in each
period
As MTT and manpower are limited, we can not execute tamping everyday.
Therefore, the limit for the number of UNITs to be tamped is determined for each
term. From the number for each term, the total length of tamping with the MTT
in a year is defined as.
(5)
wmkj Amk
m M, k K
j
iii) The period in which tamping can be executed for each UNIT
For each UNIT, the period in which tamping can be executed is determined
as a rule, by taking seasonal restrictions, business policy and repairing the MTT
into consideration.
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w
j J (m,k ) R mkj
1
j
0
J1={UNIT No. given with this constraint}
R={The period in which tamping can not be executed for the UNIT No.j }
(6)
iv) Upper bound for the number of times of tamping for each UNIT
(7)
w mkj 1
j Ui
mk
v) Tamping method
For all UNITs, tamping can be scheduled only when the MTT is allocated
to the depot including the UNIT.
(8)
wmkj z md 0
m M , d D , j Ui
k
vi) The area where the MTT can move and work in each term
The MTT should not work for two UNITs being far away from each other
in a term. We set an allowed area for MTT operation considering the layout of
maintenance depots in each term.
B wmkj
wmkj
j
j J
2
B=Maximum value of
j Jj
2
B
w mkj , m
M, k
K
j
J2 ={UNIT No. for which tamping can not be executed in the same term
with the UNIT No.j }
(9)
vii) MTT allocation to the appointed depot in the appointed month
When a large track renewal is scheduled, MTTs are often used for the
renewal. In the month when the renewal is scheduled, therefore, the MTT should
be allocated to the appointed depot.
(10)
zmd 1
m, d {Appointed depot d and month m}
viii) Upper bound for the surface irregularity
We determine the upper bound of the surface irregularity as a rule, taking
running safety and riding quality into consideration. Then the surface irregularity
of any lot must not exceed the bound throughout the year.
mcj 1
x 1k
kcj 1
w
y 1 mcj yj
w xkj
1 j J3 , k
K
J3={UNIT No. of which the surface irregularity exceed the bound} (11)
ix) The UNIT for which tamping must be executed next year
The UNIT including a number of lots for which tamping should be
executed in terms of long-term economical benefit must be tamped next year.
mk
w mkj
1
j J4
J4={UNIT No. for which tamping should be executed next year}
(12)
3.3 Objective function
The main purpose of tamping for surface irregularities is to secure the running
safety and good riding comfort of vehicles. When we evaluate the mean value of the
surface irregularity in a year under the constraints, therefore, it is desirable to take into
consideration the influence of a surface irregularity on the vibration of vehicles and
conditions of train operation. From such a viewpoint, the objective function of this
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scheduling model is to minimize the mean value of the weighted surface irregularity in a
year. The objective function is expressed by the following formula.
m 1 3
k
j
Sr w
)
xvj
myj
m
m
j x 1v 1
j ky 1
S : Mean value of standard deviation of weighted surface irregularity
S
C1
C2 (3
C , C : Constant
1 2
j
Sr w
min.
(13)
j
S r : The quantity of improvement by MTT (Unit j)
This function is equal to the following formula.
m 1 3
3
4
j m x 1v 1
j
Sr w
xvj
k
jmky 1
j
Sr w
myj
max.
(14)
Application to an actual railway network
In this section, we apply the scheduling model to the actual data in the field.
4.1 Illustration of the actual railway network
In this application, an MTT is shared by two maintenance depots and we make an
optimal annual MTT tamping schedule by considering various constraints given by the
features of railway division (RD) I and RD II. The layout of the depots is as shown in
Figure 4.
Layout of the depots
4.2 Computational results
As to the monthly schedule for the MTT, a comparison of the solution and the
actual data is as shown in Table 1. The solution agrees well with the actual data. And
the condition of surface irregularity that may be brought about by following the solution
and the actual data is as shown in Table 2. At both RDs, following the solution, the
standard deviation of surface irregularity is smaller (throughout the year : -4%, end of
the year : -8%) than that in the case where the actual data is followed.
In addition to the standard deviation of the surface irregularity, the number of lots
of which surface irregularity exceeds the criteria (in terms of riding quality) is smaller
(throughout the year : -12%, end of the year : -56%) than that in the case where the
solution is followed. Thus, it seems that the solution can realize a better condition than
actual data.
As seen from the above results, we can execute efficient maintenance activities
with the model.
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Monthly schedule for MTT operation
Results of solving the model
5
Conclusions
We have described the modeling of optimal decision-making for the annual
tamping schedule with an MTT and confirmed the usefulness of the model. We
obtained the following results.
We built a mathematical programming model to make an optimal annual tamping
schedule of an MTT being shared by several track maintenance depots.
The model provides us with an optimal monthly MTT operation and optimal
tamping schedule while taking into account various constraints.
We confirmed that the model solution keeps the standard deviation of surface
irregularity smaller than that in the present state by using the actual maintenance data.
BIBLIOGRAPHY
M. Miwa, T. Ishikawa, T. Oyama: “Basic Study to Construct the Model for
Deterioration and Restoration of Track States, (in Japanese) ” J-Rail`98, November 1998.
Brown, R. G.: “Statistical Forecasting for Inventory Control,” 1959.
M. Miwa, T. Ishikawa, T. Oyama: “Modeling the Transition Process of Track
Irregularity and its Application to the Multiple Tie Tamper Operation (in Japanese),” JRail`99, Tokyo, December 1999.
M. Miwa, T. Ishikawa, T. Oyama: “Modeling the Transition Process of Railway
Track Irregularity and its Application to the Decision Making for Maintenance Strategy,”
WCRR`99, Tokyo, November 1999.
M. Miwa, T. Ishikawa, T. Oyama: “Modeling the Transition Process of Railway
Track Irregularity and its Application to the Optimal Decision-Making for Multiple Tie
Tamper Operations,” Railway Engineering 2000, London, July 2000.