Int. J. Production Economics 128 (2010) 404–412
Contents lists available at ScienceDirect
Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
Rolling stock maintenance strategy selection, spares parts’ estimation, and
replacements’ interval calculation
Yung-Hsiang Cheng a,n, Hou-Lei Tsao b
a
b
Department of Transportation and Communication Management Science, National Cheng Kung University, No.1, University Road, Tainan City 701, Taiwan, ROC
Department of Logistics Management, National Kaohsiung First University of Science and Technology, 2 Juoyue Road, Nantz District, Kaohsiung 811, Taiwan, ROC
a r t i c l e in f o
a b s t r a c t
Article history:
Received 22 December 2007
Accepted 5 July 2010
Available online 27 July 2010
The purpose of this paper is to permit an approach for: (1) selecting a maintenance strategy for rolling
stock and (2) obtaining possible spare parts’ quantities and replacement intervals for the components of
rolling stock. The methodology adopts an analytic network process (ANP) technique for the strategy
evaluation, because ANP considers the important interactions among evaluation factors. The ANP’s
result decides upon a proper rolling stock maintenance strategy formed by various combinations of
preventive maintenance (PM) and corrective maintenance (CM). The ratio PM/CM, obtained by ANP, can
help to predict spare parts’ quantities of the components of rolling stock. The empirical result also
indicates that preventive maintenance should be much more valued than corrective maintenance. In
addition, safety is considered the most crucial factor for the selection of a rolling stock maintenance
strategy. The fruit of this study serves as a reference for railway system operators when evaluating their
rolling stock maintenance strategy and also when estimating their spare parts’ quantities and
replacement intervals for specific components of the rolling stock.
& 2010 Elsevier B.V. All rights reserved.
Keywords:
Rolling stock maintenance
Maintenance strategy
Spare parts
Replacement interval
1. Introduction
The maintenance of rolling stock can be categorized into two
types: corrective maintenance and preventive maintenance. The
time intervals at which preventive maintenance is scheduled are
dependent on both the life distribution of the components and the
total cost involved in the maintenance activity, but corrective
maintenance cannot be avoided when component failure component occurs. The total cost of maintenance depends on the
percentages in performing preventive maintenance and corrective
maintenance. In general, more frequent preventative maintenance drives up the total maintenance costs for rolling stock. On
the other hand, proper preventative maintenance can potentially
reduce the risks associated with rolling stock mechanical failure.
Thus, railway system operators are constantly left weighing the
safety risks against the maintenance costs. In contrast to
maintenance strategy selection in the manufacturing industry,
the maintenance of rolling stock also impacts passenger comfort.
Because preventative maintenance and corrective maintenance
affect these three factors (safety, comfort, and cost), railway
system operators must establish a maintenance strategy that
strives for an optimum balance. Given this, a method that defines
a proper rolling stock maintenance strategy is invaluable to
n
Corresponding author. Tel.: +886 6 2757575x53227; fax: + 886 6 2753882.
E-mail address: [email protected] (Y.-H. Cheng).
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijpe.2010.07.038
system operators (railway companies), system safety supervisors
(governments), and system users (passengers).
The selection of a suitable rolling stock maintenance strategy is
complicated. The system operator needs to simultaneously
consider non-metric evaluation factors (safety, quality) and
metric evaluation factors (maintenance cost, inventory cost,
shortage cost) to select an appropriate strategy. In addition, the
strategy selection should consider important interactions among
evaluation factors. Therefore, this study adopts an expert decision
approach – the analytic network process (ANP) technique – which
jointly considers non-metric and metric variables all together to
determine an appropriate maintenance strategy. In addition,
spare parts’ quantities and replacements’ interval estimation are
conducted using the Weibull distribution based on the preventative maintenance (PM)/corrective maintenance (CM) ratio obtained by expert decisions in the ANP analytical result. The results
of this study serve as a reference for railway system operators
when evaluating their rolling stock maintenance strategy and also
when estimating their spare parts’ quantities and replacement
intervals for specific components of the rolling stock.
2. Literature review
Many studies on manufacturing system maintenance and
replacement problems exist (Mccall, 1963; Barlow and Proshan,
1975; Pierskalla and Voelker, 1976; Sherif and Smith, 1981;
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
Jardine and Buzacott, 1985; Valedz-Flores and Feldman, 1989;
Cho and Parlar, 1991; Ruhul and Amanul, 2000; Yun and Ferreira,
2003; Berthaut et al., in press; Muchiri et al., in press; Sun and Li,
2010), as well as thousands of maintenance and replacement
models have been proposed. Most previous research studies on
maintenance models have centered around total cost (Ruhul and
Amanul, 2000; Yun and Ferreira, 2003). In comparison with
manufacturing system maintenance, studies on rolling stock
maintenance are relatively rare. Chaudhuri and Suresh (1995)
developed an algorithm for determining the best type of
maintenance, period length, and replacement policy using fuzzy
set theory, but did not consider safety, which could be crucial for
rolling stock maintenance. Wang et al. (2007) evaluated different
maintenance strategies based on a fuzzy analytical hierarchy
process, however, the AHP’s evaluation did not reflect on the
possible interaction among the evaluation factors.
In consideration of spare parts, Chelbi and Aı̈t-Kadi (2001)
proposed a jointly optimal periodic replacement and spare parts
provisioning strategy. They evaluated the performance of this
strategy in terms of total average cost per time unit over an
infinite horizon. Yun and Ferreira (2003) described the development of a simulation model to assess the inventory requirements
of alternative rail sleeper replacement strategies. Almeida (2001)
presented multi-criteria decision models for two maintenance
problems: repair contract selection and spare provisioning. In the
repair contract problem the model incorporates consequences
modeled through a multi-attribute utility function. The consequences consist of contract cost and system performance,
represented by the system interruption time. Two criteria (risk
and cost) are combined through a multi-attribute utility function
in the spare provisioning decision model.
As for the replacement interval, Chaudhuri and Suresh (1995),
Cassady et al. (1998), and Nakagawa (1989) conducted research
on the principles of maintenance cost minimization to determine
the best replacement cycle. They considered maintenance costs
resulting from preventative maintenance and system damage.
Huang et al. (1995) and Ruhul and Amanul (2000) suggested that
system damage rates increase with amortization. When damage
maintenance costs exceed preventative maintenance costs, a
suitable preventative maintenance period will minimize total
maintenance costs. This study follows related literature that
assumes the Weibull distribution be applied to rolling stock
component amortization to obtain a more reasonable result.
Most research studies regarding maintenance strategies and
replacement policies have developed models focusing on cost
consideration. However, it should be better to consider factors
such as safety when developing a model to assist in selecting a
maintenance strategy (Wang et al., 2007). Therefore, this study
incorporates relevant safety factors by adapting the ANP method
which allows for the consideration of important interactions
among the evaluation to select an appropriate rolling stock
maintenance strategy.
Experts were surveyed through the use of questionnaires in two
phases. Phase 1 interviewed the maintenance staff on site to establish
the ANP questionnaire decision hierarchy framework for phase 2.
Phase 2 interviewed maintenance managers to obtain an appropriate
rolling stock maintenance strategy for deciding upon the preventive
and corrective maintenance ratio and also the weight for the
evaluation factors. In addition, the multi-attribute utility theory is
adopted to estimate the possible spare parts’ quantities in the various
planning horizons. Finally, the Weibull distribution is applied to
determine the life of components in order to obtain optimal
replacement intervals. Fig. 1 presents the analytical procedure.
In this study the maintenance strategy must consider the
interaction between selected factors. For example, safety and cost
are the two essentials factors affecting a rolling stock strategy
selection; however, one must be sacrificed to some degree to favor
the other. The major drawback of the analytic hierarchy process
(AHP) lies in the assumption that independent conditions exist.
This is oftentimes an unrealistic premise, and thus ANP is a more
appropriate technique for rolling stock maintenance selection.
3.1. The analytic network process
The analytic network process (ANP) is a multi-criteria theory
of measurement used to derive relative priority scales of absolute
numbers from expert judgments, which also belong to a
fundamental scale of absolute numbers. These judgments represent the relative influence of one of the two elements over the
other in a pair-wise comparison process on a third element in the
system, with respect to an underlying control criterion. Through a
supermatrix, whose entries are themselves matrices of column
List evaluation factors affecting maintenance strategy through
literature review and experts’ opinions
Use factor analysis to condense evaluation factors into in a smaller
set of higher level factors and establish an evaluation hierarchy
framework of ANP questionnaire
An appropriate maintenance strategy is obtained through ANP
technique through expert’s decision choices
The ratio between preventive maintenance and corrective
maintenance and the weight of evaluation factor are obtained by
ANP empirical result
3. Research design and methodology
This study applies expert decision methodology to select an
appropriate maintenance strategy from various possible strategies of different PM and CM ratios. Following this selection, the
estimation of spare parts’ quantities is established using a multicriteria decision model based on the ratio of preventative and
corrective maintenance obtained previously in the ANP result. In
addition, the rolling stock component replacement interval is
estimated using the Weibull distribution assumption for the
lifetime of the component. The objective of this study is to address
a group of identical units of rolling stock.
405
The ratio between PM and CM are inputs for the estimation of
spare parts’ quantities
The rolling stock’s component of replacement interval is estimated
by making an assumption through the Weibull distribution for the
component life cycle
Fig. 1. Research analysis procedure.
406
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
priorities, the ANP synthesizes the outcome of dependence and
feedback within and between clusters of elements (Saaty, 2004).
The ANP is the general form of the analytic hierarchy process
(AHP) (Saaty, 1980), but the drawback of it lies in certain
limitations when the complexity of decision problems increases
and interactions among decision criteria and sub-criteria are not
implicitly covered. The ANP is proposed (Saaty, 1996) to overcome
the problem of interdependence and feedback between the criteria
or alternatives. The ANP methodology has also been applied in
areas including priority ranking, substitution production, project
selection, optimal scheduling, production planning, performance
evaluation, strategic decision forecasts, and risk evaluation (Meade
and Presley, 2002; Presley et al., 2007; Liberatore, 1987; Lee and
Soung, 2000; Meade and Sarkis, 1999; Sarkis and Talluri, 2002;
Sarkis and Sundarraj, 2002; Sarkis (2003a, 2003b); Sarkis et al.,
2007; Karsak et al., 2002; Momoh and Zhu, 2003).
The important phases of the ANP compare criteria in the whole
system to form the supermatrix. This is executed through pairwise comparisons by asking ‘‘how much importance does an
evaluation factor have compared to another evaluation factor
with respect to our research goal?’’ The relative importance value
can be determined using a scale of 1–9, representing equal
importance to extreme importance (Saaty, 1980, 1996). The
general form of the supermatrix can be described as follows:
columns represent the components of the clusters. The bottom
rows and columns represent the alternatives. For supermatrix
formation, the pair-wise comparisons are conducted for the
purpose of weighting the clusters and estimating the direction
and importance of influence between elements to form a supermatrix. The supermatrix allows a resolution of the effects of
interdependency that exists between the elements of the system.
An ANP model is mathematically implemented following a threestep supermatrix calculation (Saaty, 2001). In the first step, the
unweighted supermatrix is created directly from all local priorities
derived from pair-wise comparisons among elements influencing
each other. The elements within each cluster are compared with
respect to influencing elements outside the cluster. This also yields
an eigenvector of influence for all clusters on each cluster.
The second step calculates the weighted supermatrix by
multiplying the values of the unweighted supermatrix with their
affiliated cluster weights. By normalizing the weighted supermatrix, it is made column stochastic. In the final step, the limit
supermatrix is processed by raising the entire supermatrix to
powers until convergence (Wolfslehner et al., 2005).
3.2. The estimation model for spare parts’ quantity
This study formulates the estimation model for spare parts’
quantity by following Almeida (2001) with some modification. A
multi-criteria decision model, U(c,a), allows the quantification of
spare parts by provisioning for a single item and taking into
account the total spare cost (C) and the risk of an item’s non-supply
(a). The assumption to formulate the estimation model of spare
parts’ quantities is close to the model formulation of Almeida
(2001). The multi-attribute utility function is assumed to be
additive as indicated in Keeney and Raiffa (1976), Almeida and
Bohoris (1995), and Almeida and Souza (1993). The meaning of the
multi-attribute utility function is derived from two main components. The first utility component is derived from minimizing the
maintenance cost (mainly from preventative maintenance). The
second utility component is derived from minimizing the probability of the shortage of stock of spare parts (much sensitive to
corrective maintenance). The basis of this assumption implies an
independence condition in preferences between PM and CM
(Keeney and Raiffa, 1976). The interaction among PM and CM is
considered by the ANP technique in the previous stage.
Maxq UðC, aÞ
where Cm denotes the mth cluster, emn denotes the nth element in
the mth cluster, and Wij is the principal eigenvector of the
influence of the elements compared in the jth cluster to the ith
cluster. In addition, if the jth cluster has no influence upon the ith
cluster, then Wij ¼ 0. Therefore, the form of the supermatrix
depends much on the variety of the structure. A pair-wise
comparison has two goals: one for weighting the clusters (i.e.,
criteria) and the other for estimating the direction and importance of influences between elements, which are numerically
pictured as ratio scales in a so-called supermatrix.
The essential step to apply the ANP approach is to construct the
hierarchical decision network. The ANP model needs the definition
of elements and their assignment to clusters, as well as a definition
of their relationships (Wolfslehner et al., 2005). After determining
all the elements that affect the decision, we group them into
clusters for each network. Comparisons of clusters, comparisons of
elements, and comparisons for alternatives are performed to
calculate the relative importance and to construct the unweighted
supermatrix. In the supermatrix, each row and respective column
represents an element of the decision network. The first row and
column represents the ‘‘goal’’ elements. The second middle levels
of elements represent the clusters. The next levels of rows and
ð1Þ
Here, U means utility, item reliability is assumed to follow a
Weibull distribution with given y and b, y represents the scale
parameter, and b represents the shape parameter. Total spare cost
C depends on the unit cost Ce and the number of spare parts q:
C ¼ qCe
ð2Þ
The lifetime distribution of spare parts and the investigated single
unit are both assumed to follow the Weibull distribution. a is derived
from the probability when the investigated single unit fails and the
spare part also fails simultaneously. Thus, we assume z¼ x+y, where x
means the event indicating the investigated single unit fails, y means
the event indicating the spare part fails, and the occurrences of these
two events are independent. The convolution is used to calculate the
integration of these two cumulative distribution functions Fx and Fy so
as to obtain the probability of the provisioning shortage a. When the
single unit fails and the spare part also fails simultaneously, the
provisioning shortage occurs, and thus:
Z Z
a ¼ FX þ Y ðaÞ ¼ PfX þ Y r ag ¼
f ðxÞgðyÞ dx dy
Z
1
Z
ay
f ðxÞgðyÞ dx dy ¼
¼
1
1
Z
xþyra
1 Z ay
1
1
f ðxÞdx gðyÞ dy
ð3Þ
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
As before, one-dimension utility functions are first obtained for
U(C(q)) and U(a(q)), and then a multi-attribute utility function
U(C(q),a(q)) is obtained. This multi-attribute utility function is
assumed to be additive:
UðC, aÞ ¼ K1 Up ðCÞ þ K2 Uc ðaÞ
ð4Þ
Here K1 is the conditional probability of preventive maintenance
and K2 the conditional probability of corrective maintenance;
K1 +K2 ¼1, where K1 and K2 value are derived from the ANP result.
Substituting (2) and (3) into (4):
UðC, aÞ ¼ K1 Up ðqCe Þ þ K2 Uc ðaÞ
ð5Þ
If (5) is applied to (1), then the optimum solution can be
obtained while the utility is maximized.
3.3. Component replacement interval estimation
3.3.1. Weibull distribution
A Weibull distribution is very useful as a failure model in
analyzing the reliability of different types of systems (Qiao and
Tsokos, 1995). Therefore, this study implements it for representing the time to failure of rolling stock components. The Weibull
distribution consists of 2 parameters, where y represents the scale
parameter and b represents the shape parameter. The failure time
follows the Weibull distribution with a distribution function:
b x
FðxÞ ¼ 1exp y
where x Z0, y 40, and b 40.
The probability density function is as follows:
b b xb1
x
f ðx Þ ¼
exp y y
y
where x Z0, y Z0, and b Z0.
The 3-parameter distribution cumulative density function is
xgb
FðxÞ ¼ 1exp y
where x Z g, y 40, b 40, and g Z0.
The difference between the 3-parameter and 2-parameter
Weibull distributions is a location parameter represented by g.
The Weibull distribution is primarily dependent on the scale and
shape parameters.
3.3.2. Weibull probability density function parameter estimation
Graphical methods are commonly used for reliability data
analyses due to their simplicity and visibility (Jiang and
Kececioglu, 1992). The Weibull probability method is a relatively
easy and frequently applied graphical approach. It can express the
exponentiated Weibull distribution as a linear function. The slope
of this linear function is the shape parameter’s estimated value.
Linear expression of the Weibull distribution is derived as follows:
1F ðxÞ ¼ eðx=yÞb
where y is the scale (characteristic life) parameter of the Weibull
distribution and b the shape parameter of the Weibull distribution.
Applying a natural logarithm in this equation gives the
following:
x b
x b
lnð1FðxÞÞ ¼ , lnð1FðxÞÞ ¼
y
y
Applying a natural logarithm again gives:
x
¼ b ln xb ln y
ln lnð1FðxÞÞ ¼ b ln
y
We make Y ¼ln[ ln(1 F(x))], where X ¼ln x
407
Rewriting the above function into a straight function, we have:
Y ¼ b Xb ln y
Fothergill (1990) used both an exact technique and MonteCarlo simulations to estimate the cumulative probability of
failure. We take Pi as the estimation value for F(x):
Pi ¼
i0:3
n þ 0:4
where Pi represents the ith observed cumulative frequency value
of n numbers of samples. Therefore, this study applies F(x) for the
value estimation followed by a frequency graphic analysis
expressed as Y ¼ln[ ln(1 F(x))], and X ¼ln x to obtain the scale
(characteristic life) and shape parameters of the Weibull distributions y and b. This study follows Huang et al.’s (1995)
mathematical model T0 ¼ 0:1 Ts y to find the optimal replacement interval T0 , where Ts is the average service time of a
component under the PM policy (MTBR, mean time between
replacement). The y value is obtained from sampling the
replacement data. Replacement time ‘‘Ts ’’ is achieved through
the b value and the ratio between corrective and preventive
maintenance costs.
4. Empirical result analysis
4.1. Selecting an appropriate maintenance strategy
This study applies the ANP technique expert questionnaire
survey to select an appropriate rolling maintenance strategy.
Accordingly, this study finds evaluation factors affecting the
maintenance strategy selection via a literature review and
experts’ opinions. These evaluation factors and higher-level
evaluation factors derived from factor analysis shape the evaluation framework of the ANP questionnaire. Finally, there are three
different maintenance strategies based on various combinations
between PM and CM.
Alternative A: the frequency ratio between PM and CM is 7:3.
Alternative B: the frequency ratio between PM and CM is 1:1.
Alternative C: the frequency ratio between PM and CM is 3:7.
This study uses the model 321 metro rolling stock current
collecting shoe as a component for analysis according to rolling
stock maintenance experts’ suggestion. In addition, the 321 metro
rolling stock collecting shoe is crucial for the daily operation of
Taipei’s mass rapid transit (MRT). The ratio of PM:CM¼7:3 is
derived from the sample component data for analysis. Of the 50
components randomly sampled, 35 samples executed preventative maintenance and 15 executed corrective maintenance.
Therefore, we derive the ratio of PM:CM ( ¼7:3) for our first
possible maintenance strategy alternative. Because the aim of this
study is to investigate the importance between preventative and
corrective maintenance in the rolling stock maintenance strategy
selection, the second possible maintenance strategy alternative
PM:CM¼1:1 is then created, indicating that preventative maintenance had the same importance as corrective maintenance in
the rolling stock maintenance strategy selection. The third
alternative is created by the ratio of PM:CM ( ¼3:7), which means
preventative maintenance is less important than corrective
maintenance. The ratio PM/CM multiplied by the weight obtained
by the ANP analysis can help predict the preparation for spare
parts’ quantities of the rolling stock component by maximizing
the multi-attribute utility.
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Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
‘‘reduce rolling stock shut-down time’’ and ‘‘reduce maintenance
time’’ as 0.626. Therefore, we group these two sub-factors
together, naming the new group ‘‘reduce maintenance cost and
shut-down time’’. The correlation between ‘‘assure worker and
staff safety’’ and ‘‘assure passenger safety’’ is 0.643. Therefore, we
group these two sub-factors together, naming the new factor
‘‘assure worker and passenger safety’’. The correlation of ‘‘reduce
maintenance cost’’, ‘‘improve staff work efficiency’’, and ‘‘improve
staff work assignment and preparation’’ are 0.643 and 0.673,
respectively. The correlation between ‘‘improve staff work
efficiency’’ and ‘‘improve staff work assignment and preparation’’
reached as high as 0.780. Therefore, we group these three subfactors into one sub-factor and call it ‘‘improve staff work
efficiency’’. The correlation between ‘‘reduce the impact in case
of emergency’’ and ‘‘improve railway system coach failure’’ is
0.704. Hence, we group these two sub-factors together and
rename it ‘‘improve rolling stock failure rate’’.
The second phase questionnaire design is based on the factor
analysis result of the first phase questionnaire. The second phase
questionnaire is based on three main evaluation factors with eight
sub-factors and three selection alternatives. This study refers to
Taipei’s mass rapid transit system on preventative and corrective
maintenance percentages for possible selection alternatives.
The following presents eight reorganized sub-factors categorized
into three main factors.
4.2. Questionnaire design and investigation
This study adopts a two-phase questionnaire design. The first
phase of the questionnaire is designed by following Shu (1999),
who considered some specific factors affecting maintenance
strategy selection. Interviews are conducted on site with the
maintenance staffs who work for various railway system
operators, including conventional railway system, mass rapid
transit systems, and high-speed rail system. A total of 78
questionnaires were returned. This study applies the factor
analysis technique to conduct data reduction and summarization,
and then extracts high-level evaluation factors to form the
decision hierarchy of the ANP questionnaire. The ANP technique
could aid in selecting the most suitable rolling stock maintenance
strategy based on experts’ choice. Table 1 presents the first phase
questionnaire distribution information.
4.2.1. Factor analysis results
This study adopts the KMO test and Bartlett test to examine
the appropriateness of 13 factors on rolling stock maintenance for
factor analysis application. The result shows a KMO value greater
than 0.7 and a Bartlett test value at 635.797, indicating the
appropriateness of factor analysis application in rolling stock
maintenance-related issues. Rolling stock maintenance factor
analysis results show evidence of total explainable variance at
71.143% and extract three high-level factors. As for the reliability
analysis result, this study conducts a Cronbach’s a value to
examine the rolling stock maintenance questionnaire reliability to
assure all measured factors are highly consistent. Table 2 presents
the factor analysis result and shows that all Cronbach’s a
individual are above 0.7.
(1) Factor 1: Quality and Efficiency
This factor combines four sub-factors: maintain high quality
maintenance, improve staff work efficiency, maintain appropriate usable spare parts, and reduce stock failure rate.
(2) Factor 2: Cost and reliability
This factor includes two sub-factors: maintain rolling stock in
good condition and reduce maintenance cost and shut-down
time.
(3) Factor 3: Safety
This factor encompasses two sub-factors: assure worker and
passenger safety and assure railway rolling stock safety.
4.2.2. ANP questionnaire design and empirical result
It is suggested that a reduction of the sub-factors be
considered when applying the ANP technique. Therefore, this
study groups highly correlated sub-factors into one factor through
correlation analysis. Empirical results show a correlation between
Table 1
First phase questionnaire return rate.
Interviewee
Distributed questionnaires
Returned questionnaires
Effective questionnaires
Ineffective questionnaires
Conventional railway maintenance staff
Taipei rapid mass transit maintenance staff
High-speed rail maintenance staff
Total
100
100
100
300
40
29
11
80
39
28
11
78
1
1
0
2
Table 2
Empirical results of factor analysis to extract evaluation factors affecting the selection of a suitable maintenance strategy.
Factor
Sub-factor
Factor loading
Eigenvalue
% of variance
Cumulative variance (%)
Cronbach’s a
Quality and efficiency
Maintain high quality maintenance
Maintain appropriate available spare parts
Improve staff work efficiency
Improve staff work assignment and preparation
Reduce the impact in case of an emergency
Improve railway system coach failure rate
0.615
0.735
0.643
0.668
0.696
0.814
6.357
48.898
48.898
0.889
Cost and reliability
Maintain rolling stock in good condition
Reduce rolling stock shut-down time
Reduce maintenance time
Reduce maintenance cost
0.436
0.812
0.894
0.721
1.814
13.957
62.855
0.820
Safety
Assure staff and personnel safety
Assure railway system safety
Assure passenger safety
0.874
0.832
0.693
1.077
8.288
71.143
0.810
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
Table 3 indicates the final factors and sub-factors affecting the
rolling stock maintenance included in the ANP model.
The second phase questionnaires were distributed to middlehigh level maintenance executives of Taipei’s mass rapid transit.
Qualified interviewees needed to have more than 10 years of
rolling stock maintenance experience.
In the ANP application the consistency of the judgments is
verified against misevaluations. If a matrix includes inconsistencies,
then the respondents are required to review their corresponding
evaluations until a reliable judgment is obtained. For an n n
pair-wise comparison matrix, the consistency index is defined
as follows:
CI ¼
lmax n
n1
Table 3
Evaluation factors and possible alternatives as the basis for the ANP method
questionnaire.
Factor
Sub-factor
409
where lmax is the largest or principal eigenvalue of n n pair-wise
comparison matrix. In this empirical result the consistency index
(CI) values are all less than 0.1. Therefore, the result of the expert
judgments for the ANP questionnaire can be accepted.
For the evaluation results, Fig. 2 shows cost and reliability,
safety, and quality and efficiency, with weight values of 0.268,
0.528, and 0.204, respectively. This result fully explains safety as a
first priority in railway rolling stock maintenance considerations.
Fig. 2 further shows that maintenance strategy A (0.607) is
significantly superior compared to the other two maintenance
strategies after the calculation of the maintenance strategy
expectation index in Table 4. Therefore, maintenance strategy
A’s preventative maintenance and corrective maintenance with
percentages at 7:3 are followed by preventative and corrective
maintenance strategy weight percentages at 1:1 (alternative B,
weight value 0.244). Finally, the preventative and corrective
maintenance strategy weight percentage is at 3:7 (alternative
C, weight value 0.149). This result explains that the majority
Alternative
Quality and Maintain high quality
Alternative A: the frequency ratio
efficiency maintenance
between PM and CM is 7:3
Improve staff work
efficiency
Maintain appropriate
usable spare parts
Reduce rolling stock failure
rate
Cost and
Maintain rolling stock in
reliability good condition
Reduce maintenance cost
and shut-down time
Alternative B: the frequency ratio
between PM and CM is 1:1
Safety
Alternative C: the frequency ratio
between PM and CM is 3:7
Assure worker and
passenger safety
Assure rolling stock safety
Table 4
Maintenance strategy expectation index.
Cj
Evaluation factors Maintenance strategy weights
Weight (Rj)
WT1
C1 0.176
0.619
C2 0.092
0.521
C3 0.294
0.615
C4 0.234
0.641
C5 0.052
0.615
C6 0.048
0.508
C7 0.042
0.576
C8 0.062
0.629
Maintenance strategy expectation
STi Ckj
WT2
WT3
ST1 Cj
ST2 Cj
ST3 Cj
0.244
0.281
0.239
0.230
0.227
0.293
0.268
0.219
index DIi
0.137
0.198
0.146
0.129
0.158
0.199
0.156
0.152
0.109
0.048
0.181
0.150
0.032
0.024
0.024
0.039
0.607
0.043
0.026
0.070
0.054
0.012
0.014
0.011
0.014
0.244
0.024
0.018
0.043
0.030
0.008
0.010
0.007
0.009
0.149
Maintain rolling stock in good condition (0.176)
Cost and Reliability
(0.268)
Rolling stock maintenance strategy
Reduce maintenance cost and shut-down time
(0.092)
PM CM = 7 3
(0.607)
Assure workers and passengers’ safety (0.294)
Safety
(0.528)
Assure rolling stock safety (0.234)
PM CM = 1 1
(0.244)
Maintain High quality maintenance (0.052)
Quality and Efficiency
(0.204)
Improve staff work efficiency (0.048)
Maintain appropriate usable spare parts (0.042)
Reduce rolling stock failure rate (0.062)
Fig. 2. ANP analysis framework and weight of the factors, sub-factors and alternatives.
PM CM = 3 7
(0.149)
410
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
of rolling stock maintenance work focuses on preventative
maintenance. It is dangerous to cease railway system operations
for the reason of rolling stock component failure. It seems that a
preventive-oriented maintenance strategy could probably assure
rolling stock safety. In addition, this empirical result obtains the
ratio PM/CM multiplied by the weights of choosing this scenario
alternative in order to obtain the K1/K2 ratio based on the expert
decisions for estimating the spare parts of the component.
Table 5
The probability of an item’s non-supply during the various planning horizons.
1
2
3
4
5
year
year
year
year
year
0 Spare part
1 Spare part
2 Spare parts
3 Spare parts
0.317345
1
1
1
1
0.0000217639
0.272266
0.999999
1
1
2.86572 10( 11)
0.00014376
0.236493
0.999278
1
3.106 10( 18)
3.209 10 9
0.000359
0.20752
0.991189
4.3. Estimation of spare parts
Through the ANP methodology, the railway rolling stock’s
preventive maintenance and corrective maintenance ratio is
obtained as 7:3, and this ratio is applied to estimate the needed
spare parts’ quantities of Taipei’s MRT rolling stock model 321
current collecting shoe. The multi-attribute utility function is
applied to estimate spare parts’ quantities.
The PM/CM ratio can vary according to the rolling stock components investigated. The scenario alternative regarding the estimated
PM/CM ratio of 7:3 is obtained through the randomly selected 50
sampled data from the historical data of the rolling stock components
investigated. This ratio can also be 6:4 or 6.5:3.5. The ratio is
principally subjected to the historical maintenance data of the rolling
stock components which we would like to investigate. If the ratio is
8:2 instead of 7:3, then this result indicates much more importance of
PM compared to CM. The ANP empirical result obtains the weight of
various alternatives (0.607 for PM/CM¼7:3, 0.244 for PM/CM¼1:1,
and 0.149 for PM/CM¼3:7). Our approach also uses this ratio
(PM/CM) of various alternatives multiplied by the weights to obtain
the K1/K2 ratio, and K1 +K2 ¼1, where K1 and K2 are derived. K1 and K2
are the key inputs in the multi-attribute utility function U(C,a) that
estimates the spare parts’ quantities. The utility function is formulated
as U(C,a)¼K1U(C)+K2U(a). We assume here that K1 and K2 reflect the
relative importance weight between the PM and CM judged by the
relevant experts. The various ratios are obtained by the ANP result,
leading to various estimations of the spare parts’ quantities.
Fig. 3 presents the relationship between accumulated failure
rate (Pi) and time (x) of the selected 50 sample units. The
estimation of the probability of the provisioning shortage a in one
year with only spare parts available is derived as follows. The
value obtained by the randomly selected 50 samples with values
b ¼7.5 and y ¼13.644 is adopted and the planning horizon is one
year (12 months), Thus, x ¼12 is used in our estimation.
The probability density function of the Weibull failure
distribution here is assumed as follows:
b b xb1
x
f ðxÞ ¼
exp y y
y
where xZ0, y Z0, b Z0, F(x)¼1 exp[ (x/y)b].
Fig. 3. The relationship of accumulated failure rate Pi and time (months).
The probability of the provisioning shortage a could be derived
by using the software Mathematica:
Z Z
a ¼ FX þ Y ðaÞ ¼ PfX þ Y r ag ¼
f ðxÞgðyÞ dxdy
Z
1
Z
ay
f ðxÞgðyÞ dxdy ¼
¼
1
1
pðx þy raÞ ¼
Z
xþyra
1
1
Z
a
Z
Z
ay
f ðxÞdx gðyÞ dy
1
12 Z y
FX ðzyÞgðyÞ dy ¼
f ðxÞgðyÞdxdy
0
0
Z
12
y b1
b
b x
b yb1 ð1ðy=yÞÞb
eð1ðx=yÞÞ dx
e
dy
¼
y
y
y y
0
0
2 Z 12 Z y
b
b
1
¼
b2
xb1 yb1 eð1ðx=yÞÞ eð1ðy=yÞÞ dx dy
7:5
0
Z
¼
y
1
7:5
y
2
0
b
2
Z
0
12
0
Z
12
b
b
ðzyÞb1 yb1 eðzy=yÞ eðy=yÞ dz dy
0
¼ 0:0000217639
The probabilities of the provisioning shortage a during the
various planning horizons are generalized by using the convolution to calculate the integration of these two cumulative
distribution functions Fx and Fy and are estimated in Table 5.
The cell in Table 5 indicates the probabilities of an item’s nonsupply for the preparation of different spare parts during the
various planning horizons (year). For example, the cell (2 years, 2
spare parts) of Table 5 reveals that the probability of an item’s
non-supply when the system operator prepares 2 spare parts
under a two-year planning horizon is 0.00014376.
4.3.1. Multiple utility function
The multiple utility function is applied to obtain the maximum
utility value after calculating the a value. The multiple utility
function is expressed as follows:
UðC, aÞ ¼ K1 Up ðCÞ þK2 Uc ðaÞ
where U(a)¼exp( A1a) and U(C)¼exp( A2C). K1 is the probability of preventive maintenance; K2 the probability of corrective
maintenance. K1 + K2 ¼1, and K1 and K2 values are derived from the
ANP result.
In the current collecting shoe cost estimation, due to various
purchase volumes of Taipei’s mass rapid transit, the current
collecting shoe costs vary is between NT$2000 and NT$4000. This
study assumed the current collecting shoe cost as U$100
(U$1¼NT$33). This study also assumes that A1 ¼16 and
A2 ¼0.002 in order to obtain the maximum utility value according
to Almeida (2001). The ANP empirical result obtains the weight of
various alternatives (0.607 for PM/CM¼7:3, 0.244 for PM/CM¼
1:1, and 0.149 for PM/CM¼ 3:7). Therefore, the K1/K2 ratio can be
obtained by 0.607 7/3 +0.244 1/1+ 0.149 3/7¼1.724, and
K1 + K2 ¼1. This study adopts K1 ¼0.633 and K2 ¼0.367 to obtain
effective values of individual spare parts by using the multiattribute utility function. Fig. 4 indicates the relationship between
preparation of the spare parts’ quantities and the utility value of
the multi-attribute utility function under various planning
horizons.
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
411
1.2
1
Utility
0.8
0.6
0.4
0.2
no spare part
1 spare part
2 spare parts
3 spare parts
0
0
3
6
9
12
15
18
21
Months
24
27
30
33
36
Fig. 4. Estimated utility values for the preparation of different spare parts under various planning horizons.
4.3.2. Optimum spare parts estimation
According to the result derived from the multiple utility
function in Fig. 4, the optimum spare part estimation under
various planning horizons can be derived. Our result indicates the
system operator needs to prepare at least one spare part when the
planning horizon is greater than 7.5 months and two spare parts
when the planning horizon is between 7.5 and 19 months. This
study follows Almeida (2001) and assumes a one-to-one facility
and spare part composition. Therefore, this study assumes that
model 321 of Taipei’s mass rapid transit adopts the one-to-one car
and current collecting shoe with one train containing 6 cars. In
other words, each train must be equipped with 6 spare parts, and
thus 36 trains require 216 spare parts when the planning horizon
is greater than 7.5 months, but less than 19 months (Fig. 4).
4.4. Optimal replacement interval
This study estimates the optimal replacement interval according to the 50 sample data obtained from Taipei’s mass rapid
transit (MRT) after conducting maintenances. The sample data are
from 50 random maintenance records of the past maintenance
history of Taipei’s mass rapid transit. The results show the average
replacement time to be 12.779 months. This study follows Huang
et al. (1995) in assuming preventive and corrective maintenance
cost ratios at 1:15. The cost of corrective maintenance is 15 times
more than preventive maintenance.
4.4.1. Parameter estimation
Random replacement data show Y ¼ln[ ln(1 F(x))], and
X¼ln x and obtains the first Weibull distribution failure function
linear equation expressed as Y ¼7.50X 19.6. Through this linear
equation, we obtain that the Weibull distribution shape parameter b and scale parameter y are 7.50 and 13.644, respectively.
Huang et al. (1995) applied values of these two parameters to
obtain the optimal replacement interval.
4.4.2. Replacement interval calculation
This study follows Huang’s et al (1995) mathematical model
T0 ¼ 0:1 Ts y to obtain MRT 321 type rolling stock collecting
current shoe’s optimal replacement interval with T0 as the
optimal replacement time. The y value is estimated from 50
random replacement data samples. Replacement time ‘‘Ts ’’ is
obtained by following Huang et al. (1995) through the b value and
ratio between corrective and preventive maintenance costs. This
study adopts the b value of 7.50 and a ratio between corrective
and preventive costs at 15:1. Through an estimation graph
proposed by Huang et al. (1995), the estimated replacement time
Ts is between 5.0 and 6.0. Therefore, optimal replacement
intervals are found to be between 6.8 and 8.2 months. Our
empirical results with respect to the estimation of spare parts’
quantities and replacement interval optimization in this present
model are considered to be reasonable after seeking the relevant
railway rolling stock maintenance experts’ opinions.
5. Conclusion and suggestions
The purpose of this paper is to present an approach for deciding
on an appropriate rolling stock maintenance strategy selection and
estimating possible spare parts’ quantities and replacement
interval for the component of rolling stock. The issue of rolling
stock maintenance strategy selection is not just a cost-oriented
issue, as it should also consider safety, cost, and quality aspects.
The trade-off relationship among these three main factors also
increases the complexity of the problem. Thus, the ANP technique
can be used to investigate the interrelationship among the factors.
The empirical result derived from the ANP can obtain the possible
ratio between preventive maintenance and corrective maintenance
in order to indicate the possible spare parts’ quantities and
replacement interval of the rolling stock components.
The empirical result based on the ANP method regarding
rolling stock maintenance strategy selection also indicates that
preventive maintenance should be much more valued than
corrective maintenance. This result is consistent with the studies
of maintenance strategies on industrial equipment (Nakagawa,
1989; Huang et al., 1995; Chelbi and Aı̈t-Kadi, 2001). In addition,
the empirical result derived from the ANP method shows that
safety is the most crucial factor for rolling stock maintenance
strategy selection. Safety here considers not only passenger
safety, but also maintenance agent safety. This finding is not
consistent with previous studies that focus on industrial facilities
and equipment maintenance (Bevilacqua and Braglia, 2000), but is
consistent with Wang et al. (2007) who indicated safety as being
the most important factor in selecting a maintenance strategy in
the chemical industry and power plants. Safety is also essential
for rolling stock maintenance. Railway systems cannot risk a
break in operations due to the failure of rolling stock components,
which could result in possible uncompensated loss of a passenger’s life. The second important factor for rolling stock maintenance strategy selection is about cost and reliability. This means
that reducing maintenance cost and avoiding trains idling in the
maintenance site are essential for rolling sock maintenance. The
third factor affecting the rolling stock maintenance strategy
choice by the experts includes maintenance quality and efficiency.
412
Y.-H. Cheng, H.-L. Tsao / Int. J. Production Economics 128 (2010) 404–412
This proposed approach herein can also assist decision makers
in the evaluation and selection of rolling stock maintenance
strategies and also in the estimation of component spare parts’
quantities and replacement interval optimization. This study
chose Taipei’s MRT rolling stock component, the model 321
current collecting shoe, as an analytical component to estimate
spare parts’ quantities under the various planning horizons and
optimal replacement interval, because the current collecting shoe
is necessary for the daily operation of mass rapid transit systems.
Most previous research studies on rolling stock maintenance
do not consider rolling stock maintenance strategy selection and
spare part estimation simultaneously. This study’s contribution is
in combining a multi-criteria decision-making method, spare part
estimation, and a replacement interval model to establish a new
systematic approach in order to obtain a suitable rolling stock
maintenance strategy and then to produce an estimation of spare
parts’ quantity and replacement interval approximation. An
advantage to this approach lies in its ability to link safety, cost,
and quality, which are the factors associated with selecting an
appropriate rolling stock maintenance strategy. Although the ANP
technique appears complicated, it can capture the complexity of
real world situations for maintenance strategy selections.
In this study, rolling stock maintenance experts suggested the
choice of components for spare part optimization and replacement
interval estimation analysis. Taipei’s MRT rolling stock model 321
current collecting shoe is taken for analysis, because of its
importance for safe operation. In the future, the choice component
for analysis can be based on a more objective approach. The critical
component of rolling stock operations may be previously determined by the analysis result of failure modes effects and criticality
analysis (FMECA) and fault tree analysis (FTA).
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